Ecole Polytechnique. Ne pas reproduire.
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* *&) * &* % ) ! '."!#$v YbdKMXW[TSTeJYqHJSCKBFIJ]cpILWuem]&FBHJ]OXFX] cQY\ILHP>KUFQemSfMNWIZY\p[C] MNOHJIJYqYUSC[lMXOgbdY\HLrXW&b YqHLSfP>ST]F
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' NKUOM W&ijOpY\HLSTKUF$MN]^MNSCOpeLSTKUF=A+KBFpemSfMNW&ILKBFpe1OpF"MNKBb Y\STFX]
ΩMN]4[¢ ]&eLcpYUP]
NMXSCbd]FQemSTKUFpe FXK\HJW
RN XYU]uP`]F/VUW&FXWIZY\[N = 1, 2
KBO dijOX]$[¢ KUFgeLOXcXcKBeL]"KNP&P>OXcWcpY\I OpFb Y\HLWIJSfY\O rpKUbdKUVBkFX] STeLK\HJILKBcQ] ]>HPKUFpMXOpPRHJ]OXIlMX][TYPZrpY\[T]OXIuA 7FFXKUHL]
x[TY qYUILSfY\X[T]M ]&eLcpYUP] pPU ]ueH ) ) MXSCIJ]4OXF cKUSTFBH0MN]
Ω
]Ht[TY7qY\IJSTYUX[T]MN]+HL]&bdcpeA\¡0Y\FQe
Ω[C]ue eLKUOpIJP]&e MN]PZrQY\[T]OXI W&U]&FBHJOX][T[T]bd]FjH FXKUF%OXFXS 1 KUIJb ]ue
]&F]&eLcpYBP>]]H qY\IJSTYUX[T]&e3MpY\Fpe3[C]1HL]bdcpe emKBFBH IL]&cXILWuem]&FjHLW]uecpYUI OXFp] 1 KUFQPRHLSTKUF%MNKUFXFpW]f(x, t)
HZY\FpMXSTe^ijOX]d[fYHJ]bdcWIZYqHLOpIL]d]&emH^OXFX] 1 KUFQPRHLSTKUF STFpP>KBFXFOX]
θ(x, t)A v3Y$ijOpY\FjHLSCHLW MN] PZrpYU[C]&OXI
]ueHdcXIJKUcKUILHLSTKUFXFp][T[C] [fYHL]&bdcQW&IJY\HLOXIJ]θ]>HdqY\ONH
cθKo
c]ueHdOXFX]$P>KBFpemHJY\FjHJ]cXr~NemSfijOX]
ijOXS0MXWc]FpMMNOgH~cQ]MN]b Y\HLWIJSfY\O YUcXc][TW] PZrQY\[T]OXI"emcW&PS QijOX]BA+ KBOXI/MNWHL]&ILbdSTFX]I/[TYHJ]bdcWIZYqHLOpIL]
θjFpKUOpe1W&P>IJSTUKUFQe [TY #$ 1"' %% KUOMN]7[TY9iBOQY\FjHLSCHLW
MX]4PZrQY\[T]OXIuAj¡0Y\FQe+OpF/BKU[TOXbd]7W&[CW&b ]&FjHJY\STIJ]VSCFQP>[TOpesMXY\FQe
ΩN[fY%YUILSfYqHJSCKBF]F/HL]&bdcpesMN]0[TY
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- -
iBOQY\FjHLSCHLW^MN]PZrpYU[C]&OXI]&emHs[C]4XST[TYUFMN]^P>]^ijOXS=]&emHscXIJKMXOXS H`cQY\I[C]uesemKBOXIZP>]&e]HlMN]^P>]^ijOXS3eLKUILHKUO"IL]&FBHJIL]%HLIZYU]&IJe+[T]&escpYUILKBSTe&AN¥`ONHJIL]&bd]FjHlMNSCH
d
dt
(∫
V
cθ dx
)
=
∫
V
f dx−∫
∂V
q · n ds @UAT@ KJo
∂V]ueH+[C]`KUIZMdMN]
V M W&[CW&bd]FjH+MN]leLOXI 1 YUP]
ds n]ueH1[fY4FXKBILb Y\[T]]>aHLW&ILST]OpIL]sOXFpS HJWlMN]
Vj]>H
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∫
∂V
q · n ds =
∫
V
divq dx.
s]VBILKBOXcpY\FjH[T]&e+MNS WIJ]FjHZeHJ]IJb ]ue1MN] @BAC@ ]>HONHLST[CSfeJY\FjH1[T]1 Y\SCHijOX]l[C]7UKU[TOXbd]`W&[CW&b ]&FjHJY\STIJ]V]ueH`ijOX]&[TPKUFpijOX]jSTFpMNW&cQ]&FpMXYUFBH`MNO/HL]&bdcpe NKBF/]&F"MXW&MNOXSCHs[¢ W&ijOpY\HLSTKUF$MN]^P>KBFpem]&ILqY\HLSTKUFMN]
[¢ WFX]&ILVBSC]c∂θ
∂t+ divq = f
@UA iBOpSpY0[CST]Od]F HJKUONHcKUSTFjH
x ∈ Ω]>H 0HJKUONH HJ]bdcpe
tAUlY\cXc][TKUFpe ijOX]s[ KUcWIZYqHL]&OXI MNSTU]IJVU]&FpP>]
]&emHlMNW>pFpScpY\Idivq =
N∑
i=1
∂qi∂xi
YU]uPq = (q1, ..., qN )t.
DG[1 YUONHdbdYUSCFjHJ]FpYUFBH IL]&[CST]I [T] *pONa MX]"PZrpYU[C]&OXI [fYHL]&bdcQW&IJY\HLOXIJ] ]>HdKUF 1 Y\SCHdYUcXcQ]&[ P>]iBO3 KBF"YUcXcQ]&[C[T]4OXFX] 1"''$ AX¡7YUFpe[C]^P&YUecXIJW&eL]FjHST[3e& YUVUSCHMN]^[fY [CKBS=MN]^¤XKBOXILST]IiBOpS=IL]&[CST]4[C] *pOXa"MN]PZrpYU[C]&OXIsMN]^b Y\FXSTkIJ]4cXIJKUcKUILHLSTKUFXFX]&[C[T]4Y\O"VUIZYUMNST]FjHsMN]4HL]&b cWIZYqHJOXIJ]
q = −k∇θ, @UA
KJok]&emH OXFp]PKUFpemHJYUFBHJ]1cKBeLS HJSCB] ijOXSNMNWc]FpM MNO9H~jc]MN]+b Y\HLWIJSfY\O3Y\cXc][TW]P>KUFQMNOpPRHJSCSCHLW
HLrX]&ILbdSfiBOp]UAX`YUcXcQ]&[CKBFpe`ijOX]4[ KBcQW&IJY\HL]OpIVBIJYBMNST]FjHs]&emHsMXW>pFXS3cpY\I
∇θ =
(
∂θ
∂x1, ...,
∂θ
∂xN
)t
.
4F PKUb9pSCFpYUFjHl[TY[CKBS MN] P>KBFpem]&ILqY\HLSTKUF @BA s]H7[fY[TKUS PKUFpemHLSCHLOXHLSTU] @UA QKBF KUNHJSC]&FjH7OXFX]W&ijOpYqHJSCKBF/cKUOpIs[TY%HL]&bdcQW&IJY\HLOXIJ]
θ
c∂θ
∂t− k∆θ = f,
KJo∆ = div∇ ]&emH`[¢ KUcWIZYqHJ]OXI[fY\cX[fYUPSC]&F"MNKBFXFXW4cpYUI
∆θ =N∑
i=1
∂2θ
∂x2i
.
DG[ 1 Y\ONH0Y7KUOXHL]I dP]>HmHJ]W&ijOpY\HLSTKUF$ijOXS=]&emH`qY\[fY\X[T]MXYUFpeHLKBONHl[T]MNKUb Y\STFX]ΩpOpFX]^IL]&[TY\HLSTKUF
Y\cXc][TW] $ %' & ' ijOXSlSCFpMXSTijOX]P] ijOXS7eL]cpYUeJeL] [TY 1 IJKUFjHLSTkIJ]KUOgYUO
Ecol
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x 7 /6*
n
Ω
∂Ω
@UAT@ ]&PRHJ]OXIsFpKUIJbdYU[OXFXSCHLW^KBILST]FjHLW0U]&IJe[ ]>aHLW&ILST]OXIuA
KUIZM∂Ω
MNOMXKUb Y\STFX] ]>HdOXFX]"Y\ONHJIL]/IL]&[TY\HLSTKUF ijOXSSCFQMNSTijOX]"ijOX][]&emH%[ W>HJY\H%STFXSCHLSfY\[sMN][fYHJ]bdcWIZYqHLOpIL]BAQ Y\I0P>KBFjB]FjHLSTKUF pKUF PZrXKUSfemSCH7[ SCFpemHJYUFjH
t = 0cKUOXI 2>HJIL]%[C]9HL]&b cQe7STFXSCHLSfY\[ ]>H
KBF/STbdcQKjem]^OpFX] 1$ $%
θ(t = 0, x) = θ0(x), @UA x
Koθ0]&emH^[TY31 KUFQPRHLSTKUFMN] MNSfeHJILSTXONHJSCKBF STFXSCHLSfY\[T]dMX]%HJ]bdcWIZYqHLOpIL]dMXY\FQe7[T] MNKUb YUSCFX]
ΩA 4F
P]ijOXS7PKUFpP]IJFX][fY P>KBFpMNSCHLSTKUF Y\OXa[TSCbdSCHL]ue P>][fY MNW&cQ]&FpM MNO PKUFjHL]aHL] cXr~NeLSTijOX]BA+yS0[C]MXKUb Y\STFX]]ueHeLOXcXcKBeLWpYUSCVBFX]I$MXY\FQeOXFHLrX]&ILbdKjeHZYqH HL]&b cWIZYqHJOXIJ]PKUFpemHJYUFBHJ] Y\[TKUIZe ijOXSCHmHJ] "b KNMNSCp]&I^[¢ W&PZrX]&[C[T] MN]&e0HJ]bdcWIZYqHLOpIL]ue [fY/HL]bdcWIZYqHJOXIL]dBWIJS p] [fY"PKUFpMXS HJSCKBFYUONa[TSTb SCHL]ue`MN]¡7SCIJSfPZrX[C]H
θ(t, x) = 0cQKBOXIHLKBONH
x ∈ ∂Ω]>Ht > 0.
@UA| yNS[T]/MXKUb Y\STFX]]ueHdemOpcXcQKjemW"YUMNSfY\QYqHLSfijOX]KBOHLrp]IJb SfijOX]bd]&FBHdSfemKB[CWMX]/[ ]aHLWIJST]OXI YU[CKBIJe[T] *pONaMN] PZrpYU[C]&OXI4emKBImHZY\FjH4YUOKUIZM ]ueH^FOX[ ]>H^[fYHJ]bdcQW&IJY\HLOXIJ] UWIJSCp]%[fY"P>KUFQMNS HJSCKBF YUONa[TSTb SCHL]ue`MN]l]OXb Y\FpF
∂θ
∂n(t, x) ≡ n(x) · ∇θ(t, x) = 0
cKUOXIHJKUONHx ∈ ∂Ω
]Ht > 0,
@UA KJo
n]&emH0[fYFpKUIJbdYU[C]%]>aHLW&ILST]OpIL]%OXFXSCHLW MN]
Ω BKUSTI7[fY/¤ STVUOXIJ]/@BAC@ RA _`FX] eLS HJOpYqHJSCKBFSTFjHL]I )
bdWuMNSTYUSCIJ] cQ]&ONH9Y\OpeJeLS YBKUSTIl[TSC]&O %[T] *pONa MX]dPZrQY\[T]OXI4emKBImHZY\FjH^Y\OKUIZM ]ueH^cXIJKUcKUILHLSTKUFpFX][YUOeJY\OXH MX]/HJ]bdcWIZYqHLOpIL]"]FjHLIJ]$[¢ ]>aHJWIJSC]&OXId]>H[ STFjHLWIJST]OXI ]>H[fY HJ]bdcQW&IJY\HLOXIJ]$UW&ILSCp]"[TYPKUFpMXS HJSCKBF"YUONa[TSCbdSCHL]&elMN]¤XKBOXILST]I
∂θ
∂n(t, x) + αθ(t, x) = 0
cKUOpIHLKUOXHx ∈ ∂Ω,
]>Ht > 0
@UA Ko
α]ueHOXFX]^PKUFpemHJYUFBHJ]7cKBeLSCHLSTU]UAOpSTeJijO= ST[ 1 YUONH`PZrXKUSfemSTI P\ ]&emHsOXFX]^MN]ueW>HJYUcQ]ueMN]4[TY%bdK )
MXW[TSTeJYqHJSCKBF FXKBOpe7YU[C[TKUFQe7eLW[T]&P>HLSTKUFXFX]&I7[fY"P>KUFQMNS HJSCKBF YUONa[CSTbdS HJ]&e^MN]d¡lSTIJSTPZrX[T]>H @BA | RA`YBe )eL]b%X[fY\FjH`]FNQF [¢ W&ijOpY\HLSTKUF Q[TYP>KUFQMNS HJSCKBFSCFpS HJSTYU[C]p]>H0[TY PKUFpMXS HJSCKBF Y\ONa$[CSTbdS HJ]&e0eLY\HLSfe 1 YUS HJ]&e
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cpY\Is[fY%HL]bdcWIZYqHJOXIL]NKUF"KUNHJSC]&FjH`[¢ W&ijOpY\HLSTKUF$MN]^[TYdPZrpYU[C]&OXI
c∂θ
∂t− k∆θ = f
cQKBOXI(x, t) ∈ Ω× R+
∗
θ(t, x) = 0cQKBOXI
(x, t) ∈ ∂Ω× R+∗
θ(t = 0, x) = θ0(x)cKUOXI
x ∈ Ω
@UA
v=]0cXILKBX[Tkbd] @UA +]&emH`MNKUFQP7PKUFpemHLSCHLOXW^M OXFX]4WuiBOQYqHLSTKUF"Y\ONaMXWIJSCBW]&e+cpYUImHJSC]&[C[T]&eb9OpFXSC]^MN]P>KUFQMNS HJSCKBFpe Y\OXa9[CSTbdS HJ]&e ]H1M OpFX]`P>KBFpMNSCHLSTKUF SCFXSCHLSfY\[T]UAB¥P&Y\OpeL]sMN][TY0cXIJW&eL]FQP>]sMN]sP>KBFpMNSCHLSTKUFpeY\ONa[TSTb SCHL]ue 3KUF MNS H9ijOX] @UA 7]&emHOXF !(# & %' & ' b Y\Sfe4KUF MNSCH9Y\OpeJemSijOX]P\ ]&emHOpF ! # & # %'$(" dPYUOpeL]4MX]^[TYdMNKBFXFXW]4STFXSCHLSfY\[T]^]F/HL]bdcpe&A & ( ' ¢ ¡0Y\FpeP>]7b KNMNk&[C]0MN]lcpILKBcpY\VjYqHLSTKUF MN]7[TY%PZrpY\[T]OpIBSC[FXKUOQe 1 YUONH+cXIJW&PSTeL]I[C]ue9OXFpS HJW&e%KUO MNSTbd]FpeLSTKUFpe%cXr~NemSfijOX]&e % [fYHJ]bdcWIZYqHLOpIL]
θe ]>aNcXIJSTb ]/]F 4]&[CSTF
K [TY
PZrpY\[T]OXIeLcW&P>SCQijOX]c]F0BKUOX[T]cpY\I ST[CKBVUIZY\bdbd]+cpYUI 4][TjSTF
J/(kg×K) [TY^PKUFpMNOQPRHLSTS HJW
HLrX]&ILbdSfiBOp] cpYUIOXFXSCHLWMN]bdYBeLeL] k]F 0BKUOX[T] bdk>HJIL]P&Y\IJILW cpY\I9jST[TKUVUIZY\bdbd]%cQY\I4][TSCF
cpY\I`eL]&PKUFpMN] Jm2/(kg ×K × s) RAQ¡% OXF$cQKBSCFjHlMN]^OX]^b YqHJrXWb Y\HLSfiBOp] XFXKBOpesY\[T[TKUFpeHJILkueemKBOXU]&FBHsKBOXX[TSC]&IlP>]uesOXFXSCHLWue Q]>Hlb 2bd]P]&e`PKUFpemHJYUFBHJ]&eX]FeLOXcXcKBeJY\FjHlijOX]
c]>HkqY\[T]FjH@
P>]&[TY IJ]ST]FjH YBMNSTb ]&FpeLSCKBFXFX]Is[T]&eVBIJYUFpMN]&OXIJecXr~NeLSTijOX]ue >A • & ( ' `KBOpesYUKBFpe+b ]&FjHLSTKUFXFXW0HJILKBSTe+H~cQ]uesMN]^P>KBFpMNSCHLSTKUFpeY\OXa[CSTbdS HJ]&eX¡lSTIJS )PZrX[C]H`]&OXb Y\FXF N¤pKUOXIJSC]&I bdYUSTe+ST[]&F]>aNSfeHJ]7M Y\OXHLIJ]&e ijOXSKUFjH+[TSC]&O/eLOXI[¢ SCFjHJWVUIZY\[TSCHLWlMN]7[TY1 ILKBFBHJSCk&IL]
∂ΩA1,SC]&Fe 0pI\KBFdcQ]&ONH+Y\SfeLWbd]FjHSTbdYUVUSTFX]IMX]&eeLS HJOpYqHJSCKBFpe Ko [T]&e+P>KBFpMNSCHLSTKUFpe YUONa
[CSTbdS HJ]&e0emKBFBH0bdW[fY\FXVBW]&e %¡7STILSfPZrX[T]>H0eLOXI∂ΩD
l]OXb Y\FpFemOpI∂ΩN
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∗
θ(t, x) = 0cKUOXI
(x, t) ∈ ∂Ω× R+∗eLSV (x) · n(x) < 0
θ(t = 0, x) = θ0(x)MpY\Fpe
Ω
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x ∈ R
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∗
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t]HV Y\[TKUIZe0ijOX] @UAT@x YU]uP
V = 0MNW&PILK TH]&FHL]bdcpe%P>]ijOXSSTFpMNSfijOX][fY + *Qk&PZrX] ,
MXOHJ]bdcpe&A7FMNS HijOX][ W&ijOpYqHJSCKBFM= YBMNU]uPRHLSTKUF]&emH ' # ]FHL]&b cQe HZY\FpMXSTe ijOX][ W&ijOpYqHJSCKBFMN]^[TY PZrQY\[T]OXIs]ueH ' # ]&F/HJ]bdcpe&A +]HmHL]^KBpeL]IJY\HLSTKUF"b YqHLrpWb YqHJSTijOX]]ueH4P>KBF 1 KUIJbd] [ STFjHLOXSCHLSTKUFcXr~eLSfiBOp] % P>]ILHJYUSCFQe`cXrXW&FXKUbdk&FX]&e0emKBFjHlILW&U]&IJeLSCp[C]ues]FHL]bdcpeM= YUONHLIJ]&eFXKBF P>KBb bd]^[fYdMNS OpeLSCKBFM OpFX]^VUKBONHmHJ]4MX]^[TYUS H`MXYUFpeOXFX]4HZYUeJem]^MN]4HJrXW >A • & ( '$ f 0 _`FX]Y\ONHJIL] MXS WIJ]FpP] 1 KUFQMXY\bd]FjHZY\[T]]FjHLIJ][C]ue WuiBOQYqHLSTKUFQe @BA $]>H @UAT@ =cKUILHL]seLOXI [C]ue cXILKBcXIJSCWHLW&e %%1 !# ("$ $ & ("$ AyOpcXcQKjemKBFpeijOX]9[C]^HJ]IJbd]9emKBOXIZP>]^]&emHlFOX[
f = 0AXDG[ ]&emH1 YUPSC[T]9MN]BKUSTIlijOX]9eLS
θ(x, t)]&emHlOXFp]9emKB[COXHLSTKUF
MX]l[ WuiBOQYqHLSTKUFMX]l[fYPZrpY\[T]OpI @BA Y\[TKUIZe UcQKBOXIHJKUONHλ > 0
θ(x
λ ,t
λ2 )]ueH+YUOpeJemS emKB[COXHLSTKUFMN]
[fYdb 2bd]9WuijOpYqHJSCKBF cQKBOXIlOpFX]9MNKBFXFXW]^STFXSCHLSfY\[T]9MNSCW&IL]&FjHL] >AQ¡7]9b 2bd]p]F eLOXcXcKBeJY\FjHlijOX][fY%jSCHL]ueLeL]
V]ueH`PKUFpemHJYUFBHJ] NeLS
θ(x, t)]&emHOpFX]^emKB[CONHJSCKBF/MN]4[ WuijOpYqHJSCKBF/M YUMNB]&P>HLSTKUF @UAT@
YU[CKBIJeθ(x
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Ecol
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V BKUSTI[fY 1 KUIJb9OX[T] @BAC@u %XST[=~YdMNKBFpP !(!#%% ' ' $ A •
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minx∈R
θ0(x) ≤ θ(x, t) ≤ maxx∈R
θ0(x)cQKBOXIHLKBONH
(x, t) ∈ R× R+,
Y\cXc][TW] !($ ! $' & & ' &gA +]>HLHL]$cXIJKUcXIJSCWHLW HJILkuedSCbdcKUILHJY\FjHJ] Y\OpeJeLSpSC]&FMXOcQKBSCFjHMN]OX]$b YqHLrpWb YqHJSTijOX]ijOX]$cXr~NemSfijOX] eL]"VBWFXW&IJYU[CSfeL]"YUONa 1 KUIJbd]&e cX[TOpe VBWFXW&IJYU[C]ueMN]^[¢ W&ijOpY\HLSTKUF$MN]P>KBFU]&P>HLSTKUF )GMNSCOQemSTKUF @UAT@&t +]>HlMN]4[ WuiBOQYqHLSTKUF$M YUMNB]&P>HLSTKUF @UAT@ RAXlKUOpe[¢ W>HJOpMNST]IJKUFpecXIJW&PSTeLWbd]&FBH`MXYUFpe[fY eLOXSCHL]BA •
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emSfijOX]&e&A v=]$XONHMN]P>]HmHJ]"W&FOXbdWIZYqHLSTKUF]&emHMN]MNWVjY\VU]&I%MNkuedb Y\STFBHJ]FpYUFjH [T]&edcXIJSTFpP>STcpY\[T]&eP>[fYUeJem]ueM W&ijOpYqHJSCKBFpeYUONagMNWIJSCBW]uecpYUImHJSC]&[C[T]&eijOX] FXKUOQeW>HJOpMNST]IJKUFpecpY\I/[TY emOXSCHL]+]H"MN]b KBFjHLIJ]I7ijOX] P>]uelWuiBOQYqHLSTKUFQe 7KBOX]FjH7OpFIU[T]SCbdcKUILHJYUFBH4MXY\FQe7MN]ue7MXKUb Y\STFX]&e7eLPSC]&FjHLSCQijOX]&eHLIJk&edMNSTU]IZe&A ¡7W&eLKUIJb Y\Sfe FXKUOpe YBMNSTb ]&FpeLSCKBFXFXKUFQeHJKUONHJ]&ed[T]&edqY\IJSTYUX[T]&e P]iBOpSscQ]&ILbd]>HMN]XaN]IHLKBONHL]ue[C]ue`P>KBFpemHJY\FjHJ]&eMN]&e`b KNMNk&[C]ueWVjY\[T]&e "@UA "!$#%'&)(+*,-&%./+KBbdb ]^FpKUOpeB]FXKBFpeMN]^[C]4BKUSTI[ WuiBOQYqHLSTKUF$MN]^[fYdPZrpY\[T]OXISTFjHL]IJSC]&FjHsPKUbdbd]4b KNMNk&[C]
MXY\FpesMX]^FXKUb%XIL]&ONacXIJKUX[Tkbd]uesMN]&eseJP>ST]FpP]&esMN]^[ SCFXVBWFXST]OpI&A 4[T[T]e W&PILSCH
∂u
∂t−∆u = f
MpY\FpeΩ× R+
∗
u = 0eLOXI
∂Ω× R+∗
u(t = 0) = u0MXY\Fpe
Ω
@UAT@
DG[1e Y\VBS HM OXFX] WuijOpYqHJSCKBF M= KBIJMNIJ]"@ ]FHL]&b cQe^]>H9M KUIZMNIJ]V]&F ]&eLcpYUP] [¢ KUIZMNIJ]d]&emH9P>][TOXSMN]&e%MNWIJSCBW]ue4cpY\ILHLST][T[C]ue^[T]&e^cX[TOpe9W[T]UW&]&e RA 7F MXSCIZYijOX]P>]HmHL]/W&ijOpY\HLSTKUF ]&emHcpY\IZY\KU[TSfiBOp] UKBSCI"cX[TOpe$[TKUSTF[fYyKBOpe ) eL]&P>HLSTKUF5@BA |XA RAs`KBOpe$YUKUFQe/MNW 7 O P>]ILHJYUSCFp]&e"cXIJKUcXIJSCWHLWue/MN]P>]>HLHL] W&ijOpYqHJSCKBF %SCIJILW&U]&IJeLSCpSC[TS HJW]FgHJ]bdcpecXILKBcpY\VjYqHJSCKBF jSCHL]ueLeL]STFNpFpSC]+]H"cXIJSTFpP>STcQ]MNO$bdY\aNSCb%OXb$A; 11 T O F6E /ILN/NF6EE I^E+Q7I^NP[ E/I:[O#E JI^N /I M W+QWE JWK3ILNM F6FTO#K+E E"H/NPOEJO%QM E EE/O QE/Y#F NMI T X NI YZ[ E ] JE TSFVN-[QM N2OJE 10 W 2[3TQM N2OE TqK323T E/[I ]54 TOF[O JNPYGTJMOE /NI O#W]76 O [O#E $MY E/O#FM NPO 80 E+F #T3K+E:9 NPO /NJFVE
Ω = (0, 1)E"Q N2OqF+[/NJFVE 2[#E
Ecol
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@ut 7 /6*
f = 0]NM Q
u(t, x)[O#E F6N-[QM N2O ILW2[ M U/ILE JE ] 4+] 6 O YZ[ QM M TO%Q 10 WP[#TQM NPO #TJI
uE+Q;EJO
MOQWPILTOQ 3TJI:ILT/NI Qx92W+Q6T MI 10 WT M QW
1
2
d
dt
(∫ 1
0
u2(t, x) dx
)
= −∫ 1
0
∣
∣
∣
∣
∂u
∂x(t, x)
∣
∣
∣
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2
dx
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K"N2OQMOY EJO%Q W/I>M T E F"[I [0, 1]9(QE E 2[#E
v(0) = 09
W/I>M $E 10 MO#WT M QW E2N$MO#K%T/ILW∫ 1
0
v2(x) dx ≤∫ 1
0
∣
∣
∣
∣
dv
dx(x)
∣
∣
∣
∣
2
dx.
6 O WP[JMI^E T JWK3ILNM F6FTO#K+EGE"H/NPOEJO%QM E EGEJOZQE/Y F E ∫ 1
0 u2(t, x) dx
]
7 , "!$#% "!,#,%
x
u(x)
f(x)
Ω
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STXIZYqHLSTKUF3Aj YUIs]>aN]bdcX[T] N]&F"MN]&ONa"MXSCbd]FQemSTKUFpeM ]&eLcpYBP>]4][T[C]4]ueHsOXF$bdKNMNk[T]4cQKBOXIsW>HJOpMNST]I[T]&elSCXIZYqHJSCKBFpelM= OXFp]9bd]b%XIJYUFX]W&[TYBeHJSTijOX]^HJ]FpMXOX] P>KBb bd][fYc]&YUOM OXFHZY\b9KUOpI >A 4FOpFX]MXSCbd]FQemSTKUF M ]uemcQYUP>]&]&[C[T]]&emH YUOpeLeLSNY\cXc][TW]+W&ijOpY\HLSTKUF%MX]&e P>KBIJMN]ue=STXIJYUFjHL]&e&A¥`O IL]&cQKje [fY%b ]&b9XIZY\Fp]7KNPPOXc]0OpF"MNKBb Y\STFX]4cX[TYUF
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u]&emHlemKB[CONHJSCKBF/]ueH`MNKBFXFXW&]^cpY\I
∂2u
∂t2−∆u = f
MpY\FpeΩ× R+
∗
u = 0eLOXI
∂Ω× R+∗
u(t = 0) = u0MXY\Fpe
Ω
∂u
∂t(t = 0) = u1
MXY\FpeΩ
@UAT@&
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uA 7FMNSTIZY"ijOX]P]>HLHL]WuiBOQYqHLSTKUF]&emHr~cQ]&ILKU[TSTijOX] UKBSCI
cp[COpe`[CKBSCF"[TY yKBOpe ) eL]&P>HLSTKUF@BA |XA RA
Ecol
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/- / @B@
; 11 T f OGFVE T3K"EREJO $MY E/O#FM NPON = 1
0 E+F 3T3K"E] O F"[/NF6E P[#E E+F N2OO#W+E+FMO/M QM T E%F
u0E"Qu1F6N2OQ E+FGX N2O3K7QM NPOF ILW2[ M U/ILE+F 9 E"Q 2[#E
f = 0T E%K Ω = R
] O ONJQEU1[O#E /I MY M QM E E u1
]9W/I>M EJI P[E
u(t, x) =1
2(u0(x+ t) + u0(x− t)) +
1
2(U1(x+ t)− U1(x − t)) ,
@UAT@&n E+FQ T0FVN-[QM N2O [O/M P[E E] TO#F T0K T#FVF6E E+F X N2O3K7QM NPOF ILW2[ M U/ILE+F]
v1 WuijOpYqHJSCKBF MN]ue4KUFpMN]ue0cpY\ILHJYUVU]%YB]&P9[¢ W&ijOpY\HLSTKUF M= YBMNU]uPRHLSTKUF @BAC@S l[TY/cXIJKUcXIJSCWHLW SCb )cQKBImHZY\FjHL]dMN] !$! ' ' ' A$4F ]> ]>H3[¢ 4 aN]IZP>SfP>]"@UA XA bdKBFBHJIL] ijOX] eJYemKB[CONHJSCKBF]&F OXFcKUSTFBH
(x, t)FX] MNW&cQ]&FpMcQYUe4MN] HLKBONHL]ue7[T]&e4qYU[C]&OXIJe4MN]ue4MNKUFXFpW]&e0STFXSCHLSfY\[T]&e
bdYUSTelem]&OX[C]&bd]FjH7MN]uesqY\[T]OXIZe`MXYUFpesOXFSTFjHL]IJqY\[T[C]^IJ]&emHLIJ]STFjH`Y\cXc][TW & $ $! $%1 KUOP UFX]9MX]9[TOXbdSCk&IL]QUKBSCI`[fY¤ STVUOXIJ] @UA >AlY\cXc][TKUFpe7ijOX]9P>]HmHJ]9cXIJKUcXIJSTW>HLW9F= ]ueHcpYUe cpYUImHZY\VUW&]1cpYUI [¢ W&ijOpY\HLSTKUF%MX][fY0PZrpY\[T]OXI cXOpSTeJijO= ST[N]&emHP>[fY\STI 7HLIZYU]&IJe3[TY/1 KUIJb9Op[C] @BAC@&x iBOp]0[fY eLKU[TONHLSTKUF"]F
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t]&F −t p[fY 1 KUIJbd]9MN][ WuijOpYqHJSCKBFFX]9PZrpYUFXVU]^cpYBeA 7FcQ]&ONHMNKUFpP+STFjHLWVBIL]&I3, [¢ W&ijOpY\HLSTKUF MN]ueKUFQMN]&e^B]IZe^[T]&e4HJ]bdcpe%cQKjemSCHLS 1 eKBOFXWVjYqHLS 1 eMN][TYb 2&b ]
bdYUFXSTkIJ]UA 7F$MNS HlijOX]^[ W&ijOpYqHJSCKBF"MX]&eKUFQMN]&e]ueH ' & !' A; 11 T T 9W/I>M $E/I 2[#E T F6N -[%QM NPO ] T[/NMOQ
(x, t)O#E JW /E/O E+F N2OO#W+E+F
MO/M QM T E%Fu0E"Qu1
2[ 0 Q7ILT E/ILF EJ[ILF T E/[ILF;F+[I E FVEPY E/OQ [x− t, x+ t]] 9WJI M $EJI(T[FVFMP[E
u(−t, x) E%FQ:F6N -[%QM NPO JE ] TOF Ω×R−∗P[/M Q*QE K 2#TO JEJI ERFM POE E T M QE+F6FVE MOJM QM T E
u1(x)]
t
x−t x+t x
(x,t)
@UA ¡7KUb Y\STFX]4KUOP UFX]^MN]^MNW&cQ]&FpMXY\FQP>]MN]^[ WuijOpYqHJSCKBF"MN]ueKUFpMX]&e&A
; 11 T , OFVE /ILN/NF6E E W/Y NPO%Q7ILE/IG[O /I>MO#K#M/E JEK"N2O#F6EJI TQM N2O JE 10 WJOEJI $M E/N2[I 10 W 2[3TQM N2O JE%FN2O JE%F ](FVTO#F[QM M F6EJI T X NI YZ[ E E"H M KM QE ]"] 6 O [OE MYGE/O#FM N2O80 E+F #T3K+E:9$NPO /NJFVE
Ω = (0, 1)E"Q NPO0F"[/NF6E
f = 0] $NM Q
u(t, x)[O#EZFVN-[QM N2O ILW2[ M U/ILEE
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@S 7 /6*
]"] 6 OYZ[ QM M TOQ 10 WP[#TQM NPO #TJI ∂u∂t
E"Q E/O MO%QW 2I@TO%Q 3T/I9I@T/NI Q x9PW"Q6T MI 10 WT M QW
0 W/O#E/I M Ed
dt
(
∫ 1
0
∣
∣
∣
∣
∂u
∂t(t, x)
∣
∣
∣
∣
2
dx+
∫ 1
0
∣
∣
∣
∣
∂u
∂x(t, x)
∣
∣
∣
∣
2
dx
)
= 0.
N2O3K -[ILE E+Q K+N2Y3T/ILE/I K"E P[/M2F6E 3T#F6FVE /NP[I 10 W 2[3TQM N2OE T K323T E/[I*]
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Y\HmHJ]STFBH`OpF"WHJY\H ' % XP\ ]&emH ) )GMNSTIL]^ijOX]u(t, x)
YUMNbd]>H`OpFX][TSCbdSCHL]u∞(x)
iBOQY\FpM[T]/HL]bdcpe
tHJ]FpM U]IZe%[¢ SCFXpFXSA yNKUOXB]FjH SC[]ueHdSCFjHJWIJ]&eJeLYUFBH MN]"P&Y\[fP>OX[T]I MNSTIL]uPRHL]&bd]FjHdP]>H
WHJY\HleHZYqHLSTKUFpFpY\STIL]BAN¡7YUFpesP>]P&YUejcKUOpI`OXF"HL]&ILbd]^emKBOXIZP>]f(x)
STFpMNW&cQ]&FpMXY\FjHlMNO"HJ]bdcpeXKUFIJW&eLKUOXHOpFX]4W&ijOpYqHJSCKBF"MXO$MX]ONaNSTkbd]^KUIZMNIJ]7]&F"]uemcQYUP>]
−∆u = fMXY\Fpe
Ωu = 0
emOXI∂Ω,
@UA \t ijOX]/[¢ KUFY\cXc][T[C]"v3YUcX[TYBP>ST]FKBOW&ijOpYqHJSCKBF MN]"v3Y\cp[TYBP>]UA 7F MNSTIJY ijOX]"P>]HmHL]"W&ijOpY\HLSTKUF]ueH]&[C[TSTcNHLSfijOX] BKUSTIcX[TOpe[TKUSTFd[TY%yKUOpe )Gem]uPRHLSTKUF"@UA|NA >AUs]&bdYUIJijOXKBFpe ijOX]`[T]lv3YUcX[fYUP>ST]Fd]ueH+YUOpeJemS[fY$U]&IJeLSTKUF eHZYqHLSTKUFpFpY\STIL] MN] [ W&ijOpYqHJSCKBFMN]&e^KBFpMN]ue @BAC@un RA=v3]v3YUcX[TYBP>ST]F STFBHJ]IJSC]&FBH9YUOpeJemSMpY\FpeMN]0HLIJk&e+FXKUb%XIJ]ONaMNKBbdYUSCFp]&eMN]&eeJP>ST]FpP]&eMN]7[ SCFXVBWFXST]OpI&A Y\I]>aN]bdcX[T] @UA \t bdK )MXW[TSTeL]7[T]4MNWcX[fYUP]bd]FjHU]&ImHJSTP&Y\[QM= OXFp]0bd]&b9XIZY\FX]7W[fYUemHLSfijOX]0eLKUOpb SfeL] %OXFX]/1 KUIZP>]lFpKUIJbdYU[C]f]>HsXaNW&]4eLOXI`eLKUF$P>KBFBHJKUOXIuA
7 , "!$#%(8*/ # !"% /v1 WuiBOQYqHLSTKUFMX]yNPZrXIJzNMNSTFXVU]&I3MNW&PILSCH [ WUKB[COXHLSTKUFMN]+[TY1 KBFpPRHJSCKBFM KUFpMN]
uM OXFX]+cpY\ILHLSfP>Op[C]
eLKUOpb SfeL]7OXF9cK\HJ]FjHLST][VAq`Y\cXc][TKUFQe ijOX]
u(t, x)]&emH OXFp]1 KBFpPRHJSCKBF%MN]
R+×RN lqY\[T]OpIJeMpY\Fpe
C]H7ijOX]9eLKUF$bdKNMNOX[T]%Y\OP&Y\IJILW |u|2 e& STFjHL]&ILcXIJk>HJ]P>KBb bd][fY MN]&FpemSCHLW%MN]9cXIJKUQY\XST[CSCHLWcKUOpIdMXW>HL]uPRHJ]IdiBOp]/[fYcQY\ILHLSfP>OX[T]"eL]HJILKBOXU]"Y\OcQKBSCFjH
(t, x)Av=]"cK\HL]&FjHLST][
V (x)]&emHdOXFX]
1 KBFpPRHJSCKBF dqYU[C]&OXIJeIJW]&[C[T]&e&AXv3Y 1 KUFQPRHLSTKUFM KUFQMN]u]&emHlemKB[CONHJSCKBF"MX]
i∂u
∂t+ ∆u− V u = 0
MXYUFpeRN × R+
∗
u(t = 0) = u0MXY\Fpe
RN
@UA N@
DG[7F= ~YcQYUeMN]P>KBFpMNSCHLSTKUFgY\ONa[TSCbdSCHL]&e/Y\cXcQY\IJ]FjHL]ueMXY\FQe @BA X@ dcXOpSTeJijOX] [ WuiBOQYqHLSTKUFgY[TST]O MXY\Fpe7HJKUONH[ ]uemcQYUP>] ijOXS F3 Y"cpYBe0MX] KUIZM>A=`W&YUFXbdKUSTFpeFpKUOpe4B]IJILKBFpe7ijO= OXF PZrXKUSCa+mIZY\SfemKBFXFpY\p[C] ,%M ]uemcQYUP>] 1 KUFQPRHLSTKUFpFX][MXYUFpe4[T]&ijOX][ FpKUOpe^PZrX]&IJPZrp]IJKUFpe7[TY$eLKU[TONHLSTKUF]FjHJIJYfFX]MX]1 YUP>HLK OXFp]P>KBFpMNSCHLSTKUF MN]MXW&P>IJKUSfeJeLYUFpP>] $[ STFNpFXSMN]
uijOXS1c]ONHde STFBHJ]IJcXIJW>HL]&IP>KBb bd]
OpFX]emKBImHJ]0MX]P>KUFQMNS HJSCKBF"YUONa[TSCbdSCHL]ue [¢ SCFXpFXSA; 1 T # O0F6E /I^N /NJFVEJEW/Y NPO%Q7ILE/IJE%F /I>MO#K#M/E+F E0K"NPOFVE/I TQM NPO JE10 W/O#E/I M E/NP[I 10 W 2[3TQM N2O E $K 2I MO JEJI ] "] $NM Q
u(t, x)[O#E F6N-[QM N2O ILWP[ M UJI^E E ] &EJOZ[O#E
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/- / @
MY EJOF7M N2O 0 E%F 3T3K"EP[/MJWK3ILNLQ E/ILF "WJI^N TJMOF7M P[E ∂u∂x
NILFP[E |x| → +∞ ] N2OQ7ILE/I P[E/N2[IQN2[QE X NPO#K"QM N2O W/I>M T E v(t) N2O0T
R(
∂v
∂tv
)
=1
2
∂|v|2∂t
,
N R W+F7M 2O#E T #TJI^QM EILW+E E E"Qv E K+N2Y E7H3E K+N2OV[ 2[#W E
v]6 O Y0[ QM M TO%Q 10 W 2[3TQM N2O
#TJIuE"Q E/O MOQWPILTOQ 3T/IILT/NI Q
x9$W"Q6T MI 10 WT M QW 0 WJOEJI $M E
∫
R
|u(t, x)|2 dx =
∫
R
|u0(x)|2 dx.
6 O YZ[ QM M TOQ 10 W 2[3TQM N2O3TJI ∂u∂t
9 YGN2OQ7I^EJI 2[#E∫
R
(
∣
∣
∣
∣
∂u
∂x(t, x)
∣
∣
∣
∣
2
+ V (x) |u(t, x)|2)
dx =
∫
R
(
∣
∣
∣
∣
∂u0
∂x(x)
∣
∣
∣
∣
2
+ V (x) |u0(x)|2)
dx.
: % #,%
v3]9eL~emHLk&bd]MN]9v3YUbdW^]&emH`OXF PYBescpY\ILHLSfP>Op[CST]I`MN]uesW&ijOpY\HLSTKUFpeleHZYqHLSTKUFpFpY\STIL]uesMN][ W&[TYBeHJS )P>SCHLW4[CSTFXWuY\IJSTeLW]0iBOpSb KNMNW&[CSfeL]FjHs[C]ueMXW1 KUIJb YqHJSCKBFpeM OpF"eLKU[TSTMX]0eLKUOpe[ r~cK\HLrpk&eL]0MN]4c]>HJS HJ]&eMNW1 KBILb YqHJSCKBFpe ]>H1MX]scQ]HLSCHJe1MXWcX[fYUP]bd]FjHJe UKBSCI [fYyKUOQe )Gem]uPRHJSCKBFd|XA pAC@cQKBOXIcX[TOpeMX]`MNW>HZY\ST[TeemOXI0[fYbdKNMNW[TSfeLY\HLSTKUF>AQ KUOpI0KBNHL]&FXSTI7[T] eL~NeHJkbd]%MX] v Y\bdW KUF emOXcpcQKjem]dijOX]%[T]demKB[CSfMN]9]ueHrXKUbdKUVBkFX]7STeLK\HJILKBcQ]7]>HsijO= ST[ ]&emH+paW4eLOXIeLKUFQKBIJMANv YcXIJSCFQP>STcpY\[T]0MXS WIJ]FpP]7YB]&P`[T]&e+bdK )MNk[T]&e cXIJW&PW&MN]&FjHJe ]&emH1ijO= SC[Xe& YUVUSCH STPSpM OXF ' ' & M W&ijOpY\HLSTKUFpe\P\ ]&emH ) )GMNSTIL]MN]cX[TOpemST]OpIJeW&ijOpYqHJSCKBFpe^P>KBOXcX[TW]ue4]FjHLIJ] ][T[C]ueA v=]eLKU[TSTMN] YUOIJ]cKBe4KNP&P>OXc]OXFMNKBbdYUSCFp]
ΩMN] [ ]&eLcpYUP]
RN AUyKBOpe [ YBPRHLSTKUFdM OXFX]1 KBIJP] f SC[XeL]sMNW1 KBILbd] q]H1PZrpYUijOX]cKUSTFBH x eL]MXWcX[fYUP]]&F x+u(x)A
v3Y 1 KBIJP]f(x)
]ueHOXFX]1 KBFpPRHJSCKBFU]uPRHJKUIJSC]&[C[T]lMN]ΩMXY\Fpe
RN jPKUbdbd]l[T]7MNW&cX[fYUP>]&bd]FjH u(x) A+]4MX]IJFXSC]&Is]&emHlemKB[COXHLSTKUF$MN]
−µ∆u− (µ+ λ)∇(divu) = fMXYUFpe
Ωu = 0
emOXI∂Ω
@UA KJo
λ]>HµeLKUFjH4MX]ONa PKUFpemHJYUFjHL]&eMNSCHL]&e^MX] v Y\bdW =P&Y\IZYUPRHJWIJSTemHLSfijOX]&e7MNObdY\HLW&ILSfY\O rXKUbdK )
VUkFp]lSfemKUHLIJKUc]lMNKUFjH]&emHP>KBFpemHLSCHLOXW7[C]4emKB[CSfMN]BAU KUOXIMN]ue1IZY\SfemKBFpebdWuPY\FpSTijOX]ue1P>]ue+PKUFpemHJYUFjHL]&eUWIJSCp]FjH
µ > 0]H
2µ+Nλ > 0Apv3Y PKUFpMXS HJSCKBF$YUONa[TSCbdSCHL]uelMN]¡7SCIJSfPZrX[C]H`cQKBOXI
uHJIJYBMNOXSCH
[C] 1 YUS H`ijOX]^[T]emKB[CSfMN]4]&emH`emOXcpcQKjemW0XaNW^]>H`STbdbdKUXST[CSfeLW4eLOXI`eLKUF"QKBIJM∂ΩA
v3] em~NemHLk&b ] @BA Y W>HLWWuP>IJS H ]&FFXKUHJY\HLSTKUFU]&P>HLKBILST][T[C]BA yNS`KUFFXK\HJ]fi]>Hui cKUOXI
1 ≤ i ≤ N [C]ue P>KUbdcKBeJY\FjHL]uedMN]
f]HuMpY\Fped[fYpYUeL]"P&Y\FXKBFXSfiBOp]$MX]
RN @UA 9]ueHW&ijOXSTYU[C]&FjH
−µ∆ui − (µ+ λ)∂(divu)∂xi
= fiMXYUFpe
Ω
ui = 0emOpI
∂ΩcQKBOXI1 ≤ i ≤ N
A3s]&b Y\IZiBOpKUFpe4ijOX] emS(µ + λ) 6= 0
=YU[CKBIJe4[T]&e4WuijOpYqHJSCKBFpe4cQKBOXI9PZrpYBijOX]P>KUbdcKBeJY\FjHL]
uieLKUFjHlP>KBOXcX[TW]ue`cpYUI`[C]HL]IJbd]9MN]9MXSCB]IJVU]FQP>]UANo1jSfMN]&b bd]&FBH Q]&F MNSTb ]&FpeLSCKBF
N = 1X[T]^em~NemHLkbd]^MN]v Y\bdW4F= Y ijO= OXFX]^eL]OX[T]^W&ijOpY\HLSTKUF"]>H`eL]^ILWuMNOXSCHsYUOv3Y\cp[TYBP>ST]F=A
Ecol
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@&x 7 /6*
7 : % #,% 7% v3]em~NemHLkbd] MN]yjHJKUB]&e4bdKNMNW[TSTeL]d[¢ W&PKUOX[T]bd]&FBH9M= OXF *pOpSTMN] SfeLijOX]&ONaSTFpP>KBbdcXIL]ueLeLSTX[C]
/c]>HLSCHL] S HJ]&eJem]BA 7F eLOXcXcKBeL] ijOX]%[C] *pOXSfMN] KNPP>OpcQ] OXF MXKUb Y\STFX]Ω]>H^ijO= ST[ YUMNrpkIJ] [TY
cQY\IJKUSpMN]0P>]&[COXS )GP>S BP\ ]&emH ) )GMNSTIL]`ijOX]7eLY^SCHL]ueLeL]`]&emH1FOX[T[C]0emOpI1[fY^cpY\IJKUS P>]0ijOXSQPKUFpMNOpS H ^OXFX]PKUFpMXS HJSCKBF/Y\OXa[CSTbdS HJ]&esMN]4¡lSTIJSTPZrX[T]>H RApyKBOpe[¢ YUP>HLSTKUF"M OpFX]!1 KUIZP>]
f(x) OpFX]!1 KUFpP>HLSTKUF/MX]
ΩMpY\FpeRN N[fY9SCHL]&eJeL] u(x) OpF/B]&PRHJ]OXI 1]>Hs[TY%cXIJ]&eJemSTKUF p(x) OXF$eLP&Y\[fY\STIL] eLKUFjHeLKU[TONHJSCKBFpeMX]
∇p− µ∆u = fMXYUFpe
Ωdivu = 0
MXYUFpeΩ
u = 0eLOXI
∂Ω
@UA
Koµ > 0
]ueH%[TYSfeLPKBeLS HJWMNO *pOpSTMN]BA s]&b Y\IZiBOpKUFpeijO= ]Fcp[COpe MN]ueNW&ijOpY\HLSTKUFpe ∇p −
µ∆u = f P>KBILIJ]&eLcQKBFpMXYUFBH [fY '(% ( '% & ' &
ST[`~ YOXFp] Y\ONHJIL]"W&ijOpY\HLSTKUFdivu = 0
Y\cpcQ]&[CW&] 1$ /1 & !$ ' ' iBOpSPKUIJIL]uemcKUFQM [TY 1'(% & ' ' >ANyNS=[TY MNSTbd]FpeLSCKBFM ]uemcQYUP>]0]&emH
N = 1N[T]
eL~NeHJkbd]`MN]lyjHJKUB]&e ]&emH+eJY\Fpe STFjHLW&I 2H1PY\I KBFdUKUSCH 1 YUPSC[T]bd]&FBH1ijOX]s[fY4S HJ]&eJem]s]ueHFOX[T[T]s]>H+ijOX][fY0cXIJ]&eJemSTKUF%]&emH OpFX]cXILSTbdS HJSCB]sMN][TY/1 KUIZP>]BAq Y\I P>KBFBHJIL]]F MNSTbd]FpeLSTKUF
N ≥ 2q[T]eL~NeHJkbd]sMN]
yHLKUB]&e+YXST]FOpFem]&Fpe %B]&FcpY\ILHLSfP>Op[CST]I UST[Q]aNSTemHL]0MN]&ePZrpY\bdcpe+MN]lSCHL]ueLeL]&e1SCFpPKUbdcXIJ]&eJemSTX[T]&eFpKUF/HLIJSCSfY\ONa cXIJ]FpMNIJ] NcQY\Is]>aN]&b cp[C]XOXF"ILKUHJYqHJSCKBFXFX]&[ >A
7 , "! #% & % 7FP>KUFQemSfMNkIJ][fYMNW71 KUIJb YqHLSTKUFW[fYUemHLSfiBOp]%M OXFX] cX[fYUijOX]9cp[TYUFX]%M WcQY\SfeLeL]OXI7cQ]HLSCHL] FpW)
VB[CSTVU]uY\X[T]sMN]qYUFBH1eL]&e1Y\ONHJIL]ue MNSCbd]&FpemSTKUFQe >ABySpKBF%FpK\HL]Ω[fY^emOpI 1 YBP>]bdKq~U]FpFX]sMN]s[TY4cp[TYBiBOp]
]Hf(x)
OXFX]'1 KUFpP>HLSTKUFMN]ΩMXY\FQe
R[TYdIJW&eLOX[CHJY\FjHJ]^FXKUIJb Y\[T]4MX]&e1 KBIJP]&eXY\[TKUIZe[TY PKUbdcQK )
eJY\FjHJ]1FXKBILb Y\[T]+MXO%MNW&cX[TYBP>]&b ]&FjHu(x)
OpF eJPYU[TYUSCIJ] ]ueH eLKU[TONHLSTKUF9MX]+[ WuijOpYqHJSCKBF%MN]ue cp[TYBiBOp]&e MNSCHL]&e`]F&*p]>aNSTKUF
∆ (∆u) = fMXY\Fpe
Ωu = 0
emOpI∂Ω
∂u∂n = 0
emOpI∂Ω
@UA qx
Ko^KUF4FXKUHL] ∂u∂n = ∇u·n YB]&P n [C]1U]&P>HL]&OXI=FXKBILb Y\[UOXFXSCHLW]ajHJWIJSC]&OXI ∂Ω
As]&b Y\IZiBOpKUFpe=ijO= ST[e& Y\VUSCHM OXFX] W&ijOpY\HLSTKUF Y\ONaMNWIJSCBW]ue4cpY\ILHLST][T[C]ue^MNOijOpYqHJILSTkbd]dKUIZMNIJ]%]&F]uemcQYUP>] Y\cXc][TW]YUOpeJemSjXS )[fY\cX[fYUPSC]&FRA\7 ]&emH cKUOXIZijOXKUSBSC[]ueH FXW&P]&eJeLYUSCIJ]1M YUKBSCI MN]OXa^PKUFpMXS HJSCKBFpe YUONa^[CSTbdS HJ]&e&A+]ue P>KUFQMNS HJSCKBFpe Y\ONa[TSTb SCHL]ue HLIZYUMNOpSTeL]FjH [¢ ]FpP&YUemHLIJ]bd]FjH MN][TY7cX[fYUijOX] cpYBe MN]MNWcX[fYUP]bd]FjHFpS3MN]4ILKUHJY\HLSTKUF$MNO$QKBIJM"MN]^[fY%cp[TYBiBOp] RAs]&b Y\IZiBOpKUFpe ijO= SC[7]&emHcQKjeLeLSTX[C] MX] 7OpemHLSCp]I[ W&ijOpYqHJSCKBFgMN]&ecX[TYBijOX]&e @UA qx cQY\IOXF
IZY\SfeLKUFXFX]&bd]FjH4YBem~bdcNHJK\HLSfijOX] "cpY\ILHLSTIMNOem~NemHLk&b ] MN]v Y\bdW @BA 0MXY\FQe4[C]uiBOp][ KUF 1 YUS HHJ]FpMXIL]/[¢ WcpYUSTeJeL]OXI MN][fY cX[fYUijOX]U]IZe .&WIJKpA DG[e& Y\VUSCH9[ M OXF ]aN]bdcX[T]/MN]"bdKMXW[TSTeJYqHJSCKBFb Y\HLrXW&bdY\HLSfijOX]UA
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/ / / @|
! ' ) /&) - % !-&* &'$ ! % * * !"* /8!,( % #%'& * #%¥-cpYUImH4MXY\Fpe0ijOX][fijOX]&e0PYBe`HJILkuelcQY\ILHLSfP>OX[TSC]&IJepST[ ]&emH7STbdcQKjeLeLSCp[C] MN] P&Y\[fP>OX[T]I7]>aNcX[TSTPS HJ])
b ]&FjH9MN]&eeLKU[TONHJSCKBFpeMN]ueMNSCW&IL]&FBHZe^bdKNMNk[T]&e^cpILWuem]&FBHJW&e^PS )GMN]&eJeLOpeADG[+]&emH9MNKUFQPdFXW&P]&eJeLYUSCIJ]M YBKUSTI4IL]uP>KUOpIJe^YUO P&Y\[fP>OX[ FOXbdW&ILSfijOX]emOpIKUIZMNSTFpYqHJ]OXI^cKUOXI^]ueHJSCbd]&I9ijOpY\[TS HZYqHJSCB]bd]FjH]>HiBOQY\FjHLSCHJY\HLSTU]bd]&FBHlP]&e7emKB[CONHJSCKBFpe&AQv=]9cXILSTFpPSCc]9MN]^HJKUONHJ]&e7[C]ue`bdW>HJrXKNMN]&e7MN]IJW&eLKU[TONHLSTKUF$FO )b W&ILSfijOX]$MN]&edWuiBOQYqHLSTKUFQe YUONa MNW&ILSTUW&]&e%cpY\ILHLST][T[C]ue ]&emH M= KBNHL]&FXSCI MN]uedqY\[T]OXIZe FjOpb W&ILSfijOX]&eMNSTeJP>IJk>HJ]&e P\ ]&emH ) )GMNSTIL] ]&F FXKBb9XIJ] pFXS ijOXS !!$ (" ]F OXF eL]Fpe%P>KUFB]FpYUX[C] cXILWuP>Sfem]&I [fY^eLKU[TONHLSTKUFd]>aXYBPRHL]BAU¡0Y\FpeP]`cXIJKNP>W&MXWsSC[ 1 Y\ONH1pSC]&F 2>HLIJ]`P>KBFpeJP>ST]FjH1MN]`MN]&ONa cKUSTFjHJe1 KUFpMXYUbd]FjHJYUONa % cXIJ]bdSTkIJ]bd]FjH3KUF FX]/PY\[fP>Op[C]cQYUe9MX]&e emKB[CONHJSCKBFpe]aXYUPRHJ]&e^b YUSTe%Y\cXcXIJK )PZrXW]ue 3MN]&ONaNSCk&bd]bd]FjH KUF ' ' [C]cXILKBX[Tkbd] ]FIJ]cXIJW&eL]FjHZY\FjH9MN]&e'1 KUFpP>HLSTKUFpe9cpY\IOXF$FXKUb%XIL]7pFXS MN]^qY\[T]OXIZe XPU ]ueH ) ) MXSCIJ]0ijOX] !#' ' ' 1 ' %' ' DG[B]>aNSfeHJ]1MN] FXKBb9XIJ]OpeL]&e=b WHLrXKNMN]ue=M Y\cXcpILKaNSTbdY\HLSTKUF0FOXbdWIJSTijOX]MN]ue3eLKU[TONHLSTKUFQe=M W&ijOpY )
HLSTKUFpe4Y\OXaMXWIJSCBW]&e7cpYUImHJSC]&[C[T]&e&A`KUOQelcXIJW&eL]FjHJKUFpe7b Y\STFBHJ]FpYUFjH0OXFp] MN]ue0cX[COQe0YUFpP>ST]FpFX]&e0]>HMN]&e4cX[TOpeeLSTb cp[C]ue =YUcXcQ]&[CW&]db WHLrXKNMN]MX]&e^MNSCW&IL]&FpP>]ue0pFXST]&e FXKUOpe4B]IJILKBFpe7cX[COQe^[CKBSCFOXFX]Y\ONHJIL]"bdW>HLrpKMX] MNSCHL]MN]uedW[TWbd]FjHJe%pFpSTe RA KBOXI emSTbdcX[CSCp]&I [fYcXIJW&eL]FjHJY\HLSTKUF FXKUOQe%FpKUOpe[CSTbdS HJKUFpe $[TY MXSCbd]FQemSTKUFOXF M= ]uemcpYBP>] UKUSTI[fYyKBOpe )GeL]&PRHJSCKBF XA XA $cKUOpI9[T]&eMNSTbd]FpeLSTKUFpeemOXcWIJST]OXIJ]&e RAN`KBOpe+F= YUQKBIJMXKUFpe+cQKBOXI[¢ STFpeHZY\FjHsiBOp]0[T]&ecXIJSTFpP>STcQ]uecXIJY\HLSfijOX]&eMN]^P>]HmHL]0bdW)HLrXKNMN] PU ]ueH ) ) MXSCIJ][fYPKUFpemHLIJOpPRHJSCKBF MN]$P>]$ijO= KBFYUcXcQ]&[C[T]/MX]&e '(" & ' ' & ( ' ' A`KUOQelIJW&eL]IJUKBFpe`cKUOXI0[T] +rpYUcXSCHLIJ]$[TY 7OpeHJS PYqHJSCKBF HJrXWKBILSfijOX]%MN]dP>]ue7eJPZrXW&bdYBe P\ ]&emH ) )MNSCIJ][ WHLOpMX]%MN][T]OpI7P>KBFU]IJVU]&FpP>] ]&FijOX][ em]&Fpe`[T]&e7emKB[COXHLSTKUFpe`YUcXcXIJKNPZrXW]ue`MNSfeLPILkHL]&elemKBFBHcXILKNPZrX]ueMN]&e`eLKU[TONHJSCKBFpe]>aXYUP>HL]uePKUFjHLSTFOX]&e RA
x
t
(t , x )n j
∆ xj
n ∆ t
@UA x .+Y\ST[C[fY\VB]7]&F$MXS WIJ]FpP]&epFpSC]ueA KBOXI MNSfeLPILWHLSfem]&I"[T] P>KBFjHLSTFjOpOXb eLcpYqHJSCK )¢HJ]bdcQKBIL]&[ KBFSTFjHLIJKMXOXS H OpF !#' '! 1
∆x > 0]HlOXF !' $ & !2'
∆t > 0ijOXS3eL]IJKUFjHs[C]uescX[TOpesc]>HLSCHL]ue`WuPZrX][T[C]uesIJ]cXIJW&eL]FjHLW&]&e
cpY\I[fY$bdW>HJrXKNMN]FOXbdWIJSTijOX]BA 7FMNW>QFXS H%OXFb Y\ST[C[fY\VB] KBOMN]&e%P>KKUIZMNKUFpFXW]ue0MXSTeJP>IJk>HJ]&eMN]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
@u 7 /6*
[ ]&eLcpYUP]0]HlMNO/HL]bdcpe UKBSCIs[fYd¤ SCVBOXIL]d@BA x (tn, xj) = (n∆t, j∆x)
cKUOXIn ≥ 0, j ∈ Z.
7FFpK\HL]un
j
[fYqY\[T]OXI9M= OXFp]/eLKU[TONHLSTKUF MNSfeJP>IJk>HL]YUcXcXIJKNPZrXW]YUOcQKBSCFjH(tn, xj)
]>Hu(t, x)[fYeLKU[TONHLSTKUF$]aNYBPRHJ] STFpP>KBFXFOX] >AQv3]cXIJSCFpPSCc]%MX][TYb WHLrXKNMN] MN]&e7MNSCW&IL]&FpP>]uepFXST]&e7]&emH7MN]
IJ]bdcX[fYUP]I%[C]ue9MNW&ILSTUW&]&ecQY\I%MX]&e MNS WIJ]FQP>]&e9pFXST]&e%]F ONHLST[CSfeJY\FjH MN]ue 1 KBILb%OX[C]ue%MX]3( Y~j[TKUIMpY\Fpes[T]&eJijOX][T[T]&esKBFFXWVB[CSTVU]^[T]&esIJ]&emHL]ueAp YUI`]>aN]&b cp[C]pKUFYUcXcXIJKPZrp]^[TY MNW&ILSTUW&]em]uP>KBFpMN]^]F]uemcQYUP>] [C]v Y\cX[fYUPSC]&F]FMNSTbd]FpeLSTKUF"OXFcpYUI
−∂2u
∂x2(tn, xj) ≈
−unj−1 + 2un
j − unj+1
(∆x)2 @UA U|
Ko/[ KUF$IL]uP>KBFXFpY fH[TY 1 KUIJb9Op[C]^MN]'( Y~[CKBI
−u(t, x−∆x) + 2u(t, x)− u(t, x+ ∆x) = −(∆x)2∂2u
∂x2(t, x)
− (∆x)4
12
∂4u
∂x4(t, x) +O
(
(∆x)6)
@UA \
yNS∆x
]&emH +mcQ]HLSCH3, X[fY 1 KUIJb9OX[T] @BA B| +]&emHlOXFp] +KUFpFX],Y\cXcpILKaNSTbdY\HLSTKUF ][T[T]]&emH`FpYqHJOXIJ][T[C]b YUSTecpYBeOXFXSfijOX] >Apv3Y 1 KUIJb9Op[C] @BA B| +]&emHlMNSCHL] PYUI]&[C[T]^]&emH`em~bdW>HJILSfijOX]4]F
jA
KUOpI`MNSTeJP>IJW>HJSTeL]Is[ W&ijOpYqHJSCKBF"MX]4PKUFU]uPRHJSCKBF ) MXS OpeLSCKBF∂u
∂t+ V
∂u
∂x− ν ∂
2u
∂x2= 0
@BA j ST[ 1 YUONHlY\OpeJeLS MXSTeJP>IJW>HJSTeL]Is[T]0HL]IJbd]MN]^P>KBFjB]&P>HLSTKUF=AX_lFX] 1 KUIJb9OX[T]4P>]FjHJILW&]4MXKUFXFX]
V∂u
∂x(tn, xj) ≈ V
unj+1 − un
j−1
2∆xDG[+FX]IJ]&emHL]cp[COpe%ijO= 1 YUSCIJ][fY$b 2bd]/PZrXKjem] cKUOpI[TYMNWIJSCBW] ]&F HL]bdcpe&A 7F Y ]FpPKUIJ] [C]PZrpKUSCaMXYUFpe[fY1 KUIJb9OX[T]MN]/MNS WIJ]FQP>]&epFXST]&e % eLKUSCH%P>]&FBHJILW&] 3eLKUSCH%MNWuP>]&FBHJILW&]UA-4aNYUbdSCFXKBFpeHJILKBSTe 1 KBILb%OX[T]&e +FQYqHLOpIL]&[C[T]&e3,ZA@UA4FcXIJ]bdST]Is[TSC]&O N[TY MXS WIJ]FpP]0pFXST]P>]FjHJILW&]
∂u
∂t(tn, xj) ≈
un+1j − un−1
j
2∆tP>KBFpMNOXSCH Y\O5eJPZrXW&bdYPKUbdcX[Tk>HL]&bd]FjH eL~jbdWHLIJSTijOX]cpYUI IZY\cXcKUILH n]H
j Y\cpcQ]&[CW
eLPZrpWb YdP>]FjHJILW4KUO ' " & ("$%(' un+1
j − un−1j
2∆t+ V
unj+1 − un
j−1
2∆x+ ν−un
j−1 + 2unj − un
j+1
(∆x)2= 0.
@BA U ¥`OpeJeLS +mFpYqHJOXIL]&[ ,]>HWSfMN]FjH eLKUSCH ) SC[ '(" & ' 1!## %'$$ ' ''$' !!$ (" ' MX][¢ W&ijOpY\HLSTKUFMN]PKUFU]uPRHJSCKBF ) MXS OpeLSCKBF @UA B UKBSCI
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / / @q
[ ]aN]bdcX[T]"FOXbdW&ILSfijOX] MN][fY¤ STVUOXIJ] @UA| !lKUOpe-7OpeHJS Q]IJKUFpe P]>HmHJ]SCFQPY\cQYUP>SCHLW MXOeJPZrXWb Y%Y\cXcpILKNPZrX]&I[TY%emKB[COXHLSTKUF]aNYBPRHJ]lMXY\FQe1[T]7v3]bdbd]7XA XA XA\ KUOpI+[ STFpemHJYUFBH BSTF )MNSfijOXKUFQe eLSTb cp[C]&b ]&FjH ijOX][fY7MNS 8P>OX[CHLWcXIJKqSC]&FjH MNOdPYUIJYBPRHLk&IL]+P>]FjHJILWMN][TY0MNSCW&IL]&FpP>]pFpSC]^ijOXS3YUcXcXIJKPZrp]7[fY MNWIJSTUW]0]F"HJ]bdcpe&A
NA`_lFMX]ONaNSTkbd] PZrXKUSCa]ueH[fY [fYMNS WIJ]FQP>]$pFXST] MNWuP>]FjHJILW&] Y\bdKUFjH KBFgIL]&b KBFjHL]$[C]HJ]bdcpe NKBF$cpY\IJ[C]4Y\OQeLeLS=MN] ' " & C; ' ((( $
∂u
∂t(tn, xj) ≈
unj − un−1
j
∆t
ijOXS=PKUFpMNOpS HlYUO"eJPZrXWb Y
unj − un−1
j
∆t+ V
unj+1 − un
j−1
2∆x+ ν−un
j−1 + 2unj − un
j+1
(∆x)2= 0.
@UA \n
XA`v3]7HJILKBSTeLSTkbd]^PZrXKUSCa]&emH`[T]em~bdW>HJILSfijOX]^MNO$cXILWuP>WuMN]FjH %X[fY MNS WIJ]FQP>]4pFXST]MNWuP>]FjHJILW&]YqY\[ KBFYqY\FpP]7MXYUFpe1[T]`HJ]bdcpe jKBFcpYUIL[T]7YUOpeLeLSQMX] ' ("$ & C; '$ !(( ' '
∂u
∂t(tn, xj) ≈
un+1j − un
j
∆t
PKUFpMNOpS H`Y\OeJPZrXWb Yun+1
j − unj
∆t+ V
unj+1 − un
j−1
2∆x+ ν−un
j−1 + 2unj − un
j+1
(∆x)2= 0.
@UA Ut
v Y9MNSC WIJ]FpP]lcXIJSCFQP>STcpY\[T]l]&FBHJIL]0P>]ue+MN]&ONaMN]IJFXST]IZe1eJPZrXWb YBe]&emHijOX] @BA Un ]ueHMNSCH & !1 P&Y\IsST[ 1 Y\ONH`IJW&eLKUOQMNIL]0OXF$em~NemHLk&b ]M= WuiBOQYqHLSTKUFQe+[TSTFXW&YUSCIJ]&ecKUOXI`P&Y\[fP>OX[T]I[T]&esqYU[C]&OXIJe(un
j )j∈Z
]&F21 KBFpPRHJSCKBFpesMN]&eqYU[C]&OXIJecXIJW&PW&MN]&FjHL]&e(un−1
j )j∈Z
XHJY\FQMNSTesijOX] @UA Ut 1]&emH`MNSCH $ !1 cpOXSTeJijO= SC[MNKUFXFp]sSCbdbdW&MNSfYqHJ]bd]FjH1[T]&e1qY\[T]OXIZe
(un+1j )j∈Z
]F 1 KUFpP>HLSTKUFMN]ue(un
j )j∈Z
Av=]MNWuPY\[fY\VB] MX]@emOXI^[ SCFpMXSTP]
n]&FBHJIL] [T]&e^eJPZrXW&bdYBe @BA Un 7]>H @UA Ut 0F3 ]ueHW&jSfMN]&b bd]&FBH
iBO3 YUcXcpY\IJ]FjHcXOXSfeLijO= KUF$c]ONH`IJWWuP>IJSCIJ]4MN]4bdYUFXSTkIJ]0WuijOXSCqYU[C]&FBHJ] @UA Ut emKBOpe[TY 1 KBILbd]
unj − un−1
j
∆t+ V
un−1j+1 − un−1
j−1
2∆x+ ν−un−1
j−1 + 2un−1j − un−1
j+1
(∆x)2= 0.
¡0Y\FQe [C]ue HLIJKUSfedeLPZrXW&b YUedijOX]FXKUOQedU]FpKUFpedMN]MNWpFXSTIST[s~ YXST]Fge 0XI OXFX]MNKBFXFXW]SCFXSCHLSfY\[T] cQKBOXI0MXWb Y\IJIL]&Il[T]&e7S HJWIZYqHJSCKBFpel]&F
n% [T]&e7YU[C]&OXIZelSTFXSCHLSfY\[T]&e
(u0j )j∈Z
eLKUFjH4MNW>pFpSC]ue cpY\I=]aN]bdcX[T] >cpYUI
u0j = u0(j∆x)
KJou0]&emH3[TYsMXKUFXFXW&] SCFpS HJSTYU[C]1MX] [¢ W&ijOpY\HLSTKUF^MN]1PKUFU]uPRHLSTKUF )
MNS OpeLSCKBF @UA B RAs]b YUIJijOXKBFpe7ijOX]dcKUOXI4[T]$+b Y\OpYUSTe3,%eJPZrXWb Y"P>]FjHJILW @BA U lSC[ ~ Y"OXFX]MNS 8POX[ HJW emOXcpcX[CW&bd]FjHJYUSCIJ] YUOMXWb Y\IJIJYUVU] %cKUOpI
n = 1KUFY/YUOpeJemS ]&eLKUSTF MN] PKUFXFpYTHJIL]
[C]ue7qYU[C]&OXIJe(u1
j )j∈Z
ijO= SC[1 Y\OXH^MXKUFpP PYU[TPOX[C]&I0YUONHLIJ]bd]&FBH cQY\I4]>aN]bdcX[T] cpYUI^Y\cXcX[TSfPYqHJSCKBFM OpF$MX]&esMN]&ONa"YUONHLIJ]&eseJPZrXWb YBe >A
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
@u 7 /6*
Z ` ~0
@UA| yXPZrXWb Y P>]&FjHLIJW7STFpemHJY\p[C]YB]&Pν∆t = 0.1(∆x)2
A
7 ,& ! /+ % / & ., "!$#%'& )(8*& % /+KBb bd]&F>KBFpe cpY\I 1 Y\STIL]ijOX][fijOX]&e HJ]&emHJe FOXbdWIJSTijOX]ue3HJILkue emSTbdcX[C]ue MXY\Fpe [T]P&YUe KJo
V = 0]Hν = 1
NP\ ]&emH ) )GMNSTIL]lijOX] / ' ' ' & ( ' & ( ' $ "%'$ 7F$PZrXKBSTeLS H`PKUbdbd]P>KUbdbd]^P>KBFpMNSCHLSTKUF"SCFXSCHLSfY\[T]^[fY 1 KBFpPRHJSCKBF
u0(x) = max(1− x2, 0).
KBOXI^cQKBOXUKBSCIPKUbdcpYUIL]&I0[T]&e9emKB[CONHJSCKBFpe4FOXbdWIJSTijOX]ue^Y\cXcpILKNPZrXW&]&e^YU]uP9[fYeLKU[TONHLSTKUF ]>aXYBPRHL] @UAT@x FXKUOpe%UKBOpMNIJSCKBFpe^HJIJYqYUSC[T[C]&I%eLOXI [C]"MNKUb Y\STFX]"SCFXpFXS
Ω = R P\ ]&emH ) )GMNSTIL]P&Y\[fP>OX[T]I
cKUOpI4PZrQYUijOX]n ≥ 0
OXFX] STFNpFXSCHLW MN]dqY\[T]OXIZe(un
j )j∈Z
b Y\Sfe7[ KUIZMNSCFQYqHL]&OXI0FX]d[C] cQ]&ILbd]HJecQYUe^PYUIeLY"bdWbdKBSCIJ] ]&emH^pFXST] !-4F cXIJ]bdSTkIJ] Y\cXcXIJKaNSCb Y\HLSTKUF FXKBOpe^IJ]bdcX[fY >KBFpe4MNKUFQP
RcQY\I`[T] +mVUIZY\FpM ,^MNKUb YUSCFX]Ω = (−10,+10)
b9OXFXS MN]9PKUFpMNSCHLSTKUFQe`Y\ONa$[TSCbdSCHL]&e7MN]9¡7SCIJSfPZrX[C]H&AlKUOpe4YBMNbd]>HmHJILKBFpe0[TY"qY\[TSfMNS HJW MN] P>]>HLHL] Y\cpcXILKaNSTb YqHLSTKUF ijOXS ]&emHP>KUFXpILbdW&]%cpYUI4[C]ue^P>KUb )cQY\IZY\SfemKBFpe4FOXbdWIJSTijOX]ueP>S ) MX]&eJemKBOpe >A3`KBOpe^paKBFpe^[T]cpYBe9M ]&eLcpYUP]
∆x = 0.05%3ST[1~ Y
MXKUFpPxjtX@qY\[T]OXIZe(un
j )−200≤j≤+200^PYU[TPOX[T]IuAU`Y\cpcQ]&[CKBFpe cKUOXIbdW&b KBSCIJ]sijOX]`[T]&e qY\[T]OpIJe
unj
P&Y\[fP>OX[TW]ue7cQY\I4[¢ KUIZMNSTFpYqHJ]OXI^eLKUFjH4]FjHJYBPZrXW]ue0M= ]&ILIJ]OXIZe0M YUILIJKUFQMNS ]H^FX] emKBFBH^MNKBFpP9cQYUe[T]&e9qY\[T]OXIZe]aNYBPRHJ]&e9MX]&e9eJPZrXW&bdYBe9MNSfeJP>IJ]>HJe % FpW&Y\Fpb KBSCFQe MXY\Fpe9[C]ue9P&Y\[fP>OX[fecXIJW&eL]FjHJW&eSfP>S P]&e%]IJIL]&OXIZeM Y\IJILKBFpMNS+eLKUFjHHJK\HZY\[T]bd]FjH%FpWVU[TSTVU]&YUX[T]&e]H%FX]"emKBFjH%]FY\OpPOXFX]b YUFXSCk&IL][TYP&Y\OpeL]9MN]uelMNSC WIJ]FjH7cXrpWFXKBb k&FX]&e7ijOX]FXKBOpe7YU[C[TKUFQesKUpeL]IJU]&I&AyOXI`HJKUONHJ]&el[C]ue`pVBOXIJ]&esFXKBOpeIJ]cpILWuem]&FBHJKUFpe [fY^eLKU[TONHLSTKUF ]aNYBPRHJ] UP&Y\[fP>OX[TW]`YB]&P[fY!1 KUIJb9OX[T]`]>aNcX[TSfP>SCHL] @UAT@x \]>H+[TY^eLKU[TONHLSTKUFYUcXcXIJKNPZrXW]0FOXbdWIJSTijOX]^P>KBFpeLSTMNW&ILW&]UAsW&VU[TKUFpe7HJKUONHMN]demOpS HJ] [T] emKBImH4MNO eLPZrXW&b Y/P]FjHLIJW @UA \ %=P>KBbdb ] FXKBOpe0[¢ YSCKBFpe4Y\F )
FpKUFpPW pP]9eLPZrpWb Yd]&emHlSTFpPYUcpY\X[T]MN]9P&Y\[fP>OX[T]I7MN]&e`eLKU[TONHJSCKBFpe`Y\cpcXILKNPZrXW&]&e`MX][¢ W&ijOpY\HLSTKUFMN]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / / @un
[TYdPZrpYU[C]&OXI&Aph7OX]&[=ijOX]^emKBS Hs[T]PZrXKUSCaMNO$cpYUeMN]4HJ]bdcpe∆tpP]4eJPZrXW&bdY ]ueH '# NP\ ]&emH )
)GMNSCIJ]/ijOX]"[fYeLKU[TONHJSCKBF FOXbdWIJSTijOX]"KBeJP>ST[C[T]$MN]/b YUFXSCk&IL]"FXKBF KUIJFXW]$emS[ KBFMNSTbdSCFOX]$[T]&eYU[C]&OXIZe%MN]ue9cQYUe
∆x]H
∆tA1+]/cprXWFXKBbdkFX]HLIJk&edPYUIJYBPRHLW&ILSfemHLSfiBOp] ]>HM Y\cpcpY\IJS HJSCKBF HLIJk&e
IJYUcXSTMX] ]ueHSC[T[TOpeHJILWlcpY\I[TY9¤ STVUOpIL]%@UA|NAUDGFQemSfeHJKUFpeeLOXI1[T]1 YUS HsijOX] ( ' ( '$' ("MN]&e+cpYUe
∆t]>H
∆xNKBFKUpeL]IJU]7P>]&e+KBeJP>ST[T[TY\HLSTKUFpe FXKBFcXr~NemSfijOX]&ejXSC]&Fe 0XI >A 7FMNSCHijOX]0[C]
eLPZrXW&b Y^]&emH+STFpP>KBFpMNSCHLSTKUFXFp][T[C]&b ]&FjH+STFpeHZY\X[T]UA_`Fp]7OpemHLSCQP&YqHLSTKUFILSTVUKBOXIL]&OpeL]s]FeL]IZYMNKBFXFXW]Y\O$PZrpY\cXSCHLIJ]^emOpSCqY\FjH BKUSTIs[C]^v3]bdbd]NNA NA >A
@~e `
@UA /yNPZrXWb Y STb cp[CSfP>SCHL]9YB]&Pν∆t = 2(∆x)2
A¥*[¢ KUcpcQKjemWMXOcXILWuP>WuMN]FjH%eJPZrXWb Y 3[T]eLPZrpWb Y$SCbdcX[TSfP>SCHL] @UA \n ^PYU[TPOX[C]MX] +mQKBFXFX]&e3,
emKB[CONHJSCKBFpe=YUcXcXIJKPZrpW]&eMX] [¢ W&ijOpY\HLSTKUF^MN][fYsPZrpY\[T]OpI ( '$( '$ ' [T]cpYUe3MN] HL]&b cQe∆t
UKBSCI[TYd¤ SCVBOXIJ]%@BA RA 4F"cpY\ILHLSfP>Op[CST]I KUF"F= KBpeL]IJU] 7mY\b Y\SfeM KjeLPSC[T[fYqHLSTKUFQe+FOXbdW&ILSfijOX]&eijOX][=ijOX]emKBS H [C]PZrXKBS aMN]ue cQYUe
∆t]>H
∆xA7F MNS H ijOX][C]seJPZrXWb Y`STbdcX[TSTPS HJ]]ueH STFpP>KBFpMNSCHLSTKUFXFX]&[C[T]bd]&FBH
eHZY\X[T]UA+KBFpeLSTMNW&ILKBFpe4b Y\STFjHL]FQY\FjH[C]eJPZrXW&bdY$]aNcX[CSfP>SCHL] @UA Ut %3MN]&e]>aNcQW&ILST]FQP>]&eFjOpb W&ILSfijOX]&e
b KBFjHLIJ]FjH 1 YBP>ST[C]&bd]FjHdijOX]eL][TKUF [T]&edqYU[C]&OXIJe MNOcpYUedMN]/HL]&bdcpe∆tMN]ue%KjeLPSC[T[fYqHLSTKUFQe9FO )
b W&ILSfijOX]&e4YUcXcpY\IZY\SfeJem]&FBH7KUOFpKUF UKBSCI4[fY/¤ STVUOXIJ]/@BA RAv3Y[CSTbdS HJ]MN]deHZY\XST[CSCHLWd]ueH!1 YUPSC[T] HLIJKUOXB]I]>aNcQW&ILSTbd]FjHJYU[C]&bd]FjH %XijOX]&[=ijOX]0eLKUSCHs[C]4PZrXKBS aMN]uecpYUe
∆t]H
∆xijOXS [TY
P>KUFQMNS HJSCKBF2ν∆t ≤ (∆x)2
@UA X@ [C]$eLPZrpWb Y]&emH eHZY\X[T] HJYUFpMNSfe%ijOX]$eLS @UA X@ F= ]&emH cpYBe9BWIJS QW] Y\[TKUIZe[T]$eLPZrXW&b Y]&emH STF )eHZY\X[T]UA 7FMNSCHlijOX]^[T]eLPZrXW&b Y%]aNcX[CSfP>SCHL]9]&emHlP>KUFQMNS HJSCKBFXFX]&[C[T]bd]FjHlemHJYUX[C]BAXv3Y PKUFpMNSCHLSTKUFMN]eHZY\XST[CSCHLW @BA p@ ]ueH '$ $ ' & ( ' ' ' !#'' ' & ! ' ' !#'' !$ $ '$ %% ' ' & ( ' A 41[C[T] 1 ONH9MNWuP>KUOpU]ILHL] ]F@un\ YqY\FjH^[ YUcXcpY\IJSCHLSTKUF MX]&e^cXIJ])
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
Ut 7 /6*
bdST]IZe KUIZMNSCFQYqHL]&OXIZe"! ^cpYUI +KBOXIJYUFjH ¤XIJSC]uMNILSfPZrpe ]>H v=] ~BA 4[T[C]$cKUILHL]$MN]cXOpSTe [T]"FpKUb MN] #%s) ' # '$% $("' ) A
t0~e `
t0~e ` n
@UA yXPZrXWb Y ]>aNcX[TSTPS HJ]YU]uPν∆t = 0.4(∆x)2
rpYUONH s]>Hν∆t = 0.51(∆x)2
pYBe >AlKUOpe Y\[T[CKBFpe 7OpeHJS Q]I XILSTkB]bd]FjH P>]>HLHL]P>KUFQMNS HJSCKBF9MN]eHZY\XST[TS HJW OpFX]YUFpY\[T~Nem]1cX[COQe3cKUOpe )
eLW]4eL]IZY9]]uPRHLOpW]^Y\O"cXIJKPZrQY\STF/PZrpYUcXS HJIL] RANsW&W&P>IJSTUKUFQe+[T]^eLPZrpWb Y9]aNcX[CSfP>SCHL]^eLKUOpe[fY'1 KBILbd]
un+1j =
ν∆t
(∆x)2un
j−1 +
(
1− 2ν∆t
(∆x)2
)
unj +
ν∆t
(∆x)2un
j+1. @UA
yNS1[fYP>KUFQMNS HJSCKBF ¤ v]ueH%UWIJSCpW]=Y\[TKUIZe @UA 0bdKUFjHLIJ]ijOX]un+1
j
]&emH9OXFX]PKUb%XSCFQY\SfemKBFPKUFU]aN]dMX]&eqYU[C]&OXIJe^YUO HL]&b cQecXIJW&PW&MN]&FBH
unj−1, u
nj , u
nj+1
HLKUOQe[T]&ePKj]78P>ST]FjHJe%MXY\Fpe[C]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / / X@
b ]&b9XIJ]9MN] MNIJKUSCHL]%MX] @BA 0 seLKUFjHlcKBeLSCHLS 1 el]H7[T]OXI4emKBbdb ]9YUONH%@ RA 4FcQY\ILHLSfP>OX[TSC]&IemS [TYMNKUFXFpW]4SCFpS HJSTYU[C]
u0]&emH`QKBILFXW&]4cpY\I`MN]&ONa/PKUFpemHJYUFBHJ]&e
m]HMHL]&[C[T]&eliBOp]
m ≤ u0j ≤M
cQKBOXIHLKBONHj ∈ Z,
Y\[TKUIZe`OXFX]dIJW&POXILIJ]FQP>] 1 YBP>ST[C] bdKUFjHLIJ]%ijOX] [C]uelb 2&b ]ue0SCFXW&VBYU[CSCHLWue7IJ]&emHL]&FjH7IJYUSC]uelcKUOpIlHJKUOpe[C]ueHL]bdcpe`OX[ HJWIJSC]&OXIZe
m ≤ unj ≤M
cKUOXIHJKUONHj ∈ Z
]>H`cKUOXIHJKUONHn ≥ 0.
@BA
v3YdcXIJKUcXIJSTW>HLW @UA 1]&bdc2uPZrX][T]9eLPZrpWb YdM KjeLPSC[T[T]IlMN]9bdYUFXSTkIJ]^FXKUF$KUIJFXW]%QSC[ ]&emHlMNKUFpPeHZY\X[T]%eLKUOpel[TY/P>KBFpMNSCHLSTKUF¤ vAv YcXIJKUcXIJSTW>HLW @UA s]ueH4Y\cXc][TW]%cXILSTFpPSCc] MXO b YqaNSCb%OXbMNSTeJP>IJ]>H%QST[ e& Y\VUSCHlMN]%[¢ W&ijOXSTqY\[T]FjH ' MNO cpILSTFpP>STc]9MNO b YqaNSTb9OXb ' cQKBOXIl[T]&eemKB[CONHJSCKBFpe]>aXYBPRHL]ueiBOp]^FXKUOpesYBKUFpeO %[fY s]&bdYUIJijOX] @UA NAT@&tpAyNOXcXcKBeLKUFpesYUO"PKUFjHLIZY\STIL]4ijOX]4[TYdPKUFpMNSCHLSTKUF ¤ vFX]^emKBS HscpYBeUWIJSCpW]PU ]ueH ) ) MXSCIJ]0ijOX]
2ν∆t > (∆x)2.
¥`[TKUIZe jcQKBOXIP>]ILHJYUSCFp]&eMNKUFXFpW]&e+STFXS HJSTYU[C]ue1[T]4eLPZrXW&b Y^]&emHSTFpemHJYUX[C] ST[ cQ]&ONH 2>HLIJ]7emHJYUX[T]lcKUOXIP>]ILHJYUSCFp]&e9MXKUFXFXW&]&e%SCFpS HJSTYU[C]ue +m]>aXP>]&cNHLSTKUFpFX][T[C]ue,1% cpY\I%]>aN]bdcX[T] eLS
u0 ≡ 0! >A IJ]FpKUFpe9[TY
MNKUFXFpW]4SCFpS HJSTYU[C]MNWpFXST]^cpY\Iu0
j = (−1)j
iBOpS=]&emHsXSC]&F$OXFXS 1 KUIJb W&bd]FjH`QKBILFpW]UAX_lFPY\[fP>Op[=emSTbdcX[C]4bdKUFjHJIL]^ijOX]
unj = (−1)j
(
1− 4ν∆t
(∆x)2
)n
iBOpS PILKBS H0]F bdKNMNOX[T]%B]IZe`[ STFNpFpS [TKUIZeJiBOp]nHL]&FpMU]&IJel[¢ STFNpFXS PYUI
1− 4 ν∆t(∆x)2 < −1
Av=]eLPZrXW&b Y%]aNcX[CSfP>SCHL]4]ueHlMNKBFpP0SCFQeHZY\X[T]emS[fYdP>KBFpMNSCHLSTKUF¤ v F3 ]ueH`cpYBeseLY\HLSfe 1 YUS HJ]UA; 11 , D3E [QE K+E"Q E7H3E/I@KM K"E E%FQ E Y N2OQ7ILE/I P[E E FVK32#W/Y T MY M KM QE ] 9T EK V = 0
9 W/I>M $E0T[#F6FM E/I>MO3KM/EP[SYGT%HMYZ[Y $M FVK/I^E+Q+] OSMY/NJFVEJE%FRK"N2O $M QM N2O#F T[%H MY M QE+F EMI M K 2 E+Q9 K:0 E%FQ*\ \ $MI^E 2[#E T X NI YZ[ E ] E%FQ T T E /N2[I 1 ≤ j ≤ J
E"QRNPO H3E
un0 = un
J+1 = 0/NP[I QN2[Q
n ∈ N] $NM Q E/[%H K+N2O#FQ6TOQE+F
m ≤ 0 ≤ MQE E+F 2[#E
m ≤ u0j ≤ M
/NP[I1 ≤ j ≤ J
] 9WJI M $EJI P[E 10 N2O /EJ[%Q /M EJOqK+T K3[ E/I JE YGTOJM U/ILE [OJM P[E E+F
un+1j
E/OqX NPO#K"QM N2O JE%Fun
j
] 0NPO%Q7ILE/I P[E /NP[I QN2[#F E+F0QE/Y#Fn ≥ 0
N2O T E/O3K"NILE E+FMOW T M QW+F
m ≤ unj ≤M
/N2[I1 ≤ j ≤ J E"Q K"E%K#MFTOF K"NPO $M QM N2O F+[I
∆tE"Q
∆x"]
yNSFXKUOQe^YUKUFQe "c]O cXIJk&e4W&[COpPSTMXW [fY$ijOX]&emHLSTKUF MN] [fY"emHJYUXST[CSCHLW MNOeJPZrXWb Y"]>aNcX[TSTPS HJ] FXKUOpe+F= YUKBFpeIJSC]&F/MNSCHsemOXIeJY9P>KBFU]IJVU]&FpP>]BPU ]ueH ) ) MNSTIJ]lemOpIeJY9P&Y\cpYBP>SCHLW %Y\cXcpILKNPZrX]&I+P>KBI )IL]uPRHL]&bd]FjH [fYemKB[CONHJSCKBF ]aNYBPRHJ]UA `KBOpe ILW&cQKBFpMNIJKUFpe%ILSTVUKBOXIJ]OpeL]bd]FjH P>]HmHJ]$ijOX]ueHJSCKBFYUOcXILKNPZrpYUSCFPZrQY\cXSCHLIJ]UAjs]b Y\IZijOXKUFQeijOX]l[fYemHJY\pSC[TS HJWl]ueH BXST]F/e 0XI UOpFX]7PKUFpMNSCHLSTKUFFXW&P]&eJeLYUSCIJ]MN]9P>KBFU]IJVU]&FpP>]bdYUSTelcpYUelemO 8eJY\FjHL]BA+KUFjHJ]FjHLKBFpesFXKBOpescKUOXI7[¢ SCFQeHZY\FjH7MN]BWIJS Q]Is]>aNcW)ILSTbd]FjHJYU[C]&b ]&FjH`[TY PKUFU]&ILVB]FpP]0MNO eJPZrXW&bdY XP\ ]&emH ) )GMNSTIL]^ijOX][TKUIZeLijOX]4[T]&escQYUe`M ]&eLcpYBP>]^]>HMN]4HL]&b cQe`MN]ST]FXFp]FjH`MN]^cX[TOpes]F$cX[TOpesc]>HJS HZes[T]&e`eLKU[TONHLSTKUFQeFjOpb W&ILSfijOX]&ePKUIJIL]uemcKUFpMpY\FjHL]ue
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
7 /6*
jiiS `
@BA /yNPZrXW&b Y%]aNcX[CSfP>SCHL]YB]&Pν∆t = 0.4(∆x)2
cQKBOXI`MNSTU]IZeL]&eqY\[T]OXIZesMN]∆xA
PKUFU]&ILVB]FjH ]>HiBOp]`[C]&OXI[CSTbdS HJ]l]&emH1XST]F[TYeLKU[TONHJSCKBF ]>aXYUP>HL] FXKUOpe+cKUOXBKUFpe BWIJS Q]I1P]7MN]&I )FpSC]&IcQKBSCFjHcXOpSTeJijO= SfP>Sp[TY4eLKU[TONHLSTKUFd]>aXYBPRHL]]ueH1MNSfeLcQKBFXSCp[C] RAByOpI[TY^¤ STVUOXIJ]4@UA 0FXKUOpe BWIJS QKUFpeFOXbdW&ILSfijOX]bd]FjHiBOp] NemS[ KBFIJY 8 FX]l[T]7cQYUeM ]uemcQYUP>]
∆x iBOpScpIL]&FpMXe+[C]ue1qY\[T]OpIJe
0.50.1
]H0.05
sY\STFpeLS ijOX][T]9cQYUelMX]HL]&bdcpe∆t]FVBY\IZMXYUFBHlPKUFpemHJYUFjHl[C]%IJYUcXcKUILH
ν∆t/(∆x)2 [C]
FpKUb9pIL] ¤ v =YU[CKBIJe7[TY"eLKU[TONHLSTKUF FjOpb W&ILSfijOX] ]&emH4MN]dcX[TOpe0]FcX[TOpe0cXIJKPZrp] MN] [fY/eLKU[TONHLSTKUF]aXYUPRHJ]UA v YP>KBb cQY\IZY\SfemKBFde& ]> ]&PRHJOX]4Y\Ob 2bd]lHL]bdcpe1pFpYU[
t = 1jMXKUFpPl[C]7FXKUb%XIL]7MN]lcpYBe
MX]%HJ]bdcpeYUOXVUbd]&FBHJ]%[TKUIZeJiBOp]%[T]dcpYUe4MN]dHJ]bdcpe∆tMNSTbdSCFOX]BA +]dcXIJKNP>WuMNW MN] ( *
' & ( ' $ $ ]ueH4HLIJk&e4eLSCbdcX[T] ]>H^KUF FX] MNKUSCH 7mYUb Y\Sfe7rpW&eLS HJ]I[¢ ONHJSC[TSTeL]I 1 YUONHL]$MN]"b ST]OXa PU ]ueH ) ) MNSTIJ]emS[ YUFpY\[T~Nem]HLrXW&KUIJSTijOX]"MN]"[TY P>KUFB]IJVU]FQP>]]ueHSTbdcQKjeLeLSTX[C]4KBO/HJILKBc"MXS 8P>ST[T] RA
7 ,& ! /+ % / & ., "!$# #%.( "!4 ]&P>HLOXKBFpesOXFp]4MX]ONaNSTkbd]emW&ILST]4M ]aNcQW&ILST]FpP]&eFOXbdWIJSTijOX]uesemOpI ( ' 1-
3 '"' @UA B "YU]&P OXFX]SCHL]ueLeL]V = 1
FXKUFFOX[T[C]BAl`KBOpe$IJ]cXIJ]FXKBFpe"[C]ueb 2&b ]ue MNKBFXFXW&]&e ijOX] cpILWuP>W&MX]bdbd]FjH ]>H FpKUOpePZrXKUSfemSfeJemKBFpe [C] eJPZrXW&bdY]aNcX[CSfP>SCHL]YU]&Pν∆t = 0.4(∆x)2
A`KUOQedIL]&VBY\IZMNKBFpe[ STF*pOp]FpP]$MX]"[fY YU[C]&OXI MN]$[TY PKUFpemHJYUFBHJ]"MX]"MNSC O )eLSTKUF
ν KBO SCFB]IZem]"MNO FXKBb9XIJ]/MX]/ W&P[C]H %emOpI [fYemHJYUXST[CSCHLW$MNOeJPZrXW&bdYpA v Y ¤ STVUOXIJ]@BA n
bdKBFBHJIL] ijOX]%[T]deLPZrXW&b Y]&emH^eHZY\X[T]9cKUOXIν = 1
SCFpemHJYUX[T]%cKUOXIν = 0.01
Q]H4ijOX] cKUOXI0[TYqYU[C]&OXIlSTFjHL]IJbdW&MNSfY\STIJ]
ν = 0.1X[T]%eJPZrXWb Y eL]b%X[C]%eHZY\X[T]b Y\Sfe`[fY emKB[COXHLSTKUF Y\cpcXILKNPZrXW&]^]&emH
[TWVBkIJ]bd]FjH4MNS WIJ]FjHJ]dMX] [fY"emKB[CONHJSCKBF]aNYBPRHJ]UA 7F PKUbdcXIJ]FpMXSC]&FijOX]dcX[TOpe0[ STFU]IZeL] MNOFpKUb9pIL]MN] WuP>[T]>H
ν]&emH%c]>HLSCH cX[TOpe[T] HJ]IJb ]/P>KUFB]&PRHJS 1s]ueH9cpILWuMNKUbdSTFpY\FjH9eLOXI9[T] HL]&ILbd]
MXS OpeLS 1A YUI PKUFpeLW&ijOX]&FBH [TY P>KBFpMNSCHLSTKUFg¤ v @UA X@ KBNHL]&FOX]$[CKBIJeJijOX][fYSCHL]ueLeL]V]ueH
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /
e$sPe ` F
e$sPe ` Pn
e$sPe ` Pv
@BA n yNPZrXWb Y]acp[CSfP>SCHL]cQKBOXI [ WuiBOQYqHLSTKUF5MX]P>KBFU]&P>HLSTKUF )GMNSCOQemSTKUFYU]uPν∆t =
0.4(∆x)2]>HV = 1
A 4F$rpYUONHν = 1
XYUO/bdST[CST]Oν = 0.1
N]H`]F$pYBeν = 0.01
A
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
\x 7 /6*
FOX[T[T] X]ueH`MN]^bdKUSTFpebdKUSTFpe]&F"qYU[TYUX[C]^YUO21 OXIl]>H bd]&eLOXIL]^ijOX]νMNSTb STFOX]UA
KUOpIdPKUbdcXIJ]FpMXIL]P]/cprXWFXKBbdkFX] ]aNYUbdSCFXKBFpe ( ' %$ iBOpSe& KU )HJSC]&FjH [fY [TSCbdSCHL]
ν = 0A `]b Y\IZijOXKUFpeHLKBONH M Y\KUIZMijOX][fYPKUFpMNSCHLSTKUF¤ v @UA X@ 4]&emH
YUONHLKBb YqHLSfijOX]bd]&FBH0eLY\HLSfe 1 YUS HJ]9emSν = 0
ijOX]&[ ijOX] emKBSCI∆t]>H
∆x P]%ijOXS eL]b%X[T]%P>KBFjHLIZY )
MXSTP>HLKUSTIJ]4YB]&P7[C]^IJW&eLOX[CHJY\H]aNcQW&ILSTbd]FjHJYU[=MNO$pYUesMN]4[fYd¤ STVUOpIL]d@UA nXA
` jSj ` F
@BAC@ut yNPZrXW&bdY ]>aNcX[TSTPS HJ]/P>]&FjHLIJWcQKBOXI9[ W&ijOpYqHJSCKBF M YUMNB]&P>HLSTKUFYU]uP∆t = 0.9∆x
V = 1
ν = 0
A KUOpI9[ WuiBOQYqHLSTKUFM YUMXU]&P>HLSTKUF PU ]ueH ) ) MXSCIJ] @BA j 4YU]uP
ν = 0 3[T]eJPZrXWb Y$]>aNcX[TSfP>SCHL]
@UA Ut 1c]OXHlem]4IJWW&PILSTIJ]un+1
j =V∆t
2∆xun
j−1 + unj −
V∆t
2∆xun
j+1. @UA \x
+]deJPZrXWb YPKUFpMNOpS H^Y\ONa KjeLPSC[T[fYqHLSTKUFQelMN] [TY/¤ SCVBOXIL]/@UAT@&t/MXY\Fpe7[T]&e0b 2bd]&e0P>KBFpMNSCHLSTKUFpe7]>a*)cWIJSTb ]&FjHJY\[T]&e9iBOp] [C]QYUeMN] [fY ¤ STVUOXIJ]$@UA nXA 7F UKBS HpSC]&F iBOp]
un+1j
F= ]&emH 7mY\b Y\Sfe iBOp][ijOX] emKBS H
∆t`OXFp] PKUb9pSCFpYUSTeLKUF PKUFU]aN]MN]
unj−1
un
j
]>Hun
j+1
A DG[ Fp]%c]ONH^MNKBFpP~YUKBSCIMX]cXIJSCFpPSCc]MNOb Y\aSTb9Opb£MNSfeLPIL]H9cKUOXI9P]eJPZrXWb Y =P]ijOXS1]ueH%OXFX]STFpMNSfPY\HLSTKUFemOpcXcX[TW)bd]&FBHZY\STIL]%MN]%eLKUF STFpemHJYUXSC[TSCHLW OXFX] cXIJ]OXB]ILSTVUKBOXIJ]OpeL]^]FeL]IZYMNKBFXFXW&]%Y\O v=]bdbd]$XA pAC@ RAv1 KBILSTVUSTFX]MN]sP>]HmHL]STFpemHJY\pSC[TS HJW]&emH1iBOp] \MXYUFpe [C]seJPZrXWb Y7]>aNcX[TSTPS HJ] @UA \x FXKBOpe YUKBFpe PZrpKUSfemSMX]`HLIZY\SCHL]&I1[T]`HJ]IJb ]0P>KBFjB]&P>HLS 1MX]7b Y\FpSCk&IL]lP]FjHLIJWUAj`KBOpe1cKUOpUKUFQe1P>]&cQ]&FpMXYUFBHMNW&P]FjHLIJ]IP>]HJ]IJbd]P>KUbdbd]^FXKBOpe[¢ YUKBFpe1 Y\SCHlcKUOXIs[fY MNWIJSTUW]4]&F/HJ]bdcpe&Ap¡7]ONaPZrXKUSCa/eLKUFjHscQKjeLeLSTX[C]ue %MXW&P>]&FjHLIJ]I U]&IJe [fY7MNIJKUSCHL]KBO%B]IZe [fY7VBY\OQPZrX]UAqv=]seLSCVBFX]sMN][TY0SCHL]&eJeL]
V]&emHXST]Fde 0pIP>IJOpP>SfY\[%
SfP>S FXKBOpe7eLOXcXcKBeLKUFQe7ijOX]V > 0
OXF YUILVBOXbd]FjH7eL~jbdWHLIJSTijOX]%]&emH7qY\[fY\X[T]%eLSV < 0
>AQ KUOpIV > 0
N[T]^MNW&P]FjHLIJ]bd]FjH MNIJKUSCHL]4]&emH`MNSCH $1 ( & 1% %XKUF"KUXHLST]FjH
V∂u
∂x(tn, xj) ≈ V
unj+1 − un
j
∆x
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / / B|
]F Y\[T[TYUFjH PZrp]IZPZrX]I+m[¢ STF 1 KUIJb YqHLSTKUF ,+]F eLOXSCqYUFBH[T]`P>KBOXIZY\FjH&AB+]`PZrXKBS a P>KUFQMNOXSCH 4OXF eLPZrXW&b YMNW&P]FjHLIJW^YYU[+mMXW&eJYUemHLIJ]ONa ,
un+1j − un
j
∆t+ V
unj+1 − un
j
∆x= 0
@UA B| iBOpS^]&emH"HJKUONH YUOpeLeLS4SCFQeHZY\X[T]ijOX] [T]eLPZrXW&b Y P>]&FjHLIJWUA`¥`OP>KBFjHLIZY\STIL] [C] & & PU ]ueH ) ) MXSCIJ] VjY\OpPZrp]lemS
V > 0 ijOXS Y%PZrX]IZPZrX]&I+m[¢ SCF 1 KUIJbdY\HLSTKUF ,`]&FIL]&bdKUFjHJYUFBH
[C]PKUOXIZY\FjHV∂u
∂x(tn, xj) ≈ V
unj − un
j−1
∆xP>KUFQMNOXSCHlY\O$eLPZrXW&b Y%]aNcX[CSfP>SCHL]^MNWuP>]&FBHJILW^YUb KBFjHun+1
j − unj
∆t+ V
unj − un
j−1
∆x= 0
@UA U iBOpSMNKBFXFX]7[C]ue1IJW&eLOX[CHJYqHZe+MX]l[fY9¤ SCVBOXIJ]9@UAT@U@BA 7FBWIJS Q]lY\SfemW&bd]FjHijOX]7[C]0eLPZrXW&b Y @BA B ]ueH
` jS~Jj ` v
@BAC@B@ $yNPZrXW&b Y ]>aNcX[TSTPS HJ]9MNWuP>]FjHJILW9YUbdKUFjHlcKUOXI7[¢ W&ijOpY\HLSTKUF M YUMNB]&P>HLSTKUF YU]uP∆t =
0.9∆xV = 1
AeHZY\X[T]eLKUOpe^OpFX]FXKBOXU]&[C[T]P>KBFpMNSCHLSTKUF ¤ v MNSC WIJ]FjHL]MX][TY cXILWuP>WuMN]FjHL]PKUFpMXS HJSCKBF¤ v @UA X@
|V |∆t ≤ ∆x. @UA j
4F$]> ]>HpKUF$cQ]&ONH`IJWW&PILSTIJ] @UA U eLKUOpe[fY 1 KUIJbd]
un+1j =
V∆t
∆xun
j−1 +
(
1− V∆t
∆x
)
unj ,
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
U 7 /6*
ijOXSpb KBFjHLIJ]ijOX]BeLSX[fY0PKUFpMXS HJSCKBF @UA j ]&emH1eLY\HLSfe 1 YUS HJ] un+1
j
]&emH1OXFp]`P>KUb%XSTFpY\SfemKBF PKUFU]aN]MX]
unj−1
]Hun
j
A YUIP>KUFQemWuiBOp]FjH [T]deJPZrXW&bdY$MNWuP>]&FBHJILW YUb KBFjH @UA U 7BWIJS p] OXF cXIJSCFQP>STcQ]MXOb YqaNSTb9OXb MNSfeJP>IJ]>HNP]0ijOXS]&FBHJIJYfFp]7eJY%emHJYUXSC[TSCHLW4P>KUFQMNS HJSCKBFXFX]&[C[T]UANv SfMNW]4MNO 1 & & ' ' %' & ' $% ' ' & ( ' A4[T[C]$]ueHcQY\ILHLSfP>OX[TSCk&IL]&bd]FjH+P>IJOpPSTYU[C]lMpY\Fpe1HLKBOpe1[T]&e+cXIJKUX[Tkbd]ue1MN]7b WuPYUFXSTijOX]0MN]&e*pOXSfMN]ue1Ko][T[C]1 ONHM= YUQKBIJM MNWuP>KUOpU]ILHL] ]FY\FXVB[TYUSTe4KBF cpY\IJ[C]MX] ' ! 3PU ]ueH ) ) MXSCIJ] MX]IL]&bdKUFjHL]&I^[C]B]FjHsKUO"[C]PKUOXIZY\FjH Nb Y\Sfe][T[C]^YUcXcpY\IZY fHsMXY\Fpe`XSC]&F"M= YUONHLIJ]&ebdKNMNk[T]&e&Av Y%PKUFpP[COpeLSTKUF/MX]0P]>HmHJ]7WHLOpMX]4eLOXI[ WuiBOQYqHLSTKUF"M YUMNB]&PRHJSCKBF]&emHsiBOp]7cKUOXI[T]4b KNMNk&[C]4MN]
PKUFU]uPRHJSCKBF ) MXS OpeLSCKBF0YB]&P1 Y\STX[T]qY\[T]OXI MN][fY`P>KBFpemHJY\FjHJ]MN]1MNSCOQemSTKUFνuST[ 1 Y\OXH Y\QemKB[COXbd]&FBH
MXW&P>]&FjHLIJ]IB]IZe[ YUb KBFjH1[C]`HL]IJbd]7PKUFU]uPRHLS 1 ]>HeLOXSTIL]`[fYPKUFpMNSCHLSTKUF$¤ v @UA j cX[TONH UHijOX]P][T[C] @UA X@ >AX¥ P>]^cXIJSCaKUF"cQ]&ONHlY\bdW&[CSTKUIJ]I[T]&eIJW&eLOX[ HZYqHZesMN]^[TYd¤ STVUOXIJ]d@UA nXA
; 1 , f 0NPO%Q7ILE/I P[#E:92F7M T K+N2O M QM NPO D ] 4GO 0 E+F6Q 3T#FRFTQM FX>TJM QE:9 E FVK32#W/Y TW%K+E/OQ7I^W TY NPO%Q ] /N2[I 10 WP[#TQM NPO 0 T E%K7QM NPO E%FQ MO#FQ6T E/N2[I T JNPOO#W+E MO/M QM T Eu0
j = (−1)j]
; 1 , T K3I>MI^Eq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∆t ≤ ∆x
]
v3]&ePKUFpP[COQemSTKUFpe MX]P>]HmHL] em]uPRHLSTKUFgeLKUFjHFXKBb9XIJ]OpeL]&e]>HBKUFjHFXKUOpILIJSCI [T]&eILW *p]>aNSTKUFpeMXOcXILKNPZrpYUSCF PZrpYUcXS HJIL]BA-(3KBONH%M Y\KUIZM HJKUOpe^[T]&e%eLPZrpWb YUe4FOXbdWIJSfiBOp]&e +mIJYUSTeLKUFXFQY\X[T]&e3,%FX]1 KBFpPRHJSCKBFXFX]&FBHscpYBe p[CKBSCF$e ]F31 Y\OXH&A 7F$IJ]FpPKUFjHLIJ]0MX]&ecXIJKUX[Tkbd]uesMN]eHZY\XST[TS HJW eLYUFpecpYUIL[T]IMX]+P>KBFU]IJVU]&FpP>] ijOXSBFpW&P>]ueLeLSCHL]FjH M= YUFpY\[T~Nem]&I=]F9MXW>HJYUSC[fe P>]&e eLPZrpWb YUe %uPU ]ueH [TYlIJYUSTeLKUF^M 2>HJIL]MX]l[ YUFpY\[T~Nem]sFOXbdWIJSfiBOp]lijOXSPKUFpPSC[TST]`KU7]uPRHJS 1 e1cXIJY\HLSfijOX]&e1]>H+WHLOpMX]&e1HLrXW&KUIJSTijOX]ueA 4FNpF j[T]&e+mKUFpe3,1eLPZrXW&b YUe=FOXbdW&ILSfijOX]&e MNKUSTU]&FjH3IL]uemc]&P>HL]&I3OXF P>]ILHJYUSCF^FpKUb9pIL]MN]1cXIJKUcpILSTW>HJW&e P>KBb bd]cQY\I]aN]bdcX[T] [C] cpILSTFpP>STc] MNO b Y\aSTb9Opb MXSTeJP>IJ]>H 1KUO[C]MXW&P>]&FjHLIJ]bd]FjH"YUbdKUFjH ijOXS0FX]eLKUFjHsijOX]7[fY9HLIZYUMNOpP>HLSTKUF Y\O/FXSTU]&YUO/MNSfeJP>IJ]>H +MN]4cXIJKUcXIJSTW>HLWue1cprj~NeLSTijOX]ue1KBOb YqHJrXWb Y\HLSfiBOp]&eMX] [ WuijOpYqHJSCKBFY\OXaMNWIJSCBW]uecpYUImHJSC]&[C[T]&e&A !'$ $ !' % /1$ & '$&1 & !(""' $ & ' !#" ' ( '& ' ! ! '& " & ( '$ ' ' & ' ' '$ %' ' ' & '%' ' & ( ' '
* ' - &* ) * ) * '. &% '."!-&* lKUOpeHJ]IJb STFXKBFpeP>]^PZrpY\cpS HJIL]0cpY\IOXF$P>]&ImHZY\STF/FpKUb9pIL]0MN]^MNW>pFpS HJSCKBFpesijOXScQ]&ILbd]HmHLIJKUFjH
YUO[C]uPRHJ]OXIlMX]%e& ~/IL]HLIJKUOXB]I`MXYUFpe`[T]UKNP&Y\XOX[fY\STIJ]0]&bdcX[CKq~BW^STPS PKUbdbd]9MXY\FQes[C]ue`KBOXIJYUVU]&eP[TYBeLeLSfiBOp]&eemOpIs[¢ Y\FpYU[C~NeL]4FjOpb W&ILSfijOX]UA
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
- j
"! #% ,/+ & % %.! $ #¢ ' G A !$ & ' & ' #G D!HG 2 D!H #G H2(G (D H(G( CD5 # AJD5(D5( D : (G D!>B ( 2((G # D C YUI]>aN]bdcX[T] [T]4v Y\cX[fYUPSC]&F @BA Ut 1]&emHOpF/cXIJKUp[Ck&b ]4Y\OXa[CSTbdS HJ]&e&AX¥PKUFjHLIZY\IJSCK U[ WuijOpYqHJSCKBFMNS WIJ]FjHJSC]&[C[T]^KUIZMNSTFpY\STIL]
dydt = f(t, y)
cKUOXI0 < t < T
y(t = 0) = y0
@UA U F= ]ueH/cpYUedOXFgcpILKBX[Ck&bd] Y\ONa[CSTbdS HJ]&e cXOpSTeJijO= WHJYUFBH/cQKjemW&]$eLOXIOpF em]&VUbd]FjH
(0, T )YU]&P
0 < T ≤ +∞ N]&[C[T]^F= YdMN]P>KBFpMNSCHLSTKUFpe +LY\O"QKBIJM ,4ijO= ]F t = 0 ]>H`cpYBe]F
t = T>A
$ # ' G !$ & $# %' " #G D!HG J ( D!H (# ( (G #G (( HA - 7G GD%AD5
t8 G (D H(G!F ,GD FG D!HG GD H( -0 = D >;(> G D GD # #GF
t = 0 D
( Gt = T
8JC YUI ]aN]bdcX[T] [¢ W&ijOpY\HLSTKUF9MXS WIJ]FjHLST][T[T]+KBIJMNSTFpYUSCIJ] @BA B =]&emH OXF%cXIJKUX[Tkbd]+MX]sYUOpPZr~ b YUSTecpYUe [T]`v3YUcX[TYBP>ST]F @UA \t ijOX][QiBOp]eLKUSCH[T]PZrpKUSCa%MN]s[fY0PKUbdcKBeJY\FjHL]MN]s[fY7qYUILSfY\X[T]sM ]&eLcpYUP]x ijOXS3KUF31 ]IZY\SCH7KBOX]Is[T]^IU[T]4MXO/HJ]bdcpe RA¡7]lFpKUb9pIL]&ONa9bdKNMNk[T]&e+emKBFBH 4[fY 1 KBSTeMX]&ecpILKBX[Ck&bd]&e Y\OXa%[TSCbdSCHL]&e+]>HMN]&e cpILKBX[Ck&bd]&eMN]
Y\OpPZr~UAX¥lSCFQemS X[ WuiBOQYqHLSTKUFMN]^[fYdPZrpY\[T]OpI @UA ]ueH`OpF"cpILKBX[Ck&bd]4MX]%Y\OpPZr~cpYUIsIJYUcXcKUILH^[TY9qY\IJSTYUX[C]7MN]`HJ]bdcpe
t]>H+OpFcXIJKUX[Tkbd]lYUONad[CSTbdS HJ]&ecpY\I+IJYUcXcKUILH ^[fYqY\IJSTYUX[T]lM ]&eLcpYUP]
xA( KUOpe0[T]&e4bdKNMNk[T]&e^ijOX] FpKUOpe4Y\[T[TKUFpe4WHLOpMNST]IMpY\Fpe4P>]PKUOXIZe0IL]&FjHLIJ]FjH4MpY\Fpe4OXFp]dMX]dP]&eMN]ONa$PY\HLWVBKUIJSC]ue+MX]^cXILKBX[Tkbd]UAv3] 1 Y\SCH7ijO= OXF$b KNMNk&[C]^b Y\HLrXW&bdY\HLSfijOX]emKBS HsOXF$cXIJKUp[Ck&b ]MX]%Y\OpPZr~KBO/OXFcpILKBX[Ck&bd]
Y\ONa [TSCbdSCHL]ue4F= STbdcX[TSTijOX]dcpYBe0YUONHLKBb YqHLSfijOX]bd]&FBH^ijO= SC[e& YUVUSfeLeL]%M= OXF*+mQKBF ,%bdKNMNk[T]UA=v1 ]a )cXIL]ueLeLSTKUF & F3 ]ueH9cQYUe]&bdcX[CKq~BW] SfP>SY\O eL]Fpe%MN][fYc]ILHLSTFX]FpP]cXr~NeLSTijOX]MXOb KNMNk&[C]7]>HsMN]4em]ue1IJW&eLOX[CHJYqHZe Bb Y\Sfe+YUOem]&FpeMN]0eJY9PKUrXW&IL]&FpP>]lbdY\HLrXW&b YqHLSfijOX]UAX+KUbdbd]lFpKUOpeY\[T[CKBFped[C]BKUSTI P>]>HLHL] PKUrXW&IL]&FpP>]$b YqHJrXWb YqHJSTijOX]$]ueHOXFp] P>KUFQMNS HJSCKBFFXW&P]&eJeLYUSCIJ]"YqYUFBHMN]cQKBOXUKBSCI+b 2&bd]l]FSfeLYUVU]IMX]&eemSTb9OX[fYqHJSCKBFpeFOXbdW&ILSfijOX]&e1]>HMX]&e+SCFjHL]&ILcpILWHJYqHJSCKBFpecprj~NeLSTijOX]ueAv=] b YqHJrXWb YqHJSTPSC]&F 0jYUP&ijOX]&e 7YUMXYUbdYUIJMY$MNKUFpFXW OXFX]MNWpFXSCHLSTKUFMN]P]iBO3 ]ueH^OXF +mQKBF ,b KNMNk&[C]p]&FcpY\IJ[TYUFjHsMX] !$ & !' OXFcXIJKUp[Ck&b ]^b YU[3cKBeLW^]&emHl[C]%P>KUFjHJIJYUSCIJ]M OpFcXIJKUX[Tkbd] XST]FcKBeLW >A 7F MNWuP>SfMN] MX]%FXKUHL]&I
f[T]&e4MNKBFXFXW&]&e [T] eL]&PKUFpMbd]b9pIL]Q[T]&e
MNKUFXFpW]&eSCFXSCHLSfY\[T]&e3[T]MNKBb Y\STFX] =]HJPUA u[fYeLKU[TONHLSTKUFIJ]&PZrX]&IJPZrpW] ]H A +[ KUcWIZYqHL]&OXI ,dijOXS
Y\VUSCHemOpIuA3DG[1e Y\VBS H^SfP>SMN] FXK\HZYqHJSCKBFpe^Y\pemHLIZY\SCHL]ue A MNWuemSTVUFpYUFjH /[fY1 KUSfe^[ WuijOpYqHJSCKBFYUONaMNWIJSCBW]ue4cpY\ILHLST][T[C]ue4]>H[T]dH~cQ]MN]PKUFpMXS HJSCKBFpe4SCFpS HJSTYU[C]ue^KUO Y\ONa[CSTbdS HJ]&e&A3v=] cpILKBX[Ck&bd] ]ueH
MNKUFpP4MN]4HJILKBOXU]&IuemKB[CONHJSCKBF"MX]
A(u) = f @BA Bn
$ #f ' G D ^B<-0/C218KD # ! ' (# ,D #D 2GG
f%2D #G (C#D H(G #G H
u D D D C#D H(G
u G (G D!G#GD 2
2(G G fC
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
U 7 /6*
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; 1 # D3E [%QJE K"E+QE"H#EJILK#M K"E E+F6Q JE Y NPO%Q7ILE/IP[E E/ILN U/Y E JE (T[#K 2 /NP[I ED/T T3KM EJORE+F6Q;YGT/NJFVW] $NM Q E JNPYGTJMOE JM MYGE/O#FM N2OO#E
Ω = (0, 1)× (0, 2π)] O K+N2O#FM U/ILE
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E JI^N UJY E JE (T[#K 2 0EJOxE"Q E /I^N UJY E T[+H MY M QE+F E/O
yF+[/M TOQ
−∂2u
∂x2− ∂2u
∂y2= 0
TO#FΩ
u(x, 0) = u(x, 2π) = 0/NP[I
0 < x < 1
u(0, y) = 0,∂u
∂x(0, y) = −e−
√n sin(ny)
/N2[I0 < y < 2π
9W/I>M EJI 2[#Eu(x, y) = e−√
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n sin(ny)sh(nx)E+F6QS[O#E F6N-[QM N2O$] N2OQ7ILE/I 2[#E T K"NPO $M QM N2O
MO/M QM T E E"Q9QNP[%QE%F;FVE+F JWJI M W+E%F;EJO x = 0K"NPO E/I JEJO%QR[OJM X NI Y WJY E/OQ E/ILF89Q6TO $M F 2[#E:9 /N2[I
QN2[Qx > 0
9 T0F6N -[%QM NPOGQ7ILN2[ W%E u(x, y) E"QQNP[%QE%FFVE+F JWJI M W+E%F O#E F6NPO%Q 3T#F/NI O#W+F 2[3TO nQE/O EJI^F 10-MO OJM^] N2O3K -[I^E]
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a∂2u
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b2 − 4ac = 0 '%:g6D$('%CdF>^) <=9*5'8GD< b2 − 4ac > 0S
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ax2 + bxy + cy2 + dx + ey + f = 0
J4-;'Fc$;)%\38V@)%Iw 'V*6)%:'7)+:4444D[ * *b2 − 4ac < 0
./)+6)/c$R*
b2 − 4ac = 0.V;e3!J$Fc$% *
b2 − 4ac > 0?
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(x, y) → (X, Y ) 'De)+ 3
Oa%)+)/c %*7. '1 *=Qb+= - \/ *% #)+'^cJ = XxYy − XyYx
G1 )%/)%BV@%%
Zz
) 4V,a4 JZ)+ 6)+$+*8b
z[V?oz;')%'@. * )% '$ Z)+
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A∂2u
∂X2+ B
∂2u
∂X∂Y+ C
∂2u
∂Y 2+ D
∂u
∂X+ E
∂u
∂Y+ Fu = G,
)1a' %6)% /A = aX2
x + bXxXy + cX2y
.B = 2aXxYx + b(XxYy + XyYx) + 2cXyYy
.C = aY 2
x + bYxYy + cY 2y
.:V a#4-|\3B2 − 4AC = J2(b2 − 4ac)
?P )+*'7. 'De)% / a%)+);c )J4\3)+ $ * - 1 4\3)+)%J<S4a#4 )+*VMX? _&Z^[R$%R) 6)% R*%R)xwL+ '7)%\3? *@. % 4d\/)+ J\3 $R*6)% Z)% U )%)%' ∂2
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∂Y 2
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∂X− ∂2
∂Y 2
.:% 4\3)"#e3!J$Fc$,\3Kb @1 4d\/)+ 9 ∂2
∂X2 − ∂2
∂Y 2
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∂x2 = 1)a#'
a = 1
b = c = d = e = f = 0P1 G)% )+6);c$%\3 J * J<uV=c3\3
b2 − 4ac = 0[
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¡0Y\FQe P]PZrQY\cXSCHLIJ]+FpKUOpe Y\FpYU[C~NeLKUFpe3[C]ue eLPZrXW&b YUe FjOpb W&ILSfijOX]&e MN]MNSCW&IL]&FpP>]ue3QFXSC]ueAqlKUOpeMNW>pFpSTeJemKBFpe [TY ' # ]>H [fY ' '%1 M OXF eLPZrXW&b Yl]H FXKBOpe bdKUFjHLIJKUFpe ijOX] cKUOXI [T]&eW&ijOpYqHJSCKBFpe3YUONa^MNWIJSCBW]uecpY\ILHLST][T[T]&e=[TSTFXW&YUSCIJ]&e `P>K]8PSC]&FBHZe PKUFpemHJYUFjHJe>[TY7eHZY\XST[TS HJW+P>KBb9XSTFXW&]%[fYdP>KBFpemSfemHJY\FQP>]^M OXFeLPZrXW&b Y9STbdcX[CSfijOX]FjHleJY 1 1 Av3]scX[fY\F MN]sP>]`PZrpYUcXSCHLIJ]]ueH[T]semOpSCqY\FjHuAUv3Y^y]uPRHLSTKUFVXA lHLIZY\SCHL][T]`PYBe MN][¢ W&ijOpY\HLSTKUFdMN]s[TY
PZrpY\[T]OXI STFBHJILKNMNOpS HJ]+Y\O +rpY\cpS HJIL]7@UAqv3Y0y]&P>HLSTKUF$XA lVUW&FXWIZY\[TSTeL][T]&e ILWuemOp[ HZYqHJe cXIJW&PW&MN]&FBHZe YUONaPYUelMN][ WuiBOQYqHLSTKUFMN]uesKUFQMN]&esKBO MN][ WuijOpYqHJSCKBF$M= YBMNU]uPRHLSTKUF3Ap_`F MN]uesXONHZelMN]%P>]9PZrpYUcXSCHLIJ]]&emH9MN] 1 KUOXIJFXSTI^OXFPYUMXIL]dMN]P>KBFpP>]&cNHLSTKUF]>H9M= YUFpY\[T~Nem]dMN]&e9eLPZrXW&b YUe4MN]MNSCW&IL]&FpP>]ue0pFXST]&ecQKBOXI0MX]&e0bdKMXk[T]&e7Q]uY\OpPKUOXc cp[COpe0VBWFXW&IJYUONaAQv=]%[C]uPRHJ]OXI0FX]dMN]IZY\SCH7cpYBe7YBKUSTIlMN] bdYU[ W>HL]&FpMNIJ]+[T]&e PKUFpP]cNHZe cpILWuem]&FBHJW&e STPS 7eLKUF%b KNMNk&[C]cXIJW1 W&ILW]>H 7PKUFpP]BKUSTI Y\STFpeLSNMN]&e eJPZrXWb YUeFjOpb W&ILSfijOX]&eKBILSTVUSTFpYUONaA( ]IJbdSCFpKUFpe^P]>HmHJ] SCFjHJILKNMNOpP>HLSTKUF]FMNSfeLYUFjH^ijOX]d[fY"b WHLrXKNMN]MX]&e^MNSCW&IL]&FpP>]ue0pFXST]&e^]ueH
OXFX]lMN]ue cX[TOpeYUFpP>ST]FXFp]&e bdW>HJrXKNMN]&e+MN]`eLSCb%OX[TY\HLSTKUFdFOXbdWIJSTijOX]`ijOXSp]&emH]FpPKUIJ]sONHLST[CSfeLW]scKUOXIP>]ILHJYUSCFp]&e Y\cpcX[CSfPY\HLSTKUFpe1PKUbdbd][TY cXIJKUcQY\VBY\HLSTKUFM KBFpMN]ue eLSTeLbdSTijOX]ue KUOW&[C]uPRHLIJKUb YUVUFXW7)HLSfiBOp]&e KBO[TY9bdW&PYUFXSfiBOp]7MN]ue *pOpSTMN]ue+PKUbdcXIJ]&eJemSTX[T]&e&AU KUOpIM Y\OXHLIJ]&eY\cXcX[TSfPYqHJSCKBFpeBP>KBbdb ][TYb WuPYUFXSTijOX]%MNOeLKU[TSTMN]%KUO P][T[C] MN]&e *pOXSfMN]&e7STFpP>KBb cpIL]ueLeLSCp[C]ue XKBF [TOXS cpILW71 kIJ]9emKBOXU]&FjHl[TYb WHLrXKNMN]/MN]&e9W[TWbd]FjHJepFXSfeA `WuY\FXbdKBSCFpe3MN]FXKUb%XIL]&ONaPKUFpP]cNHZe]&F MNS WIJ]FQP>]&epFXST]&eem]IJ]>HJILKBOXU]&FjHlMXY\Fpe`HLKBONHL]ue`[T]&elYUONHLIJ]&e`b WHLrXKNMN]uelFOXbdW&ILSfijOX]&e&Ap¥`STFpeLS [T]&eleJPZrXW&bdYBesFOXbdW)ILSfijOX]&eMNO +rpY\cpS HJIL]4%PKUb%XSCFp]IJKUFjHMN]ueW[TWbd]FjHZe1QFXSTecKUOpI[fY MNSTeJP>IJW>HJSTeJYqHJSCKBF/eLcpYqHJSTYU[C]0]>HMN]&e^MNSC WIJ]FpP]&e7pFXST]&e4cKUOXI^[fY"MNSTeJP>IJW>HJSTeJYqHJSCKBF HJ]bdcQKBIL]&[C[T]UA=v YVBWFXW&IJYU[CSCHLW ]>H^[fY$emSTbdcX[CSfP>SCHLWMN]4[TY bdW>HJrXKNMN]MN]&esMXS WIJ]FpP]&epFpSC]uesbdK\HLSTU]^MNKBFpP4emKBF]acKBeLSCHLSTKUF$MNW>HZY\ST[C[TW]4]FMNW&XONHlMN]P>]>H`KBOXIJYUVU]BA
p@
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
0 7 / / /
! % * * ! * $ ) % - '.! $ &* ) ' ' ) * % / %s % & % #,% (8* lKUOpe FpKUOpe [TSTb SCHLKBFpe 4[fY^MNSTb ]&FpeLSCKBF OpFdM= ]uemcpYBP>]s]HFXKUOpeIL]&FjBKq~UKBFpe 0[fYyKUOQe )Gem]uPRHJSCKBF
XA XA %cKUOXI`[T]^PYBesMN]^cX[TOpemST]OpIJesMNSTbd]FpeLSTKUFpe`M= ]uemcpYBP>]BAX`KUOQesP>KUFQemSfMNWIJKUFQe+[ W&ijOpYqHJSCKBF"MX]^[TYPZrQY\[T]OXIsMXYUFpe[C]MXKUb Y\STFX]4QKBILFpW
(0, 1)
∂u
∂t− ν ∂
2u
∂x2= 0
cQKBOXI(x, t) ∈ (0, 1)× R+
∗u(0, x) = u0(x)
cQKBOXIx ∈ (0, 1).
NAT@
KBOXI MNSfeJP>IJW>HLSfeL]I3[T]+MXKUb Y\STFX](0, 1)×R+ uKUFSTFjHLIJKNMNOXSCH OXFcQYUe3M= ]uemcpYBP>] ∆x = 1/(N+1) >
0 YB]&P
NOXF"]FjHJSC]&IcKBeLSCHLS 1 ]>HsOXF"cpYUeMN]0HJ]bdcpe
∆t > 0]>HsKUF"MNW>QFXS Hs[T]&eFXK]&OpMXeM OXF
b YUSC[T[TYUVU]0ILW&VUOX[TST]I
(tn, xj) = (n∆t, j∆x)cKUOXI
n ≥ 0, j ∈ 0, 1, ..., N + 1.7F"FXKUHL]
unj
[TY qY\[T]OpIM= OXFp]emKB[CONHJSCKBF/MNSfeJP>IJk>HL]^YUcXcXIJKPZrpW]4Y\OcKUSTFjH(tn, xj)
N]>Hu(t, x)
[fYeLKU[TONHJSCKBF/]aNYBPRHJ]4MX] NAT@ >AXv3YdMNKBFXFXW]4STFXSCHLSfY\[T]^]&emHsMXSTeJP>IJW>HJSTeLW]4cpYUI
u0j = u0(xj)
cQKBOXIj ∈ 0, 1, ..., N + 1.
v3]&ePKUFpMNSCHLSTKUFQe Y\ONa9[TSTb SCHL]ueMN] XAC@ cQ]&OXU]&FBH 2HLIJ]sMN]scX[TOpemST]OpIJe H~c]&eqbdYUSTe[C]&OXIPZrXKUSCaF3 STFjHL]IJSC]&FjHcQYUesMXYUFpe[TYdMNWpFXSCHLSTKUFMN]ue`eLPZrpWb YUe&ADP>S XFXKUOQeONHLST[CSfeLKUFpesMN]uesP>KBFpMNSCHLSTKUFpesYUONa[TSTb SCHL]ue`MN]¡7SCIJSfPZrX[C]H
u(t, 0) = u(t, 1) = 0cQKBOXIHLKBONH
t ∈ R+∗
ijOXS em]0HLIZYUMXOXSTeL]FjHs]Fun
0 = unN+1 = 0
cKUOXIHJKUONHn > 0.
YUI7P>KBFpeLW&ijOX]FjH PZrQYUijOX]9cQYUelMX]HL]&bdcpe0FXKUOpe0YUKBFpe /PY\[fP>Op[C]&Il[T]&e7YU[C]&OXIZe(un
j )1≤j≤NijOXS 1 KUIJbd]FjH1OXFB]&P>HL]OpI1MN]RN ABlKUOpe+MNKUFpFXKUFpe1b Y\STFBHJ]FpYUFjH1cX[TOpemST]OpIJeeJPZrXW&bdYBecKBeJemSTX[T]&ecKUOpI9[ WuiBOQYqHLSTKUF MX][TYPZrpY\[T]OXI NAT@ >A( KUOQe9P>]ue9eJPZrXWb YBeemKBFjH9MNW>pFpSTe%cpY\I
NW&ijOpY\HLSTKUFpe
]F PZrQYUijOX] cKUSTFBHxj1 ≤ j ≤ N
iBOpS+c]IJbd]>HmHJ]FjH MX]/PYU[TPOX[T]I[T]&e% qY\[T]OXIZeun
j
A ¥`O+rQY\cXSCHLIJ]d@0FpKUOpesYBKUFpeMNW 7 dcpY\IJ[CW4MNO ' " & !#1
un+1j − un
j
∆t+ ν−un
j−1 + 2unj − un
j+1
(∆x)2= 0
NA cKUOpI
n ≥ 0]Hj ∈ 1, ..., N NYUSCFpeLS=ijOX]MNO ' " & & !#
un+1j − un
j
∆t+ ν−un+1
j−1 + 2un+1j − un+1
j+1
(∆x)2= 0.
NA
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /-
DG[X]&emH 1 YUPSC[T]`MN]sUW&ILSCp]&IiBOp]s[C]`eJPZrXWb Y4STbdcX[CSfP>SCHL] NA ]ueH]]uPRHLSTU]&bd]FjHXST]FMNWpFXS UP\ ]&emH ) )GMNSCIJ]iBO3 KBF%c]ONHPY\[fP>Op[C]&I [T]&e qYU[C]&OXIJe
un+1j
]&F 1 KBFpPRHJSCKBF MX]&eun
j
%\]&F%]]H\ST[ 1 YUONHSCFB]IZem]&I[TY b YqHJILSfP>]0HLIJSfMNSTYUVUKBFpY\[T]0P&Y\IJILW&]0MX]7HZY\ST[C[T]
N
1 + 2c −c 0−c 1 + 2c −cA A A A A A A A A
−c 1 + 2c −c0 −c 1 + 2c
YU]uPc =
ν∆t
(∆x)2,
NA x
MNKUFjH^SC[ ]ueHYUSTeLW MN]dUWIJSCp]I4[T]dP&Y\IZYUP>HLkIJ]%MXW>pFXS1cQKjemSCHLS 1 3MNKBFpP9STFU]&IJeLSCp[C]BA 41F 1 YUSTeJY\FjH^OXFX]P>KUb%XSTFpY\SfemKBF/PKUFU]a]4MN] NA +]>H NA NcKUOpI
0 ≤ θ ≤ 1NKUF"KUNHJSC]&FBH`[T]
θ ' ("$ &
un+1j − un
j
∆t+ θν
−un+1j−1 + 2un+1
j − un+1j+1
(∆x)2+ (1− θ)ν
−unj−1 + 2un
j − unj+1
(∆x)2= 0.
NA| ,+ST]Fe 0pIKUF IL]HLIJKUOXB][C]deJPZrXWb Y ]acp[CSfP>SCHL] XA leLS
θ = 0 ]>H4[T] eJPZrXWb YSCbdcX[TSTPS HJ] NA
emSθ = 1
Av=]θ)GeLPZrpWb Y NA| %]&emH SCbdcX[TSfP>SCHL]MNk&e ijOX]
θ 6= 0A KUOXI [fY YU[C]&OXI
θ = 1/2
KUF KUNHJSC]&FjH9[C] ' ("$ & # % J ("' A _`FYUONHLIJ]eLPZrpWb Y$SCbdcX[TSfP>SCHL] MNSCH emSCacQKBSCFjHJeN]&emHlMNKBFXFXW4cpY\I
un+1j+1 − un
j+1
12∆t+
5(un+1j − un
j )
6∆t+un+1
j−1 − unj−1
12∆t
+ν−un+1
j−1 + 2un+1j − un+1
j+1
2(∆x)2+ ν−un
j−1 + 2unj − un
j+1
2(∆x)2= 0.
NA
; 11 f 0NPO%Q7ILE/I 2[#E ESFK32#W/YGT ] O 0 E+F6Q I>M E/O 80 T[Q7I^E P[#E Eθ\FVK 2WJYGTT E%K
θ = 1/2− (∆x)2/12ν∆t]
( KUOQe [T]&e eJPZrXW&bdYBe ijOXSNcpILWuP>k&MX]FjH eLKUFjH MNSCHJe ' ' PY\I ST[Te FX] 1 KBFBH STFBHJ]IJU]&FXSCIiBOp] MN]&ONaSCFQMNSTP]&e4MN]%HL]&b cQeA 7Fc]OXH0XST]F e 0XI0PKUFpemHLIJOXSTIL]%MN]&e4eJPZrXWb YUe7b9Op[ HJSCFXSTU]uY\ONa %[C]uescX[TOpescKUcXOX[fY\STIJ]&eseLKUFjH 9HJILKBSTeFpSCB]&Y\OXa A 41FcX[COQe`MNOeLPZrpWb Y STFpeHZY\X[T] `MN]^sSTPZrQY\IZMXemKBFjOYUO+rpYUcXSCHLIJ] @NKUF$P>SCHL]^[T] '(" & ' ( Z %
un+1j − un−1
j
2∆t+ ν−un
j−1 + un+1j + un−1
j − unj+1
(∆x)2= 0,
NA [C] ' ("$ & $ &
3un+1j − 4un
j + un−1j
2∆t+ ν−un+1
j−1 + 2un+1j − un+1
j+1
(∆x)2= 0.
NA lKUOpe9UKBSC[ "]F 1 YUP]MN]]&Y\OQP>KUOpc HJILKBcMN]eLPZrpWb YUe !-4 H9[fY$[CSfemHL]P>S )GMN]&eJemOQe^F= ]&emHcpYUe
]>aNrpY\OQeHJSCB] !u_lF MN]&e XONHZe MN][¢ Y\FQY\[T~eL]1FOXbdW&ILSfijOX]+qY 2>HJIL]MN]sP>KBbdcpY\IJ]I ]>HMN]semW&[C]uPRHJSCKBFXFX]I[C]ueb ]&SC[T[T]OXIZeseLPZrpWb YUeseLOXSTYUFjHsMX]&e`PILSCHLk&IL]ueMX]^cXILWuP>SfemSTKUF MX]P>K10XHNKUOMN]4IJKUXOQeHJ]&eJem]BA
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
Ux 7 / / /
& ( '$ f y SC[~YOpF em]uP>KBFpMbd]b%XIL]f(t, x)
MpY\Fpe%[¢ W&ijOpY\HLSTKUF MX]/[fYPZrpYU[C]&OXI NAT@ YU[CKBIJe7[T]&e4eLPZrpWb YUe0em] bdKNMNS Q]FjH^]FIJ]bdcX[fY &Y\FjH .&WIJK"YUO eL]&PKUFpMbd]&b9XIJ]%cQY\I4OXFX]YUcXcXIJKaNSCb YqHJSCKBF P>KBFpemSfemHJY\FjHJ] MN]
f(t, x)Y\OcKUSTFjH
(tn, xj)As YUI/]a]&bdcX[C]`emS4KBFPZrXKUSfeLS H
[ Y\cXcXIJKaNSCb Y\HLSTKUFf(tn, xj)
X[T]eLPZrpWb Y9]aNcX[CSfP>SCHL] NA MN]&jST]FjHun+1
j − unj
∆t+ ν−un
j−1 + 2unj − un
j+1
(∆x)2= f(tn, xj).
• & ( '$ f f v=]ue9eJPZrXWb YUe9P>S ) MN]ueLeLOpe^KBFjHOXFX]WuP>IJS HJOXIJ] cX[COQeKUObdKBSCFpe%P>KBb cQYUPRHJ] PU ]ueH ) ) MXSCIJ]ijO= ST[ 1 KUFjHSTFBHJ]IJU]&FXSCI OXFdFXKBb9XIJ]pFXS UcX[TOpe KBO bdKBSCFpe IJ]&emHLIJ]STFjH\MN]sqYU[C]&OXIJe
unj
Av Y/PKU[T[C]uPRHJSCKBFMN]ue4P>KUOpcX[C]ue
(n′, j′)ijOXS STFjHL]&ILST]FXFX]&FjH^MpY\Fpe0[ WuiBOQYqHLSTKUF MNSfeLPILkHL]dY\OcQKBSCFjH
(n, j)]ueHY\cXc][TW '1 MNOeJPZrXWb Y HL]IJbd]7YUFXVU[fY\Sfe1ijO= KUF c]ONH]&eJeJY~U]I+MN]`HJIJYBMNOXSTIL]`cQY\I
' '$!$!( >A 4FVUW&FXWIZY\[ cX[COQe[C]emHL]&FpP>ST[=]&emHs[TYUILVB] cX[TOpes[T]^eLPZrXW&b Y%]ueHlPK10NHJ]ONa/]>HlMNS 8P>ST[C]dcpILKBVUIZY\bdbd]I ]FcQY\ILHLST] P&Y\OpeL]MN]ue +]]HJe`MX]^QKBIJM , pPU ]ueH ) ) MNSTIJ]0MN]ue`PYBesKJoP]ILHJY\STFpeMX]&esP>KBOXcX[T]&e
(n′, j′)eLKUILHL]&FBHsMNOMNKBbdYUSCFp]4MX]PY\[fP>Op[ >A •
& ( '$ f T 7FcQ]&ONHIJ]bdcX[fYUP>]&I [C]ueP>KBFpMNSCHLSTKUFpeY\ONa[CSTbdS HJ]&eMX] ¡lSTILSfPZrX[T]>HMXYUFpe NAT@ cpYUIlMN]ue`P>KBFpMNSCHLSTKUFpe`YUONa/[TSTb SCHL]uelMN]%`]OpbdYUFXF QKUO$XST]F MN]ue`P>KBFpMNSCHLSTKUFpe`YUONa$[CSTbdS HJ]&eMX]cQW&ILSTKNMNSfP>SCHLW ]FjHLIJ]&e7Y\ONHJIL]ue >A+KUbdbd]FKUFpe`cpY\IlMXW&P>IJSTIL]MN]&ONa$bdYUFXSTkIJ]&e`MNSC WIJ]FjHL]uelMN]MXSTeJP>IJW>HJSTeL]I[T]&e`PKUFpMNSCHLSTKUFQesMN]^`]OpbdYUFXF
∂u
∂x(t, 0) = 0
]>H ∂u∂x
(t, 1) = 0.
( KUOXHlM YUQKBIJM jKBF$cQ]&ONH`W&PILSTIL]un
1 − un0
∆x= 0
]H unN+1 − un
N
∆x= 0
ijOXS7cQ]&ILbd]H/M W[TSCbdSTFX]I[T]&eqY\[T]OXIZeun
0
]Hun
N+1
]H/MN] Fp]PYU[TPOX[C]&IijOX] [T]&eNqY\[T]OpIJe
(unj )1≤j≤N
A1+]>HLHL]MNSfeLPILWHLSfeLY\HLSTKUF MN]"[fY PKUFpMNSCHLSTKUFMN]`]OpbdYUFXF F3 ]ueH ijOX]"MXOcXIJ]bdST]IKBIJMXIL]BApyS=[T]eLPZrXW&b Y%]ueH0MNOMN]ONaNSTkbd]^KUIZMNIJ] XP][fY%]&FXVU]&FpMNIJ]^OXFX]^c]ILHL]MN]^cpILWuP>SfemSTKUF$cXIJk&eMXO"KUIZMAQ7 ]&emH`cQKBOXIZiBOpKUS KUF$cXIJKUcKBeL]7OpFX]Y\ONHJIL]^MNSfeJP>IJW>HLSfeJYqHLSTKUF MNOMN]&ONaNSCk&bd]^KUIZMNIL]
un1 − un
−1
2∆x= 0
]>H unN+2 − un
N
2∆x= 0
ijOXSN]ueH cX[TOpe cXIJW&PSTeL] ub Y\Sfe FXW&P]&eJemSCHL][¢ Y7KBONH MN] +mcQKBSCFjHJe QP>HLS 1 e,x−1
]HxN+2
A 7F W[TSTb STFX][T]&e qYU[C]&OXIJe
un−1
]>Hun
N+2
\P>KBILIJ]&eLcKUFpMXYUFjH 0P>]&e cKUSTFjHJe QP>HLS 1 e \]H SC[NIJ]&emHL]b Y\STFBHJ]FpYUFjHN+2qYU[C]&OXIJe dP&Y\[fP>OX[T]I eLYBKUSTI
(unj )0≤j≤N+1
A¡% YUONHLIJ]^cpYUImH N[C]uesP>KUFQMNS HJSCKBFpesY\OXa[CSTbdS HJ]&e`MX]^cQW&ILSTKNMNSTPS HJWe W&PILSTU]&FjH
u(t, x+ 1) = u(t, x)cKUOXIHJKUONH
x ∈ [0, 1], t ≥ 0.
41[C[T]&eem]0MNSfeLPILWHLSfem]&FBH+cpY\I+[C]ue1W&VBY\[TSCHLW&eun
0 = unN+1
cKUOpI1HJKUONHn ≥ 0
B]H+cX[TOpe+VUW&FXWIZY\[T]bd]FjHun
j = unN+1+j
A •
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /- j|
"! : -! ( % % ,/ .( !,ST]F e 0XI N[C]ue1 KUIJb9OX[T]&elMN]&e`eJPZrXW&bdYBesP>S ) MX]&eJemOpeFp]9emKBFjHcQYUe`PZrXKBSTeLST]&esY\O$rpYBeLYUIJM %X]&[C[T]&e
ILWuemOX[CHL]&FjH M OXFX]+YUcXcXIJKaNSCb YqHJSCKBF0MX][¢ W&ijOpY\HLSTKUF^cpYUI MNW&U]&[CKBcXcQ]&bd]FjH3MN]( Y~[CKBI=P>KBb bd]FpKUOpe[¢ YUKBFpe ]>aNcX[TSTijOXWsY\O+rpYUcXS HJIL]4@BAq KBOXI 1 KUIJb Y\[TSTeL]I P]>HmHJ]`Y\cXcpILKaNSTbdY\HLSTKUF9MX]s[¢ W&ijOpY\HLSTKUF%YUONaMNWIJSCBW]uecpYUImHJSC]&[C[T]&e+cpY\IMN]ueMXS WIJ]FpP]&e1pFXST]&eKUFSTFjHLIJKNMNOXSCH+[fY9FXKUHLSTKUFMX] '' %1 ]>HMN] !1' A(,ST]F iBOp]+cKUOXI [ STFpemHJYUFBH FXKBOpe FX]P>KUFQemSfMNWIJKUFQe ijOX][¢ W&ijOpY\HLSTKUF MN]+[fY0PZrpY\[T]OXI NAT@ &FpKUOpe Y\[T[CKBFpe MNKUFXFp]I3OpFX]MXW>pFXSCHLSTKUF MN]1[fYlPKUFpeLSfeHZY\FpP]1qY\[fY\X[T]cKUOXI F= STb cKUILHL]ijOX][T[C]W&ijOpYqHJSCKBFY\ONa MNW&ILSTUW&]&e cpYUImHJSC]&[C[T]&e+ijOX]`FXKBOpeFXKUHLKBFpe
F (u) = 0ABs]&b Y\IZiBOpKUFpeijOX]
F (u)]&emH
OXFX]^FXKUHJYqHJSCKBF$cQKBOXIsOXFX] 1 KBFpPRHJSCKBF$MX]u]>HlMN]eL]&elMNWIJSCBW]ue+cQY\ILHLST][T[C]ue]F"HLKBONHlcKUSTFBH
(t, x)A
¡l] b Y\FXSTkIJ]VUW&FXWIZY\[T]OXF eLPZrXW&b YY\ONa MNSC WIJ]FpP]pFXST]&e7]&emH^MNW>pFpS cQKBOXI7HLKUOQel[T]&e7SCFQMNSTP]&ecQKjeLeLSCp[C]ue
n, jNcpY\Is[fY 1 KUIJb9OX[T]
F∆t,∆x
(
un+mj+k m−≤m≤m+, k−≤k≤k+
)
= 0 NA n
KJog[T]&e]&FjHLST]IZem−,m+, k−, k+ MNWpFXSfeLeL]FjH[fY [fY\IJVU]&OXIMNOemHL]&FpP>ST[0MXO eLPZrXW&b Y BKUSTI[TY
s]b Y\IZijOX] NANA >A $ , E G A*GH - 3C$18D (D*(G $D5(GD ( >= D!HG ( 2 D!H
F (u) = 0 # D5(#D5 C#D!HG
u(t, x).GD
#C>B 2D D D!HG (('$ $ 1%'$( E >A GH
F∆t,∆x
(
u(t+m∆t, x+ k∆x)m−≤m≤m+, k−≤k≤k+
)
, NAT@&t
D G ( #G : G D ( D ; (t, x) A(
∆tD
∆xD5G 2GD (
G 2 G 2G DC C (GJ(D % E D $ ; >= (
pGK 2 D ;,= (
qGJD
>= F# D! F(G D!# <- 3C/268JD5G ( F O(
(∆x)p + (∆t)q)
∆tD∆x
D5G 2GD ( JC
& ( ' # DG[ 1 Y\ONHcXIJ]FpMXIL]7VBY\IZMN]lMpY\Fpe+[TY 1 KUIJb9Op[C] NA n OXFp]7c]>HJS HJ]4YUb9XSTVUO HJWiBOQY\FjH 7[fY4MNW>QFXS HJSCKBF MNO eLPZrXW&b YXA 4Fd]> ]>H \KUFdc]ONH HJKUO7KBOXIZe b9OX[CHLSTcX[TSC]&I F= STbdcQKBImHJ]`ijOX][T[C]1 KUIJb9OX[T] cpY\I3OXFX] cXOXSfeJeLYUFpP>]eLO 8eLYUbdb ]&FjH=W[T]UW&]MN]
∆t]>H
∆xMN]1bdYUFXSTkIJ] lP>]ijOX] [ ]&ILIJ]OXI
MN]4HLIJKUFpP&YqHJOXIL]7HL]&FpMN]^B]IZe .WIJKpAp+]&[TYdIJ]FpMXIJYUS H`P>KBFpeLSTemHJYUFBHF= SCbdcKUILHL]ijOX]&[3eLPZrpWb Y !N KUOXIWS HJ]I9P]>H9STFpPKUFUW&FXSC]&FjH=KBF eLOXcXcKBeL]IZY"HLKBO7KUOpIJeijOX] [fY 1 KBILb%OX[C]
F∆t,∆x(un+mj+k ) = 0Y WHLW$W&PILSCHL]MN]"HJ][T[C]b Y\FpSCk&IL]$ijOX] cKUOpI OXFX]1 KUFpP>HLSTKUFILW&VUOX[TSTkIJ]
u(t, x)ijOXSsF= ]&emHdcQYUe
emKB[CONHJSCKBFMN]l[ W&ijOpYqHJSCKBFMN]`[fY9PZrpYU[C]&OXI \[TY[CSTbdS HJ]7MN]7[¢ ]IJIL]&OXI1MN]lHLIJKUFpP&YqHJOXIL]sF3 ]ueHcpYUe1FjOp[C[T]UA•+KUFpPILkHL]bd]&FBH KUF PY\[fP>Op[C]+[¢ ]IJIL]&OXI MX]1HJILKBFpPY\HLOXIJ]+M OXF eLPZrXW&b Y`]F%IL]&bdcX[TY YUFBH
un+mj+k
MpY\Fpe[TY 1 KUIJb9Op[C] XA n cpY\I
u(t+m∆t, x+ k∆x)A+KUbdbd]Y\cXcp[CSfPY\HLSTKUF$MN]4[TY ¡7W>pFpS HJSCKBF XA XA x
FXKUOpesYU[C[TKUFQeb KBFjHLIJ]I[C]^[T]bdbd]^emOpSCqY\FjHuA) & & " E HD5 - 3CA38 D FG $D GD $<; >= ( / G D D 3JG 2%C C (G E $D 2 7( F2 ,(GD GD , F ( D
ν∆t/(∆x)2 = 1/6 ( F FEKD $ ;<>= ( 3 G D5 D G %C
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
B 7 / / /
& ( '$ f ¡7YUFpe[TY%MN]OXaSTkbd]0cXrXIZYUeL]7MN]0[ W&FXKUFpPW0MXO/v=]&bdb ] NANA ^KBF/Y[TWVBkIJ])bd]&FBH4bdKNMNS QW [fYMXW>pFXSCHLSTKUF MN] [TY"P>KBFpeLSTemHJYUFpP>]%]F eLcW&P>SCQYUFBH4[T]%IZY\cXcKUILH7]&FjHLIJ]
∆t]H
∆x[TKUIZeJiBO3 ST[Te HL]FQMN]FjH1B]IZe .W&ILKQAj+]&PSpc]IJb ]HMN]`HL]FpSCIP>KUbdcNHJ]lM WB]FjHLOX]&[C[T]&e+P>KBb c]FQeLY\HLSTKUFpe]&FjHLIJ]7HJ]IJbd]&e`YUcXcpY\IZY\SfeJeLYUFBHMXY\FQe[¢ ]IJIL]&OXI`MN]4HJILKBFpPY\HLOXIJ]UA 4F$cXIZYqHJSTijOX]NKUF$KUQem]&ILB]7]]uP )HJSCB]bd]FjH7MX]4HL][T[T]&elYUbdW[TSCKBIJY\HLSTKUFpeMN][fYdcXIJW&PSTeLSCKBFemS KUFYUMNKBcNHL]^[T]KUF$IZY\cXcKUILH`]FjHLIJ]^[T]&ecQYUe
∆t]H
∆xA •
& & "' yNKUSCHv(t, x)
OpFX]1 KUFpP>HLSTKUF MN]`P[TYBeLeL] C6 A\ Y\IMNW&U]&[CKBcXcQ]&bd]FjH1MN]( Y~j[TKUIYUONHLKBOXI`MNO"cQKBSCFjH(t, x)
pKUF$PYU[TPOX[T]0[ ]IJIL]&OXIsMN]4HLIJKUFpP&YqHJOXIL]4MNOeJPZrXWb Y XA
v(t+ ∆t, x) − v(t, x)∆t
+ν−v(t, x−∆x) + 2v(t, x)− v(t, x+ ∆x)
(∆x)2
=(
vt − νvxx
)
+∆t
2vtt −
ν(∆x)2
12vxxxx +O
(
(∆t)2 + (∆x)4)
,
Kovt, vx
MNWuemSTVUFp]FjH[C]ueMNW&ILSTUW&]&e cpYUImHJSC]&[C[T]&e1MN]vANyS
v]&emHOXFX]leLKU[TONHJSCKBFdMX]`[¢ W&ijOpY\HLSTKUF MN]`[fY
PZrQY\[T]OXI NAT@ UKUF KUNHJSC]&FBH1YUSCFQemSpYUSTeLWbd]&FBH [fY^P>KBFpemSfemHJY\FQP>]sY\STFpeLSXiBOp]s[TY4cXIJW&PSTeLSTKUF 4[¢ KUIZMNIJ]7@]&FdHL]&b cQe]>H ^]Fd]&eLcpYUP]UAByNSpKUF eLOXcXcKBeL]`]FdcX[TOpe+iBOp]
ν∆t/(∆x)2 = 1/6\Y\[TKUIZe [C]ue HL]&ILbd]ue
]&F∆t]>H`]&F
(∆x)2em]^eLSCbdcX[TS Q]FjHlPYUI
vtt = νvtxx = ν2vxxxxA
yNPZrXWb Y 41ILIJ]OXI`MX]7HJILKBFpPY\HLOXIJ] yjHZY\XST[CSCHLW4 aNcX[TSTPS HJ] XA O
(
∆t+ (∆x)2) emHJY\p[C]
L2 ]H L∞ emSP>KBFpMNSCHLSTKUF¤ v
2ν∆t ≤ (∆x)2DGbdcX[TSTPS HJ] XA O(
∆t+ (∆x)2) emHJY\p[C]
L2 ]H L∞
+IZY\FX )G`SfPZrXKU[femKBF XA | O(
(∆t)2 + (∆x)2) emHJY\p[C]
L2
YB]&Pθ = 1/2
θ)GeJPZrXWb Y XA | O
(
∆t+ (∆x)2) emHJY\p[C]
L2 eLS3PKUFpMNSCHLSTKUF YB]&P
θ 6= 1/2 ¤ v
2(1− 2θ)ν∆t ≤ (∆x)2yNPZrXW&b Y ddcKUSTFjHJe XA O(
(∆t)2 + (∆x)4) emHJY\p[C]
L2
¡7Op¤pKUILH )G¤XIZY\FXB][ XA O(
( ∆t∆x)2 + (∆x)2
) emHJY\p[C]L2
%0]&Y\I NA O(
(∆t)2 + (∆x)2) emHJY\p[C]
L2
[email protected]]OXIZeMN]0HLIJKUFpP&YqHLOpIL]0]>HlemHJYUXSC[TSCHLWMN]^MNSTU]&IJeseJPZrXWb YBe+cKUOpIs[¢ W&ijOpY\HLSTKUF$MN]4[TYPZrQY\[T]OXI
; 1 f f 2N2[IK323T3K3[O E+F FVK 2WJYGT#F E T NP[F6\ FVE%K"QM N2O P] P]:9 W/I>M $E/I P[#E 10 EJI ILE/[IE Q7ILN2O3K+TQ7[I^E E%FQ JM E/O P[ Q/E TOO#N2O3K"W TOF E T ET[ P]] O I^EJYGT/I P[EJILT P[E QNP[F9K+E+FFK32#W/YGT#F FVN2OQ K"NPOF7M FQ6TOQF;FT[%XK"E -[/MJE [ 3NI Q*\ I@TO7E ]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /-
& % !& %'#% ",/+ %./¡0Y\FQe1[T]0+rpYUcXS HJIL]9@`FXKUOQeYUKBFpe WBKNiBOpWs[TYemHJYUXST[CSCHLWlMX]&eeLPZrXW&b YUeMN]7MNS WIJ]FQP>]&epFXST]&e
eLYUFpel]&FMNKBFXFX]I0OXFX]dMNWpFXSCHLSTKUFcpILWuP>Sfem]BA( KUONH^Y\O cp[COpe4YBKUFpesFpKUOpe7]>aNcX[TSTijOXWdijOX] FOXbdW)ILSfijOX]bd]FjH [ STFpemHJYUXSC[TSCHLW/eL]b YUFXS 1 ]ueHJ]cpYUI MX]&e%KBeJP>ST[C[fYqHJSCKBFpe^FXKBF QKBILFpW]&e%MN]"[TY eLKU[TONHJSCKBFFjOpb W&ILSfijOX]UA DG[l]&emHMNKUFpP/HJ]bdcpeMN] MXKUFXFX]&I OXFX] MNWpFXSCHLSTKUFgb YqHJrXWb Y\HLSfiBOp]$MX][TY emHJY\pS )[CSCHLWBA KUOpI P][fYFXKUOQe%YBKUFpe^]&eLKUSTF MN]"MNW>QFXSCI OXFX]/FXKUIJbd]cKUOXI%[TY emKB[COXHLSTKUFFOXbdWIJSTijOX]un = (un
j )1≤j≤NA `KBOpeIJ]cpIL]&FXKUFpe[C]ueFXKBILbd]&e9P>[fYUeJemSfijOX]&eeLOXI
RN ijOX]FpKBecKUFQMNWIJKUFpeemSTbdcX[C]&bd]FjH`cpY\Is[T]4cpYUesM= ]uemcpYBP>]
∆x%
‖un‖p =
N∑
j=1
∆x|unj |p
1/p cKUOXI1 ≤ p ≤ +∞, XAC@B@
KJo [T]"P&YUe%[CSTbdS HJ]p = +∞ MNKUSCH 2HLIJ]"PKUbdcXIJSTe MXYUFpe9[T]$em]&Fpe ‖un‖∞ = max1≤j≤N |un
j |A
s]b Y\IZijOXKUFQe`ijOX][fY FXKUIJbd]9Y\STFpeLS MNWpFXST]%MNW&cQ]&FpM MN]∆x
dHJIJYB]IZe[TY cKUFQMNWIZYqHJSCKBF"b YUSTeY\OpeJemS HLIZYU]&IJe=[¢ ]FjHLST]I
NPYUI
∆x = 1/(N+1)A %0IUP] s[fYscKUFpMNW&IJY\HLSTKUF^cpYUI
∆xu[fY`FXKBILbd]
‖un‖p]&emH0SfMN]&FBHJSTijOX] /[TY/FXKBILbd]
Lp(0, 1)cQKBOXI4[C]ue1 KBFpPRHJSCKBFpe4P>KBFpeHZY\FjHL]ue0cpY\I0bdKUIZP>]&YUONa
emOXI9[C]ue9eLKUOpe )STFjHL]&ILqY\[T[T]&e[xj , xj+1[
MN][0, 1]
A yKUOpU]FjH =KUF [ YUcXcQ]&[C[T]IZYMNKBFpP +mFXKUIJbd]Lp ,ZA
4F$cXIJY\HLSfijOX]4KUF"ONHLST[CSfeL]emOXILHLKBONHs[C]uesFXKBILbd]&ePKUIJIL]uemcKUFpMpY\FjHYUONaqY\[T]OpIJep = 2,+∞ A
$ 0 G FEK J FG F A GH D (D '# # ,G ( ‖ ‖ 2 A*GH ( - 3C/2/8 = $D5 #G ,FG D5(GD5 K > 0G 2 G 2(G D
∆tD
∆x- (
F # ,D5G 2GD 8D
‖un‖ ≤ K‖u0‖ (# D5(#D n ≥ 0, XAC@
(D J(GG G D!Hu0C
+ - 3C/(38 G = CH (# <2 (∆t
D∆x
D GD!; D G G CD (G(D@ E KD 1$$ & $' C
& ( ' OXSfeJiBOp]HLKBONHL]ue[T]&e%FXKUIJbd]&eeLKUFjH9WuijOXSCqYU[C]&FBHJ]&e%MXY\FpeRN [T][C]uPRHJ]OXIHLIJKUcdIJYUcXSfMN]`cKUOXIJIJYUS H1PILKBSCIJ]sijOX]`[fYeHZY\XST[TS HJW`cpY\I1IJYUcXcQKBImH OXFX]`FXKBILbd]sSTbdcX[CSfijOX]l[fY^emHJY\pS )
[CSCHLW^cpYUIIJYUcXcQKBImH %HLKBONHL]ue[C]ueFXKUIJbd]&e&A + Y\[TrX]OpIL]&Opem]&bd]FjHST[F= ]F$]&emHsILST]F"]>H`ST[]>aNSfeHJ]^MN]&eeLPZrXW&b YUe`ijOXS emKBFBHlemHJYUX[T]&escpYUI`IZY\cXcKUILH OXFX]9FXKUIJbd]^bdYUSTelcpYUe`cpY\I`IZY\cpcQKBImH dOXFp]%Y\OXHLIJ] UKBSCI cp[COpe[CKBSCFd[ ]a]&bdcX[C]`MXOdeJPZrXW&bdY4MN]`v Yqa*) ]&FpMNIJK\YU]uP1[T]&e 4 aN]IZP>SfP>]&e XA pA 7]H NA XA RA4F]]H [C]%cQKBSCFjH^P>IJOpP>SfY\[ MXY\FQel[fY¡7W>pFpS HJSCKBF NANA ]ueH^ijOX]9[fYb Y7KBIJY\HLSTKUF]ueH0OXFXS 1 KUIJb ]cpY\IIJYUcXcKUILH
∆xYU[CKBIJe1b 2&bd]0ijOX]0[C]ueFXKUIJbd]&eMNW>pFpSC]ue+cQY\I NAT@U@ 1MXWc]FpMN]&FjHMN]
∆xA •
$ ¢ G FE ( .( G A*G H D (D % :( #
F∆t,∆x(un+mj+k ) = 0
# 2 A*G D D CG F D%; #GD!un+m
j+k
Cv Y/emHJYUXST[CSCHLWdM OXFeJPZrXW&bdY[CSTFXWuY\STIL] /MX]ONa FXSTU]uY\ONa]ueH0HLIJk&e/1 YUPSC[T] STFjHL]IJcXIJW>HJ]IuA4F
]> ]>H cpYUI%[TSTFXW&YUILSCHLW/HLKBONHdeJPZrXW&bdY[CSTFXWuY\STIL] MN]&ONa FXSTU]&YUONacQ]&ONH e W&PILSTIL]"emKBOpe%[TY1 KUIJb ]P>KUFQMN]FpeLW]
un+1 = Aun, NAT@
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
B 7 / / /
KoA]&emHOXF KUcWIZYqHL]&OXI^[TSCFXWuY\STIL] OXFX] b YqHJILSfP>]3MNSCHL]M S HJWIZYqHJSCKBF4MN]
RN MXYUFpeRN A3 Y\I]aN]bdcX[T] NcKUOXI`[T]^eLPZrpWb Y ]>aNcX[TSTPS HJ] NA +[fY bdY\HLIJSTP]
AYUONH
1− 2c c 0c 1− 2c cA A A A A A A A A
c 1− 2c c0 c 1− 2c
YB]&Pc =
ν∆t
(∆x)2,
NAT@x
HZY\FpMXSTeijOX]cKUOpI[C] eJPZrXW&bdYSTbdcX[CSfP>SCHL] NA %[fY b YqHLIJSfP>]A]&emH[ STFU]IZeL]$MX][TY b YqHJILSfP>]
NA x >A=¥[¢ Y\SfMN]MN]P]>HmHJ] b YqHLIJSfP>] M S HJWIZYqHLSTKUF KUF Yun = Anu0
YqHLHL]&FBHJSCKBF =[fY"FXK\HZYqHLSTKUFAn MNWuemSTVUFp]lSfP>S [TY%cXOXSfeJeLYUFpP>] n )k&b ]4MN] A N]HcQY\IP>KUFQemWuiBOp]FjH[fY9emHJYUXST[CSCHLW4MNO$eLPZrXW&b Y]&emHWuijOXSCqYU[C]&FBHJ]
‖Anu0‖ ≤ K‖u0‖ ∀n ≥ 0, ∀u0 ∈ RN .DGFjHJILKNMNOXSfeJY\FjHs[TY FXKBILbd]4b YqHLIJSfP>ST][T[C]^eLOXQKBIJMXKUFXFXW&] UKBSCIs[fYd¡lWpFXSCHLSTKUF @ pAC@BAC@
‖M‖ = supu∈RN ,u6=0
‖Mu‖‖u‖ ,
[fYdeHZY\XST[TS HJWMNOeLPZrXW&b Y9]ueH`WuiBOpSCqY\[T]FjHJ] ‖An‖ ≤ K ∀n ≥ 0,
XAC@| ijOXS3U]OXHsMXSCIJ]^iBOp]^[TYdeLOXS HJ]MN]&ecpOXSTeJeJY\FpP]&esMN]
A]&emH`KUIJFXW]BA
2 % & L∞ v Y/emHJYUXSC[TSCHLWd]FFXKUIJbd]
L∞ ]&emH4HLIJk&e0[TSCW&] YB]&P[T]dcXIJSCFpPSCc] MNObdY\aNSCb%OXb MNSfeLPIL]H4ijOX]FpKUOpesYBKUFpeMNW 7 %OY\O +rpYUcXS HJIL]d@BAX`Y\cXc][TKUFQes[TYdMNWpFXSCHLSTKUFMN]^P>]^cpILSTFpP>STc]UA& ¢Q G FE G A*G H A < !$1! ' & & ' & $'1( (# D5(#D
n ≥ 0D D5(#D
1 ≤ j ≤ N G
min
(
0, min0≤j≤N+1
u0j
)
≤ unj ≤ max
(
0, max0≤j≤N+1
u0j
)
D AJ2GG F GD H(Au0C
& ( '$ f X ¡0Y\FQe3[fY`¡7W>pFXSCHLSTKUF NANAT@U@ [T]&e3SCFXW&VBYU[CSCHLWue=HJSC]&FXFX]FjH P>KUbdcNHJ]1FXKBFem]&OX[T])bd]&FBHMNObdSTFXSTb9OXb]>HsMNOb Y\aSTb9OpbMX]
u0 b Y\Sfe+YUOpeJemSMN] .&WIJK9ijOXS]&emH+[fYqY\[T]OXI+SCbdcKBeLW]YUOKUIZMcQY\I4[C]ue4P>KUFQMNS HJSCKBFpe^Y\OXa [CSTbdS HJ]&e^MN] ¡7SCIJSTPZrp[C]H&A +][fY/]&emH4FpW&P>]ueLeJY\STIJ]%eLS [TY"MNKUFpFXW]STFXSCHLSfY\[T]
u0 FX]BWIJS p]cpYBe [C]ueP>KBFpMNSCHLSTKUFpe Y\OXa9[CSTbdS HJ]&eMX]`¡lSTILSfPZrX[T]>H P]siBOpSXF= ]&emH cQYUe ]>aNSTVUW ]H`SCFONHJSC[T]MXYUFpe[C]^P&YUesP>KBFjHLIZY\STIL]BA •+KBb bd]0FXKUOQe+[ YUKUFQeO/YUO$+rpYUcXS HJIL] @ UKBSCI @BA 1]>H[¢ 4 aN]IZP>SfP>] @UA xpAT@ B[fY9BWIJS PY )
HJSCKBFMNO/cpILSTFpP>STc]MNO$bdY\aNSCb%OXb*MNSfeLPIL]H`cQ]&ILbd]HsMX]MNWbdKUFjHJIL]&I[T]^[T]bdbd]emOXSTqY\FjH&A
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /- Bn
) & & ¢ " FE< CHD J- 3CA38JD D G G ( L∞ *D #G D
FG D!HG 2ν∆t ≤ (∆x)2
D (D $:D C " E CHD5- 3C8D D5 GG (
L∞ (D D5 ∆t
D@ = 2∆x
-0G D *= D G FG D!HGG GD D A58JC
; 11 f T 0NPO%Q7ILE/I 2[#E E FVK 2WJYGT JE :I@TO"\9M K32#N FVN2O ] T E%K θ = 1/2 E+FQ
FQ6T E EJOZO#NI Y EL∞ F7M
ν∆t ≤ (∆x)29/E"Q 2[#E E FVK 2WJYGT E [ 3NI^Q*\ ILTO 7E P] 4&E%FQFQ6T E
E/O O#NI Y EL∞ F7M
2ν∆t ≤ (∆x)2
2 $ & L2 ¡7]FXKBb9XIJ]OXaeLPZrXW&b YUeFX]^BWIJS p]&FjHcQYUe[T]^cXILSTFpPSCc]^MNO$bdY\aNSCb%OXb MNSfeJP>IJ]>Hsb Y\SfesemKBFBH
FXW&YUFXbdKUSTFpe1MN] +KUFQe,leJPZrXWb YUe&A\ KUOpI+P>]&ONa*)[ BSC[1 Y\OXH+UW&ILSCp]&I[TY%eHZY\XST[TS HJWlMXY\FQe1OXFp]7Y\OXHLIJ]FXKUIJbd]`iBOp]`[TYFXKUIJbd]
L∞ ABv Y4FXKUIJbd] L2 em]scXI32>HJ]sHLIJk&eXSC]&F 4[ WHLOpMN]0MN]`[fY^eHZY\XST[TS HJWlVUIUP][ KBONHLST[3HJILkuelcpOXSTeJeJY\FjH0MN]%[¢ Y\FQY\[T~eL]9MN]d¤XKBOXIJSC]&I7ijOX]9FpKUOpe7cXIJW&eL]FjHLKBFpe`b Y\STFjHL]&FpY\FjH&A KUOXIP>] 1 Y\STIL]qFXKUOQe emOXcpcQKjemKBFpe MNW&eLKUIJb Y\Sfe ijOX][T]&e P>KBFpMNSCHLSTKUFpe YUONa9[TSCbdSCHL]ue cQKBOXI [¢ W&ijOpY\HLSTKUFdMN][TYPZrpY\[T]OXI eLKUFjHMN]&e ' %' & ' ! ijOXSXe& WuP>IJSCB]FjH
u(t, x+1) =u(t, x)
cQKBOXI HLKUOXHx ∈ [0, 1]
]H HJKUONHt ≥ 0
Au KUOXI [C]ue eLPZrXW&b YUe=FOXbdW&ILSfijOX]&e&]&[C[T]&e P>KBFpMNOXSfeL]FjHY\ONaW&VBYU[CSCHLWue
un0 = un
N+1
cQKBOXIHLKBONHn ≥ 0
N]H`cX[TOpesVUW&FXWIZY\[T]bd]&FBHun
j = unN+1+j
AXDG[3IL]ueHJ]MNKUFpP dPYU[TPOX[T]I
N + 1qY\[T]OXIZe
unj
A¥ PZrQYUijOX]9B]&P>HL]OpI
un = (unj )0≤j≤N
KUFYUeJemKNP>ST] OXFX] 1 KUFpP>HLSTKUFun(x)
3PKUFpemHJYUFjHL] cpY\Ib KBIJP]&YUONa jcWIJSTKMXSTijOX]^MN]^cWIJSCKNMN]d@pMNW>QFXSC]^eLOXI
[0, 1]cpY\I
un(x) = unj
eLSxj−1/2 < x < xj+1/2YU]&P
xj+1/2 = (j + 1/2)∆xcKUOXI
0 ≤ j ≤ Nx−1/2 = 0
]HxN+1+1/2 = 1
A ¥lSCFpeLSMNW>pFpSC]U[fY!1 KUFpP>HLSTKUF
un(x)Y\cXcpYUImHJSC]&FjH
L2(0, 1)A 7IUM= YUcXILkue [ YUFpY\[T~Nem]sMN]l¤pKUOXIJST]I qHLKBONHL]
1 KUFpP>HLSTKUFMN]L2(0, 1)
c]OXH4eL]%MNWuP>KBb cKBeL]I7]F OpFX]%eLKUbdbd]%MN] ¤XKBOXILST]I BKUSTIl[T]9HJrXWKBILk&b ]xpA|NAT@ MN] >AX[TOpescXIJW&PSTeLWbd]FjHsKUF$Y
un(x) =∑
k∈Z
un(k) exp(2iπkx), XAC@u
YU]&Pun(k) =
∫ 1
0un(x) exp(−2iπkx) dx
]H`[TY 1 KBILb%OX[C]^MN]4[fY\FQPZrX]IJ][∫ 1
0
|un(x)|2 dx =∑
k∈Z
|un(k)|2. NAT@
s]b Y\IZijOXKUFQesiBOp]^b 2&bd]9emSun ]&emH`OXFX] 1 KBFpPRHJSCKBF$ILW&][T[C]X[C]ue`P>K]78 PSC]&FjHJe un(k)
MN]9[TY eLWIJSC]MN]¤XKBOXILST]IseLKUFjHlP>KBbdcX[C]aN]&e&AX_`FX]^cXIJKUcpILSTW>HJW7STbdcQKBImHZY\FjHL]^cKUOXIs[fYdemOpS HJ]MN]^[TY%HJIJYUFpe 1 KBILbdW&]MN]¤XKBOXILST]IsMN]ue 1 KUFpP>HLSTKUFpe`cQW&ILSTKNMNSTijOX]ue]&emHs[TYdeLOXSCqYUFBHJ]-%pemSKBF$FXK\HJ]
vn(x) = un(x+ ∆x)
Y\[TKUIZevn(k) = un(k) exp(2iπk∆x)
A4acp[CSfijOXKUFpelb Y\STFBHJ]FpYUFjHl[TYdbdWHLrXKNMN]9eLOXI7[¢ ]>aN]bdcX[T]9MNO eJPZrXW&bdYd]aNcX[CSfP>SCHL] NA >AQ¥ U]uP
FXKBeFXKUHJY\HLSTKUFpeNKUF"cQ]&ONH`IJWW&PILSTIJ]0P]eLPZrXW&b Y cQKBOXI0 ≤ x ≤ 1
un+1(x) − un(x)
∆t+ ν−un(x−∆x) + 2un(x) − un(x+ ∆x)
(∆x)2= 0.
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
xjt 7 / / /
YUI`Y\cXcX[TSfPYqHJSCKBF"MX]0[fY HLIZY\FQe 1 KBILbdW]4MN]^¤XKBOXIJSC]&INST[=SC]&FjH
un+1(k) =
(
1− ν∆t
(∆x)2(− exp(−2iπk∆x) + 2− exp(2iπk∆x))
)
un(k).
¥lONHLIJ]bd]&FBHlMXS H
un+1(k) = A(k)un(k) = A(k)n+1u0(k)YU]uP
A(k) = 1− 4ν∆t
(∆x)2(sin(πk∆x))2.
KBOXIk ∈ Z
p[T]P>K]8PSC]&FBH0MN]¤XKBOXILST]Iun(k)
]&emHlKUIJFXW^[CKBIJeJijOX]nHJ]FpM$B]IZe[¢ STFNpFXS emS ]>H
eL]Op[C]&b ]&FjHseLS=[T] 1 YUP>HL]OpI`M YUbdcX[CSCQP&YqHLSTKUF"UW&ILSCp] |A(k)| ≤ 1XP\ ]&emH ) )GMNSTIL]
2ν∆t(sin(πk∆x))2 ≤ (∆x)2. NAT@&
yNS[TY$PKUFpMNSCHLSTKUF ¤ v @UA X@ SA ]BA2ν∆t ≤ (∆x)2
]ueH9eJYqHJSTe 1 Y\SCHL]=Y\[TKUIZe0[¢ SCFpWVBYU[CSCHLW NAT@& ]ueHIJYUSC]lijOX]&[TijOX]0emKBS H[C]7bdKMX]7MN]0¤XKUOpILST]I
k ∈ Zj]>HcpY\I+[TY'1 KUIJb9OX[T]lMN]0[fY\FpPZrX]&IL]&[pKBF]F
MXW&MNOXSCH
‖un‖22 =
∫ 1
0
|un(x)|2dx =∑
k∈Z
|un(k)|2 ≤∑
k∈Z
|u0(k)|2 =
∫ 1
0
|u0(x)|2dx = ‖u0‖22,
P]ijOXS=F= ]&emHlIJSC]&F$M= YUONHLIJ]ijOX]^[TY emHJYUXST[CSCHLWL2 MNO eLPZrpWb Y ]>aNcX[TSTPS HJ]UAQyS [TY PKUFpMXS HJSCKBF ¤ vF3 ]ueHlcQYUeleJYqHJSTe 1 Y\SCHL]p[T]9eJPZrXWb Yd]&emHlSTFpemHJY\p[C]BA4F]]HSC[ eLO 8dH7MN]%PZrpKUSfemSTI
∆x W&U]&FBHJOX][ )
[T]bd]&FBHeLO 8eLYUbdb ]&FjH4c]>HJS H 4]>Hk0
eLO 8eLYUbdb ]&FjH4VBIJYUFpMl]>HOXFp]dMXKUFXFXW&]dSCFXSCHLSfY\[T]Y~BYUFjHOpFX]7eL]OX[T]4P>KUbdcKBeJY\FjHL]7MN]7¤pKUOXIJSC]&I1FXKBFFOX[T[C]
u0(k0) 6= 0YU]uP
πk0∆x ≈ π/2 b KNMNOp[CK
π
MX]4HL][T[T]0b YUFXSCk&IL]^ijOX] |A(k0)| > 1A 7F$Y MNKUFQP0MXWbdKUFjHLIJW4[C]4[T]bdbd]emOpSCqY\FjHuA
) & & ¢ , "E CHD - 3CA38 D D5 G G ( L2
D #GD AJ(G (D H(G
2ν∆t ≤ (∆x)2D (D $:D C
¡7]^[TYdb 2&b ]!1 Y >KUF"KUF$qYdMNWbdKBFBHJIL]&I[TY emHJYUXSC[TSCHLW^MNOeLPZrpWb Y SCbdcX[TSTPS HJ]UA) & & ¢ # " FE< HD5K- 3C8KD*D AG G (
L2C
& ( '$ f " KBOXI+[T]&eeJPZrXWb YUe1]>aNcX[TSTPS HJ] NA ]>H+STbdcX[TSTPS HJ] NA [TY%P>KUFQMNS HJSCKBFMN]emHJYUXST[CSCHLW
L2 ]&emH+[fYb 2bd]4iBOp]0P][T[C]0MN]0emHJYUXST[CSCHLW L∞ Ap+][fYF= ]&emH+cQYUe1HLKUO 7KUOXIZe1[T]4PYUe1cQKBOXIM= YUONHLIJ]&eseJPZrXWb YBeA •& & "' _`F IZY\SfeLKUFXFX]&bd]FjHlY\FpYU[CKBVUOX]P][TOXS3OXHLST[CSfemW%cQKBOXI`[T]%eJPZrXWb Yd]acp[CSfP>SCHL]PKUFpMXOXS H XcKUOXI
0 ≤ x ≤ 1
un+1(x) − un(x)
∆t+ ν−un+1(x−∆x) + 2un+1(x) − un+1(x+ ∆x)
(∆x)2= 0,
]H`cpY\I`YUcXcX[TSTP&YqHJSCKBF/MN]^[fY%HLIZY\Fpe 1 KUIJbdW]4MN]¤XKBOXILST]I
un+1(k)
(
1 +ν∆t
(∆x)2(− exp(−2iπk∆x) + 2− exp(2iπk∆x))
)
= un(k).
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /- xQ@
¥`ONHJIL]&b ]&FjHlMNS H
un+1(k) = A(k)un(k) = A(k)n+1u0(k)YU]uP
A(k) =
(
1 +4ν∆t
(∆x)2(sin(πk∆x))2
)−1
.
+KUbdbd] |A(k)| ≤ 1cQKBOXIHLKBONH`bdKNMN]^MN]¤XKBOXILST]I
kX[fY 1 KUIJb9OX[T]^MN]4[fY\FpPZrX]&IL]&[cQ]&ILbd]HlMN]
P>KUFQP>[TOXIL] [fY emHJYUXST[CSCHLWL2 MXO$eJPZrXW&bdYpA
& ( ' ¢ v Y\FpYU[C~NeL]MN]+¤pKUOXIJSC]&I=IJ]cKBeL]1emOpI3[T]1PZrXKUSCa^MN]&e P>KBFpMNSCHLSTKUFpe Y\ONa4[CSTbdS HJ]&eMN]7cWIJSTKMXSTPS HJWUA 7Fc]ONH`Y\OQeLeLSQ[ ]> ]&PRHJOX]IseLS [¢ W&ijOpY\HLSTKUFYUONaMNWIJSCBW]ue1cpYUImHJSC]&[C[T]&eY[TST]O/eLOXIHLKUOXH
RY\O[TST]OMN]
[0, 1] KBF YY\[TKUIZe`Y\ YUSCIJ]OXFp]9STFBHJWVBIJYU[C]MN] ¤XKBOXILST]IlcX[COXH \H4ijO= OXFX]
emW&ILST]7MN]4¤XKBOXIJSC]&I >AlW&YUFXbdKUSTFpe jST[F3 ]ueHcpYUe+HJILkue1IJW&YU[CSfeHJ]7MN]4cpYUIL[T]IMN]4eLPZrXW&b YFOXbdWIJSTijOX]emOXI HJKUONH
RcpOXSTeJijOX]P>]&[TYSTb cp[CSfijOX]$OXFFpKUb9pIL]"SCFXpFXS`MN]$qY\[T]OXIZe
unj
PZrQYUijOX]"cpYUe MN]HL]bdcpe
nY\[TKUIZeijO= OpF"KBIJMXSCFpY\HL]&OXIFX]^cQ]&ONHlijOX]4HLIZY\SCHL]&IsOXF"FXKUb%XIL]0pFpS3MN]^qY\[T]OXIZe&A
v Y/emHJYUXST[CSCHLWL2 c]ONH9YUOpeLeLS em] MNW&b KBFjHLIJ]I4MXY\Fpe0[T] PYUe0MN] P>KBFpMNSCHLSTKUFpe4YUONa [CSTbdS HJ]&e^MN]¡lSTILSfPZrX[T]>HuAQDG[1 Y\OXH4YU[CKBIJe`YBMXY\cXHL]I0[T]&e7STMNW&]&e0MN] [¢ Y\FpYU[C~NeL]9MN] ¤XKUOpILST]IuAQ YUIl]>aN]&b cp[C] P]%ijOXS
IL]&b cp[TYBP>]4[TY HJIJYUFpe 1 KUIJb W&]^MN]¤XKBOXILST]I`MpY\Fpe`P]PYBe]&emH`[TY MNWuP>KBb cKBeLSCHLSTKUF$emOXIlOXFX]^pYBem]MN]U]&P>HL]&OXIJe`cXILKBcXIJ]&esMN]^[fY b YqHJILSfP>]M SCHLWIZYqHJSCKBF NAT@ ijOXS=c]IJb ]H7MN]cQYUeJem]&I`MNOU]uPRHL]&OXI
unY\O"U]&P>HL]&OXIun+1 A •
& ( ' ¢*0 ; ' ' /'$ ! '$ !$ ( ') ( IJYBMNOXSfemKBFpeemKBOpe1 KUIJb ]"MN]+IJ]&P]>HmHJ],[fYbdWHLrXKNMN]$MN]"[¢ Y\FpYU[C~NeL]MN]$¤XKBOXILST]I cKUOXIdcXIJKUOXB]I%[TY eHZY\XST[CSCHLW
L2M OpF$eJPZrXW&bdYpA 7F/STF7]uPRHL]MpY\Fpe[T]eLPZrpWb Y OXF$b KNMN]^MN]¤pKUOXIJST]Iun
j = A(k)n exp(2iπkxj)YU]&P
xj = j∆x,
]>H`KUF$]&F$MXW&MNOXSCHl[fYdqY\[T]OXI`MNO 1 YBPRHJ]OXI`M Y\bdcX[TSCQPY\HLSTKUFA(k)
AQlY\cXc][TKUFpesijOX]XcKUOXI`[ SCFpe )HJY\FjH =FXKUOQe4FXKUOpe^eLKUbdbd]&e4[TSTb SCHLWYUOP&YUe^eJPYU[TYUSCIJ] P\ ]&emH ) )GMNSTIL]dijOX]
A(k)]&emHOXF FXKUb%XIL]
P>KUbdcX[T]>aN]lMpY\FpeCA 7FY\cpcQ]&[C[T] 1$ '# $2+ ' & % [¢ SCFpWVBYU[CSCHLW
|A(k)| ≤ 1cQKBOXIHLKBONH`bdKMX]
k ∈ Z. XAC@un
yS [TY$PKUFpMNSCHLSTKUF MN] emHJY\pSC[TS HJWMN] KUF `]&OXb Y\FXF]&emH9eLY\HLSfe 1 YUS HJ] YU]uP9W&U]FjHJOX][T[C]&bd]FjHMN]&eIL]ueHJILSfPRHJSCKBFpe`eLOXI
∆t]>H
∆x Y\[TKUIZes[C]%eLPZrXW&b Yd]&emH7eHZY\X[T]cKUOXI7[TYdFpKUIJb ]
L2 emS FXKUFST[3]ueHSCFpemHJYUX[T]UA41FVUW&FXWIZY\[ pOpFeLPZrpWb YemHJY\p[C] ]>H^P>KBFpeLSTemHJYUFBH ]&emH0PKUFU]&ILVB]FjH BKUSTIl[fYyKBOpe )GeL]&PRHJSCKBF
NANA x RA 4FcXIZYqHJSTijOX]XOXFeJPZrXW&bdYSCFpemHJYUX[T]]&emHlHJK\HJYU[C]&bd]FjH +mSCFONHJSC[TSTeJY\p[C] ,ZA 41F]> ]>H Qb 2&b ]emS=KBFcpY\ILHlM OXFX]9MXKUFXFXW&]^SCFpS HJSTYU[C]9eLcW&P>SfY\[T]bd]&FBH`cpILW&cpY\IJW]^MN]b YUFXSCk&IL] P]9ijO= Y\OpPOXFMN]&eb KNMN]uedMN]¤XKUOpILST]I SCFQeHZY\X[T]&e FX]eLKUSCH ]aNPS HJW/cQY\Id][T[C] [C]uedSCFXW&S HZY\X[T]&e ]IJIJ]OXIZe9M Y\IJIJKUFpMNSUKUFjH^P>IJW]&I4MX]&e^P>KBbdcQKjeLYUFBHJ]&e0FXKBF FOX[T[C]ue XST]F ijOX] HLIJk&e4c]>HJS HJ]&e 4MN]d[TY$eLKU[TONHLSTKUF emOpI4P]&eb KNMN]ue SCFpemHJYUX[T]&e&A v YP>IJKUSfeJeLYUFpP>]]aNcQKBFX]FjHLST][T[T]"MN]ue9bdKNMN]ue%STFpemHJYUX[C]ue9]&FjHLIZY FX]/ijO= Y\cXIJk&eem]&OX[C]&bd]FjH+ijOX][fijOX]&ecpYUe1]FdHL]&bdcpe1P]&e"+mc]>HLSCHJe3,`bdKMX]&e1MX]SC]&FXFX]&FBH2+W&FXKUIJbd]&e3,]>HcQKB[C[TOX]&FBHP>KUbdcX[Tk>HJ]bd]FjH`[T]4IL]ueHJ]4MX]^[TYdeLKU[TONHLSTKUF"FOXbdWIJSTijOX]BA •; 11 f , 0NPO%Q7ILE/I2[#E E
θ\FVK32#W/Y T P] E+FQRF6Q6T E0E/O ONI Y E
L2MO3K"NPO $M QM N2OOE \
E/Y EJO%Q FM1/2 ≤ θ ≤ 1
9E"Q F6N2[#F T0K+N2O M QM NPO PD2(1− 2θ)ν∆t ≤ (∆x)2
F7M0 ≤ θ < 1/2
]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
x 7 / / /
; 1 f # 0NPO%Q7ILE/I 2[#E EGFK32#W/YGT /NMOQF P] E+FQ MO#K+N2O M QM NPOO#E EJY E/OQ&F6Q6T EEJO ONI Y E
L2]
03q(-/mno0= &*6)+ )%J *3 )+*4- q) 6)/c-4.9 *'a#\/ )# 4-%uY/? Y3? fSZ)%RG'% <J?o )%G'*Z4- Z*'De4Z) - 6);cZ$J ) % ‖ ‖ *&$%RJ T > 0
&<JZ '6)%3K(T ) > 04$)+3
∆t
∆x8\3
‖un‖ ≤ K(T )‖u0‖ $%:% 0 ≤ n ≤ T/∆t,
\/K\3*-:) %4;-)%u0 ? &*;(a# J4-%Z$ Hb )R* ' *)1a' ' '%1 '7)+.)+<J .J$%9) *% @1 4\3)"%
∂u
∂t− ν
∂2u
∂x2= cu
$(t, x) ∈ R
+ × R,
\/@./)+ '%e)%% 3&P1 '%/v(t, x) = e−ctu(t, x)
./*;6)% 8bR@1 4\3)+% ;)G'De)%JV ' $%
c > 0*, )+ 3&+6)%P./)G*%J
u'9<J$%3 3Z D[V?/a#'
J4-% ( )96);c-4.")9'- (6);c-4&% Z)% da/3 @1 4)%-4
|A(k)| ≤ 1 + C∆t$J:9 /J
k ∈ Z.
T )+*' J * '-4 :4*w^4;: PbG ) 4%Y/? Y3? f J ) 6);c-4? •
"! "% / 7%.!,( %'#,% (+* lKUOpe+YUKUFQe bdYUSCFjHJ]FpYUFBH HJKUOpe1[C]ue KUOXHLST[Te1cQKBOXI1MNW&b KBFjHLIJ]I[fY^P>KBFjB]IJVU]&FpP>]MN]&e+eLPZrpWb YUe
MX]/MNSCW&IL]&FpP>]ueQFXSC]ueA v3]IJW&eLOX[ HZYqH cXILSTFpPSCcpYU[MX]/P>]HmHJ]"eLKUOpe )Gem]uPRHLSTKUF]ueH [C](rXWKBILk&bd]/MN]v Yqa ijOXSY 8 ILbd] ijOX] cKUOXI^OpFeJPZrXWb Y"[TSCFXWuY\STIL] 1"' '%1 1'% & !#( '$ ($ A v YcKUILHLW&]%MN] P>]9IJW&eLOX[CHJY\H0MNW&cpYUeJem]9]F51 YUS H^MN]9]&YUOpP>KBOXc [fYbdW>HJrXKNMN]%MX]&eMXS WIJ]FpP]&e pFpSC]ueA KUOpI%HJKUONHJ]$b WHLrXKNMN]FOXbdW&ILSfijOX] MNSCW&IL]&FpP>]ue pFXST]&e W[TWbd]FjHJe QFXSTe]HJPUA [fYPKUFU]&ILVB]FpP]0eL]9MNW&b KBFjHLIJ]^]F P>KBF7OXVBOpY\FjHlMN]&ONa$YUILVBOXbd]FjHJe.%eHZY\XST[TS HJW]>H7PKUFpeLSfe )HZY\FpP] [T]OpIJe7MNW>pFpS HJSCKBFpe`cXIJW&PSTeL]&elqY\IJSC]&FBHlM= OXFp]9bdW>HJrXKNMN] [ Y\ONHJIL] RAQ¡% OXF cKUSTFBH4MN]OX]cpIJY\HLSfiBOp] N[T]'(rXWKBILk&b ]4MN]v Yqa]ueHHLIJk&eIZYUeJeLOXIJYUFjH %peLS[ KBF/ONHJSC[TSfem]^OXFeJPZrXWb YdP>KBFpeLSTemHJYUFBH SC[fe4emKBFjH0P>KBFpemHLIJOXS HZelcKUOpI4P][fY]&FVUW&FXWIZY\[ ]H4ijOX] [¢ KUFF= KBpeL]IJU]9cQYUe0M KjeLPSC[T[fYqHLSTKUFQe`FO )bdW&ILSfijOX]&e P\ ]&emH ) )GMNSTIL]dijO= ST[1]ueH9emHJYUX[T] Y\[TKUIZe0[TYeLKU[TONHLSTKUF FOXbdWIJSfiBOp] ]&emHcXILKNPZrX]MX] [TYeLKU[TONHJSCKBF/]aNYBPRHJ] [C]eJPZrXW&bdY P>KBFU]IJVU] RA "$ & f f ) $) + (D
u(t, x) (C#D H(G G D #C>B @ >= D!HG
FE(# J- 3C/8 -0 (FG D!HG J D5 H 8JC + (Dun
j
J(C#D H(GG # H $ VBD ,D5G ( #GFE,( G (A GH (F 2GG F GD H(Au0
j = u0(xj)C ' G. ( JJE D G ( ; 2 G(( (G$D5(G D DD5 # #G G ( ‖ ‖ C ( FEJD G ( 7GD G
∀T > 0, lim∆t,∆x→0
(
suptn≤T
‖en‖)
= 0, XA Ut
en
(D5# ! # 2 A*G , ( ((GD5en
j = unj − u(tn, xj)
C
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /- x
C "A FED $ ;>= ( p G 2 D ;>= ( qG D5 ( # D #D D5
T > 0 $D #G G D GD5
CT > 0D5
suptn≤T
‖en‖ ≤ CT
(
(∆x)p + (∆t)q)
. NAN@
& ( ' `KUOQe^F= YUKUFQe4cpYUe^]&FpP>KBIL]MXWbdKUFjHLIJW [ ]aSfemHL]FQP>] ]>H%[¢ OXFXSfP>SCHLW"MN] [TYemKB[CONHJSCKBFMN]l[ W&ijOpYqHJSCKBFMN]l[fY9PZrQY\[T]OXI NAT@ YU]uP`MN]ue1P>KBFpMNSCHLSTKUFpeY\OXa [TSCbdSCHL]ue+MN]0¡lSTIJSTPZrX[T]>HKUO9cWIJSTKMXSTijOX]ue >Aq KBOXI [¢ STFpeHZY\FjH FXKUOQe1 YUSTeLKUFQe MXKUFpP+[¢ r~cQKUHLrXkuem]MN][¢ ]>aNSfeHJ]FpP]+]H [¢ OXFXSfP>SCHLWM OpFX]0HL][T[T]4eLKU[TONHJSCKBF Y\STFpemS=ijOX]^MN]^eJY9IJWVBOX[fY\IJS HJW b Y\SfeFXKUOQeU]IJIJKUFpe+Y\O +rpY\cXSCHLIJ]4 ijOX]P>]4ILWuemOp[ HZYqHs]&emH`IJYUS=MN]4bdYUFXSTkIJ]0VBWFXW&IJYU[C]BA • & "' KUOXI0emSTbdcX[TS p]&IQKUF eLOXcXcKBeL]ijOX]9[T]&e`PKUFpMXS HJSCKBFpelYUONa"[TSTb SCHL]ueleLKUFjH7MN]¡lSTILSfPZrX[T]>HuA=v3Y/b 2&bd]dMXWbdKUFpemHLIZYqHJSCKBF]ueHY\OQeLeLS YU[TYUX[T]%cKUOXI^MN]ue^P>KBFpMNSCHLSTKUFpe4Y\OXa[CSTbdS HJ]&eMN]4cQW&ILSTKNMNSTPS HJW7KBO/MN]ueP>KUFQMNS HJSCKBFpeY\ONa[TSTb SCHL]uesMN]^`]&OXb Y\FXF ]F$eLOXcXcKBeJY\FjHP]&eMN]&ILFXSTkIJ]&eMNSTeJP>IJW>HJSTeLW]ue%YB]&P[C]"b 2bd]$KUIZMNIL]"MN]"cXILWuP>SfemSTKUFiBOp]"[T]$eLPZrXW&b Y RA _lFeJPZrXWb Y [CSTFXW&YUSCIJ]MN]ONa"FXSTU]&YUONac]ONH7e& W&P>IJSTIL]4emKBOpe[fY 1 KBILbd]^P>KBFpMN]&FpemW&] NAT@ SA ]BA
un+1 = Aun,
KJoA]&emH/[fY b YqHJILSfP>] M SCHLW&IJY\HLSTKUF PYUILIJW]MX] HJYUSC[T[T]
N>AyNKUSCH
u[fY eLKU[TONHJSCKBF eLOXcXcKBeLW]
emO 8eJY\bdbd]FjHIJWVUOp[CSTkIJ] 0MN][ W&ijOpYqHJSCKBFMN] [TYPZrQY\[T]OXI NAT@ RA 7F FXK\HJ]un = (un
j )1≤j≤NYU]&Pun
j = u(tn, xj)A+KBb bd]4[T]eLPZrXW&b Y9]ueHlPKUFpeLSTemHJYUFjHNST[]>aNSTemHL]^OpF"B]&P>HL]OpI
εnHL]&[3ijOX]
un+1 = Aun + ∆tεnYU]uP
lim∆t,∆x→0
‖εn‖ = 0, NA
]>H+[fY9PKUFU]&ILVB]FpP]`MN]εn]ueHOXFXS 1 KUIJb ]7cQKBOXIHJKUONH[T]&e1HL]bdcpe
0 ≤ tn ≤ TAjyS [T]7eJPZrXWb Y^]ueH
cXILWuP>Sfe [¢ KUIZMNIJ]p]F]&eLcpYUP]]>H [ KBIJMNIJ]
q]F HL]bdcpe Y\[TKUIZe ‖εn‖ ≤ C((∆x)p + (∆t)q)
A4F$cQKjeLYUFBH
enj = un
j − u(tn, xj)KUF"KUNHJSC]&FjH`cpY\I`eLKUOpemHLIZYUP>HLSTKUF"MN] NA NAT@
en+1 = Aen −∆tεn
M Ko/cQY\IsIJW&P>OpILIJ]FpP]en = Ane0 −∆t
n∑
k=1
An−kεk−1. NA
7I[fY%emHJYUXSC[TSCHLW4MNO"eJPZrXW&bdY%U]&ONHsMNSCIJ]4iBOp] ‖un‖ = ‖Anu0‖ ≤ K‖u0‖ cKUOXIHLKBONHL]4MNKBFXFXW]SCFXSCHLSfY\[T] =PU ]ueH ) ) MXSCIJ]%ijOX] ‖An‖ ≤ KKo [fY"P>KUFQeHZY\FjHL]
KFX] MNWc]FpM cpYBe0MX]
nA=¡% Y\ONHJIL]
cpY\ILHe0 = 0
XMNKUFQP NA +MXKUFXFX]
‖en‖ ≤ ∆t
n∑
k=1
‖An−k‖‖εk−1‖ ≤ ∆tnKC(
(∆x)p + (∆t)q)
,
P>]ijOXSMNKUFXFp][¢ STFXWVjY\[TS HJW NAN@ 4YU]uPd[TY PKUFpemHJYUFjHL]CT = TKC
A=v Y MNWbdKBFpeHJIJY\HLSTKUFMN] NA\t +]&emHseLSTb ST[fY\STIL]BA
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
xBx 7 / / /
& ( '$ f f v3](rXW&KUIJkbd]1MN]v3Yqa NANA\t]ueH ]F 1 YUS H YU[TYUX[T]cQKBOXI HLKUOXHL]1WuijOpYqHJSCKBFYUONa MXWIJSCBW]&ecpY\ILHLST][T[C]ue [TSCFXWuY\STIL]BA\DG[QYBMNbd]>H1OpFX]`IJW&P>STcXIJKNiBOp]`Y\O em]&FpeKo OXFeJPZrXWb Y4[TSCFXWuY\STIL]PKUFpeLSfeHZY\FjH MN]&ONa"FpSCB]&Y\OXa"ijOXS P>KUFB]IJVU]4]&emHlFXWuP>]&eJeJY\STIL]&b ]&FjHsemHJYUX[T]UAQ`]b Y\IZijOX]IlijOX]9[TYSCHL]ueLeL]MN]PKUFU]&ILVB]FpP] MXYUFpe NAN@ 7]ueH9]aXYUPRHJ]bd]FjH[fY$cXIJW&P>SfeLSCKBF MXO eJPZrXW&bdYpA-4FNQF ST[]ueHsKUF$MN]4FXK\HJ]IsijOX]^P>]HmHL]4]ueHJSCb YqHJSCKBF NAN@ F= ]&emHsqY\[fY\X[T]0ijOX]^eLOXIOXF"SCFjHL]&ILqYU[C[T]7KUIJFXWMX]9HJ]bdcpe
[0, T ]bdYUSTe0ijO= ][T[C] ]&emH0STFpMNW&cQ]&FpMXYUFBHJ] MXOFXKBb9XIJ]%MN] cKUSTFBHZe7MX] MNSfeJP>IJW>HLSfeJYqHLSTKUF
NA •
(8* & ! "%.-80BOpeJiBO3 SfP>SFXKBOpesYUKBFpecXIJSCFpPSCcQY\[T]bd]FjHsY\FpYU[C~NeLW0MN]ueseLPZrpWb YUe MN]OXa/FXSTU]uY\ONa NP\ ]&emH ) )
MXSCIJ]MN]ue9eJPZrXWb YUe9iBOpS1IJ][TSC]&FBH%[C]ueqY\[T]OXIZeMN]un+1 P][T[C]ueMN] un em]&OX[T]bd]FjH&A 7FcQ]&ONHcQY\I 1 Y\SCHL]bd]&FBH^]&FjSfeLYUVU]&I4MX]&eeJPZrXWb YBe4b9OX[CHLSTFXSTU]&YUONa ]>H]&FcQY\ILHLSfP>OX[TSC]&I^FXKUOQeYUKBFpe4MNW 7
STFjHLIJKNMNOXSCHMN]&e eJPZrXWb YBe HJILKBSTedFpSCB]&Y\OXa KJoun+1 MNWc]FQMMX] un ]>H un−1
P>KUbdbd][T]&eeJPZrXW&bdYBedMN] sSfPZrpY\IZMXeLKUF MN]¡7Op¤XKBImH ) ¤pIJYUFXU]&[ KUOMN] %0]uY\I RA 4 aXY\bdSTFXKUFQe P>KUbdbd]FjH [T]&eIJW&eLOX[CHJY\HJe cXIJW&P>WuMN]FjHZe eL]"VUWFpWIZY\[TSTeL]FjH YUONa eJPZrXWb YUe b%OX[ HJSCFpSCB]&Y\OXa FXKBOpedFXKUOQe%[TSTb SCHLKBFpecQY\I`eLKUOpPS=MN]^P>[fY\ILHLW^Y\ONa"eLPZrXW&b YUe %HLIJKUSfeFXSTU]&YUONaRAv Y¡lWpFXSCHLSTKUF XA XA $MN] [fY emHJY\pSC[TS HJWM OXF eLPZrpWb Y$]&emHSCFQMNWc]FpMpY\FjHL]MX]emKBF FXKUb%XIL]
MX]FXSTU]&YUONaA ( KUONHJ]1 KUSfep[ STFjHL]IJcXIJW>HZYqHLSTKUFMN]9[TYemHJYUXST[CSCHLW]&F$HL]&ILbd]9MN]9bdY\HLIJSTP]M S HJWIZYqHLSTKUF]ueH^OXFc]OcX[COQe^P>KUbdcX[TSfiBOpW] cQKBOXI^OXF eLPZrXW&b Y[TSCFpW&Y\STIJ] HLIJKUSfe0FXSCB]&YUONaA 41F]]H
un+1MXWc]FpM/[CSTFXWuY\STIL]&b ]&FjHMN]un ]>H un−1 XMNKBFpP`KBFFX]0cQ]&ONHscpYUe+W&PILSTIJ]l[fYIJ][fYqHLSTKUF NAT@ RA Y\IPKUFjHLIJ] XeLSKUF"cKBeL]
Un =
(
un
un−1
)
, NAqx
YU[CKBIJeST[=]>aNSfeHJ]^MN]ONa"b YqHLIJSfP>]&eM KUIZMNIJ]NA1]HA2HL][T[T]&e`ijOX]
Un+1 = AUn =
(
A1 A2
Id 0
)
Un, NAU|
Ko%[fY4b YqHJILSfP>]sM S HJWIZYqHJSCKBFA]ueH1MNKBFpPsMN]HZY\ST[C[T]
2NAj+KBb bd]cXIJW&PW&MN]&b bd]&FBH
Un = AnU1]H`[TYdemHJYUXSC[TSCHLW^]&emHsW&ijOXSTqY\[T]FjHL]
‖An‖ = supU1∈R2N ,U1 6=0
‖AnU1‖‖U1‖ ≤ K ∀n ≥ 1.
¡7]4[TY b 2bd]4b Y\FXSTkIJ]l[fY b WHLrXKNMN]^M Y\FQY\[T~eL]7MX]4¤pKUOXIJST]Ie W>HJ]FpM"Y\ONaeJPZrXW&bdYBe HJILKBSTeFpSCB]&Y\OXa"VBIBP>] [TYFXK\HZYqHLSTKUFU]&P>HLKBILST][T[C] NAqx RA 4FVBOXSfem] M ]aN]bdcX[T] FXKUOpe0MNW&b KBFjHLIJKUFpeOpF"IJW&eLOX[CHJY\HcpIL]ueLeL]FjHLS=YUO+rQY\cXSCHLIJ]d@UA) & & p " FEKGD -0/CA348KD GD5 G G
L2C
& & "' ¥ U]uP`[T]&eFXK\HZYqHJSCKBFpeOpemOp][T[C]ue1[T]^eLPZrXW&b Y @BA U 1e W&PILSCHcKUOXIx ∈ [0, 1]
un+1(x)− un−1(x)
2∆t+ ν−un(x−∆x) + 2un(x)− un(x+ ∆x)
(∆x)2= 0,
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /- x|
]>H`cpYUI`Y\cXcX[TSfPYqHJSCKBF/MN]^[fY%HLIZY\Fpe 1 KUIJbdW]4MN]¤XKBOXIJSC]&I
un+1(k) +8ν∆t
(∆x)2(sin(πk∆x))2un(k)− un−1(k) = 0.
¥`ONHJIL]&b ]&FjHlMNS H
Un+1(k) =
(
un+1(k)un(k)
)
=
( − 8ν∆t(∆x)2 (sin(πk∆x))2 1
1 0
)
Un(k) = A(k)Un(k),
]>HUn+1(k) = A(k)nU1(k)
ADP>S A(k)
]&emH OXFX]b YqHJILSfP>]M KUIZMNIJ] Y\[TKUIZedijOX]$cQKBOXI [C]ueeLPZrXW&b YUe MN]&ONa$FXSCB]&YUONa/P\ W>HZY\SCHlOXF eJPYU[TYUSCIJ]UAN KUOpI
k ∈ ZQ[T]^U]uPRHJ]OXI
Un(k)p]>H0MNKUFpP[C]
P>K]8P>ST]FjHsMN]4¤XKUOpILST]Iun(k)
N]&emH+KUIJFXW0[CKBIJeJijOX]nHJ]FpM/U]&IJe1[¢ STFNpFXS=eLS ]>Hsem]&OX[C]&bd]FjHeLS [TY
bdY\HLIJSTP]^M YUbdcX[CSCQP&YqHLSTKUF"UW&ILSCp]
‖A(k)n‖2 = supU∈R2,U 6=0
‖A(k)nU‖2‖U‖2
≤ K ∀n ≥ 1, XA U
KJo ‖U‖2 ]ueH9[fYFXKBILbd]]&OpP>[TSfMNSC]&FXFX]MpY\Fpe R2 A YUI%PKUFpeLW&ijOX]&FBH 3eLS1[ STFXWVjY\[TS HJW XA U 4]ueHjIZY\ST]sijOX][fijOX]semKBS H[T]sbdKMX]`MN]`¤XKBOXILST]Ik ∈ Z
\cpY\I [fY/1 KUIJb9OX[T]`MN]s[fY\FQPZrX]IJ][NKBF%]&FdMXW&MNOXSCH‖un‖22 =
∑
k∈Z
|un(k)|2 ≤ K∑
k∈Z
(
|u0(k)|2 + |u1(k)|2)
= ‖u0‖22 + ‖u1‖22,
P\ ]&emH ) )GMNSCIJ]`[fY9emHJYUXSC[TSCHLWL2 MNO"eLPZrpWb YXAj¥[ STFU]&IJeL] je SC[]>aNSfeHJ] k0
HJ][ijOX] ‖A(k0)n‖ F= ]ueHcpYUesKUIJFXW[TKUIZeJiBOp]
nHJ]FpM$B]IZes[¢ STFNpFXS QYU[CKBIJe]&F PZrXKUSfemSfeJeLYUFBH`PKUFU]&FpY\X[T]bd]&FBH`[fY MNKBFXFXW]
SCFXSCHLSfY\[T]$YU]&POXFeL]OX[bdKNMN]u0(k0)
Y\STFpeLSijOX]u1(k0)
KUFKUNHJSC]&FjH%[ STFpemHJY\pSC[TS HJWL2 MXOeLPZrXW&b YXA
+KBbdb ]/[TYb YqHLIJSfP>]/M= YUb cp[CSCQPY\HLSTKUFA(k)
]&emHdeL~b WHLIJSTijOX]ILW&][T[C] KBF Y [fY cXIJKUcXIJSTW>HLW‖A(k)‖2 = ρ(A(k))
]>H ‖A(k)n‖2 = ‖A(k)‖n2KJo
ρ(M)MNWuemSTVUFX]9[C]IZY~UKBF"eLc]&PRHJIJYU[ MN]9[TY
bdY\HLIJSTP]M UKUSTI3[T]v=]bdbd]l@ XAT@UA >A&¡7KUFpP u[ SCFXW&VBYU[CSCHLW NA\ ]ueH eLY\HLSfe 1 YUS HJ]+eLSB]H eL]OX[T]bd]&FBH
emSρ(A(k)) ≤ 1
AUv3]&e qY\[T]OpIJe cXIJKUcpIL]ue MX]A(k)
eLKUFjH [T]&e IZYUP>STFX]ue MNOdcQKB[C~F Bb ]sMNO MN]&ONaNSCk&bd]MN]VBILW
λ2 +8ν∆t
(∆x)2(sin(πk∆x))2λ− 1 = 0
iBOpSQYUMXb ]HHLKBO7KUOpIJe+MN]ONadIZYUP>STFX]ue ILW&][T[C]ueMNSfeHJSCFQPRHL]ue1MNKBFjH1[C]7cXIJKMXOXS H+YUONH −1AB Y\I1PKUFpeLW)
iBOp]FjHB[¢ OXFX]lMX]&eMN]&ONadIJYBP>STFX]&e ]ueH1cp[COpe1VUIZY\FpMX]ijOX]9@ emHLIJSTP>HL]bd]&FBH ]F qYU[C]&OXI1Y\QemKB[COX]\]>HMNKUFpP
ρ(A(k)) > 1AU Y\I1PKUFpeLW&ijOX]&FjHU[T]leJPZrXWb Y^P]FjHLIJW`]&emH1STFpP>KBFpMNSCHLSTKUFXFp][T[C]&b ]&FjH1SCFQeHZY\X[T]
]F$FXKUIJbd]L2 A
& ( ' , v3Y b WHLrXKNMN]$MN]"[¢ Y\FQY\[T~eL]$MN]"¤pKUOXIJST]IdiBOp]/FXKBOpe U]&FXKUFpe M ONHLST[TSTeL]IMXY\Fpel[TY/MNWbdKBFpeHJIJY\HLSTKUF MNO v=]bdbd] XA XA d]&emH7OXFcQ]&O cX[TOpe7PKUbdcX[TSTijOXW&]%MXYUFpe`[T]%P&YUe7MN]&eeLPZrXW&b YUeb%OX[CHLSTFXSCB]&YUONaijOX] MXY\Fpe/[T] PYBe MX]ONagFXSTU]uY\ONa UKBSCI/[TYs]b YUIJijOX] XA XAC@u RAv=KUIZeJiBO3 KBFSTF7]&P>HL]MXYUFpe[C]eJPZrXW&bdY%OXF$bdKNMN]MN]¤XKBOXIJSC]&IKUF"KUXHLST]FjH
(
un+1j
unj
)
= A(k)n
(
u1j
u0j
)
exp(2iπkxj)
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
xj 7 / / /
KoA(k)
]&emH`MNW&eLKUIJb Y\SfeOXFX] & % M YUbdcX[CSCQP&YqHLSTKUF ]HFpKUFcp[COpeOXF31 YUP>HL]&OXIseLP&Y\[fY\STIL] RA7F$Y\cpcQ]&[C[T] 1$ $*'# $ -g ' & % [TY PKUFpMXS HJSCKBF
ρ(A(k)) ≤ 1cKUOpIHLKUOXH`b KNMN]
k ∈ Z, NAB
Koρ(A(k))
]ueH0[C]%IJY~BKUF$emc]&P>HLIZY\[ MN]9[fY b YqHJILSfP>]A(k)
A3+KBb bd]cKUOpIlOXFX]%b YqHLIJSfP>]9ijOX]&[ )PKUFpijOX]
BKBF"Y
‖B‖ ≥ ρ(B)]>H ‖Bn‖ ≥ ρ(B)n,ST[`]ueH/P[TYUSCIijOX][fYP>KBFpMNSCHLSTKUFgMN]emHJYUXST[CSCHLW MN] KBF `]&OXb Y\FXF]&emHOXFX] 1$ $
' ' % MN] eHZY\XST[CSCHLWL2 MNOeJPZrXWb Y MNKBFpP9MN] P>KBFjB]IJVU]&FpP>] RAQv3KUIZeLijOX]^[fYb Y\HLIJSTP] A(k)]ueH^FXKBILb YU[C]]&[C[T]dUW&ILSCp] ‖A(k)‖2 = ρ(A(k))
]H ‖A(k)n‖2 = ‖A(k)‖n2 BKUSTI0[T] v=]&b bd]
@ XAT@UA MNKBFpP [TY$PKUFpMNSCHLSTKUFMN] KUF `]&OXb Y\FXF XA j l]&emH4FpW&P>]ueLeJY\STIJ]%]H^eLO 8eJY\FjHL] FXKBOpeYBKUFpe0]&O[TY +LPZrpY\FQP>],d[CKBIJe^MX][TYMXWbdKUFpemHLIZYqHJSCKBFMXO v=]bdbd] NANA$MN]dHJKUb%Q]&I9MXY\FQeP>]P&YUe1 YUKBIJYUX[C] RAj+]&cQ]&FpMXYUFBH jeLS
A(k)F= ]&emH+cQYUe+FXKUIJb Y\[T] BYU[CKBIJe1]F/VUWFpWIZY\[p[fY9PKUFpMXS HJSCKBFMN]
emHJYUXST[CSCHLW^MN] KUF$`]&OXb Y\FXF"F= ]&emH !'-' ' ' % ]>HsSC[$1 YUONH1 Y\STIL]4OXFp]4YUFpY\[T~Nem]7Q]uY\OpPKUOXccp[COpelMNW[TSTP&YqHJ]4MX]
A(k) ]>H`FXKUHJYUb bd]&FBH`MN]eJYdMNSfY\VUKBFpY\[TSfeLY\HLSTKUFKBO"FpKUF>A •
& ( '$ f f # v3]/(rXW&KUIJkbd]`MN]0v3Yqa NANA\t^eL]`VBWFXW&IJYU[CSfem]`eJY\FQe1MNS 8P>OX[CHLW4Y\ONa eJPZrXW)b YBe b9OX[CHLSTFXSTU]&YUONa%eLSXKUFdPZrXKBSTeLS H [fY4FXKUIJbd]
L2 AUv Y4bdW>HLrpKMX]`MN]sMNWbdKUFQeHJIJY\HLSTKUF ]&emHSCFpPZrQY\F )VBW]-%X]&[C[T]4ONHLST[CSfeL]^[¢ Y\FpYU[C~NeL]0MX]¤XKUOpILST]Is]H[fYdFXKUHJYqHJSCKBF/B]&P>HLKUIJST][T[C] XA \x RA • & ( '$ f f" ( KUOXH+P>]lijOX]sFpKUOpe1U]FpKUFpeMX]lMNSTIL]`eLOXI[fYemHJY\pSC[TS HJW`]>H+[TY^PKUFU]&ILVB]FpP]MX]&edeLPZrpWb YUe%b9OX[CHLSTFXSTU]uY\ONa eL]/VBWFXW&IJYU[CSfeL]SCbdbdW&MXSTY\HL]bd]&FBHYUONa eLPZrpWb YUe%cQKBOXI MN]&edeL~Ne )HJkbd]&eM W&ijOpY\HLSTKUFpe&Aj¡0Y\FQe+P]0MN]&ILFpSC]&I+PYBe jKUF/Y9Y\OQeLeLSQY\ YUSCIJ] 9OXFX]lWuP>IJS HJOXIJ]lU]uPRHJKUIJSC]&[C[T]lMN][fY%IL]&[TY\HLSTKUF/MX]7IJW&POXILIJ]FQP>] NAU| ]H %OXFX]4b YqHJILSfP>]4M Y\bdcX[TS PYqHJSCKBF Y\O[TST]O$M OpF21 YBPRHL]&OXIeJPYU[TYUSCIJ] >A •; 1 f " 0NPO%Q7ILE/I 2[#E E FK32#W/YGT EET/I P] 9E+F6QMO3K"NPO $M QM N2OO#E E/YGE/OQFQ6T ERE+QN2O3K K"NPO E/I JEJO%QEJOSO#NI Y E L2
]
; 1 f 0NPO%Q7ILE/IP[#E E9FVK 2WJYGT E [ 3NI Q*\ ILTO 7E P] 4 E%FQMO3K"N2O $M QM N2OOE E/Y EJO%QF6Q6T E E/O O#NI Y E
L2] N2OQ7ILE/I 2[#E:9F7MNPO X>TJM Q QE/O PI^E
∆tE+Q
∆x EJI^F EGQE E0Y TO/M UJI^E P[E E I@T/NI^Q
∆t/(∆x)2ILE+FQE /NI OW 9T NILF E FVK 2WJYGT E [ 3NI Q*\ I@TO7E 3E%FQ;K+N2O EJI E/OQ+] O
$M Q P[ 0 ME+FQ*K"N2O $M QM N2OOE E/Y EJO%QRK"NPO E/I JEJO%Q+]
%'( & # %.! "!,!,%.&v Y b WHLrXKNMN] MN]&eMXS WIJ]FpP]&edQFXSC]uee& WHL]FQMeJY\FQeMNS 8 POX[CHLWYUONa cXIJKUX[Tkbd]ue ]FcX[TO )
eLST]OXIZe1MNSTbd]FpeLSTKUFpe+M ]uemcQYUP>]BA+KBFpemSfMNW&ILKBFpe cpYUI1]>aN]&b cp[C]`[ W&ijOpYqHJSCKBFMN]`[fY9PZrQY\[T]OXI]&FMN]&ONaMXSCbd]FQemSTKUFpelM ]uemcQYUP>] [T]9PYBe`MN]^HJILKBSTesKBOcX[TOpelMNSTbd]FpeLSTKUFpe`M= ]uemcpYBP>]^F= ]&emHlcpYBe`cX[TOpe7P>KUb )cp[CSfijOXW 3MXO bdKUSTFpe4]&F HJrXWKBILST] 0MXYUFpe4[T]dMXKUb Y\STFX]dIL]uPRHZY\FXVBOX[TYUSCIJ]
Ω = (0, 1) × (0, L)YU]uP
MX]&esP>KBFpMNSCHLSTKUFpesYUONa[TSCbdSCHL]ue`MN]¡7SCIJSTPZrp[C]H
∂u
∂t− ν ∂
2u
∂x2− ν ∂
2u
∂y2= 0
cKUOXI(x, y, t) ∈ Ω× R+
∗
u(t = 0, x, y) = u0(x, y)cKUOpI
(x, y) ∈ Ωu(t, x, y) = 0
cQKBOXIt ∈ R+
∗ , (x, y) ∈ ∂Ω.
NA\
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /- xN
KBOXIlMNSfeJP>IJW>HLSfeL]I`[T]9MNKBbdYUSCFp]ΩpKUFSTFjHLIJKNMNOXSCH7MN]OXa"cpYBe`M ]&eLcpYUP]
∆x = 1/(Nx + 1) > 0]>H∆y = L/(Ny + 1) > 0
YU]uPNx]HNyMN]ONa"]FjHLST]IZecKBeLS HJS 1 e RA¥3B]&P0[C]^cQYUe`MN]4HJ]bdcpe
∆t > 0NKUF$MNW>QFXS H0Y\STFpemS[T]&eFpKj]&OpMXesM OXFb Y\ST[T[TYUVU]0ILW&VUOX[TSC]&I UKBSCIs[fYd¤ SCVBOXIL] XAC@
(tn, xj , yk) = (n∆t, j∆x, k∆y)cKUOXI
n ≥ 0, 0 ≤ j ≤ Nx + 1, 0 ≤ k ≤ Ny + 1.7FFXKUHL]
unj,k
[TYqY\[T]OpI M OpFX] emKB[COXHLSTKUFMXSTeJP>IJk>HJ] YUcXcXIJKNPZrXW]Y\O cKUSTFBH(tn, xj , yk)
`]>Hu(t, x, y)
[fYdemKB[COXHLSTKUF"]>aXYUP>HL]^MN] XA U RA
∆ xjx
y
∆ j k(x , y )yk
XAC@ 2+ YUSC[T[TYUVU]4M OXF$IL]uPRHJYUFXVU[T]4]F$MNSCW&IL]&FpP>]uepFXST]&e&Av3]&e`PKUFpMXS HJSCKBFpesY\ONa/[CSTbdS HJ]&e`MN]^¡7SCIJSfPZrX[C]Hlem]0HJIJYBMNOXSfem]&FBH NcQKBOXI
n > 0N]&F
un0,k = un
Nx+1,k = 0, ∀ k, ]>H unj,0 = un
j,Ny+1 = 0, ∀ j.v3YdMNKUFpFXW]4STFXS HJSTYU[C]^]ueHlMXSTeJP>IJW>HJSTeLW]0cpY\I
u0j,k = u0(xj , yk) ∀ j, k.
v3Y VUW&FXWIZY\[TSTeJYqHJSCKBFY\OP&YUeXSfMNSCbd]&FpemSTKUFpFX][3MXO ' ("$ & !1 ]ueHsW&jSfMN]&FBHJ]un+1
j,k − unj,k
∆t+ ν−un
j−1,k + 2unj,k − un
j+1,k
(∆x)2+ ν−un
j,k−1 + 2unj,k − un
j,k+1
(∆y)2= 0
NA\n cQKBOXI
n ≥ 0j ∈ 1, ..., Nx
]>Hk ∈ 1, ..., Ny
AQv Yem]&OX[T]9MNSCW&IL]&FpP>]%FXK\HZY\X[T]YU]uP4[C] PYUeOXFXSfMNSCbd]&FpemSTKUFpFX][=]ueHs[T]PYUIJYBPRHLk&IL]0MN]OXa21 KUSfecX[TOpe`eLWBkIJ]0MN]^[fYdP>KBFpMNSCHLSTKUF ¤ v1A; 11 f 0 0NPO%Q7ILE/I 2[#E E FVK 2WJYGT E"H M KM QE P] 0E%FQ FQ6T E E/O ONI Y E
L∞ >E+QY/Y E P[ 0-M W/I>M $E E /I>MO#K#M/E P[SYGT%HMYZ[Y ;F6NP[F T0K+N2O M QM NPO D
ν∆t
(∆x)2+
ν∆t
(∆y)2≤ 1
2.
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
xj 7 / / /
lKUOpe$ST[C[TOpemHLIJKUFpe$[T]eLPZrpWb Y ]aNcX[CSfP>SCHL] NA\n Y\OpijOX]&[^FXKUOpe$Y 7KUONHJKUFpe"OXF HL]&ILbd]MN]PKUFU]uPRHJSCKBFcpY\I7[TY/¤ SCVBOXIL] NA ijOXS3IJ]cXIJW&eL]FjHJ][TYP>KUFB]&PRHJSCKBF )GMNS OpeLSCKBFM OpFX] +mQKjeLeL], [C]PK]8P>ST]FjH`MN]MNSCOQemSTKUF$qY\ONH
0.01]H`[TY S HJ]&eJem]
(1., 0.)RA
e$s2nu `
e$s2nu
NA yNPZrXW&b Y/]aNcX[CSfP>SCHL]dcKUOXI4[ WuijOpYqHJSCKBFMN] PKUFU]uPRHLSTKUF ) MNSC OpemSTKUF ]F MN]OXa MNSTb ]&F )eLSTKUFpe.%XMNKBFXFXW&]^SCFXSCHLSfY\[T] rpYUONH s]>H`emKB[COXHLSTKUF pYBe >A¡7]^b 2bd] XKBFY%[T] '(" & & !
un+1j,k − un
j,k
∆t+ν−un+1
j−1,k + 2un+1j,k − un+1
j+1,k
(∆x)2+ν−un+1
j,k−1 + 2un+1j,k − un+1
j,k+1
(∆y)2= 0.
XA Bt `]b Y\IZijOXKUFpeijOX]4[C]^eJPZrXWb Y%SCbdcX[TSfP>SCHL]^FXWuP>]&eJeLS HJ] cQKBOXI`PYU[TPOX[T]I
un+1 ]&F21 KBFpPRHJSCKBF"MX] un [fY ILWuemKB[COXHLSTKUF$M OpF$eL~NeHJkbd]4[CSTFXWuY\STIL]^eL]FpeLSCp[C]&b ]&FjHcp[COpelP>KUbdcX[TSfiBOpW4ijO= ]F$OXFp]MNSCbd]&FpemSTKUF
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ / /- xjn
M ]uemcQYUP>] [TY9emSCHLOpY\HLSTKUFeL]IZY\SCH1]FQP>KUIJ]scXSTIL]s]&F HLIJKUSfeMNSTbd]FpeLSTKUFpe RAj`Y\cXc][TKUFQe1ijO= ]FMNSTbd]F )emSTKUF OXF BST[QeLO 8dHM STFjB]IZem]&I OXFX]7bdY\HLIJSTP]sHLIJSfMNSTYUVUKBFpY\[T]UA\lKUOpe+Y\[T[CKBFpeBKUSTI1ijO= ]&FMNSTb ]&FpeLSCKBFMN]ONad[fY^b YqHLIJSfP>]lY^OpFX]lemHLIJOpPRHJOXIL]lb KBSCFQe1eLSCbdcX[T]UAjv SCFpPKUFXFOX]7MNSTeJP>IJk>HJ]
unj,k
]&emH1STFpMNSfP>W]7cpY\IMN]ONa]&FjHLST]IZe
j]Hkjb YUSTe]F"cXIZYqHJSTijOX]7KUF"ONHLST[CSfeL]7OXF$eL]OX[ STFpMNSfP>]4cQKBOXIseHJKNPZU]I
un eLKUOpe+[TY1 KUIJb ]7M OpF U]&P>HL]&OXI+MXYUFpe[ KUIZMNSCFQYqHL]&OXIuAU_`FX]lbdYUFXSTkIJ] eLSCbdcX[T]`]H+]8P&YUP>] MN]`IZY\FpVU]IMpY\FpeOXFem]&OX[U]uPRHL]&OXIs[T]&eSTFpP>KBFXFOX]&e
unj,k
]&emHlM W&P>IJSTIL]un = (un
1,1, ..., un1,Ny
, un2,1, ..., u
n2,Ny
, ..., unNx,1, ..., u
nNx,Ny
).
s]b Y\IZijOXKUFQe%ijO= KUFYIZY\FpVUW"[C]ue%STFpPKUFXFOX]ue.+LP>KB[CKBFXFX]"cpY\I PKU[TKUFXFX] , b Y\SfedijO= KBFYUOXIZY\SCHY\OpeJemS XST]F cXO[C] 1 Y\STIL] +m[TSCVBFX]cpYUIl[TSCVBFX],]F+mMNW&ILKBOX[fY\FjH3,M Y\KUIZM$[¢ SCFQMNSTP]
jcX[TONH UH0ijOX]
kNx]ueH+[C]sFXKBb9XIJ]sMN]lPKU[TKUFXFX]ue ]>H
NyP][TOXSpMN]l[CSTVUFX]ue >AU¥ U]&PsP]>HLHL]lPKUFU]&FBHJSCKBF [C]7eLPZrXW&b Y
SCbdcX[TSTPS HJ] XA Bt +IL]uijOXSC]&ImHs[ STFU]&IJeLSCKBF/MN]4[fYdb YqHLIJSfP>]4em~bdW>HJILSfijOX]0HLIJSTMNSfY\VBKUFpYU[C] +mcpY\IsX[TKNPe3,
M =
D1 E1 0E1 D2 E2A A A A A A A A A
ENx−2 DNx−1 ENx−1
0 ENx−1 DNx
KJo"[C]uesX[TKP&esMNSfY\VUKBFpY\OXaDjemKBFjHsMX]&esb YqHJILSfP>]uePY\IJIJW]&eMN]0HZY\ST[C[T]
Ny
Dj =
1 + 2(cy + cx) −cy 0−cy 1 + 2(cy + cx) −cyA A A A A A A A A
−cy 1 + 2(cy + cx) −cy0 −cy 1 + 2(cy + cx)
YU]&Pcx = ν∆t
(∆x)2]>Hcy = ν∆t
(∆y)2]H [C]uedX[CKNP&e%]aHLIZY )GMNSTYUVUKBFpY\ONa
Ej = (Ej)t eLKUFjH MN]&ebdY\HLIJSTP]&esP&Y\IJILW&]&eMN]4HJYUSC[T[C]
Ny
Ej =
−cx 0 00 −cx 0A A A A A A A A A
0 −cx 00 0 −cx
.
¥`O4HLKUHJYU[\[TYb Y\HLIJSTP]M]ueH=c]FjHJYBMNSfY\VUKBFpY\[T] ]>H3eL~b WHLIJSTijOX]BAu+]c]FQMXY\FjH=[T]&e3PSCFQilMNSfY\VBKUFpYU[C]ue
FX] emKBFBH4cpYBe0PKUFjHLSTVUO&e P] iBOpS ]FjHLIZY FX] OXFX] Y\OXVBbd]FjHJY\HLSTKUFPKUFpeLSTMXWIZY\X[T] MXOPK10NH^MN]d[TYILWuemKB[CONHJSCKBF M OXFeL~emHLk&bd] [TSTFXW&YUSCIJ] YUeJemKNP>STW
M BKUSTI0[ Y\FXFX]aN]deLOXI^[ YUFpY\[T~Nem] FOXbdWIJSTijOX]
bdY\HLIJSTPSC]&[C[T]]>HFXKUHJY\bdbd]FjH [T]&e s]b YUIJijOX]ue+@ XAT@UAN@+]>Hs@pAC@BA xQ@ RAqv Y7eLS HJOpYqHJSCKBF eL]IZY\SCH ]FpPKUIJ]cXSCIJ]4]F"HLIJKUSfesMNSTbd]FpeLSCKBFpe&A
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
|Ut 7 / / /
; 1 f 0NPO%Q7ILE/I P[E E FK32#W/YGT JE 2ET3K"E/Y TO%\ T3K32X NI
un+1/2j,k − un
j,k
∆t+ ν−un+1/2
j−1,k + 2un+1/2j,k − un+1/2
j+1,k
2(∆x)2+ ν−un
j,k−1 + 2unj,k − un
j,k+1
2(∆y)2= 0
un+1j,k − u
n+1/2j,k
∆t+ ν−un+1/2
j−1,k + 2un+1/2j,k − un+1/2
j+1,k
2(∆x)2+ ν−un+1
j,k−1 + 2un+1j,k − un+1
j,k+1
2(∆y)2= 0.
E%FQ /ILW%K#M F 0 NI PI^E ZE/O E+F #T3K+E E+Q9QE/Y#FE+Q MO#K+N2O M QM NPOO#E EJY E/OQF6Q6T EGE/O O#NI Y EL2
/N2[IE+F K+N2O M QM NPOF T[+H MY M QE+F JE /WJI M N $M KM QW TOF K323T P[E $MI^EK7QM N2O "]¥5P&Y\OpeL]MN]9eLKUFPK10NHlMN]%PY\[fP>Op[=W[T]BW XKBF"IJ]bdcX[fYUP]emKBOXU]&FjH[T]9eJPZrXWb Y STb cp[CSfP>SCHL]%cpY\I
OpFX]1VBWFXW&IJYU[CSfeLY\HLSTKUF`cp[COpeLST]OXIZe3MNSTbd]FpeLSTKUFpe M ]&eLcpYUP]1MN]+eJPZrXW&bdYBe=OXFpSTMNSTbd]FpeLSTKUFXFX]&[TeuKUNHJ])FOX]1cpY\I=OpFX] HL]uPZrXFXSfijOX]MN] "' %$ ' MXS HJ]1Y\OpeJeLSUMN]1eLWcpYUIJY\HLSTKUF4M KUcWIZYqHL]&OXIZe KBO '! ]FdY\FXVB[TYUSTe RAqv SfMNW]]&emH MN]ILWuemKBOpMNIJ] Y\O [CST]OdMN][¢ W&ijOpY\HLSTKUF%XSTMXSCbd]FQemSTKUFXFp][T[C] NA\ XY\[CHL]&ILFQYqHLSTU]&bd]FjH[T]&elMN]ONa/W&ijOpYqHJSCKBFpeOXFXSfMNSTbd]FpeLSCKBFXFX]&[C[T]&e
∂u
∂t− 2ν
∂2u
∂x2= 0
]H ∂u
∂t− 2ν
∂2u
∂y2= 0
MXKUFjH1[fY +bdKq~B]FXFX] ,IJ]&MNKBFXFX] NA\ RA\ Y\I+]>aN]bdcX[T] B]F ONHJSC[TSfeLYUFBHMXY\FQePZrpYUijOX]7MNSCIJ]&P>HLSTKUF OXFeJPZrXW&bdY MN]%+IZY\FX )G`SfPZrXKU[feLKUFcKUOXIsOpF"MN]&bdS=cpYUesMX]7HJ]bdcpe
∆/2NKUF"KUNHJSC]&FjHOpF ' ("$ &
"' % ( '
un+1/2j,k − un
j,k
∆t+ ν−un+1/2
j−1,k + 2un+1/2j,k − un+1/2
j+1,k
2(∆x)2+ ν−un
j−1,k + 2unj,k − un
j+1,k
2(∆x)2= 0
un+1j,k − u
n+1/2j,k
∆t+ ν−un+1
j,k−1 + 2un+1j,k − un+1
j,k+1
2(∆y)2+ ν−un+1/2
j,k−1 + 2un+1/2j,k − un+1/2
j,k+1
2(∆y)2= 0 NA X@
v1 YqYUFBHZY\VB]MN]1P>]1H~cQ]MN]+eJPZrXWb Ys]ueH iBO3 ST[jeLO 8dH lPZrQYUijOX]1MN]&bdSBcpYBe3MN]1HL]&b cQe M SCFU]&IJeL]IOpFX]7b Y\HLIJSTP]`HLIJSfMNSTYUVUKBFpY\[T]`MN]7H~jc]0OXFXSfMNSCbd]&FpemSTKUFpFX][ PU ]ueHsMNKBFpP`OXF"PYU[TPOX[Qc]O"PZrX]I RA 4FHJILKBSTeMNSTbd]FpeLSTKUFpeSC[=eLO 8dH`MN]!1 Y\STIL]7HLIJKUSfe+HLST]IZe ) cpYUeMN]0HJ]bdcpe]>Hs[T]&ecXIJKUcXIJSTW>HLWueMNO"eJPZrXW&bdYeLKUFjHsSTFpPZrpY\FpVUW]ueAQ+]^eJPZrXWb Y ]&emH`FXKBF$eL]Op[C]&b ]&FjHsemHJYUX[T]^bdYUSTesPKUFpeLSfeHZY\FjH`YU]&P7[ WuiBOQYqHLSTKUFpSTMNSTbd]FpeLSTKUFXFX]&[C[T] XA U >A; 1 f N2OQ7I^EJI 2[#E E FVK 2WJYGT E MILE%K"QM N2O#FT QEJI O#W+E%F P] E+FQ /ILW%KM F 0 NI PI^E EJOSE+F #T3K+E E"Q QE/Y#F E"Q0MO#K+N2O M QM NPOO#E EJY E/OQRFQ6T E E/O O#NI Y E
L2 /NP[I JE%FZK"N2O $M QM N2O#F
T[%H MY M QE%F E /W/I>M N M K#M QW TO#F K323T 2[#E MILE%K"QM N2O "]+KBFpP>[TOXKUFQesP>]>HLHL]eL]&P>HLSTKUF"cpY\I`ijOX]&[TijOX]uesP>KUFQemSfMNWIZYqHJSCKBFpecXIZYqHLSfijOX]&e dcXIJKUcKBeMN]^[fY bdW)
HJrXKNMN]`MN]ueMNSCW&IL]&FpP>]ue pFXST]&e&AByKUF YqY\FjHZY\VU]+cXIJSCFpPSCcQY\[X]ueHeJY0eLSTb cp[CSfP>SCHLWlYUOpeJemSNXST]F PKUFpP]c )HJOX][T[T]`iBOp]`MN]sbdSTeL]s]FdK]OXIJ]SCF 1 KBILb Y\HLSfiBOp]UA 4[T[C]scpILWuem]&FBHJ]`P>]&cQ]&FpMXY\FjH OXFP]ILHJY\STF FXKUb%XIL]MX]lMNW1 YUONHJe1iBOpS BcQKBOXI1MN]`FpKUb9pIL]&ONa%cXIJKUp[Ck&b ]ue P>KBb cp[C]a]ue U[COXS 1 KUFjH1cXIJW1 W&IL]&IM Y\ONHJIL]ue bdW)HJrXKNMN]&esPKUbdbd]0P][T[C]^MN]ueW[TWbd]FjHJeQFXSTe UKBSCI[T]&e`+rpY\cpS HJIL]ue%]>H` RAX_`Fp]4MX]&ecXIJSCFQP>STcpY\[T]&e
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/- |X@
NA `Y 8 FX]&b ]&FjHMN]0b Y\ST[C[fY\VB]l]&F"MNSC WIJ]FpP]&e1pFXST]&e %N[TY .KBFX]7]&FjHLKUOpILW&]l]&emHsP>][T[T]7Ko[¢ KUF"U]&ONH`cX[TOpesMN]^cXIJW&PSTeLSCKBF=A
[CSTbdS HZYqHLSTKUFQe7MN]%[TYb WHLrXKNMN]%]&emH0ijO= ][T[C]%FX]'1 KBFpPRHJSCKBFXFX]9ijOX]%cQKBOXI7MX]&e7bdYUSC[T[fY\VU]ue`IJWVBOX[CST]IZeMNS HZe %$'%( ' AXDG[3F3 ]ueH`cpYBeHLKBO7KUOpIJe 1 YBP>ST[C]9MX]^cpYU]&IsOXFMNKUb Y\STFX]ijOX]&[TPKUFpijOX]MN][¢ ]&eLcpYUP]dcpY\IMN]ue4b Y\ST[C[T]&e^IJ]&P>HJYUFXVUOX[fY\STIJ]&e"! Y\IYUSC[T[C]&OXIZe ST[F= ]&emHcpYUe4cKBeJeLSCX[T] MN] IZY 8 FX]I[CKNPYU[C]&bd]FjH[C]bdYUSC[T[fY\VU]cKUOpIYUKBSCI OXFp] bd]ST[C[T]OXIJ] cXIJW&PSTeLSTKUF]FgOXFg]&FpMNIJKUSCHcXIJW&P>SfeMXOMNKUb Y\STFX]MX]/PYU[TPOX[A3DG[1]ueH cQKjeLeLSTX[C]/MN].1 YUSCIJ]qY\IJSC]&I9MXYUFpe9PZrpYBijOX]MNSTIL]uPRHJSCKBF[C]cQYUe9M= ]ue )cpYUP]`b Y\SfeP>]HmHL]7qY\IJSTY\HLSTKUFd]&emH1OXFXS 1 KUIJbd]lMXYUFpe[T]&e+MNSCIJ]&P>HLSTKUFpe1cQ]&ILc]FQMNSTPOX[fY\STIL]ue
∆x]H
∆ycQ]&OXU]&FBHsPZrpYUFXVU]&I+[T]7[TKUFpV MN]ueYqaN]&ex]>HyIJ]&eLcQ]uPRHLSTU]&bd]FjHb Y\SfeP>]>HLHL]4qY\IJSTY\HLSTKUF]ueHOXFXS )
1 KUIJb ]MXYUFpe [C]ue MNSCIJ]&P>HLSTKUFpe KBImHJrXKUVBKUFpYU[C]ue BKUSTI [TY0¤ SCVBOXIL] NA >A\_`F IJY 8 FX]bd]FjH MN]b Y\ST[C[fY\VB]]FMNSCW&IL]&FpP>]ue0pFXST]&e^cXIJKNMNOXSCH9MNKUFQP MN]&e^b YUSC[T[C]ue4W[fY\FpPW]ue4[CKBSCFMN] [fY&.KUFp]M STFjHLW&I 2H&A= Y\IY\ST[C[T]OXIZej[fY9HLrXW&KUIJSC]4P>KBb bd]4[fY9cXIZYqHJSTijOX]4MN]uesMNS WIJ]FQP>]&e+pFpSC]uesem]^P>KBbdcX[CSfijOX]FjHsemSTFXVBOX[CSTkIJ])b ]&FjH7[TKUIZeLijOX][T]&e0P>K]78 PSC]&FjHJelMpY\Fpe7[C]ue`WuiBOQYqHLSTKUFQe7YUONa$MXWIJSCBW]&elcpY\ILHLST][T[T]&eleLKUFjHlqYUILSfY\X[T]&e]>H`[TKUIZeLijOX]0[C]uecXIJKUX[Tkbd]&eseLKUFjHsFXKBF )[TSTFXW&YUSCIJ]&e&A
* ) * "!$# -# "%.( !lKUOpedPKUFpeLSTMXWIJKUFpe9[¢ W&ijOpY\HLSTKUF M YUMXU]&P>HLSTKUF]&F OXFX]$MNSTbd]FpeLSCKBF M ]&eLcpYUP]/MXYUFpe%[C]$MNK )
bdYUSCFp]^QKBILFXW(0, 1)
YU]uPlOXFp]S HJ]&eJem]^P>KBFpemHJY\FjHJ]V > 0
]H7MN]&elP>KUFQMNS HJSCKBFpe`YUONa[TSCbdSCHL]&e7MN]cQW&ILSTKNMNSTPS HJW
∂u
∂t+ V
∂u
∂x= 0
cQKBOXI(x, t) ∈ (0, 1)× R+
∗u(t, x+ 1) = u(t, x)
cKUOXI(x, t) ∈ (0, 1)× R+
∗u(0, x) = u0(x)
cQKUOpIx ∈ (0, 1).
NA
7F%MNSfeLPILWHLSfem]+HLKBO7KUOpIJe [¢ ]&eLcpYBP>]+YB]&P1OXF%cQYUe∆x = 1/(N+1) > 0
N]&FBHJSC]&I cQKjemSCHLS 1 ]>H [T]
HL]bdcpeYB]&P∆t > 0
]HKBFFXK\HJ](tn, xj) = (n∆t, j∆x)
cQKBOXIn ≥ 0, j ∈ 0, 1, ..., N + 1
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
| 7 / / /
unj
[TY qY\[T]OXI`M= OXFp]emKB[CONHJSCKBFMNSTeJP>IJk>HJ]^Y\cXcXIJKNPZrXW]4Y\O$cKUSTFjH(tn, xj)
Q]>Hu(t, x)
[TYdeLKU[TONHLSTKUF]aXYUPRHJ]MN] NA RAv3]&edP>KBFpMNSCHLSTKUFpe YUONa [TSCbdSCHL]&eMN]$cQW&ILSTKNMNSTPS HJWP>KUFQMNOXSfem]&FBHYUONa WVjY\[TS HJW&eun
0 = unN+1
cQKBOXI HJKUONHn ≥ 0
\]>H+cX[COQeVUW&FXWIZY\[T]bd]&FBHun
j = unN+1+j
AB Y\I1PKUFpeLW&ijOX]&FBH \[ STF )PKUFXFOX]7MNSTeJP>IJk>HJ] PZrpYBiBOp]`cpYUe1MN]`HJ]bdcpe]ueHOXF U]uPRHJ]OXI
un = (unj )0≤j≤N ∈ RN+1 AB`KBOpeMXKUFXFXKBFpe iBOp][fiBOp]&e eLPZrXW&b YUe=cKBeJeLSCX[T]&e cKUOXI [¢ W&ijOpY\HLSTKUF9M= YBMNU]uPRHLSTKUF NA RAu¥`Od+rQY\cXSCHLIJ]l@
FpKUOpeYUKBFpeMNW 7 %PKUFpemHJY\HLW`[T]7b YUOXqY\Sfe1P>KBbdcQKBImHJ]bd]FjH+FOXbdW&ILSfijOX]7MNO '(" & !#1
un+1j − un
j
∆t+ V
unj+1 − un
j−1
2∆x= 0
XA cKUOpI
n ≥ 0]Hj ∈ 0, ..., N AQv3] P&Y\IZYUP>HLkIJ]^STFpeHZY\X[T]%MX] P]%eJPZrXWb Y]ueH4P>KBFNpIJb W%cpY\I0[T][T]bdbd]eLOXSCqYUFBHuA
) & & f ¢ E CHD GD <- 3C 8 D@FG $D GD@ ( = F(D H(G = ((D H(G - 3C38 FF$;= ( /KG D5 D 3 G 2 $ G (G (D H(GG AG DGD G G
L2C
& & "' ¥ [ YUSTMX]M OXF MNWB][TKUcXc]bd]&FBH MN]2( Y~j[TKUIYUONHLKBOXI9MNO cQKBSCFjH(tn, xj)
KBFUW&ILSCp]1 YBP>ST[C]&b ]&FjH ijOX][T]/eJPZrXW&bdY ]&emH P>KBFpeLSTemHJYUFBH 3cXIJW&PSTe [ KBIJMNIJ] @]&FHL]&bdcpe9]H ]&F ]&eLcpYBP>]UAQ Y\I7YUFpY\[T~Nem]MX]%¤XKBOXIJSC]&IQKUFW>HJOpMNST]9[fYemHJYUXSC[TSCHLW
L2 A ¥3B]&P^[T]&e7FXK\HZYqHJSCKBFpe`MN]%[TYyNKUOpe ) eL]&P>HLSTKUF NANA N[T]&esP>KBbdcQKjeLYUFBHJ]&eMN]¤pKUOXIJSC]&Iun(k)
MN]un UW&ILSCp]&FBH
un+1(k) =
(
1− iV∆t
∆xsin(2πk∆x)
)
un(k) = A(k)un(k).
7F"UW&ILSCp]^ijOX]4[C]^bdKNMNOX[T]MNO1 YUP>HL]&OXI`M Y\bdcX[TS QP&YqHJSCKBF/]ueHsHJKUO7KBOXIJecX[TOpesVBIJYUFpM"iBOp] @
|A(k)|2 = 1 +
(
V∆t
∆xsin(2πk∆x)
)2
≥ 1,
YB]&P7SCFpWVBYU[CSCHLW^emHLIJSTP>HL]^MNkue`ijOX]2k∆x
F= ]&emH`cpYUe]&FBHJSC]&I&AX¡7KUFQP7[T]eJPZrXWb Y ]&emHsSCFpemHJYUX[T]UA
7Fdc]ONH1WuP>IJSCIJ]OXFX]sB]IZemSTKUF SCbdcX[TSfP>SCHL]lMNOdcXIJW&PW&MN]&FjH1eLPZrXW&b Y0ijOXSQMN]ST]FjH1eHZY\X[T] %UPU ]ueH[T] ' " & & !# 1 (
un+1j − un
j
∆t+ V
un+1j+1 − un+1
j−1
2∆x= 0.
XA Ux ; 1 T 0NPO%Q7ILE/IP[#E E:FVK 2WJYGT MY M K#M QE9K"E/OQ7ILW ] E+F6QK"N2O#FM F6Q6TO%Q T EK 10 WP[#T\QM NPO 0 T EK7QM NPO ] 9 /I^WKM F 10 NI PI^E0E/OZQEJYF E"Q E/O E+F 3T3K"E 9 MO3K"N2O $M QM N2OOE E/Y EJO%QF6Q6T E EJO ONI Y E
L29JNPO#K K+N2O EJI E/OQ+]
yNS=[¢ KUF/HLST]FjHlYUpemKB[COpb ]&FjH IJ]&emHL]&I`P>]FjHJILW4]>Hs]acp[CSfP>SCHL]X[C] ' ("$ & $) $ J "'2un+1
j − unj+1 − un
j−1
2∆t+ V
unj+1 − un
j−1
2∆x= 0
XA j| ]ueH`OpF"eJPZrXWb YdeLSCbdcX[T] XIJKUpOpeHJ] Nb Y\SfecpYBeHLIJk&ecXIJW&PSTe&A
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/- |
) & & f KFE 2< H HFE - 3C8 D D5 G G ( L2
(<FG D!HG
|V |∆t ≤ ∆x.+ ( D
∆t/∆xD 2 G D GD
∆tD
∆xD5G 2GD ( F %DFG $D GD (F>= (D H(G = 2 D!HG - 3C38 D $ ;<>= ( F/KG 2D D +C
9 ( (GFGD D FG D!HGG GD%(G 7G DC & "' YUI`Y\FpYU[C~NeL]^MN]^¤XKUOpILST]IsKBF"Y
un+1(k) =
(
cos(2πk∆x)− iV∆t
∆xsin(2πk∆x)
)
un(k) = A(k)un(k).
v=]4bdKMXOX[C]MXO31 YBPRHJ]OXI`M Y\bdcX[TSCQPY\HLSTKUF"]&emHlMNKUFXFpW0cQY\I
|A(k)|2 = cos2(2πk∆x) +
(
V∆t
∆x
)2
sin2(2πk∆x).
7F"BWIJS Q]4MXKUFpP4ijOX] |A(k)| ≤ 1cQKBOXIHLKBONH
keLS=[fYdP>KUFQMNS HJSCKBF |V |∆t ≤ ∆x
]&emHleJYqHLSfe 1 Y\SCHL] HJY\FQMNSTe ijO= SC[N]aNSTemHL]MN]&e bdKNMN]&e STFpemHJY\p[C]ue
kHL]&[Te ijOX] |A(k)| > 1
emSFXKBF=Aqv=]eLPZrpWb Yl]ueH MNKBFpPP>KUFQMNS HJSCKBFXFX]&[C[T]bd]FjHseHZY\X[T]UAj KUOXIWHLOpMNST]I[fY9PKUFpeLSTemHJYUFpP>]BKBF]]uPRHLOp]7OXF"MNWB][TKUcXc]bd]&FBHMN]'( Y~[TKUIY\ONHJKUOXI`MX]
(tn, xj)cQKBOXI`[fYdemKB[CONHJSCKBF
u%
2u(tn+1, xj)− u(tn, xj+1)− u(tn, xj−1)
2∆t+ V
u(tn, xj+1)− u(tn, xj−1)
2∆x=
(ut + V ux) (tn, xj)−(∆x)2
2∆t
(
1− (V∆t)2
(∆x)2
)
uxx(tn, xj)+O(
(∆x)2 +(∆x)4
∆t
)
. NA U
+KUbdbd] [ ]IJIL]&OXI0MX]dHLIJKUFpP&YqHLOpIL] P>KUFjHJSC]&FBH^OXF HL]&ILbd]d]F O((∆x)2/∆t) [T]eJPZrXWb Y/F= ]ueHcpYUe7P>KUFQemSfeHZY\FjHleLS∆tHL]&FpMU]&IJe .WIJK cX[TOpe`SCHL] iBOp]
(∆x)2Ap Y\I0P>KUFjHJIL]XSC[ ]&emH7PKUFpeLSTemHJYUFjH
]>H cXIJW&P>Sfe M KUIZMNIJ] @eLSs[C]$IZY\cXcKUILH∆t/∆x
]ueHP>KBFpemHJY\FjHuA KBOXI KUNHJ]FXSTI [fY PKUFU]&ILVB]FpP]KUF IJ]cpIL]&FpM[fY MNW&bdKUFpemHLIZYqHJSCKBF MNO (rXWKBILk&b ]"MN]v3Y\a XA XA UtXA v ]IJIL]&OXI
en ]ueHdHJKUO7KBOXIJebdY 7KUIJW]0cpY\Is[ ]IJIL]&OXIsMN]4HLIJKUFpP&YqHJOXIL]]>HlMNKBFpP0STPS
‖en‖ ≤ ∆tnKC((∆x)2
∆t+ ∆t
)
.
yS KUFVjY\IZMN]`pa]0[T]7IZY\cXcKUILH∆x/∆t
[ ]&ILIJ]OXI]&emHsMNKUFpP7b Y7KBILW7cpY\IOXFX]4P>KBFpeHZY\FjHL]/1 KUSfe∆tiBOpSHL]&FpM"XSC]&F/B]IZe .W&ILK M= Ko"[fYdP>KBFjB]IJVU]&FpP>]BA
& ( 'f f v=] eLPZrXW&b YMX]v3Y\a )G¤XIJST]&MNIJSTPZrQe%F3 ]ueHcQYUePKUFpeLSTemHJYUFjH eHJILSfPRHJK em]&FpeLOemOXSTqY\FjH[fY^¡lWpFXSCHLSTKUF NANA x >AU`WuY\FXbdKBSCFpeSC[Q]&emH1P>KUFQMNS HJSCKBFXFX]&[C[T]bd]FjH+PKUFpeLSfeHZY\FjH ]>H+PKUFXB]I )VU]FjHuApDG[-1 YUONH7P]c]FpMXYUFjH1 Y\STIL]Y\HmHJ]FjHLSTKUF ijOX]9eLS3KBFcXIJ]FpM"[T]cpYUe7MN]^HJ]bdcpe
∆t]&Y\OQP>KUOpc
cX[COQec]>HLSCHsiBOp]lP>]0iBOpSp]&emH1c]IJbdSTe+cpY\I1[TYPKUFpMXS HJSCKBF"¤ v MN]7emHJYUXST[CSCHLWU[fYP>KBFjB]IJVU]&FpP>]seL]IZYHLIJk&e[T]FjHL]BA 4F$cXIZYqHLSfijOX]4[C]^eJPZrXWb YdMN]v Yqa*) ¤pILST]&MXILSfPZrpe+F= ]ueH`cQYUeIJ]&P>KBbdbdYUFpMNWBA •
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
|\x 7 / / /
_lF$eJPZrXW&bdYdP]FjHLIJW N]acp[CSfP>SCHL]XcX[TOpescXIJW&PSTe]&emH`[T] '(" & ) $ $$
un+1j − un
j
∆t+ V
unj+1 − un
j−1
2∆x−(
V 2∆t
2
)
unj−1 − 2un
j + unj+1
(∆x)2= 0.
XA yXY MNWIJSCqY\HLSTKUFF= ]&emH`cpYUelSCbdbdW&MXSTY\HL] Y\OpeJeLS=FXKUOQes[TYcXILWuem]&FjHLKUFQe]F MNWHJY\ST[A 7FP>KBb bd]&FpP>]cQY\IsW&PILSTIJ]7OXFMNW&U]&[CKBcXcQ]&bd]FjH %[ KUIZMNIL] %]F"HJ]bdcpesMN]^[fYdemKB[CONHJSCKBF/]aXYUPRHJ]
u(tn+1, xj) = u(tn, xj) + (∆t)ut(tn, xj) +(∆t)2
2utt(tn, xj) +O
(
(∆t)3)
.
41F ONHLST[CSfeJY\FjH1[ WuiBOQYqHLSTKUF M YUMXU]&P>HLSTKUFdKUFdIJ]bdcX[fYUP]s[C]ue1MNW&ILSTUW&]&e ]FdHJ]bdcpe1cpY\I1MX]&eMNW&ILSTUW&]&e]&F$]&eLcpYUP]
u(tn+1, xj) = u(tn, xj)− (V∆t)ux(tn, xj) +(V∆t)2
2uxx(tn, xj) +O
(
(∆t)3)
.
41FNpF XKBF$IL]&b cp[TYBP>]0[C]ue`MNWIJSTUW]ue]F$]&eLcpYUP]7cQY\I`MN]ue1 KUIJb9OX[T]&eP]FjHLIJW]uesM KBIJMXIL]
u(tn+1, xj) = u(tn, xj)− V∆tu(tn, xj+1)− u(tn, xj−1)
2∆x
+(V∆t)2
2
u(tn, xj+1)− 2u(tn, xj) + u(tn, xj−1)
(∆x)2+O
(
(∆t)3 + ∆t(∆x)2)
.
7F IJ]>HJILKBOXU]dXST]F[T]eJPZrXWb YMN]v Yqa*) ]&FpMNIJK\ ]&FFpWVU[TSTVU]&YUFjH[C]ue4HL]&ILbd]&e%MNO HLIJKUSfeLSCk&b ]KBIJMXIL]0]>Hs]F"IJ]bdcX[fY YUFjH
u(tn, xj)cpY\I
unj
AXs]b YUIJijOX]&IijOX]NcpY\IIZY\cXcKUILHsY\ONacpILWuP>W&MX]FjHJeeJPZrXW&bdYBe UKUF YMNSfeLPILWHLSfemW +LeLSCb%OX[ HZY\FXW&bd]FjH3,s[C]ueMNW&ILSTUW&]&e ]&F ]&eLcpYUP]s]>H1]&FdHL]bdcpe+MN]`[ WuijOpY )HJSCKBFM YUMNB]&P>HLSTKUF=AQ YUI0PKUFpemHLIJOpP>HLSTKUF Q[C] eLPZrXW&b YMN] v3Yqa*) ]&FpMNIJK\]&emH7cXIJW&PSTe [ KBIJMNIJ] ]&F"HL]bdcpe`]>H`]F$]uemcpYBP>]BA 7F$c]ONH`bdKBFBHJIL]&I`ijO= ST[=FX]^BWIJS Q]7cpYBe[C]^cpILSTFpP>STc]MNO$bdY\aNSCb%OXbMXSTeJP>IJ]>H BKUSTI [¢ 4 aN]IZP>SfP>] XA pA RAq Y\IPKUFjHLIJ] ST[X]&emH1eHZY\X[T]]FdFXKBILbd]
L2 ]>H1MXKUFpPP>KBFU]IJVU]&FjHeLKUOQe[TYdP>KBFpMNSCHLSTKUF ¤ v |V |∆t ≤ ∆xA
; 1 T f 0NPO%Q7ILE/I P[E EFK32#W/YGT JE D/T%H#\ EJO PILN E+F6Q0FQ6T EE+Q K"NPO E/I JEJO%Q E/OO#NI Y E
L2FM |V |∆t ≤ ∆x
]
; 1 T T 0NPO%Q7ILE/I P[#E EZFVK 2WJYGT JESD/T%H#\ I M E 2I>M K32#F /ILW+F6EJI E EJI MO3KM/EP[qYGT%HM \Y0[Y M FK3ILE"Q F7M T K+N2O M QM NPO PD |V |∆t ≤ ∆x
E+F6Q0FVTQM F6X>TJM QE:9Q6TO $M F 2[#E EFK32#W/YGT ED/T%H#\ E/O 2ILN O#E E /ILW+F6EJI E 3T#F FT[%XFM V∆t/∆x T[%Q −1, 0,
N2[1]
; 1 T , 0NPO%Q7ILE/IP[E E FVK32#W/Y T JE D/T%H#\ E/O 2ILN P] 4(E+F6Q E F6EJ[+FK32#W/YGT /I^WKM F 10 NI PI^E E/OZE%F 3T3K"E E"Q QE/Y F P[/M$F6NM Q 2[ Q/E
un+1j = αun
j−1 + βunj + γun
j+1,
N α, β, γ
W /EJO E/OQ;FVE/[ E/Y EJO%Q JEV∆t/∆x
]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/- |B|
+KBbdb ]9FXKUOpel[¢ YUKBFpe`MNW 7 MNSCH7Y\O +rQY\cXSCHLIJ]@ QOXFX]SfMNW&] 1 KUFpMpY\bd]FjHJYU[C]9cQKBOXI`KUXHL]FpSCIMN] +mQKBFpe,9eJPZrXWb YBescQKBOXIl[ W&ijOpYqHJSCKBF M YUMNB]&PRHJSCKBF XA ]&emH7[T] 1 & & A`KUOQesMNKUFXFpKUFpe[fY 1 KUIJb ]4VBWFXW&IJYU[C]4MNO '(" & 1 &
un+1j − un
j
∆t+ V
unj − un
j−1
∆x= 0
eLSV > 0
un+1j − un
j
∆t+ V
unj+1 − un
j
∆x= 0
eLSV < 0.
NA U
7F Y"MNW 7 /OYUO +rpYUcXS HJIL]/@ ijOX]d[T] eJPZrXW&bdY/MNW&P]FjHLIJWdY\bdKUFjH4]&emH4emHJYUX[C] ]F FXKUIJbd]L∞emS[TY P>KBFpMNSCHLSTKUF¤ v |V |∆t ≤ ∆x
]&emH eJYqHJSTe 1 Y\SCHL]BA +KBb bd]/SC[]&emHdPKUFpeLSfeHZY\FjH9]>HdcXIJW&PSTeM KBIJMXIL]@%]F ]&eLcpYUP]9]>H7HL]&b cQe ST[ PKUFU]&ILVB]^]F FXKUIJbd]
L∞ M= YUcXILkuel[T]HJrXWKBILk&bd]%MN]dv3Y\a Av=]4b 2bd]^ILWuemOp[ HZYqHs]&emH`IJYUS]&F$FXKUIJbd]
L2 YU]&P7[fYdb 2&bd]4PKUFpMXS HJSCKBF ¤ v1A; 11 T # 0NPO%Q7ILE/I P[E E FVK 2WJYGT E7H M K#M QE W%K+E/OQ7I^W TY NPO%Q P] RE%FQGK+N2O#FM FQ6TOQT EK 10 WP[#TQM NPO 80 T EK7QM N2O ] 9/ILW%KM F 10 NI PI^E E/O E+F 3T3K"E E"Q QE/Y F 9 F6Q6T E E+QK"N2O E/I JE/OQ;EJO ONI Y E L2
F7M T K"N2O $M QM N2O D |V |∆t ≤ ∆xE+FQ FVTQM F6X>TJM QE]
& ( 'f , KUOXI$MN]uecXIJKUp[Ck&b ]ueFXKBF ) [CSTFXW&YUSCIJ]&e Ko [TY S HJ]&eJem]VMNW&cQ]&FpM ]&[C[T])
b 2&bd]$MX]"[ SCFpPKUFXFOX]u ]HFXK\HZY\bdbd]FjH cKUOpIMN]&edbdKNMNk[T]&e M= WuP>KUOp[C]&b ]&FjHJe MN]&*pOXSfMN]ue
[C]$eLPZrpWb Y MXW&P>]&FjHLIJW"YUb KBFjH ]ueHdIJYUSCbd]FjH emOpcQW&ILST]OXI YUONa Y\ONHJIL]ueA DG[s]&emH [fY emKBOXIJP]/MN]FXKUb%XIL]&OpeL]&e4VUW&FXWIZY\[TSfeLY\HLSTKUFpe Q]uY\OpPKUOXc cX[TOpePKUbdcX[T]>aN]&eijOX] [ KUIJSCVBSCFpYU[ BKUSTIP>]eLO7]>H[C]9PKUOXIZe T@& RA 4FcpYUImHJSTPOX[TSC]&IXpSC]&FijOX][T]9eLPZrpWb YdKUIJSCVBSCFQY\[=eLKUSCH7em]&OX[T]bd]FjH7M KUIZMNIJ] @ QSC[YUMNbd]>HlMX]&eqY\IJSTYUFjHL]&eM KUIZMNIJ] XA •
yNPZrXWb Y yjHJYUXST[CSCHLW 4IJIJ]OXIsMN]4HJILKBFpPY\HLOXIJ]4 aNcX[TSfP>SCHL]P>]&FjHLIJW NA SCFQeHZY\X[T] O
(
∆t+ (∆x)2)
DGbdcX[TSTPS HJ]P>]FjHJILW NA \x eHZY\X[T]L2 O
(
∆t+ (∆x)2)
v3Y\a*) ¤pILST]&MNIJSfPZrpe NA B| eHZY\X[T]L2 ]>H L∞ eLS O
(
∆t+ (∆x)2
∆t
)
P>KBFpMNSCHLSTKUF ¤ v |V |∆t ≤ ∆xv Yqa*) ]&FpMNIJK\ XA eHZY\X[T]L2 eLS O
(
(∆t)2 + (∆x)2)
P>KBFpMNSCHLSTKUF ¤ v |V |∆t ≤ ∆x¡7W&P]FjHLIJW^Y\bdKUFjH XA B eHZY\X[T]L2 ]>H L∞ eLS O
(
∆t+ ∆x)
P>KBFpMNSCHLSTKUF ¤ v |V |∆t ≤ ∆x
XA sWuemOXbdW^MN]uescXIJKUcXIJSCWHLWueMX]MNSCB]IZeeLPZrXW&b YUecKUOXIs[ WuijOpYqHJSCKBF"M YUMXU]&P>HLSTKUF KBOXI PKUbdcpY\IJ]I P>]&e MNSTU]IZe3eJPZrXW&bdYBe=M OXFcKUSTFjH MN]OX]+cXIZYqHLSfijOX] OXF9P>KBFpP>]&cNH c]ILHLSTFX]&FBH
ijOXKUSfijOX]!1 KUIJbd][ ]&emHsP][TOXS3M W&ijOpY\HLSTKUF"W&ijOXSTqY\[T]FjHL]BA
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
|U 7 / / /
& f # ' G A ( '$% ( ' % "= #G E = F(D H(G D5G KG (#D5(G D ( BA D (H D!H G -0 = D >;(>( F KD = ( F(G GD8< = # 2D (G (D # ( E C
e ji
e j$u
NA x ^DGF*pOX]&FpP>]sMN][fY4¤ v$emOpI [fY0MNS OpeLSCKBF%FOXbdW&ILSfijOX]MXO eJPZrXWb Y7MN]sv3Yqa*)G¤XIJSC]uMNIJSTPZrpe rpY\OXH ]>HlMXO$eJPZrXW&bdYdMNWuP>]&FBHJILW4Y\bdKBFBH pYUe RA( KUOpe"[T]&eeJPZrXWb YBe/ijOX] FXKBOpe"U]FpKUFpe"MN] UKBSCI$emKBFjH$PKUFpeLSTemHJYUFjHJe&As+]&cQ]&FpMXY\FjH `emS4KUF
Y 7KUONHJ] ^[ WuiBOQYqHLSTKUFd[fY^cpYUImHJSC]scXIJSTFpP>STcpY\[T]lMN]l[¢ ]IJIL]&OXIMN]sHJILKBFpPY\HLOXIJ]`M OXFeJPZrXWb Y UY\[TKUIZe P>]eJPZrXW&bdY`]&emH FXKBF9eL]OX[T]bd]FjH ]FpPKUIJ]+P>KBFpeLSTemHJYUFBH YU]&P1P>]>HLHL]FXKUOpU][T[T]W&ijOpY\HLSTKUF +mW&ijOXSTqY\[T]FjHL] , b YUSTe0]&emH^b 2bd] eHJILSfPRHJ]bd]FjH4cX[COQe4cXILWuP>Sfe0cQKBOXIP>]HmHL]dWuiBOQYqHLSTKUF W&ijOXSTqY\[T]FjHL]BA 41F M= YUONHLIJ]&eHJ]IJbd]&e[C] eLPZrXW&b Y ]ueH-+cX[TOpePKUFpeLSfeHZY\FjH3,YU]&P"[ WuijOpYqHJSCKBFWuijOXSCqYU[C]&FBHJ]ijO= YU]&P"[ WuijOpY )
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/- |j
HLSTKUFgM KUIJSCVBSCFX]BA IJ]FXKBFped[¢ ]>aN]&b cp[C]MXOeJPZrXW&bdY MN] v Yqa*)G¤XIJSC]uMNILSfPZrpe XA j| %cKUOXI[ W&ijOpY )HLSTKUF M YUMNB]&P>HLSTKUF %M= YUcXILkue XA B X[TYcpY\ILHLST]^cXIJSCFpPSCcQY\[T]9MN]9eLKUF$]IJIJ]OXIlMX]^HLIJKUFpP&YqHJOXIL]^]ueH− (∆x)2
2∆t
(
1− (V ∆t)2
(∆x)2
)
uxxAQ Y\I4P>KBFpeLW&ijOX]FjH p[ WuiBOQYqHLSTKUFW&ijOXSTYU[C]&FjHL]%MXOeLPZrpWb YMN] v3Y\a )
¤XIJSC]uMNILSfPZrpe]&emH∂u
∂t+ V
∂u
∂x− ν ∂
2u
∂x2= 0
YU]&Pν =
(∆x)2
2∆t
(
1− (V∆t)2
(∆x)2
)
. NA Un
+]>HmHJ]dW&ijOpYqHJSCKBFWuiBOpSCqY\[T]FjHJ]%qY"FXKUOQe0MXKUFXFX]&I^MN]&e0IJ]FpeL]STVUFX]&bd]FjHJe0cXIJW&PSC]&ONaemOpI0[T] P>KBb )cQKBImHJ]bd]FjH1FOXbdWIJSfiBOp]`MNOeLPZrpWb YXA 4F ]> ]>H B[T]leJPZrXWb Y^MN]lv Yqa*)G¤XIJSC]uMNILSfPZrpe ]ueH1OpFX]`KUFXFX]Y\cXcXIJKaNSCb Y\HLSTKUF [ KBIJMXIL] 0MN]d[¢ W&ijOpY\HLSTKUFMN] P>KUFB]&PRHJSCKBF )GMNS OpeLSCKBF XA Bn `KJo[C]dP>K]8 )P>ST]FjH`MN]MNSCOpeLSTKUF
ν]&emHc]>HJS H UKBSCIJ]0FjOp[3emS[fYdP>KBFpMNSCHLSTKUF ¤ v ]ueHs]aNYBPRHJ]bd]FjH`eLY\HLSfe 1 YUS HJ]
S¢A ]UA∆x = |V |∆t >A `]b Y\IZijOXKUFpe^ijOX]/emS+[C]cQYUeMN]HL]bdcpe9]&emH9cXIJSTe^HLIJKUcc]>HLSCH [T]P>K]8 )P>ST]FjHdMX]/MNSCOQemSTKUF
νc]ONH 2>HJIL]HLIJk&e VUIZY\FQM ]>Hd[C]$eLPZrpWb Y b Y\OpYUSTe PYUIHJILKBc cKUILHLW[TY
MNS OpeLSCKBF BKUSTI^[TY¤ STVUOXIJ] NA x >A3v=]PK]8P>ST]FjH9MN]MXS OpeLSCKBFνMN] [¢ W&ijOpY\HLSTKUF W&ijOXSTqY\[T]FjHL]
]&emH%Y\cXc][TW '' ' & ( '$ A y ST[+]ueH9VBIJYUFpM =KBF MNSCH ijOX][T]/eJPZrXWb Y$]ueH '"' KUO MNSfeJemSTcpYqHJS 1 RAUv3]`P>KUbdcKUILHL]&b ]&FjH H~cXSfijOX]sM OXF eLPZrXW&b Y0MXS OpeLS 1 ]&emH1eLY7HJ]FpMXYUFpP>] 4WHJY\[T]IY\ILHLSCQP>ST][T[C]&bd]FjH [C]ue MNKUFXFpW]&e STFXSCHLSfY\[T]&e Y\OdP>KBOXIZe MXO%HJ]bdcpe&A\v=]ue eLPZrXW&b YUe HLIJKUcdMNSCOpeLS 1 e emKBFBHMNKUFpP4MN] +mb Y\OXqYUSTe3,0eJPZrXWb YUe&A; 11 T " 0NPO%Q7ILE/I P[E 10 WP[#TQM NPO WP[/M T E/OQE P[SFVK 2WJYGT W%K+E/OQ7I^W TY NPO%Q P] E+FQ
∂u
∂t+ V
∂u
∂x− |V |
2(∆x − |V |∆t) ∂
2u
∂x2= 0.
v3]%eJPZrXWb Y MNWuP>]&FBHJILW%Y\bdKUFjHl]ueH4Y\OpeJemS MNS OpeLS 1 eJY\O 1eLS [fYP>KBFpMNSCHLSTKUF ¤ v]ueH0]>aXYUP>HL])b ]&FjHeJYqHLSfe 1 Y\SCHL] jSA ]BA
∆x = |V |∆t RA*4FHLKBONHP&YUeB[T]7PKj]78P>ST]FjHMN]4MNSCOQemSTKUFMX]l[ WuiBOQYqHLSTKUFW&ijOXSTYU[C]&FjHL]Fp]HL]&FpM cQYUe%U]IZe[ SCFNQFXSsemS[C]"cpYBe9MN]/HL]&b cQeHJ]FpM U]IZe .&WIJK ∆x
paW P>]ijOXS1]ueHOpFX]FX]HmHL]YUbdW[TSCKBIJY\HLSTKUFcpY\IIZY\cpcQKBImHY\OeJPZrXWb Y$MN]v Yqa*)G¤XIJSC]uMNILSfPZrpe BKUSTI^[TY¤ STVUOpIL] NA x >A++]>H ]> ]>HMX] MNS OpeLSCKBFFOXbdW&ILSfijOX]$]&emHSC[T[COQeHJILW$cpYUI [TY ¤ SCVBOXIJ] XA x KJo[ KBFILWuemKBONHs[¢ W&ijOpY\HLSTKUF$M YBMNU]uPRHJSCKBF/eLOXIsOXF$SCFjHJ]IJYU[C[T]^MN]4[CKBFXVUOp]OXI^@^YU]uP7MX]&esP>KBFpMNSCHLSTKUFpesYUONa[CSTbdS HJ]&eMN]scWIJSTKMXSTPS HJW \OXFp]`MNKUFXFpW]SCFpS HJSTYU[C]`eLSCFOpeLK MXYU[C]uOpF cpYUe M ]&eLcpYBP>]
∆x = 0.01qOXFX]
jSCHL]ueLeL]V = 1
]HOXFHL]bdcpe^pFpY\[T = 5
A 7F P>KBb cQY\IJ] MN]&ONa qYU[C]&OXIJe4MNO cpYBe^MN]dHL]&bdcpe∆t = 0.9∆x
]H∆t = 0.45∆x
A; 11 T 0NPO%Q7ILE/I P[#E 10 WP[3TQM N2O WP[/M T E/OQE 2[ZFK32#W/YGT JE D/T%H#\ E/O 2ILN P] 4E+FQ
∂u
∂t+ V
∂u
∂x+V (∆x)2
6
(
1− (V∆t)2
(∆x)2
)
∂3u
∂x3= 0.
+KBbdb ]4[T]^eLPZrXW&b Y9MN]^v3Y\a*) ]FpMXILKU"]ueHscXIJW&PSTeM KUIZMNIJ] N[ W&ijOpYqHJSCKBFWuiBOpSCqY\[T]FjHJ]7FX]P>KUFjHJSC]&FBH4cpYBe7MN] HJ]IJb ] MN] MNSCOpeLSTKUFb YUSTe0OXFHL]IJbd] MXOHJILKBSTeLSCk&bd]9KUIZMNIJ] MNSCH '! ' As]b Y\IZijOXKUFQe0ijOX]d[T] P>K]8P>ST]FjHMN]&YUFjH^P] HL]IJbd] MNSTeLc]IZemS 1+]ueH^OXF KUIZMNIJ]%cp[COpe4c]>HJS H9ijOX][C]0P>K]8P>ST]FjH+MX]7MNSCOQemSTKUFMX]&e+W&ijOpYqHJSCKBFpeWuijOXSCqYU[C]&FBHJ]&e+MN]&eeLPZrpWb YUeMXS OpeLS 1 e&Ap7 ]ueHcQKBOXI )iBOpKUSP>]H]]HlMNSfemc]IZemS 1 Fp]7c]ONHleL]4UKBSCI B]&F/VBWFXW&IJYU[ NijOX]4emOpIsOXF"eJPZrXW&bdY%FXKBF/MNSCOQemS 1Apv=]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
|U 7 / / /
PKUbdcKUILHL]bd]&FBH+H~cXSTijOX]0M OXF/eJPZrXWb Y%MNSTeLc]IZemS 1=]&emHijO= ST[QcXIJKNMNOXSCHMX]&e+KBeJP>ST[C[fYqHJSCKBFpe[TKUIZeLijOX][fY eLKU[TONHLSTKUF$]&emH7MNSfeJP>KUFjHJSCFOX] UKUSTI`[fY ¤ SCVBOXIL] XA | RA 41F]> ]>H[T]4HL]IJbd]9MNSfemc]IZeLS 1 bdKMXS p]9[TYSCHL]ueLeL]MX]+cXIJKUcQY\VBY\HLSTKUFMN]ue KUFpMN]ue3cp[TYUFX]&e KUO%bdKMX]&e MN]¤XKUOpILST]I MN][TY0emKB[COXHLSTKUF cpY\ILHLSfP>OX[TSTk)IJ]bd]&FBH`MN]uesbdKNMN]&esMN] 1 IJW&ijOX]&FpP>]4W&[C]&UW] pYU[CKBIJeijO= OXF"HL]&ILbd]^MNSCOpeLS 1 Fp] 1 Y\SCHliBO3 Y\HmHLW&FOX]IeLKUF$Y\bdcX[TSCHLOpMN] UKBSCI[ 4 aN]&IJPSTP] NA XA RA
e j us
e j us2n
XA | +KUbdcpYUIJYUSTeLKUF MN]ue1eJPZrXWb YUe+MN]7v Yqa*)G¤XIJSC]uMNILSfPZrpeUMN]0v3Yqa*) ]&FpMNIJK\ U]HMNWuP>]&FBHJILWYUbdKUFjHscQKBOXIsOXFX]MNKBFXFXW&]0STFXSCHLSfY\[T]emSTFOpeLK MXY\[T] rpY\OXH KUO"]FPILW&FX]&YUO QYUe RA¥`pFM SC[T[TOpeHJIL]&IFXKUHLIJ]scXILKBcQKje FXKUOQecXIJW&eL]FjHLKBFpe MN]ue1PYU[TPOX[fe ]> ]&P>HLOXWueemOpI1OXF STFjHL]IJqY\[T[C]
MX]%[TKUFXVBOX]OpI @dYU]uP9MN]ue4P>KUFQMNS HJSCKBFpe4Y\ONa[TSCbdSCHL]ue^MN]dcQW&ILSTKNMNSTPS HJW OXFcpYUe4M ]&eLcpYBP>]∆x =
0.01qOXFdcpYBeMN]HL]&bdcpe
∆t = 0.9∗∆x \OXFp]sjSCHL]ueLeL] V = 1]HOXFdHL]&b cQe pFpYU[
T = 5AU¡7]OXa
H~c]&e MN] PKUFpMXS HJSCKBFpedSCFpS HJSTYU[C]ue emKBFBHdHJ]&emHLW]ue % HLKBONHM Y\KUIZM OXFp] P>KUFQMNS HJSCKBFSTFXSCHLSfY\[T]"HLIJk&e
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/- |Un
ILW&VUOX[TSCk&IL]BOpF/eLSCFOpecXOXSfe+OpFX]0PKUFpMXS HJSCKBFSTFXS HJSTYU[C]4MNSfeLPKUFjHLSTFOX] jOXF$P>IJWFX]uY\O UKBSCI[TY ¤ SCVBOXIJ]NA| RAXv3]&e`eJPZrXW&bdYBecXIJW&P>Sfe d[ KBIJMNIJ]d@^emKBFBHlP[TYUSCIJ]bd]&FBH`MNSC OpemS 1 e$%pSC[fesW&PIJYBem]&FjH[fYdemKB[COXHLSTKUF=Av=]7eJPZrXW&bdY9MN]7v Yqa*) ]FQMNILKU/cXIJW&PSTe [ KBIJMXIL] ]&emH1HLIJk&e1QKBFcKUOXIOXFX]0emKB[CONHJSCKBF ILW&VUOX[TSTkIJ]bdYUSTedKjeLPSC[T[T]"cKUOpI [C]P>IJWFp]&Y\OPYUI SC[s]ueHMNSfeLcQ]&IJeLS 1Av3]P>KUFQP>]cXH M= WuiBOQYqHLSTKUFW&ijOXSTqY\[T]FjHL]cQ]&ILbd]>HlMX]4PKUbdcXIJ]FpMXIL]^P>]uecXrXW&FXKUbdkFp]&eFOXbdWIJSTijOX]ueA; 11 T 0 NM Q10 W 2[3TQM N2O
∂u
∂t+ V
∂u
∂x− ν ∂
2u
∂x2− µ∂
3u
∂x3= 0
/NP[I(x, t) ∈ R× R+
∗
u(t = 0, x) = sin(ωx+ φ)/N2[I
x ∈ R,
T EK V, ν, µ, ω, φ ∈ R] 0NPO%Q7ILE/I P[#E FVTZFVN-[QM N2O E+F6Q
u(t, x) = exp(−νω2t) sin(
ω(x− (V + µω2)t) + φ)
>N2OZT PY E"Q*Q7I@TZFVN2O [O/M K#M QW "] 6 OWP[/MILE 2[#E T M &[#FM N2OZTQ*QWJO[#E 10 TY M Q7[ EE TZFVN-[QM N2O 9Q6TO $M F 2[#E T $M F /EJI^F7M N2O Y N M $E T M QE%F6F6E E /ILN3T TQM N2O$]
; 11 T O W O/M Q E FVK 2WJYGT FVT[QE"\Y N2[QN2O ET%X*ILN 9E/O TO TJM F
un+1j − un−1
j
2∆t+ V
unj+1 − un
j−1
2∆x= 0.
Q7[ $M E/I T K+N2O#FM FQ6TO3K"EqE"Q 10 E/I I^EJ[I EQ7I^NPO#K%TQ7[ILE E K"E FVK32#W/Y T$] N2OQ7I^EJI #TJIRTO#T #F6E JE3N2[I M EJI 2[ 0 ME+F6Q;F6Q6T E F6NP[F T K"N2O $M QM N2O D |V |∆t ≤M∆x
T E%K M < 1]
; 11 T O JW OJM Q E FK32#W/YGT JE :I@TO"\9M K32#N FVN2O
un+1j − un
j
∆t+ V
un+1j+1 − un+1
j−1
4∆x+ V
unj+1 − un
j−1
4∆x= 0.
Q7[ $M E/I T K+N2O#FM FQ6TO3K"EqE"Q 10 E/I I^EJ[I EQ7I^NPO#K%TQ7[ILE E K"E FVK32#W/Y T$] N2OQ7I^EJI #TJIRTO#T #F6E JE3N2[I M EJI 2[ 0 ME+F6Q MO3K"NPO $M QM N2OOE E/Y EJO%Q:F6Q6T E]
"!$#% "! #% lKUOpe PKUFpeLSTMXWIJKUFpe[ W&ijOpYqHJSCKBF9MN]ue KBFpMN]ue MXYUFpe [C]MNKUb Y\STFX]+QKBILFXW
(0, 1)YU]&P1P>KUFQMNS HJSCKBFpe
Y\ONa[TSTb SCHL]ue`MN]^cWIJSCKNMNSfP>SCHLW
∂2u
∂t2− ∂2u
∂x2= 0
cKUOpI(x, t) ∈ (0, 1)× R+
∗
u(t, x+ 1) = u(t, x)cKUOXI
(x, t) ∈ (0, 1)× R+∗
u(t = 0, x) = u0(x)cQKUOpI
x ∈ (0, 1)
∂u
∂t(t = 0, x) = u1(x)
cKUOXIx ∈ (0, 1).
NA xBt
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
Bt 7 / / /
¥ U]uP"[C]ue b 2bd]&edFXKUHJY\HLSTKUFpe ijOX]cXIJW&PW&MN]&bdb ]&FjH[ STFpPKUFXFOX]MNSfeLPILkHL] PZrpYBiBOp]"cQYUe MN]HJ]bdcpe ]&emHdOXF B]&PRHJ]OXI
un = (unj )0≤j≤N ∈ RN+1 A v=]ue PKUFpMXS HJSCKBFpe%YUONa [TSCbdSCHL]&edMN]"cW)IJSTKMXSTPS HJWdPKUFpMXOXSTeL]FjH^Y\ONa WVBYU[CSCHLWueun
0 = unN+1
cKUOpI4HLKUOXHn ≥ 0
]>H^cX[TOpe4VUW&FXWIZY\[T]bd]FjHun
j = unN+1+j
AU+KBb bd][C]ue PKUFpMXS HJSCKBFpe Y\ONa^[TSCbdSCHL]&e Fp]1XaN]&FBH cpYBe3[fYlqYU[C]&OXI MX]uY\OXa]>aHLIJW)
bdSCHLWuelMN]%[¢ SCFjHJ]IJYU[C[T](0, 1)
c]FpeL] . [¢ STFBHJ]IJcXIJW>HJY\HLSTKUF$]F$HJ]IJbd]&elMX]9P>KBIJMN]9jSTXIZY\FjHL] X[fYeLKU[TONHJSCKBF
ucQ]&ONH+Fp]lcpYBe IL]ueHJ]IKUIJFXW&]`]FdHJ]bdcpejP]lijOXSQPKUbdcX[TSTijOX]`[ W>HLOQMN]7MN]l[TYemHJYUXST[CSCHLW
MX]&eseJPZrXWb YUeFOXbdWIJSfiBOp]&e&Aj YUI]aN]bdcX[T] XeLSu0 ≡ 0
]Hu1 ≡ C
MXYUFpe(0, 1)
N[TY emKB[COXHLSTKUF$MN] NA xBt ]&emH
u(t, x) = CtA KUOpI+W&[CSTbdSCFX]&IsP>]>H]> ]>H KUF1 YUS HsMNKUFQP`[ r~cQKUHLrXkuem]4ijOX]l[fYSCHL]ueLeL]
STFXSCHLSfY\[T]^]&emH`MN]^bdKq~U]FpFX]7FOX[T[T]∫ 1
0
u1(x) dx = 0. XA xQ@
KBOXI7[ W&ijOpYqHJSCKBF MN]ue0KUFpMN]ue NA xBt s[T] eJPZrXWb YrpY\XSCHLOp][ ]&emH0[T]θ ' ("$ & % cQKBOXI
n ≥ 1]Hj ∈ 0, ..., N
un+1j − 2un
j + un−1j
(∆t)2+θ−un+1
j−1 + 2un+1j − un+1
j+1
(∆x)2
+(1− 2θ)−un
j−1 + 2unj − un
j+1
(∆x)2+θ−un−1
j−1 + 2un−1j − un−1
j+1
(∆x)2= 0
NA x
YB]&P0 ≤ θ ≤ 1/2
AUv=KBIJeJijOX]θ = 0
KBF KUNHJSC]&FBHOXFeJPZrXWb Y4]acp[CSfP>SCHL]UHZY\FpMNSfe1ijOX]l[C]0eLPZrXW&b Y]ueH`STbdcX[TSTPS HJ]emS
θ 6= 0AXv=]ue`P>KBFpMNSCHLSTKUFpeSTFXSCHLSfY\[T]&eseLKUFjHscXILSfeL]&e]F$P>KBbdcNHL]^cpYUI
u0j = u0(xj)
]>H u1j − u0
j
∆t=
∫ xj+1/2
xj−1/2
u1(x) dx,
P]7ijOXS VBYUIJYUFjHLSCH+ijOX]7[fYSCHL]ueLeL]lSTFXSCHLSfY\[T]7MNSfeLPILkHL]7UWIJSCp]7YUOpeLeLS[TY%P>KUFQMNS HJSCKBF XA xQ@ >AN+KBb bd]PZrQYUP>OpFX]+MN]ue MNS WIJ]FQP>]&e pFpSC]ue P]FjHLIJW]ue ijOXSNYUcXcXIJKPZrp]FjH [T]&e MNWIJSTUW]ue eL]&PKUFpMX]&e MXY\Fpe NA x ]ueHM= KBIJMNIJ] [T]
θ)GeLPZrpWb Y P>]FjHJILW XA x %]&emHcXIJW&PSTe [ KBIJMXIL] ]F]uemcpYBP>]$]>HHL]bdcpe&A
`]b Y\IZijOX]I ijOX]P>]$eLPZrpWb Y]&emHdSTFqY\IJSTYUFBH eLSKBFPZrQY\FXVB]/[T]em]&Fpe MNOHL]bdcpe P>]ijOXSs]ueHPKUbdcpY\HLSTX[C]/YU]&Pd[fYcXIJKUcpILSTW>HJWMN]IJWB]IZemSTXST[CSCHLW ]&FHL]&b cQe9MN]/[¢ W&ijOpY\HLSTKUF MN]ueKUFQMN]&e3OX][TKUIZeMN]^[fYdyNKUOpe ) eL]&P>HLSTKUF@UA XA >A) & & f " +
1/4 ≤ θ ≤ 1/2 θ FEJGD - 3C 38KD G (G (D H(G G GD
D5 G G ( L2 C + 0 ≤ θ < 1/4
D D A ( K(G D!HG
∆t
∆x<
√
1
1− 4θ,
D GD ∆t/∆x > 1/
√1− 4θ
C
& & "' +KUbdbd]0cXILWuP>WuMN]bdbd]FjH+KBFONHJSC[TSTeL]0[¢ Y\FpYU[C~NeL]7MN]0¤XKBOXILST]IcKUOXIKUNHJ]FXSTI
un+1(k)− 2un(k) + un−1(k) + α(k)(
θun+1(k) + (1− 2θ)un(k) + θun−1(k))
= 0,
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/- p@
YU]&Pα(k) = 4
(
∆t
∆x
)2
sin2(πk∆x).
DG[3e Y\VBS HsM OXF eJPZrXWb Y 9HJILKBSTeFpSCB]&Y\OXa/ijO= KBF/IJWWuP>IJS H
Un+1(k) =
(
un+1(k)un(k)
)
=
(
2−(1−2θ)α(k)1+θα(k) −1
1 0
)
Un(k) = A(k)Un(k),
]>HUn+1(k) = A(k)nU1(k)
A3v=]ue^qY\[T]OXIZe4cXIJKUcXIJ]&e(λ1, λ2)
MN] [TY$b Y\HLIJSTP]A(k)
eLKUFjH^[C]ueIJYBP>STFX]&eMNO$cKU[T~jF Ubd]^MNOMN]ONaNSTkbd]MN]&VUIJW
λ2 − 2− (1− 2θ)α(k)
1 + θα(k)λ+ 1 = 0.
v=]^MNSfeLPILSTbdSCFpYUFjHsMX]P>]>HLHL]^WuiBOQYqHLSTKUF"]&emH
∆ = −α(k)(4− (1− 4θ)α(k))
(1 + θα(k))2.
v W>HLOQMN]9MN][fY emHJY\pSC[TS HJW9MNO eJPZrXWb Yd]ueHlSfP>S3HLIJk&e`MXW[TSTP&YqHL]%PY\IA(k)
F= ]ueH7cpYUelOXFX]b YqHJILSfP>]FXKUIJb Y\[T]%]H ‖A(k)n‖2 6= ρ(A(k))n KJo ρ(A(k)) = max(|λ1|, |λ2|)
]&emH[C]dIZY~UKBF eLcQ]uPRHLIZY\[MN]
A(k)A 7FeL]4PKUFjHL]&FjHL]^MNKUFQP^MN]4UWIJSCp]Is[fYdP>KUFQMNS HJSCKBF $1 ' ' MN]^emHJY\pSC[TS HJWMN] KUF
`]OpbdYUFXF ρ(A(k)) ≤ 1
UKBSCI^[fYs]b YUIJijOX] NANAqx ]>HKBF IL]&FjBKUST] "[ 4a]&IJPSTP] NA XAT@U@cQKBOXI OpFX]PKUFpMXS HJSCKBF%eLO 8eLYUFBHJ]UAqyS
∆t/∆x > 1/√
1− 4θuOpF PZrXKBS a 7OpMXSTPSC]&ONa9MN]
k HJ][NijOX]
sin2(πk∆x) ≈ 1 P>KBFpMNOXSCH
∆ > 0\]>H1MpY\Fpe P]P&YUe [T]&e MN]&ONa%IZYUPSCFp]&e
λ1]Hλ2eLKUFjH ILW&][T[C]ue
MN]%cpILKNMNOXSCH4WVBYU[ @UA v OXFX]dMN]ue0MX]ONa ]ueH!1 KUIZP>W&b ]&FjH4emHLIJSTP>HL]&b ]&FjH0cX[TOpe0VUIZY\FQMN]9ijOX]"@%]&FYU[C]&OXIlYUpemKB[COp]
ρ(A(k)) > 1X]H7[T]9eJPZrXWb Yd]&emHlMNKBFpP^STFpeHZY\X[T]UA yS
∆t/∆x < 1/√
1− 4θ
Y\[TKUIZe∆ ≤ 0
cQKBOXI^HJKUONHk3]>H[T]&eMN]&ONa IZYUPSCFp]&e^eLKUFjHP>KUbdcX[T]>aN]ue^P>KUF 7OXVUOXW&]&eMX] b KNMNOp[C]
WVBYU[ @UA YUI0PKUFpeLW&ijOX]&FBH ρ(A(k)) = 1
]H0[fY"P>KUFQMNS HJSCKBFMX] emHJYUXSC[TSCHLW MN] KUF `]OpbdYUFXF]&emHleLY\HLSfe 1 YUS HJ]UA
; 11 T $MO/MI T W/YGN2O#FQ7I@TQM N2O P[ZD3E/Y0Y E ] ] E/ORK+T K/[ TOQA(k)n
9#E+Q9Y NPO%Q7ILE/I TZFQ6TJM M QW 2[ZFK32#W/YGT0F6NP[F K"NPO $M QM N2O PD 2I3K"E P] "]; 11 T X O K"NPOF7M JUJI^E E K%T#F MY M QE 2[ D3E/YZYGE P] P] +9:K 0 E+F6Q*\ \ $MI^E
∆t/∆x =1/√
1− 4θT EK 0 ≤ θ < 1/4
] N2OQ7ILE/I P[E Eθ\FVK32#W/Y TGK+E/OQ7I^W P] &E+FQ;MO#FQ6T E TO#F9K"E
K+T#FEJO WJI M 2TOQ 2[#E unj = (−1)n+j(2n− 1)
E+F6QR[O#E FVN-[QM N2O ILE/YGTJI P[#E P[ 0 M/F 0 T $M Q 80[O#EMOF6Q6T/M M QW X>TJM E [JM FP[#E T K3ILNM FVFVTO3K"E JE
unE%FQ MOWTJMI^E E"QGONPOZE7H /N2O#E/OQM E E "]
lKUOpedST[T[COpemHLIJKUFQe%P]&e eJPZrXWb YUe%cpYUI [fY¤ STVUOpIL] NA eLOXId[TYBijOX][T[C]$eLKUFjH IJ]cpILWuem]&FBHJW]ue9[T]&eILWuemOX[CHJY\HJe7KUNHJ]FOpe4YU]uP[T] eJPZrXW&bdY"P>]&FjHLIJW9]>aNcX[TSfP>SCHL]d]>H4[C]
θ)GeLPZrpWb YSTbdcX[TSTPS HJ] cQKBOXI
θ =0.25
>A=v=]&e9PY\[fP>Op[TeeLKUFjH]]uPRHLOpW&eeLOXIOXFSTFjHL]&ILqY\[T[T]MN] [CKBFXVUOp]OXI @YB]&PdMN]&e9P>KUFQMNS HJSCKBFpeY\ONa9[TSTb SCHL]ueMN]scWIJSCKNMNSfP>SCHLWqOXFdcpYUe M= ]uemcpYBP>]
∆x = 0.01OXFdcpYUe MX]HL]bdcpe
∆t = 0.9∗∆x ]>H1OXF HJ]bdcpe QFpY\[T = 5
AUv3Y^PKUFpMXS HJSCKBFpe STFXS HJSTYU[C]u0]ueH+OXF emSTFOpe \HJY\FQMNSTe+ijOX]
u1]ueH+FjOp[C[T]UA
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
7 / / /
2u
NA yXPZrXWb YUecKUOXIl[¢ W&ijOpY\HLSTKUF$MN]&eKBFpMN]&e&A
lKUOpe7YUKUFQejO[TKUIZe`MN]9[¢ 4 aN]IZP>SfP>]@BA pA xijOX]9[¢ W&ijOpY\HLSTKUFMN]&elKUFpMX]&e NA xBt sUWIJSCp]^OXFX]cpILKBcXILSTW>HJW0MX]P>KUFQem]&ILqYqHJSCKBF/MN]^[ WFX]&ILVBSC]PU ]ueH ) ) MXSCIJ]0ijOX]XcKUOXIHJKUONH
t > 0
E(t) = E(0)YU]uP
E(t) =
∫ 1
0
∣
∣
∣
∣
∂u
∂t(t, x)
∣
∣
∣
∣
2
dx+
∫ 1
0
∣
∣
∣
∣
∂u
∂x(t, x)
∣
∣
∣
∣
2
dx.
DG[ ]ueH`eLKUOXB]FjHMNWuemSTIZY\X[T]7ijO= OXF$eLPZrXW&b YFOXbdWIJSTijOX]0UW&ILSCp] ]aNYBPRHJ]bd]FjHKUO/YUcXcXIJKaNSCb YqHJS )B]bd]FjH OXFX]B]IZemSTKUFMNSfeJP>IJk>HL]%MN]9P>]HmHJ]9P>KBFpem]&ILqY\HLSTKUFMN][ W&FX]IJVUST]UAp KBOXIl[T]
θ)GeJPZrXWb YdKUF
STFjHLIJKNMNOXSCH (&'
En+1 =
N∑
j=0
(
un+1j − un
j
∆t
)2
+ a∆x(un+1, un) + θa∆x(un+1 − un, un+1 − un)
YB]&Pa∆x(u, v) =
N∑
j=0
(
uj+1 − uj
∆x
)(
vj+1 − vj
∆x
)
.
+[fY\STIJ]bd]FjHEn+1 ]ueHOpFX] Y\cXcpILKaNSTbdY\HLSTKUF O(∆x + ∆t)
cXILkue +MN][¢ WFp]IJVUST]]>aXYUP>HL]E(tn+1)
AXlKUOpe[fY\SfeLeLKUFQe+YUO/[T]&P>HL]OpI[T]^emKBSCF"MN]^MNWbdKBFBHJIL]&I+OpFX]4cXILKBcXIJSCWHLW4MN]^P>KUFQem]&ILqYqHJSCKBFMX]^[¢ WFX]&ILVBSC]4MNSfeLPILkHL]BA; 1 T N N2OQ7I^EJI P[#E E
θ\FVK32#W/Y T K"E/OQ7ILW P] qK"NPOFVE/I E 10 WJOEJI $M E$M FVK/I^U+QE:9
K:0 E%FQ*\ \ $MILE P[EEn = E0
/N2[I QN2[Qn ≥ 0
]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/-
_lFX]Y\ONHJIL] 1 Y >KBF9MN]+MXW>pFXSTI MN]&e eLPZrXW&b YUe=cKUOpI [ WuijOpYqHJSCKBFMN]&e KUFpMX]&e ]&emH MN]P>KUbdbd]FQP>]IcpY\I/ILW&W&P>IJSTIL] NA xBt dPKUbdbd]OXFem~NemHLkbd]M= WuiBOQYqHLSTKUFQeMNO cpIL]&b ST]IKUIZMNIL]BA1DGFjHLIJKNMNOXSfeLYUFjHv = ∂u
∂t
]>Hw = ∂u
∂x
NA xBt +]ueHsW&ijOXSTqY\[T]FjH
∂
∂t
(
vw
)
=
(
0 11 0
)
∂
∂x
(
vw
) cKUOXI(x, t) ∈ (0, 1)× R+
∗
v(t, x+ 1) = v(t, x), w(t, x + 1) = w(t, x)cKUOpI
(x, t) ∈ (0, 1)× R+∗
w(t = 0, x) =∂u0
∂x(x)
cQKUOpIx ∈ (0, 1)
v(t = 0, x) = u1(x)cKUOXI
x ∈ (0, 1).
NA x
7FcQ]&ONHsMNKUFXFp]I1OpFX]lSTFjHL]IJcXIJW>HZYqHLSTKUF cprj~NeLSTijOX]lKUObdW&P&Y\FXSfijOX]7MX]7P>]ueFXKBOXU]&[C[T]&e1qY\IJSTYUX[C]ueAySubdKMXW[TSTeL]dOXFMNWcX[fYUP]bd]FjH P][TOXS1MN]d[fY$P>KUIZMN] SCXIZY\FjHJ] cpYUI0]aN]bdcX[T] Y\[TKUIZe
v]ueH
OXFX]9SCHL]ueLeL]9]Hw]ueH0OXFX]dMNW1 KBILb Y\HLSTKUF=Av=]%eL~NeHJkbd]9MN]dMN]&ONaW&ijOpYqHJSCKBFpe NA x sY\cXcQY\IZY fH
P>KUbdbd]0OXFX]0VUWFpWIZY\[TSTeJYqHJSCKBFdMX]7[ WuijOpYqHJSCKBF/M YUMNB]&P>HLSTKUF=A 7Fc]ONHsY\STFpeLSMNW>pFpSCIOXF"eLPZrXW&b YMN]4H~cQ] ) $ J $("'
1
2∆t
(
2vn+1j − vn
j+1 − vnj−1
2wn+1j − wn
j+1 − wnj−1
)
− 1
2∆x
(
0 11 0
)(
vnj+1 − vn
j−1
wnj+1 − wn
j−1
)
= 0, NA xUx
KUO"XSC]&F$OXFeLPZrXW&b Y MX]7H~c] ) $ $$
1
∆t
(
vn+1j − vn
j
wn+1j − wn
j
)
− 1
2∆x
(
0 11 0
)(
vnj+1 − vn
j−1
wnj+1 − wn
j−1
)
NA xj|
+∆t
2(∆x)2
(
0 11 0
)2( −vnj−1 + 2vn
j − vnj+1
−wnj−1 + 2wn
j − wnj+1
)
= 0.
; 11 T *, N2OQ7I^EJI 2[#E E FK32#W/YGT JE D/T%H#\I>M EPI M K 2F P] E+F6QGF6Q6T E E/O O#NI YGEL2FVN2[#F T K"NPO $M QM N2O PD
∆t ≤ ∆x9 E"Q P[ 0-M$E+F6Q /I^WKM F 10 NI 2ILE 0EJO E%F 3T3K"EZE+Q QE/Y#FRF7M
E I@T/NI Q∆t/∆x
E+FQ TJI W K"NPOF6Q6TO%Q NI^F 2[#E∆tE"Q
∆xQE/O JEJO%Q EJI^F"WJI^N ]
; 11 T # N2OQ7I^EJI P[E E FK32#W/YGT E D/T%H\ E/O PI^N ] GE+F6Q F6Q6T E EJO O#NI YGEL2F6NP[F T0K+N2O M QM NPO D
∆t ≤ ∆x9$E+Q P[ 0 ME+FQ/ILW%KM F 10 NI PI^E E/O0E%F 3T3K"E E"Q;QE/Y#F%]
+KBbdb ]cKUOXI [¢ W&ijOpY\HLSTKUF M YBMNU]uPRHJSCKBF &OpFX]1SfMNW]1 KUFQMXY\bd]FjHZY\[T]1cKUOXI KUXHL]FpSCI MN] +KUFpe3,eLPZrXW&b YUe]&emH[T] $ & & AQ+]&cQ]&FpMXY\FjH jSfP>SST[=e& YUVUSCHM OXFeL~emHLk&bd]0MN]^MN]&ONaW&ijOpYqHJSCKBFpe]>HSC[F3 ]ueHscpYBeP[TYUSCIMN]^eJYUKUSTI+ijOX]&[C[T]4]&emH[fY%jSCHL]ueLeL]4iBOpScQ]&ILbd]>H`MN]4eJYUKUSTI+MpY\FpeiBOp][=eL]Fpe`MXW&P>]&FjHLIJ]IuA 41F31 YUS H XST[3emO 8 HlMX]MNSTYUVUKBFpY\[TSTeL]I[fY%b Y\HLIJSTP]
J =
(
0 11 0
)
cQKBOXI+KBNHL]&FXSTIMN]&ONaW&ijOpY\HLSTKUFpe+M YUMNB]&PRHJSCKBFMNWuP>KUOpcX[CW&]&e+iBOpS eLKUFjHMNW&P]FjHLIJW]ueSTFpMNWc]FQMN]b )b ]&FjH]H`MNS WIJ]bdbd]FjHs[ OXFp]4MX]0[ Y\ONHJIL]BAQ+]0eLKUFjH`MNKUFQPl[T]&eqY\[T]OpIJecXIJKUcXIJ]&eMN]4[fY%b Y\HLIJSTP]
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
Ux 7 / / /
J1]>H −1
]F[ KNP&P>OXIJIL]&FpP>] ijOXS17KBOX]FjH1[T]lIU[T]lMN]l[TY9jSCHL]ueLeL] B]>HKUFMNWuP>]FjHJIL]lPKUbdcKBeJY\FjHL]cQY\IP>KUbdcKBeJY\FjHL]7MXY\Fpe+[fY%MNWuP>KBb cKBeLSCHLSTKUFeLOXIOXFX]lQYUeL]7MN]0U]uPRHL]&OXIZe1cXIJKUcpIL]ue1MN]4P>]HmHL]7b Y )HJILSfP>]BAU+]1H~c]MN]seJPZrXWb Yl]&emH eL~NeHJWb YqHJSTijOX]&b ]&FjH ONHJSC[TSfemWcQKBOXI [C]ue em~NemHLkbd]ue r~c]IJQKB[CSfijOX]&e]H ]F cpY\ILHLSfP>Op[CST]I cQKBOXI7[fY"MN~FpY\bdSfiBOp] MN]ue0VBY . BKUSTI T@&YUOpijOX][ FXKBOpelIJ]FBKq~UKUFQescQKBOXIcp[COpelMN]^MNW>HZY\ST[Te RA
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
! #"$&%'()*,+-*,+ .0/214365798;:<16=>5?/
@BADCFE9GIHJG&KLADM6NPORQSHTC6U0VLEC6U0VLENWC0OSXQSHYESE,U0CLEZ\[^] A_CLAD[W`4E,Hba\AcOSK6Xa\ADORNed0VLHgf4HYE9hjilk!m4nporq<s'tm6kvuxw'hjycoz<hj|tJ~!m6ypnpor|r|tgw!|n&l~?||ro~!nYoilk'|r_U_NWQ[WA@9XLC6NPORNWU_C_4_I0CQS_[WHB_XIC6XQ&AD[WHBGHYEXpdjVLAcOSNPU0CLEH[W[PNWM4ORNedjV6HYE9GIU_QSQSHYERMFU0CLf4HCjOZf6HYEaU4f4[WHYEMLKj`4ERNWdjV6HpEE>O&AcORNWU_CLCLADNWQRHpEGD] HYE,O,Zcf4NPQSHNWCLf4XYMFHYCLf6ADCjO&Ef6VORHYaMFE-U0VLEAD[W[PU0CLE#aU0C0OSQRHYQdjV6H[PHpEgMLQRU06[PYaHYEADV4[WNaNPORHpETE,U0CjObLNPHYCM<U0ERXYEbM<U_V6Q#GHYEbH0 fl MlHY[P[WNPM6ORNed0VLHYEYvGD] HYE,O,Zcf4NWQRHd0Vv] HY[P[WHYETA0f4aHIO,OSHCjO#V6C6HE,U0[PV4OSNPU0CV6C6NedjV6H_6HOf4XM<HCLfLADCjOGIU_CjOSNPC6aHCjOf4HpEf6U_C6C6XYHYEY6 ] ADM6MLQRU4G&K6HBd0VLHBCLU_VLEA_[P[WU_CLEE,V6NWQRH\HpE>OADM6M<H[WXH¡m6~'~!ycq¢¤£'|¥z6m4ycorm6nYorq<s!s'|r|<v@9NeERU_CLE#ORU_V6OJf4HERV6NPORH¡djV6H¡[] NPCjOSXQS¦IOJf4HGIHIORORHADM6M6QSU4G&K6H¡f4XYMLA_ESERH_vHOf6H¡[WU_NWC'[WHGYA_f4QSH¡f4HpEgH0 f M§HY[P[WNPM6ORNed0VLHYEJHIOJa¦YaH[WHGA0f4QSHfl] A_CLAD[W`4E,HgaADORK6XYa\AcORNedjV6HT¨>M6VLQRHI©#A_VLdjV6H[!C6U0VLE9CLU_VLE;QRHpE>OSQRHYNP0C6U_CFEM<U_V6QB[] NWCLE>O&ADCjOYªCHI«¬«HIOYC6U0VLE2QRHYM6QRHYCLf4QSU_CLE2GIHO,ORHbADMLM6QRU4G&K6H9¤A_QRNeAcOSNPU0C6C6HY[P[WH9M<U_V6Q[WHYE2M6QRU06[WaHYEfl] X_U0[PV6ORNWU_CHYCORHaMLE¡H_ fl Mvf4HO>`M<HMLADQ&AD<U_[WNWdjV6HU0VK`MFHYQR<U_[WNed0VLHplHO#H[W[PHERHQ&AGIQSVLGINeAD[WHJM<U_VLQgGIU0aJM6QRHYCLf4QSH[WAbaXIOSK6U4f4HCV6aXQSNWdjV6Hf6HYE-XY[PXYaHYCjOSE§LCLNWEdjV6HCLU_VLE-f4XY_HY[PU0M6MFHYQRU0CLE-ADV®2KLA_M6NOSQRH¯ 6°-ADQBADNW[P[WHVLQSEYLGIHO,OSHgADM6MLQRU4G&K6HTA_f6aHO9V6C6HgNPCjORHYQRMLQRXOSAcOSNPU0CM6K`ERNed0VLHTU_VaXYGYADC6NedjV6HbORQSYECLAcOSV6QRHY[P[WH_ª±V6OSADCjO#f4NWQRHdjV6HJ[WH[PHpGORHYV6QbC6HMFHYV4OTMLA_E9«rADNWQRHJ[] XYGU_C6U0aNPHf4H[WA¡M6QRXpE,HYCjOSAcOSNPU0Cd0VLNvE,VLNOf4H#GHIO,OSH#ADM6MLQRU4G&K6HBcADQSNWADORNWU_C6C6HY[P[WH²±9VGIU_VLQSEbf4H\GIH\G&KLA_M6NPORQSHJHOTf6HYETERV6NPcA_C0O&Eª[^] H4HaM6[WHMLQRU_ORUDO>`M<Hf] Xpd0VFAcORNWU_CADV6³f4XIQRNW_XYHYEJMLADQRORNWH[W[PHpEJf6H¡O>`M<HHY[P[WNPM6ORNed0VLHERHQ&A[WH 'ADM6[eA_GNPHYC´M<U_V6Q[WHYdjV6H[C6U0VLEXIOSVLf4NWHQSU_CLEJ[PHM6QRU06[WaHTADV4¡[WNWaNPORHpEERV6NPcA_C0O
−∆u = ff6ADCLE
Ωu = 0
E,V6Q∂Ω
rµ6Wp¯
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¯0¯ !"#$%
U'&C6U_VLENPaM<U0ERU_CLEf6HYEGU_CLf4NPORNWU_CFEADV4[WNPaNPORHYE9f4Hg@;NPQSNWG&KL[PHOJC6U0VLEQSHC_Uc`0U_CLEZ[eA)(U_VFE>ERHYGIORNWU_C 0 µL µ MFU0V6QgV6CLHM6QSXYERHCjO&AcORNWU_C´f6HGIHaUf6[WHp-@BADCLErµLP¤ΩHYE,OJV6CxU_V60HQROJf4H[^] HYERMLA_GH
RN ∂ΩHYE,OE,U0C¡<U_Q&f U0V«QSU_CjORNWQSHpI
fHpE>OV6CERHYGU_CLfaHYag6QSHV6C6Hbf4U0C6C6XHbf4VMLQRU06[PYaHpHO
uHpE>O[^] NWCLGU_C6CV6H0+*NWHCE,LQYC6U0VLEf4U_C6CLHQSU_CLEADV®2KLADMLNOSQRH#Jf4HbC6U_aJ6QRHYV4A_V4ORQSHYEHI4HaM6[WHYEf] Xpd0VFAcORNWU_CFEADV4f4XYQRNW_XYHYEMLA_Q,OSNPHY[P[WHYEf6HO>`MFH HY[P[WNWM4ORNedjV6H³djV6N9M<HVL_HCjOEY] XIORVFf4NPHYQ_Q-,_GHBZ[^] A_M6M6QSUG&KLHBcA_QRNeAcOSNPU0C6C6H[W[WH_ vH;M6[eADCf4HbGIHbG&KLADM6NPORQSH9HpE>O[WHBERV6NW¤A_CjOY4@;A_CLE[WA.(HpGORNWU_Cµ60/TC6U0VLE2QSA_M6MFHY[PU0CLEd0VLH[ed0VLHYE«U0QRaJV6[WHYEfl] NWC0OSX0QSADORNWU_CMLADQ2MLADQRORNWH_0f4NOSHYE21 qLy43k'|jtbw!|65yc||js2jM6V6NeEC6U0VLE?f4XLC6NeERERU_CFEGIHdjV] HYE,OV6C6H71 qFy83k!rm4nporq<s´z6m4ycorm6nYorq<s!s'|r|< 'A9(HYGIORNWU_Cµ6 µgHpE>O9GU_CLESA_GQRXYH;ADVnY£!hqLy4:+3´|w!|<;2m4u>=@? orBAFycm3 djV6N9ERHQ&A [^] U_V4OSNP[HpERERHCjOSNPHY[M<HQSaHIOROSADCjO¡f4Hf4XYaU_CjORQSHQ\f4HpEQRXpE,V6[POSADOSEf] HNeE,ORHCFGIHgHObf] V6CLNWGNOSXf4HERU_[WV4ORNWU_CLE;f4HJ«U_QSagV6[eAcOSNPU0C¤A_QRNeAcOSNPU0C6C6HY[P[WH_ªU0VLE90HQSQRU0CLEdjV6HM<U_VLQ;M<U_V60U_NWQ;A_M6M6[WNWdjV6HYQBGH#ORKLXU_QSaHJNP[§HYE,O;NWC6XY[PVLGIOSA_6[PHf4Hf4H0U_NWQBA_LADCLf6U_C6C6HYQ9[^] HpE,MFA_GIH
C1(Ω)f6HYE«U0CLGOSNPU0CLEGIU0CjORNWCjLaHYCjO9f4NDCªXQSHCjOSNWA_6[PHpEADVM6QSUDLO9f4HgESA#¨>0XC6XYQSA_[PNeERADORNWU_C©j[] HYE,MFA_GIHTf4HE(4U_<U_[WH
H1(Ω)®2U0CLGI[WV6U_CFE¡GIHO,OSHNWCjORQSUf6VLGOSNPU0CHC aHCjOSNPU0C6CLADCjO[] HI4NeE>OSHCLGHfl] ADV4OSQRHpEaXIOSK6U4f4HYEf4HQSXYERU_[WV4OSNPU0Cxf4HpEXYdjVLAcOSNPU0CLE\ADV6 f4XQSNP0XHpEJMFADQRORNWH[W[PHpEaA_NWE\djV6NERU_CjO\aU_NWCLEM6V6NeESERA_C0OSHYEU_VML[PVLEBGIU0aM6[PNedjV6XHpE;djV6HJ[] ADMLM6QRU4G&K6HTcA_QRNeAcOSNPU0C6C6H[W[WH¡C6U0VLE;QRHYCj0Uc`_U0CLEZ[^] HYCLGI`4G[PU0MFXpf4NPH6FP4GIH[WHT[WHYGOSHV6QGV6QSNPHYV4lPW4HIO9GU_V6Q&AD0HV4L
)*,+-*KJ L53NM 8O!P!16=>52/ :QORPS+S=RT8U 'A«U_QSagV6[eAcOSNPU0C ¨,G[WA0ERERNWdjV6HI©\f6H³rµLP¤'djV6NM<U_V6QSQSA_NO#MFADQ&A8VeORQSHg¨>CLADORV6QSH[W[WH©ZM6QSHaNPYQRHV6H06HYE,O;f4H#ERV6M6M<U0ERHQ;E,V WESADaaHCjO9f4H#QSX0V6[eADQSNOSXBM<U_VLQ[WA\ERU_[WV4OSNPU0C
uAcLCdjV6H#[WHYEXpdjVLAcOSNPU0CLEgf6H³rµLP¤bADNWHCjOgV6C´ERHCLEgHCOSU_V4OgM<U_NWCjOJf4H
ΩU0V´f4H
∂Ω$XA_M6MFHY[PU0CLEbOSU_V4Ofl] A_FU0QSfdjV6HY[WdjV6HpEC6UDO&AcOSNPU0CLEfl] HYERMLA_GHYEf4Hb«U_CFGORNWU_CFEQRXY_V6[WNPYQRHpE
Yh[Z!s!onporq<s\%]K^_]K^a`>bdcfeΩ g h b g i4jlk enm j RN o Ω pqsrljkut.j e g kvjIw"xhyh bde j C(Ω) zkvjlpK|Nj- e~c i4jlt9jlh e o C(Ω) R jup~|Nq-j m jup"r b h eKcBb hp b h e~c hNgNjlp m q4hNp Ω zkvjlp~|j- eKc i4jlt9jh e o m qdhp Ω w`Qb4cfe g hjh e~c jk k ≥ 0 wxhh bde j Ck(Ω) zkvjlp~|j- eKc i4jlt9jh e o Ck(Ω) R jlp~|q'@j m jup2r b h e~cBb hp
k r bdc p b h e~c hN t9jh e2m' k c i4qjlp m qdhp Ω z~k@jup~|Nj@ e~c i8jt.jh e o m qdhp Ω w9C6HtYq<rk'nYoqFs ¢_m6tYtporilk!| U0C MLA_QR[WHA_VLESE,Nbf4HtYq<rk'nYoqFs1 qLy¤np|f4H rµLP¤HpE>OV6C6HERU_[WV4OSNPU0C
u ∈ C2(Ω) ∩ C(Ω)<GIH#djV6N'NPaM6[WNWdjV6HJd0VLHT[PHgERHYGU_CLfaHaJ6QRH
ff4U0NO9A_M6MLADQRORHYC6NWQZ
C(Ω)9®2HIORORH³«U_QSagV6[eAcOSNPU0C G[WA0ERERNed0VLH MFUjE,H³aA_[PKLHV6QSHVLERHaHYC0OV6C GIHYQ,O&ADNWCC6U_aJ6QSH³f4HMLQRU06[PYaHYE²(4ADCFEJQSHCjOSQRHYQJf6A_CLEg[WHf6XIOSA_NP[^-E,NW_CFAD[WU_CLEJd0VLH_§E,U0VLEg[eA³ERHVL[PHKj`M<UDOSK6YERH
f ∈C(Ω)
vNW[C] HI4NeE>OSHHC_XYC6XQ&AD[§MLA0ETf4HE,U0[PV6ORNWU_Cf4HGI[eA_ESE,HC2 M<U_VLQ¡rµLP¤;ERN[eAf4NPaHYCLE,NWU_Cf] HpE,MLA0GIHHYE,O-M6[WVLE-_Q&ADCFf4HdjV6Hf6HV4
N ≥ 2IcC«rADNPOY_NP[4HNeE,ORH6NWHCV6C6HERU_[WV4ORNWU_CvDGIU_aaHCLU_VLE\[WH 0HQSQRU0CLEM6[PVFE[WU_NWCa\ADNeE\H[W[WH C] HYE,O¡MLA_E¡f4H GI[eA_ESE,H
C2 H[W[WH HYE,OV6CM<HVaU_NWCLEQSX0V6[WNPYQRHTESADV4«-E,Nv[WAf6U_C6C6XYHfHpE>OML[PVLE9QRXY_V6[WNWQSHbdjV6H
C(Ω)< H#GYA_Ef4HT[eA\f4NWaHCLERNPU0CV6Cf] HpE,MLA0GIHg
N = 1HYE,O2MLA_Q,OSNWGV6[WNPHYQ?M6V6NeESd0Vv] NW[FHpE>O2«rA_GNP[WH;f4H9ORQSU_V60HQ?f6HYEE,U0[PV4OSNPU0CLE?G[WA0ERERNed0VLHYEr_U0NPQ-[^] HYQSGNWGHµ6W_WpIcaA_NWEC6U_VLE_HQSQSU_CLE!C6XpADC6aU_NWCLEdjV6H0Da¦aH9f6A_CLE-GIH;GA_E!«rA¤0U_Q&AD6[WH_GHIORORHb«U_QSagVL[WADORNWU_CG[WA0ERERNed0VLHBHpE>OM<HVGIU0aaU4f4H0@BADCLE9[WA\ERV6NPORH0LM<U_V6QXORVLf6NPHYQgrµLP¤LCLU_VLEQSHaM6[eA_GHQSU_CLEESA«U_QSagV6[eAcOSNPU0CG[WA0ERERNed0VLHTMLADQVLC6Hb«U_QSagV6[eAcOSNPU0C4f4NPORHTcADQSNWADORNWU_C6CLH[W[PH04FHpADVLGU_V6MM6[WVLEA¤cADCjOSA__HYVLE,H0
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#"% ¯ G)*,+-*) 7U :>PS 7U O!P 7;=RM Uv/S=,5?/ 8;/7 U$S "P-:>U
?CV6CLHf4NWaHCLERNWU_Cf] HpE,MLA0GIHN = 1
lERNΩ = (0, 1)
<[WHM6QRU06[WaHJA_V4 [PNWaNOSHYErµLP¤f4HNWHCjO
−d2u
dx2= f
M<U_VLQ0 < x < 1
u(0) = u(1) = 0.
rµ60/_®2HBMLQRU06[PYaHBHpE>OOSH[W[PHYaHCjO9E,NWaM6[WH#d0Vv] NW[A_f4aHOV6C6H#ERU_[WV4ORNWU_CHI4M6[WNWGNOSH²9u|jyc¢_o¢_| \]B^]B^
f ! "#" $!% [0, 1]9'& "(%)%*,+.-/ 0132456"78 "
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u(x) = x
∫ 1
0
f(s)(1− s)ds−∫ x
0
f(s)(x− s)ds <>"% x ∈ [0, 1].rµ6 µ0
°!U0V6QT[WH\QRHpE>OSHf4HGIHO,ORHERU_VFE>E,HpGOSNPU0CC6U0VLETAD[W[WU_CLEbU0V66[WNPHYQ#[WA«U0QRaJV6[WHH4M6[PNeGINPORH µ6 µ0bdjV6NC] AMLA0EORU0V'A>U_V6Q&Efl] XYdjV6NWcAD[WHCjOMFU0V6Qf4HpEM6QRU06[WaHYEM6[WVLEGU_aM6[WNWdjV6XpE?CJV6CLHf4NWaHCLERNWU_CJf] HpE,MLA0GIH0Y[^] A_M6M<H[W[WADORNWU_CB¨,XYdjVLADORNWU_CgA_V4#f4XQSNW_XHpEvMFADQRORNWH[W[PHpE©?M<HQ&fgf4HERAA>VFE>OSHYESE,H2M6V6NeERdjV6H0¤GU_aaHNW[jCv] `TAML[PVLE!djV] V6C6H2ERHVL[PH2¤A_QRNeADL[PH0YU0CTMFHYV4O!ML[PVLE§ERNPaM6[WHaHYC0OMLADQS[PHYQ¨Rfl] XYdjVLAcOSNPU0C f4NC<XYQRHYC0OSNPHY[P[WH#U0QSf4NWCLA_NPQSH©<®2HM<HCFf6ADCjOY<[^] Xpd0VFAcORNWU_Cxrµ60/_C] HpE>OBMLA_E;V6C6HXYdjVLAcOSNPU0CTf4NDCªXQSHCjOSNPHY[P[WH'¨,VLE,VLH[W[PHI©-A_VbERHCFEU&b[eAE,U0[PV6ORNWU_C#f4U0NOvESAcOSNWE,«rADNWQRHf6HYEvGU_CLf6NOSNPU0CLEL¨RADV4f4HV4b<U_V4O&E©M6[WV4OCB_O!djV] V6C6H?GU_CLf6NOSNPU0CTNPC6NPORNeAD[WHHYCTV6C6H2ERHVL[PHHORQSXaNOSX?f4H[^] NPCjORHYQRcA_[P[WH[0, 1]
®;] HpE>O[eZM6QRXpGINeE,XYaHCjO[WAf4NCªXQSHCLGHTHCjORQSHbV6CM6QRU06[WaHTADV4[PNWaNOSHYE#A¤_HYGbf4HpEGIU0CLf4NPORNWU_CLE¨,A_V4f4HYV4¡<U_V4O&E©,HIOV6CM6QSU_L[PYaHTf4HJ®A_VLG&K` rA¤0HYGBV6C6H#GU_CLf4NPORNWU_CNWC6NOSNWA_[PHTHYC¨,V6CERHV6[FU0V4O>©,D[4HpE>O§NWC0OSXQSHYESERA_CjO!GIHYMFHYCLf6A_C0O-f4H_U0NPQ!MFU0V6Q&d0VLU_N^Ya¦aHHCf4NPaHYCLE,NWU_CJV6Cc[WHYE!aXIOSK6U4f4HYEGI[eA_ESE,NedjV6HYElfl] XYdjVLADORNWU_CLE'f4NC<XYQRHYC0OSNPHY[P[WHYEU_Q&f4NPCFADNWQRHpEªC6H?ERU_CjOMLA_EORQSYEGU_aaU4f4HYElM<U_VLQvXIOSVLf4NWHQrµ60/_rHIOgERU_CjOTORU_OSA_[PHYaHYCjO#NPC6U0MFXYQSA_CjORHYEbHYCf4NPaHYCLE,NWU_CERV6M<XQSNPHYV6QRH¤°§U0V6QTV6CMLADQ&ADaORQSH
m ∈ RU_CGIU_CFE,Nef4QSH[WH;MLQRU06[PYaH;f4HT®ADVLG&K`MFU0V6Q[PHB vADML[WA0GINWHC¡A¤0HYGf4U0C6C6XYH9NWC6NOSNWA_[PHBHC
E
−d2u
dx2= f
M<U_V6Q0 < x < 1
u(0) = 0, dudx (0) = m.
rµ6 Fj@9H«rAGIU0C¥XYNWf4HYCjORH NW[9HNeE,ORHV6C6H³V6C6Ned0VLH³E,U0[PV6ORNWU_Cf4H µ6 FjIH?NW[;ERV WOfl] NWC0OSX0QRHYQ¡GIHO,ORHXYdjVLAcOSNPU0C¡[WNPCLXYADNWQSHJU0VML[PVLE0XC6XYQSA_[PHYaHYCjOfl] V4ORNW[WNWERHQ[WH9OSK6XU0QRYaHBfl] HI4NeE>OSHCLGHBf6HT®A_VLG&K`0 NWMLERG&KLNOKJp5D[C] HpE>OMFA_E\f4V¥ORU0V4OGI[eADNWQY-MFADQGIU0CjORQSH_-djV6H[WAERU_[WV4ORNWU_C f4HrµL FGIULCLGNWf6HA¤_HYGGIHY[P[WHf4HrµL /0E,NHY[P[WHHI4NeE>OSHp2 vAd0VLHYE,ORNWU_C d0VLNE,HM<U0ERHHpE>Of4HESA¤_U0NPQE] NW[HI4NWE,ORHV6CMLADQ&ADaIOSQRH
mOSH[?djV6H\[eAERU_[WV4ORNWU_Cf4H³µ6 Fj;_XYQRNPLH\ADVLESERN
u(1) = 0HIOgERU_NPOgf4U_CFGV6C6HE,U0[PV4OSNPU0Cf6H rµ60/_I!®;] HYE,O#[PH\M6QSNWCLGINWMFHf6H\[WA 3´hjnp£'qw!|w'k nYoydjV6NM<HQSaHOgf4HQSXYERU_VLf6QRH0ADVLESE,NlLNPHYCfl] V6CM<U_NWC0O9f6HBV6HBOSK6XU0QRNedjV6HTd0VLHBCV6aXYQRNedjV6H_4[WHTM6QSU_6[WaHTADV4[WNWaNPORHpE#rµL /0
D ORXQ&AcOSNP0HaHCjOpjU0C¡M6QSXYf6NOV6C6HbcAD[WHVLQf4Hm
rU_COSNPQSHTf4VMFU0NPCjO E 4U0C¡NWCjOR0QRHB[WHBMLQRU06[PYaHf4HB®ADVFG&Kj`rµ6 FjrU_CGYAD[eGIV6[WH[WABOSQSA>A>HYGIORU_NWQSHf6VORNWQ&I_M6V6NeE?ERH[WU_C[WHQRXpE,VL[O&AcOu(1)
U0C\GIU_QSQSNP0H[WA;cAD[WHV6Q-f4Hmp?CJM6Q&AcOSNWdjV6HGD] HYE,O-V6C6HaXIOSK6U4f4H2M<HVHlWGYA_GIHdjV6N6A;[] NPCFGIU_C0XC6NWHCjO§aA>A>HV6Qf4HTC6HbMLA_EERHT_XYC6XQ&AD[WNeE,HYQ2HYCf6NPaHCFE,NWU_CE,VLMFXYQRNWHV6QSH_
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¯ !"#$%
'A¡GIU0CLGI[WVLERNPU0CHYE,ObdjV] NW[v«rA_V4Obf6HYE;aXORK6U4f4HpE9MLQRU0M6QRHpE9A_V4MLQRU06[PYaHYEBADV4[PNWaNOSHYEbdjV6NCv] U0C0OQSNPHYCZJ0U_NWQA¤_HYGBGIHY[P[WHYEADO,O&A_G&K6XYHYEADV6¡M6QSU_6[WaHYEf4HJ®A_VLG&K`_¡ B T!$ b%'$# "I"
vHgM6QSNPCFGINWMFHf4Hg[^] A_M6M6QSU4G&K6H#cADQSNWADORNWU_C6CLH[W[PH#M<U_VLQ9[eAQSXYERU_[WV4ORNWU_C f4HpE9XpdjVLAcOSNPU0CLE9A_V4f6XIQSNW_XHpEMLADQRORNWH[W[PHpEHYE,Of6HBQSHaM6[eA_GHQ[^] Xpd0VFAcORNWU_CMLA_QVLC6HB«U_QSagV6[eAcOSNPU0C¡XYdjV6NWcAD[WHCjORH06f4NPORHbcAcQSNeAcORNWU_CLC6H[W[PH0LU04ORHYCjVLH#HCNWCjORX0QSA_CjO9[] XYdjVLADORNWU_CaJV6[OSNPML[PNWXHJMLADQVLC6H#«U0CLGOSNPU0C³djV6H[eGIU0CLdjV6H_f6NOSHORHpE>Op®2U_aaHNW[LHYE,O?C6XpGIHpERESADNWQRHf4HMLQRU4GIXpf4HQ2ZTf4HYE-NWCjORXY_Q&AcORNWU_CFE§MLADQMFADQRORNWHf6ADCLE[] XIO&AcL[PNeERERHaHYC0Of4H¡[eA«U0QRaJV6[WADORNWU_CcADQSNeAcORNWU_CLC6H[W[PH0vC6U0VLEgGU_aaHCGU_CLEgMLADQf4U_CLC6HQd0VLH[ed0VLHYEQSXYERV6[POSADOSEHYESERHCjORNWH[eEZ\GIHTE,V>A>HIOY)*KJ*,+ L53NM 8O!U$S´7U 3NU%Uv/
@BADCLE\ORU_V6ORHGIHO,ORH ERU_VFE>E,HpGOSNPU0CΩHpE>O\V6C¥U_VL_HQROf4H[] HYERMLA0GIH
RN <U_QSC6XU_V¥C6U_C<f6U_CjO[WHbFU0QSfU0V[eA#«QSU_CjOSNPYQRH¤HpE>OC6U_ORX∂Ω
69U_VLEERV6M6M<U0ERU_CLEADVFERERNld0VLHΩHYE,OVLCU0V6_HYQ,Oych+A<k'or|jyf4HGI[eA_ESERH C1 ' vAf4XIFC6NOSNPU0CM6QRXpGINeE,Hfl] V6CU_V60HQROTQRXY_V6[WNPHYQbHYE,OTf6U_C6C6X\ML[PVLEbFA_EfLADCLEb[eA@9XLC6NPORNWU_Cµ60/44ªa\ADNeETERAGIU0C6CLA_NWESERA_CLGIHJC] HYE,OgADLERU_[WV6aHCjObMLA_EbC6XpGIHpERESADNWQRHJMFU0V6Q[eA<U_C6C6H#GU_aM6QSXK6HYCLERNPU0Cf6HT[WA\ERV6NPORHgf4H#GH#GIU0V6QSEY"D[vERV WO?A>VLE,ORHJf4H#ERA¤0U_NWQd0Vv] V6CU0V6_HYQ,OQSX0V6[WNPHYQHpE>O k b p@p b t b mbTV6CU_V60HQROf6U_CjO[PHT<U_Q&fHYE,OV6C6HTK`MFHYQSERV6QR«rA_GIHVLC6H#cADQSNPXORXTf4Hf6NPaHCFE,NWU_C
N −1§QSX0V6[PNWQSH__HIO2djV6H9GHIO?U0V6_HYQ,OHpE>O2[PU4GA_[PHYaHCjO2ERNOSV6X9fl] V6C¡ERHV6[<GIU_ORXf4HBERA«QSU_CjOSNPYQRH0;Cf4XILCLNOAD[WU_Q&E'[WABs'qFy43´m6|#|julnYhjycor|kvyc|#A_V#<U_Q&f
∂ΩGIU0aaHXIO&ADCjO![WH_HpGORHYV6QVLC6NOSX
n = (ni)1≤i≤NC6U_QSa\AD[HC ORU_V6OM<U_NWCjOA_V M6[eADC OSA_C6_HYC0O\f4H
ΩHOM<U_NWCjOSADCjO_HYQSE[^] HIORXYQRNWHV6Q;f4H
Ω_U0NPQ[eA!NW_V6QSH_Wp<@;A_CLE
Ω ⊂ RN U0CC6UDOSHdx
[eA\aHYERV6QSHT_U_[WV6aNed0VLH_U0VaHYERV6QRHf6H¡ HYFHpE,0V6H¡f4H¡f6NPaHCFE,NWU_CN(V6Q
∂Ω'U_CC6UDOSH
ds[eAaHYERV6QRHE,V6QR«rA_GNWdjV6H0U0V³aHYERV6QSHf6H vH<HYER_V6Hf4Hf4NPaHYCLE,NWU_C
N − 1E,VLQ;[eA¡cA_QRNWXIOSX
∂Ωl HQRXpE,VL[O&AcO;MLQRNWCLGINWMLA_[f6HJGIHO,OSHERU_VLE,E,HpGORNWU_CHpE>OB[PHgORK6XYU_QSaHJERV6NW¤A_CjO;djV6HgC6U0VLEBA_f4aHIORORQSU_CLEJ0U_NWQ9[WHgOSK6XU0QRYaH
G4 ¯ /fLADCLEEF /[G H'M<U_VLQ;V6CLHgf4XaU0CLE>OSQSADORNWU_C djV6N'«rADNPOBA_M6MFHY['Zf4HYE;ADQS_V6aHCjO&Ef4Hg0XU0aXORQSNPHf6NDCªXQSHCjORNWH[W[WH_6U0V[WHBORK6XYU_QSaH#4 FL fLADCLEsF ¯ H £'hqFy8:+3´| \]%]B^! 'qLy43k'|w!|5yc||s"y`>bdcfe
Ω g h b g i8jk e k # g c jk m jnlq4p-plj C1 w `Qb4cfew g hjQr b h eKcBb h m j C1(Ω) $npug || b k e b kuh "m qdhpj%rljkut
Ω w&%" b k-pjfji k c ' jqr b kutg jm j(kvj-jh∫
Ω
∂w
∂xi(x) dx =
∫
∂Ω
w(x)ni(x) ds,µ6_
b*)ni jlp e q i ,+ t9j b t| b pq4h e j m jEq)h b kut9q4 jj- e k c jlg k@j g h cfe m j Ω w
.|+3´m4ycilk'| \]%] @9NWQSHdjV] V6C6H2«U_CFGORNWU_CQSX_VL[PNWQSHwA;ERU_CE,VLM6MFU0Q,O§<U_QSC6XfLADCLE![WH2«HYQRaX
Ω0HV4Obf4NPQSHgdjV] H[W[PHJE] ADC6CV6[WHgZ\[^] NPC4FC6N!ERNv[WH#«HYQRaX#Cv] HpE>O9MFA_E<U_QSC6X0/;C f4NPOBA_VLERERN'djV6Hg[WA«U0CLGOSNPU0C
wA V6C ERV6M6M<U_QROGIU0aMLA_GIOf6ADCFE
ΩrADO,ORHYCjORNWU_C HGIHY[WA C] NWaM6[WNWdjV6HMLA_Ed0VLH
wEY] ADC6CV6[WHTE,V6Q9[PHT<U_Q&f∂Ω
I6CMLA_Q,OSNWGV6[PNWHQp4[^] K`MFU_ORK6pE,H#f6V10KLXU_QSaHTµ60/4WTZM6QSU_M<U0Ef4VERV6M6M<U_QRO9<U_QSC6Xgf4HJ[WA\«U_CLGIORNWU_CwfLADCLE
ΩHpE>O;NPCV4OSNP[WHJERNv[^] U0V6_HYQ,O
ΩHYE,O9FU0QRCLX_(4N
ΩC] HYE,OMFA_E<U_QSC6X_6GHIO,OSHTK`jM<UDOSK6YERHTA_ESE,VLQRHbdjV6HT[PHpENPCjOSX_Q&AD[WHYEf6A_CLETrµL 0E,U0C0OLCLNPHpE •
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# !% ¯
ωi−1
ω0
y′
Q
φi
ωi
Ω
ωi+1
yN
∂Ω
Q+
ªµ6W@9XLC6NPORNWU_Cf4HT[eAJQSX0V6[eADQSNOSXBfl] V6CU_VL_HQROY vH 0K6XYU_QSaH#µ60/4W#Af4HgC6U_aJ6QRHYV4GU_QSU_[W[WA_NPQSHYEdjV6N!E,U0CjOORU0VLE9f4HpE9GU_CLERXYdjV6HYCLGIHpENPaaXpf4NeAcORHpEf4HT[eA«U_QSagV6[WHTf4HBQRHYHCrµL 0L vHB[WHYGIORHYV6Q9djV6Nl_U0VLf4Q&AgXpGIU0C6U_aNeE,HYQERAaXaU_NWQSHC6HTQRHORNWHCLf6QSAf4U_CFGBdjV6HT[eAJ«U_QSagV6[WHTf4HBQSHHCrµL 0ª²
qFycqFrm6oryc| \]%] \ ! 'qLy43k'|w Worsªnph[AFycm4nYoqFs¥~§m4y~§m4y¤nYo| " `Qb4cfeΩ g h b g i4jk e k g c jkm jq8p-pj C1 w `Qb4cfe u j e v m jg - r b h eKcBb hp m j C1(Ω) $6pug || b k e b kuh m qdhp jrljkut
Ω w%" b k-pEjfjup i k c ' jlh e qr b kut g j m c h e# kvq e~cBb h|qdk|Nq4k e~c j
∫
Ω
u(x)∂v
∂xi(x) dx = −
∫
Ω
v(x)∂u
∂xi(x) dx +
∫
∂Ω
u(x)v(x)ni(x) ds.µ6 ¯
Yh+3´q<s'tYnycm4nporq<s#] D[vE,VWO9f6HTM6QRHYCLf4QSHw = uv
fLADCLE[WH 0K6XU0QRYaHbµL /6P0
qFycqFrm6oryc| \]%] `>bdcfeΩ g h b g i4jlk e k # g c jk m jq4p@pj C1 w `>bdcfe u g h>jr b h e~cBb h m j C2(Ω)
j e v g hj#r b h e~cBb h m j C1(Ω) o e b g e jup m jg - $ pug |'| b k e b kuh Em qdhpsjrljkut Ω w %" b k-p jfjlp
i k c ' jh e qr b kutg j m c h e kvq eKcBb h|Nq4k2|Nq4k e~c j∫
Ω
∆u(x)v(x) dx = −∫
Ω
∇u(x) · ∇v(x) dx +
∫
∂Ω
∂u
∂n(x)v(x) ds,
µ6 GDb) ∇u =
(
∂u∂xi
)
1≤i≤Njlp e j7i4j- e jlg k kvq mdc jlh em j u o j e ∂u
∂n = ∇u · n wYh+3´q<s'tYnycm4nporq<s#];CA_M6M6[WNWdjV6Hb[WHJ®2U_QSU_[W[WA_NPQSH;µL /6 µZ
vHIO ∂u
∂xi
HIOU_CERU_aaHBHYCi
Yh[Z§s'ornYorq<s\%] Q] xh m4cfe g g h b g i4jk e Ωm j RN jup e k # g c jlk m j q8p-pj Ck z!q4i4j-g h
jlh eKc jk k ≥ 1 p c j- c p e j7g h h b t9kvj ' h c#m b g i8jk e p (ωi)0≤i≤Ie jl0p gNj
ω0 ⊂ Ω, Ω ⊂ ∪Ii=0ωi, ∂Ω ⊂ ∪I
i=1ωi,
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G E !"#$%
j e gNj o | b g k q gNj i ∈ 1, ..., I zi b4c kq c g kvj'wu o c _j - c p e jg hj"q@|'| c -q e~cBb h c j@ e~c i8jφi
m j l q8p-pj Ck o m j ωim q4hNpsv jlhNpljt9lj
Q =
y = (y′, yN) ∈ RN−1 × R, |y′| < 1, |yN | < 1
,
m'b h e R c h_i8jk-pj.jup e qdg+p-p c#m jq4p@pj Ck o j ee jfj gNjφi(ωi ∩ Ω) = Q ∩
y = (y′, yN) ∈ RN−1 × R, yN > 0
= Q+,φi(ωi ∩ ∂Ω) = Q ∩
y = (y′, yN ) ∈ RN−1 × R, yN = 0
.
Ω Ω
LµL / \@;HV4\HHYaM6[PHpEfl] U0V6_HYQ,O2C6U_CQRXY_V6[WNPHYQ,HU_VL_HQROFESERV6QRXBZ#jADVLG&K6H00U0V6_HYQ,OA¤_HYGVLCM<U_NWCjO9f4HbQRHY6QSU_VLESE,HYaHCjOZf4QSU_NPORH0.|+3´m4ycilk'| \]%]*NPHYC¡djV6H[eA !NP0V6QRH;µ6WQRHYM6QSXYERHCjORHV6C¡U_V60HQROQRXY_V6[WNPHYQ?djV6N<HYE,O?FU0QRCLX_[eA#@;XILCLNOSNPU0Cµ60/4bEY] A_M6M6[WNWdjV6H9A_VLESE,N<Zgf4HpEU_VL_HQROSE?C6U_C\<U_QSC6XpEj vA#@;XILCLNOSNPU0Cµ60/4BC] HIG[PV6OMLA0EE,HYV6[PHYaHCjO[WHYEU0V6_HYQ,O&E¡f4U0C0O[WH<U_Q&f C] HYE,OMLA_EV6C6HERV6Q,«rA0GIHQRXY_V6[WNPYQRH0?a\ADNeEHY[P[WH\HI6G[PV4OADVLESE,N?[PHpETU_V60HQROSE#djV6N?C6H¡ERU_CjO#MLA_E#[PU4GA_[PHYaHCjOgE,NPORVLX¡fl] V6C´E,HYV6[?GUDORXf4H\[PHYV6Q«QSU_CjOSNPYQRH0! 'A !NW_V6QSHµL /³GIU_CjOSNPHYC0O\f4HYV4´HI4HaM6[WHYEJO>`M6NedjV6HYEfl] U_V60HQROJC6U0C QSX0V6[PNWHQdjV6NMLQRXpE,HYC0OSHCjOTV6C6HERNPCL_V6[eADQSNOSXNPQSQRXYaXYf4NeAD6[WH_lERU_NPO#HC<U_V4Ogf4H<ERERV6QSH_ERU_NPObHYCV6CMFU0NPCjOgf4HQSHLQRU0VLERERHaHYC0Opv®2HYEbHI4HYaML[PHpEbC6H\E,U0C0OTMLA0ETf4HYE¨,NWCj0HCjORNWU_CFE;a\ADORK6XYaADORNedjV6HYE>© H[^] U0V6_HYQ,O<ERERV6QSXHYE,OgO>`M6NedjV6HaHCjOV4ORNW[PNeERXM<U_V6QJXIOSVLf4NWHQ[PHpEgM6QSU_L[PYaHpEgf4H¡FESE,V6QSHYEgHC´aXpGADCLNWdjV6Hf6HYE9E,ORQSVLGIORV6QSHYEY/;CM<HV4OBC6XYA_C6aU_NWCLE0XC6XYQSA_[PNeERHQV6CM<HV [eAG[WA0ERERH#f4HYE;U_VL_HQROSEQRXY_V6[WNPHYQSEA_V4U_V60HQROSE¨>QSX0V6[WNPHYQSE9MLADQbaU_Q&GIHpADV4j©FZ¡GU_CLf6NOSNPU0CdjV6HGIHYEBaU_Q&GIHpADV4f4HJ«QRU0CjORNWQSHYEBE,H¨,QSHYGIU0[P[WHCjO,©HYC«U_QSaA_CjO2f4HYEADCL_[WHYEf4NCªXQSHCjOSEf4H E GA0E2fl] V6CM<U_NWCjO2f4H;QRHY6QRU0VLESE,HYaHYCjO&!U_Vf6H
2πGA0E9fl] V6C6HJFESE,V6QSHpI 0'U_VFE;GHYEBf4XIO&ADNW[WEBf4XMFA_ESE,HYC0OB[WA_QR0HaHCjO9[WHJGYA_f4QSHgf4HGIHGIU0V6QSEYHOBCLU_VLE;QRHYC_Uc`_U0CLE[WHg[WHYGIORHVLQbZ¡[WAXHYaA_QSdjV6H FL µ6 GM<U_V6QBV6C6HADV4OSQRHJHI4M6[WNWGYAcOSNPU0C³E,VLQBGHYEMLQRU06[PYaHYEf4HbQSX_VL[WA_QRNPORX0 •
;u|jyc¢_or¢0| \]%]B^ ;9 % :9) 7 2@% & 7 :9) 5% #) +.- / 147 2:@% & 7 :9 (∫
Ω
divσ(x)φ(x) dx = −∫
Ω
σ(x) · ∇φ(x) dx +
∫
∂Ω
σ(x) · n(x)φ(x) ds,
φ "'( C27 2 %9 C1(Ω) ! σ "'( " 27 >% !@% )787 #$9) C1(Ω)9
!<<>@%>@%. ;# 92 7 )% & ; Ω /
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# !% G49u|jyc¢_o¢_| \]%] 9 & ) " N = 3 9;3 7 % >62 >7)9 , ! " 9) Ω 92R3
9φ = (φ1, φ2, φ3)
9 ! && 7 2: ! " 9 Ω 92 R3 9;3 < 2)%
rotφ =
(
∂φ3
∂x2− ∂φ2
∂x3,∂φ1
∂x3− ∂φ3
∂x1,∂φ2
∂x1− ∂φ1
∂x2
)
.
"% φ # ψ 9 "'( " 27 >% (@% >787 ,9) C1(Ω)9 !<<>@% >@% ;492 7 :>% & ;
Ω9 9);9" % 9 7 2@% & 7 :9) 5% #> +.-/ 1
∫
Ω
rotφ · ψ dx −∫
Ω
φ · rotψ dx = −∫
∂Ω
(φ× n) · ψ ds.
)*KJ*KJ L53_M 8ORP'16=,5?/P§3L=!P!16=>52/9/U$ORO!U°§U0V6Q'ERNPaM6[WNPLHQ![WAM6QRXpE,HYCjOSAcOSNPU0CIC6U0VLE!E,V6MLMFUjE,U0CLEdjV6H?[^] U_V60HQRO
ΩHYE,OvFU0QRCLXHIO!QRXY_V6[WNPHYQYHIOBdjV6HJ[PHJE,HpGIU_CFfaHYag6QSH
ff4HrµLP¤HYE,OBGU_CjORNWCV E,V6Q
Ω< vH#QRXpE,VL[O&AcO;MLQRNWCLGINWMLA_['f4HGIHIORORHE,U0VLE>ERHYGOSNPU0C¡HpE>O[eAM6QRU0MFUjE,NPORNWU_CE,VLNPcADCjOSH_
by¤qF~?q<tYornYoqFs \%] Q] `Qb4cfeu g h>jr b h> e~cBb h m j C2(Ω) w `Qb4cfe X R jup~|Nq-j m ' h c |qdk
X =
φ ∈ C1(Ω)e jl gNj φ = 0 p-g k ∂Ω
.
%" b k-p u jup e g hj"p b g eKcBb h m gE|k b l + t9j7q4g -9 c t cfe jlp z 'wuEp c j e pjg jt.jh e p c u q@||Nq4k e~c jh e$ X j e i k c ' j7v q4 cfe
∫
Ω
∇u(x) · ∇v(x) dx =
∫
Ω
f(x)v(x) dx | b g k e b g e jr b h eKcBb h v ∈ X. rµL # qd cfe z [w ujlp e q@|'|j jq 1 qLy43k'm4nYoqFs z6m4ycom4nYoqFs!s'|r| m g|k b l + t9j q4g - c t cfe jlpz [w-w.¡|+3´m4ycilk!| \%] Q] aC NPCjOSXQS¦IO³NWaaXpf4NeAcOf6H [eA«U_QSagVL[WADORNWU_C cADQSNWADORNWU_C6CLH[W[PH µ6 HpE>Od0Vv] HY[P[WH AV6C ERHCLEERN[eAE,U0[PV6ORNWU_C
uHYE,O¡ERHV6[WHaHCjOV6CLH«U_CLGIORNWU_Cf4H
C1(Ω)GIU0C0OSQSA_NPQSHIaHYCjOTZ[eA«U0QRaJV6[WADORNWU_C¨RGI[eA_ESERNWdjV6HI©rµ6Wp;djV6N-QSHYdjV6NWHQROBdjV6H
uADM6MLA_Q,OSNPHYC6C6HZ
C2(Ω) ;CM6QRHpERERHCjO§f4U_CLGf4X6A,Z;djV] NP[HpE>O§M6[WVLE-E,NWaM6[WHf4HQSXYERU_VLf4QSH;µ6 vd0VLHBµ6WpvM6V6NeERdjV] U_CJHYE,O!aU_NWCLEHI4NP0HYA_C0OERV6Q[WAQSX0V6[eADQSNOSXBf4Hb[eAE,U0[PV4OSNPU0C@BADCFE[WAJ«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WH\rµL 4[eAg«U0CLGOSNPU0C
vHYE,O9ADM6M<H[WXH 1 q<s'¢_nYorq<s nY|tYn_ vAB«U_QSagVL[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WHHYE,O2A_VLERERN6MLA_Q,«U0NWE?ADM6M<H[WXH«U0QRaJV6[eAcORNWU_CJ«rADNW6[WH9f4VMLQRU06[PYaHADV4\[WNPaNPORHYEBrµLP¤CaXpGA_C6NWdjV6H00[eAT«U0QRaJV6[eAcORNWU_C\cA_QRNeAcOSNPU0C6C6H[W[WHHYE,OGU_C6CV6HBE,U0VLE?[WH9CLU_af4H¨,M6QSNPCLGNPM<Hbf4HYE?ORQ&A¤cADV4NPQRORV6HY[WE>©_jC¡M6K`4ERNWdjV6H00U0CMLADQS[WH;ADVFERERNFfl] XYdjVLADORNWU_C¡f6H;6NW[eADCU0Vf4Hb«U_QSagV6[WHTf4HbQRXpGINWM6QSUGNOSX_ vU_Q&ERdjV] U_C³M6QRHYCLf
v = uf6A_CLErµ6 IFU0C³U04ORNWHCjOTGIHd0Vv] NW[§HYE,ObGU_C_HYCjVfl] A_M6M<H[WHQbV6C6Hh[A<m6onYh¡w Whjs!|jy4AFo|F0d0VLNLHI4M6QSNWaH9_XCLXQ&AD[WHaHCjO[^] XY0AD[WNPORXHYC0OSQRHVLC6H9XYC6HQS_NWHE>OSU4G_XHfLADCLE[PHTf4U0aA_NPCLH
Ω[WHbORHQSaHBf6HT0ADVFG&K6H;f4H\µ6 ,2HOV6C6HbXC6HYQR0NPHBM<UDORHYCjORNWH[W[PHTA_ESERUGNPXYH;Z
fr[PHORHQSaHTf4H#f4QSU_NPORHTf4HrµL ,I •
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G/ !"#$%
Yh+3´q<s'tYnycm4nporq<s#]_(4NuHYE,O?E,U0[PV6ORNWU_Cf4VM6QRU06[WaHADV4g[WNWaNPORHpErµLP¤DU0CJaJV6[PORNWM6[PNWH[^] XpdjVLAcOSNPU0CMLADQ
v ∈ X HOU_CV4ORNW[WNWERHT[WAJ«U0QRaJV6[PHTfl] NWC0OSX0QSADORNWU_CLEMLA_QMLADQRORNWHYEf4V³®2U0QRU0[P[eADNWQRHBµ60/4 F∫
Ω
∆u(x)v(x) dx = −∫
Ω
∇u(x) · ∇v(x) dx +
∫
∂Ω
∂u
∂n(x)v(x) ds.
;Qv = 0
E,VLQ∂Ω
M6V6NeERdjV6Hv ∈ X 6f4U0CLG
∫
Ω
f(x)v(x) dx =
∫
Ω
∇u(x) · ∇v(x) dx,
djV6NlCv] HpE>OQRNWHCfl] ADV4OSQRHTdjV6HB[WAg«U_QSagV6[WHrµ6 I XXYGNPM6QSU4djV6HaHCjOpERNu ∈ X 0XQSNLHrµL 4HCV6ORNW[PNeERA_CjO¨RZ[] HC0HQ&E©;[eAJ«U_QSagV6[WHTfl] NPCjORXY_Q&AcOSNPU0CMLA_QMLADQRORNWHbM6QRXpGIXpf4HCjORHbU0CU04ORNWHCjO
∫
Ω
(
∆u(x) + f(x))
v(x) dx = 0M<U_V6QOSU_V4OSHB«U_CLGIORNWU_C
v ∈ X.
®2U0aaH(∆u + f)
HYE,OgV6C6H¡«U_CLGIORNWU_C´GIU0CjORNWCjVLH_v0Q@,0GIH\ADV´ vHaaHµL /6 U0C´GIU_CFGI[WV4OJdjV6H−∆u(x) = f(x)
MFU0V6Q-ORU0V4Ox ∈ Ω
D°-ADQADNW[W[PHYV6QSEYDGIU0aaHu ∈ X cU0CQSHIOSQRU0V6_H[eABGU_CLf6NOSNPU0CA_V4#[WNPaNPORHYE
u = 0E,V6Q
∂ΩDGD] HYE,O,Zcf4NWQRHd0VLH
uHpE>OERU_[WV4ORNWU_Cf4VJMLQRU06[PYaHA_V4#[WNPaNPORHYEµ6WpI
;2|+3 3´| \%] Q] `>bdcfeΩ g h b g i8jk em j RN w `>b4cfe g(x) g hj2r b h eKcBb h b h eKc h_gNj m q4hNp Ω w `_c
| b g k e b g e jr b h eKcBb h φ m j C∞(Ω) $.pug |'| b k e b t|Nq em q4hp Ω o b hq∫
Ω
g(x)φ(x) dx = 0,
qd b k-p7 qr b h> e~cBb h g jlp e h_g fj m q4hNp Ω wYh+3´q<s'tYnycm4nporq<s#]'(4V6M6M<U0ERU_CLEdjV] NW[_HI4NeE>OSHV6CbM<U_NWC0O
x0 ∈ ΩOSH[jd0VLH
g(x0) 6= 0 (4A_CLElM<HQRORHf6H;0XC6XYQSA_[PNPORX00U0CMFHYV4OE,VLM6MFUjE,HYQdjV6H
g(x0) > 0rERNPCLU_C¡U0CM6QSHCLf −g Ij°-A_QGU_CjORNWCV6NOSX_NW[H4NWE,ORH#VLC MFHORNPO;0U_NeE,NWCLAD0HTU_V60HQRO
ω ⊂ Ωf4H
x0OSH[§djV6H
g(x) > 0M<U_V6Q9ORU0V4O
x ∈ ω >(U_NPOA_[PU0QSEV6CLHb«U_CLGIORNWU_CORHYE,OM<U0ERNOSNP0H_4C6U0CCV6[W[WH_φZ\ERV6M6M<U_QRONPCFGI[WVLEf6ADCFE
ω ;CA
∫
Ω
g(x)φ(x) dx =
∫
ω
g(x)φ(x) dx = 0,
djV6NvHYE,OVLC6H#GIU0C0OSQSA0f4NeGORNWU_CA¤_HpG9[^] K`M<UDORKLYERHbERV6QgL@;U_CLG
g(x) = 0MFU0V6QORU0V4O
x ∈ Ω
.|+3´m4ycilk'| \]%]B^ ?CC6UDO&AcORNWU_CGIU0aMLA_GIORHTU_CM<HV4O9QSXXpGIQSNPQSHb[WAJ«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C4CLH[W[PHµ6 E,U0VLE[eAg«U0QRaH H4OSQRU0V6_HYQu ∈ X ORHY[vdjV6H
a(u, v) = L(v)MFU0V6QORU0V4ORHb«U0CLGOSNPU0C
v ∈ X,A¤0HYGa(u, v) =
∫
Ω
∇u(x) · ∇v(x) dx
Ecol
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# !% GDµHIO
L(v) =
∫
Ω
f(x)v(x) dx,U&a(·, ·) HYE,O-V6C6H«U0QRaH6NP[WNWC6XYA_NPQSHERV6Q X HIO
L(·) HYE,O-V6C6H«U_QSaH[PNWC6XYA_NPQSHE,V6Q X 0®;] HYE,OE,U0VLEGIHIORORHb«U0QRaH#ADFE>OSQSA_NOSHbdjV6HTCLU_VLEQSXYERU_VLf6QRU0CLEbrA¤_HpGbd0VLH[ed0VLHYEK`MFU_ORK6pE,HpES2[eA«U_QSagV6[eAcOSNPU0C¤A_QRNeAcOSNPU0C6C6HY[P[WHBfLADCLE[eAM6QSUG&KFADNWC6HBERHYGIORNWU_C • ?] Nef4XYHM6QSNPCLGNPMFAD[WH?f4H? Pm4~!~'ycq¢¤£!|bz6m4ycorm4nporq<s's!|r|HYE,O'f4H?aU0CjORQSHQ[^] H4NWE,ORHYCLGIHHOv[^] V6CLNGINPORXf4H[eAERU_[WV4ORNWU_Cf4H\[eA«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WHµ6 IvGIH¡djV6N?HCjORQ&A8V¬C6HQ&A[PHa¦YaHQRXpE,V6[POSADOgM<U_V6QJ[] XYdjVLADORNWU_CrµLP¤TZ GA_VLERHf4H¡[eA°QSU_M<U0ERNOSNPU0Cµ60/4 GC HlCªHIOY!C6U0VLEgA_[P[WU_CLE_U_NWQ\djV] NW[HNeE,ORHV6C6HOSK6XU0QRNWHZ[eA«U_NeE\E,NWaM6[WHHOM6V6NeERESADCjOSHM<U_VLQADCLA_[P`4ERHQ[PHpE«U_QSagV4[WADORNWU_CLEcADQSNeAcORNWU_CLC6H[W[PHpE9XYA_C6aU_NWCLE\GIHO,ORHOSK6XU0QRNWHC6H«U_CFGORNWU_CLC6H djV6H E,N[^] HYERMLA_GHfLADCLE[PHpd0VLH[U0C G&KLHQ&G&K6H[WAERU_[WV4ORNWU_C HO¡f6ADCLE[PHpd0VLH[9U0C M6QRHYCLf¥[PHpE«U_CFGORNWU_CFEORHpE>O&E rf6A_CLE\[WHYEC6UDO&AcORNWU_CFETM6QSXYGIXpf4HCjOSHYEYl[] HYERMLA_GHXBHYE,O#V6CHpE,MLA0GIHf4H9NP[W<HQROY'GHdjV6N?C] HYE,O#MLA0ET[PHGA_EMFU0V6Q
X = v ∈ C1(Ω), v = 0ERV6Q
∂Ω agV6C6Nf4V´MLQRU4f4V6NPOJESGA_[WA_NPQSHg¨>CFAcORVLQRHY[ ©\MFU0V6QJGHM6QRU06[WaH_6 'AM6QRNWCLGNPMLA_[PHTf4NWGIV6[PORXJf6ADCLE[^] ADM6M6[WNeGAcOSNPU0Cf4HT[] ADMLM6QRU4G&K6HBcADQSNWADORNWU_C6CLH[W[PHTERHQ&Af4U_CLGbdjV] NP[«rADVLf4Q&AV4ORNW[WNWERHQVLCA_V4ORQSHbHYERMLA_GHTd0VLHXLZERA¤0U_NWQ2[^] HpE,MFA_GIHTf4HE(4U_<U_[WH
H10 (Ω)d0VLNHYE,O6NPHYCV6CHYERMLA_GHbf6H9NP[W<HQROg_U0NPQ[WHJ®2KLA_M6NOSQRH F
9u|jyc¢_o¢_| \]%] \ ! 9)% 7 >2<7 2' > 2 :#"'9 2 7 & 9)> & 2
−∆u = f 92 Ω∂u∂n = 0 !% ∂Ω.
rµ6 0 u ="'( 9 C2(Ω) / ="(% >%?*" u =678 9 <>%7 & =2 7 & #+.-/ 1 #6)7 & > u 2<< 2)% > C1(Ω) # ;>% 7 ; 27 ;
∫
Ω
∇u(x) · ∇v(x) dx =
∫
Ω
f(x)v(x) dx <>% ! " v ∈ C1(Ω).µ6W E
9);9" % *" #"'9 ;!6C2 % 9 ) # 9 C"78 " 92 C2(Ω) 9) + -/ 1#*" ∫Ω f(x)dx = 0 /9u|jyc¢_o¢_| \]%] ! 9)% 7 ;* 2 9) <7 2'*
∆ (∆u) = f 92 Ωu = 0 !% ∂Ω∂u∂n = 0 !% ∂Ω
µ6W_¤ ) X 7 ##< 2'# 9# "'( v 9 C2(Ω) )787 # *" v ! ∂v
∂n 27 )5#% ∂Ω / u "'( 9 C4(Ω) / (% >% * u # 4C"78 " 9 <>% 7 & 2 7 & 4+ -/1 # C>7 & ) u 2<<'2>% > X # ;>% 7 ; 27 ;
∫
Ω
∆u(x)∆v(x) dx =
∫
Ω
f(x)v(x) dx <>"% " "'( v ∈ X. rµLP8/_
Ecol
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repr
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re.
G'F !"#$%
¡, T§$ $I"#! )*)*,+ P-73U P"S0143P-=1
9U_VLEf4XYGQRNW_U0CLEV6C6HOSK6XU0QRNWH ADLE,ORQ&ADNPORHM<U_V6Q¡U_4OSHC6NWQ[^] HNeE,ORHCFGIHHIO[] V6C6NeGINPORX³f6H[WAERU_[WV4OSNPU0Cf] V6CLH«U_QSagV6[eAcOSNPU0CcADQSNeAcORNWU_CLC6H[W[PH\f6A_CLETV6CHpE,MLA0GIHf4H9NP[W<HQROY;CC6U_ORHVV6CHpE,MFA_GIHJf4H 9NP[W<HQROBQSXHY[!f4HJM6QSUf6V6NOTESGAD[eADNWQSH 〈, 〉 HIOTf4HJC6U_QSaH ‖ ‖ >(VLNPcADCjOb[WA6XHa\A_QSdjV6HµL /6P E CLU_VLEGU_CLERNWf6XQSU_CLEV6C6HB«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WHbf4VO>`jM<H H
ORQSU_V60HQu ∈ V ORHY[vdjV6H
a(u, v) = L(v)MFU0V6QORU0V4ORHb«U0CLGOSNPU0C
v ∈ V. µ6WYµj vHYEK`MFU_ORK6pE,HpEE,VLQ
aHIOLERU_CjO_
L(·) HpE>O2V6C6H«U0QRaH[WNPC6XpADNWQRHGU_CjORNWCV6H9ERV6Q V 0GD] HYE,O,Zcf4NWQRHdjV6H v → L(v)HpE>O?[WNWC6XYA_NPQSHf4H
Vf6A_CLE
RHONP[H4NWE,ORH
C > 0ORHY[d0VLH
|L(v)| ≤ C‖v‖ MFU0V6QORU0V4O v ∈ V ;
/4a(·, ·) HYE,O2VLC6H«U0QRaH9LNP[WNPCLXYADNWQSH9E,VLQ V GD] HYE,O,Zcf4NWQRHdjV6H w → a(w, v)
HYE,O2V6C6H9«U_QSaH[PNWC6XpADNWQRH#f4HVf6A_CLE
RMFU0V6QORU0V4O
v ∈ V LHIO v → a(w, v)HYE,OV6C6Hb«U0QRaHT[WNPC6XpADNWQRHTf4H
Vf6ADCFE
RM<U_V6QOSU_V4O
w ∈ V µ6a(·, ·) HYE,OGU_CjORNWCV6H_6G_] HpE>OR ZD f6NPQSHBdjV] NW[HI4NeE>OSH M > 0
OSH[vdjV6H|a(w, v)| ≤M‖w‖ ‖v‖ M<U_V6QOSU_V4O w, v ∈ V ;
rµLPFjFLa(·, ·) HYE,O¢0q|jyc¢_oz<|³rU_VH[W[PNWM4OSNWdjV6H¤LG_] HpE>OR ZD f4NWQSHBdjV] NP[vHI4NWE,ORH ν > 0
OSH[vdjV6Ha(v, v) ≥ ν‖v‖2 MFU0V6QORU0V4O v ∈ V. rµLP¤_
®2U0aaH9C6U_VLE[PH;_HQSQSU_CLE-A_VGU_V6Q&Ef4HBGIHO,ORH;E,U0VLE>ERHYGOSNPU0CcOSU_V4OSHYE[WHYEKj`M<UDOSK6YERHYE2GINP f4HpERERVLEERU_CjOTC6XpGIHYESESADNWQRHpE;M<U_VLQ#MFU0V6_U0NPQbQSXYERU_VLf6QRHrµLPpµ0IlCMLADQRORNeGIVL[PNWHQpl[WAGUjHYQSGNPNPORX\f4Ha(·, ·)HpE>OHpERERHCjOSNPHY[P[WH_
£'hqFy8:+3´| \] \]B^!;2m4u>=@? oBAFycm3 "`Qb4cfeV g h jup~|Nq-j m j c @jk e k j o L(·) g hjr b kut9j
c h q c kvj. b h eKc h_gNjspug k V o a(·, ·) g hjr b kut.j. c c h q c kvj9 b h e~c hNgNj b jkv c i8jEpug k V w %" b k-p qr b kutg q e~cBb h i8qdk c q e~cBb hNh>jlf j z [w u q m t9j e g h>j9g h c gNj.p b g eKcBb hw js|> g+p@j e~e jp b g e~cBb hm' |Njh m b h e~c hN t9jlh em jEq"r b kut9j7 c h q c kvj L wYh+3´q<s'tYnycm4nporq<s#]-°§U_VLQ;OSU_V4O
w ∈ V ª[] ADMLM6[PNeGADORNWU_C v → a(w, v)HIObV6C6HJ«U_QSaHJ[WNWC6XYA_NPQSHGU_CjORNWCV6H³ERV6Q
VHMLADQGIU_CFE,Xpd0VLHCjOY2[PH 0K6XU0QRYaH /4W_W f4HQRHYM6QSXYERHCjOSADORNWU_Cf4HXNPHpE6JHYCjORQ&A8V¬C6HTd0Vv] NW[lHI4NeE>OSHTV6CX[WXaHCjOf4H
VLC6UDOSX
A(w)6ORHY[vdjV6H
a(w, v) = 〈A(w), v〉 M<U_V6QOSU_V4O v ∈ V.°-A_Q?ADNW[P[WHVLQSEYc[WAbLNP[WNPCLXYADQSNPORXf4Ha(w, v)
NPaM6[WNed0VLHXNWf6HaaHCjO?[eAb[PNWC6XYA_QRNPORXf6H[] ADMLM6[PNeGADORNWU_Cw → A(w)
L@9HBM6[WVLEY4HC¡MLQRHYCLADCjOv = A(w)
L[WAGIU0C0OSNPCV6NPORXrµLPFj?f4Ha(w, v)
aU0C0OSQRHbdjV6H‖A(w)‖2 = a(w,A(w)) ≤M‖w‖‖A(w)‖,
Ecol
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re.
0 ," "aB !5" G_GD] HYE,O,Zcf4NPQSH9djV6H ‖A(w)‖ ≤M‖w‖ HIOf4U_CLG w → A(w)
HYE,OGU_CjORNWCV6H_+CLHBA_V4ORQSHBA_M6M6[WNWGYAcORNWU_C¡f4V0K6XU0QRYaH# /6P0P f4H9QSHM6QSXYERHCjO&AcORNWU_C\f4HsXNWHYECJ9NWaM6[PNedjV6H;djV] NW[FHNeE,ORH9VLC¡X[WXaHYC0Of4HV4C6U_ORX
fORHY[vdjV6H ‖f‖V = ‖L‖V ′
HIOL(v) = 〈f, v〉 MFU0V6QORU0V4O v ∈ V.
'NWCLA_[PHYaHYCjOY6[WHbM6QRU06[WaHb¤A_QRNeAcOSNPU0C6C6HY[rµ6WYµjHpE>OXpdjV6NPcA_[PHYC0OZIHOSQRU0V6_HYQ
u ∈ V OSH[vdjV6HA(u) = f.
rµLP ¯ °!U0V6Q-f4XaU0C0OSQRHYQv[WHORK6XYU_QSaH?NW[4C6U_VFE«rA_V4Of4U_CLG2aU_CjORQSHQ§djV6H[] U_M<XQ&AcOSHV6Q
AHpE>O-6N A>HYGIORNP«<f4H
Vf6A_CLE
VGIHbd0VLNFNWaM6[PNedjV6HB[] HI4NeE>OSHCLGH;HO[^] VLC6NWGNOSXBf6H
uHOdjV6HBE,U0CNWCj0HQ&E,H;HYE,OGU_CjORNWCVrGIH#djV6NlMLQRU0V6_Hb[eAf6XM<HCLf6A_CLGIHTGU_CjORNWCV6HTf4H
uMLADQQ&ADMLMFU0Q,OZ
LI 'A\GIUHQ&GINWNOSXµ6Wp02f6H
a(w, v)aU_CjORQSHBdjV6H
ν‖w‖2 ≤ a(w,w) = 〈A(w), w〉 ≤ ‖A(w)‖‖w‖,GIHTdjV6Nvf4U0C6C6Hν‖w‖ ≤ ‖A(w)‖ M<U_VLQORU_V6O w ∈ V, µ6W8G_GD] HYE,O,Zcf4NPQSHd0VLH
AHYE,O NPC>A>HYGOSN«>;°§U_V6Q aU0C0OSQRHYQ djV6H
AHYE,O³E,V6Q.A>HYGIORNP«>;G_] HpE>OR ZD f4NWQSHdjV6H
Im(A) = VrGIHdjV6NLC] HpE>OMFA_E-XYjNef4HYC0O?ERN
VHpE>O?f4Hf6NPaHCFE,NWU_CNWC4LC6NWHpIDNP[<E,VWO?f6HaU0CjORQSHQd0VLH
Im(A)HYE,O«HYQRaXTf6A_CLE
VHO9djV6H
Im(A)⊥ = 0 CHlCªHIOY6fLADCLEGHbGYA_EU_C_U0NOdjV6HV = 0⊥ = ( Im(A)⊥)⊥ = Im(A) = Im(A)
ªGHgdjV6N!M6QRU0V6_Hg6NPHYC³djV6HAHYE,OBERV6Q.A>HYGIORNP«>
(U_NPOA(wn)
V6C6HgERV6NPORH#f6A_CLEIm(A)
djV6NvGIU0C_HQS_HB_HYQSEbf6A_CLE
VLC_HYQ,OSVf6H¡rµLP4GD2U_CA
ν‖wn − wp‖ ≤ ‖A(wn)−A(wp)‖d0VLN6ORHYCLf_HYQSEJYXQSUTd0VFADCLfnHIOpOSHCLf4HYCjO?_HYQSE§[^] NWC4LCLN@9U0CLG
wnHYE,O?V6C6H;E,V6NPORHf6HB®ADVLG&K`f6ADCLE2[^] HpE,MLA0GIHBf4H 9NP[WFHYQ,O
VGD] HYE,O,Zcf4NWQRH;d0Vv] HY[P[WH;GU_C_HYQR0H_HQ&E2V6C6H9[WNWaNPORH
w ∈ V 4±[WU_Q&EMLADQGIU0CjORNWCjVLNOSXBf4HAU_CHCf4Xpf4V6NPOdjV6H
A(wn)GIU0Cj0HQS_H;_HQ&E
A(w) = b6G_] HpE>OR ZD f4NWQSH;djV6H
b ∈ Im(A)HIO
Im(A)HYE,OJf4U0CLG«HYQRaX_!@g] ADV6ORQSHMLADQROY!E,U0NO
v ∈ Im(A)⊥ [WA³GIUHQ&GINWNOSXrµ6Wp0f4Ha(w, v)
NPaM6[WNed0VLH#d0VLHν‖v‖2 ≤ a(v, v) = 〈A(v), v〉 = 0,GD] HYE,O,Zcf4NPQSHJdjV6H
v = 0HIO
Im(A)⊥ = 0 ªGIH\djV6N-M6QSU_V60HJdjV6H A HYE,ObLN A>HpGORNP«>$(U0NOA−1E,U0CNWC_HYQSERH HL[^] NWC6XY0AD[WNPORXrµLP4GDA¤0HYG
w = A−1(v)M6QSU_V60HbdjV6H
A−1 HYE,O;GIU0CjORNWCjVvFf4U0CLGb[WAE,U0[PV4OSNPU0Cuf6XM<HCLfGIU0CjORNWCjLaHYCjO9f4H
f
.¡|+3´m4ycilk!| \%] \%] a(N[^] HpE,MLA0GIH¡f6H 9NP[WFHYQ,OVHpE>O\f4Hf4NPaHYCLE,NWU_C´LC6NWHGIHdjV6NC] HYE,OGHIMFHYCLf6ADCjO A,ADa\ADNeEJ[PHGA0EgM<U_V6Q[WHYEADM6M6[WNeGAcOSNPU0CLEd0VLH¡C6U0VLEjNeERU_CLE&![WAf4XaU0CLE>OSQSADORNWU_C f6V
0K6XU0QRYaHµL µLPf4H; vAcNW[P0QSA_a E,HERNWaML[PNPLH;GU_CLERNef4XQ&AD6[WHaHYC0OpcCHC<HOY_HYCf4NWaHYCLERNPU0CLC6NWHORU_V6ORHYE[WHYE¡ADM6M6[WNeGAcOSNPU0CLE[PNWC6XpADNWQRHpE\E,U0C0OGU_CjORNWCV6HYEHO[] NPC>A>HYGOSNPNPORX´µ6W8GDf4HAHpE>OXYdjV6NW¤A_[PHYCjObZE,U0C³NWCj0HQ&E,NW6NW[PNPORX0 ;C³_U_NPOb6NPHYCf6A_CLEBGIHGA_ErGU_aaHfLADCLEB[WHGYA_E;_XCLXQ&AD[¬d0VLHT[] K`jM<UDOSK6YERH#f4H#GUjHYQSGNPNPORX#f4HT[eA«U_QSaHT6NW[PNWC6XYA_NPQSH
a(w, v)HpE>ONWCLf4NeE,M<HCFERA_6[PHTMLV6NWESdjV6HGD] HYE,OgH[W[PHdjV6Nf4U_CLC6H\[] NWC'A>HYGIORNWjNPORXf4H
A$XHYaA_QSdjV6U0CLEbM<U_V6QTFC6NPQdjV6H_!E,N
V = RN vV6C6H«U_QSagV6[eAcOSNPU0C cADQSNWADORNWU_C6CLH[W[PHC] HpE>OdjV6H[] XYGQRNPORVLQRH0 〈Au, v〉 = 〈f, v〉 M<U_V6QOSU_V4O v ∈ RN fl] VLCERNWaML[PH#ER`4E>OSaHb[PNWC6XYA_NPQSHAu = f
•
Ecol
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Ne
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re.
G ¯ !"#$%
.|+3´m4ycilk'| \] \] \C6HADV6ORQSH¡f4XaU0CLE>OSQSADORNWU_C rV6C M<HVxaU0NPCFE#OSHYG&K6C6NedjV6H¡a\A_NWEJd0VLNGYAcaU0VLHV6C M<HVx[PHpEA_QR0V6aHCjOSEgHYESE,HYCjORNWH[eESgf4V 0KLXU_QSaH¡µL µLPf4H vAcNW[P0QSA_a HYE,O[WAERV6NWcADCjORH0 ;C¥f4XYa\ADQSQRHGIU0aaHM6QSXYGXYf4HYaaHYC0O A>VLESdjV] Z[eA«U_QSagV6[eAcOSNPU0C µ6W ¯ Jf4V M6QSUDL[PYaH0°§U_V6QgaU0CjORQSHQT[^] HNeE,ORHCFGIHHIOg[^] V6C6NeGINPORXf4H[eAE,U0[PV6ORNWU_Cuf4H µ6W ¯ IlU_CNWC0OSQRU4f4VLNOVLC6H#ADM6ML[PNeGADORNWU_CA4W\C6H
Tf6HVfLADCLE
VLf4XIFC6NPHbMLA_Q
T (w) = w − µ(
A(w) − f) A¤_HpG
µ =ν
M2,
f6U_CjO;U0C¤A¡aU_CjORQSHQ9djV] H[W[WHgHYE,OBE,ORQSNeGORHYaHCjOBGU_CjORQ&A_GIOSADCjOSH_LGHJdjV6NvMLQRU0V6_H#[^] HI4NWE,ORHYCLGIHJHIO[^] V6C6NeGINPORX#f4Hu ∈ V ORHY[vdjV6H
T (u) = urf] U'&[WHTQRXpE,VL[O&AcO&I4CHlCªHIOp6U_CA
‖T (v)− T (w)‖2 = ‖v − w − µA(v − w)‖2= ‖v − w‖2 − 2µ〈A(v − w), v − w〉+ µ2‖A(v − w)‖2= ‖v − w‖2 − 2µa(v − w, v − w) + µ2‖A(v − w)‖2≤ (1− 2µν + µ2M2)‖v − w‖2≤ (1− ν2/M2)‖v − w‖2.
•9C6H«U_QSagV6[eAcOSNPU0C cADQSNWADORNWU_C6CLH[W[PHMFUjERERYf4HERU_V60HCjOJV6CLHNWCjORHYQRM6QSXIO&AcOSNPU0C´M6K`4E,NedjV6H_§HCMFADQRORNeGIV6[WNPHYQERN<[WAg«U_QSaH96NW[PNWC6XpADNWQRH;HYE,OER`jaXORQSNWdjV6H0j?C¡HlCªHIOf6A_CLE2GHBGYA_EY0[eAgERU_[WV4OSNPU0C¡f4HB[WA«U0QRaJV6[eAcORNWU_CcADQSNWADORNWU_C6CLH[W[PHrµLPpµ02QRXpAD[WNWERHB[PH73´os'oK3k3 w Wk's!|hs'|jy4AFo| ¬OSQRpECLAcOSV6QSH[W[PHHYCM6K`ERNed0VLHBU0VHYCaXpGA_C6NWdjV6H¤
bycqF~?q<tYornYorq<s \%] \%] xh plj|q'@j p b g+p jlp | bde + pjlp m g -b k + t9j [w 'w m j q - c kvqdt.w2xh pug || b pjEjh |Dg+p gNjsqr b kut.j c c h q c k@jjlp e p t e k c gNj a(w, v) = a(v, w)| b g k e b g e v, w ∈ V w `>b4cfe J(v) R hjk c j m' ' h c j| b g k v ∈ V |Nq4k
J(v) =1
2a(v, v)− L(v).
rµ6W `Qb4cfe
u ∈ V qp b Dg eKcBb hg h c gNj m j q.r b kutg q e~cBb hai4q4k c q eKcBb hNhjfj z 'w lw %" b k-p u jlp eqdg+p-p c v g h c gNj| b4c h em j7t c h c t g t m j7v h>jlk c j o N jlp e $ m4c kvj gNj
J(u) = minv∈V
J(v).
" c |>k b gNjt.jh e o p c u ∈ V jlp e g h)| b4c h em j.t c h c t g t m j R hjk dc j J(v) o qd b k-p u jlp e qp b g eKcBb h g h c gNj m jEqr b kutg q e~cBb h i4q4k c q eKcBb hNh>jlfjz [w uwYh+3´q<s'tYnycm4nporq<s#](4N
uHpE>OTERU_[WV4ORNWU_Cf4HJ[eA¡«U_QSagV6[eAcOSNPU0C³cADQSNWADORNWU_C6CLH[W[PHrµ6WYµjªU_Cf4X0HI[WU_MLMFHr_Q-,_GIHbZ[WAER`aXORQSNPHTf4H
a
J(u+ v) = J(u) +1
2a(v, v) + a(u, v)− L(v) = J(u) +
1
2a(v, v) ≥ J(u).
Ecol
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repr
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0 ," "aB !5" G'G
®2U_aaHu + v
HpE>OTdjV6H[eGIU0CLdjV6HJf6A_CLEVuaNWC6NWaNeERHJ6NWHC[] XCLHQS_NWH
Jf6ADCFE
V>XXYGINWM6QSUDd0VLHaHCjOYFE,U0NO
u ∈ V ORH['d0VLHJ(u) = min
v∈VJ(v).°!U0V6Q
v ∈ VU_C f4XILCLNO\V6C6H«U0CLGOSNPU0C
j(t) = J(u + tv)f4H
Rf6A_CLE
RrNP[EY] AD_NPO\fl] VLCMFU0[P`C"B0aHBf4Vf6HV44NWaH;f4HY_QSX9HYC
tL®2U0aaH
t = 0HYE,O2VLC¡aNPCLNPaJV6a f4H
jjU_CHC¡f4Xpf4V6NPOd0VLH
j′(0) = 0d0VLN<MLADQ9V6C³GA_[WGV6['ERNWaML[PH0FHpE>O9H6A_GOSHaHCjO9[eA«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6CLH[W[PHrµ6WYµj
.¡|+3´m4ycilk!| \%] \%]U0VLE#0HQSQRU0CLEbM6[WVLEJ[PU0NPC´A_Vx®2KLADM6NPORQSH djV6H0v[WU_Q&ERdjV6H\[eA«U0QRaH6NW[WNC6XYA_NPQSHaHpE>O#ER`aXORQSNWdjV6H0<NW[-HI4NeE>OSHV6CA_V4ORQSHADQS_V6aHCjOTdjV6HJ[WH0K6XU0QRYaHµL µLPf4H\ vAc
NP[W_Q&ADa M<U_VLQGIU_CFGI[WV6QRHbZJ[] HI4NeE>OSHCLGHBHO[] V6C6NeGINPORX#fl] V6C6HTERU_[WV4ORNWU_Cf4H\rµLPpµ0Ij?CHC<HOY4U0Cf4XaU_CjOSQRHYQSAJf4NPQSHYGIORHYaHYCjO[^] HI4NWE,ORHYCLGIHTfl] V6CV6C6NedjV6HbMFU0NPCjO9f4HbaNWC6NPaJV6a f6HB[^] XC6HYQR0NPHJ(v)
C_HYQ,OSV f4H#[eA°QRU0MFUjE,NPORNWU_CµL µL FFLGIHY[WA\f4XYaU_CjORQSHT[] HI4NeE>OSHCLGH#HIO;[] V6C6NeGINPORXgf6H#[WA\ERU_[WV4OSNPU0Cf4HT[WAJ«U0QRaJV6[WADORNWU_C¤A_QRNeAcOSNPU0C6C6HY[P[WH_ •
)*)*KJ , nO,=R:>P'16=,5?/ P-8 7P nO!P-:l=!Uv/2ERESA¤`_U0CLETfl] ADM6M6[WNed0VLHQg[WH0K6XYU_QSaHµ6 µ6Wf4H vAD NW[W_Q&ADa Z[eA«U_QSagVL[WADORNWU_C¤A_QRNeAcORNWU_C6C6HY[P[WHrµL 9f4V vA_M6[eA_GINWHCA¤_HpGJGU_CLf4NPORNWU_CFEBA_V4 [WNPaNPORHYETf6H@;NWQRNeG&K6[WHIOY'®2HY[P[WHIGIN-EY] XpGIQSNO6NPHYCE,U0VLE[WAJ«U0QRaH\rµLPpµ0A¤_HpG
a(u, v) =
∫
Ω
∇u(x) · ∇v(x) dxHIO
L(v) =
∫
Ω
f(x)v(x) dx,U&GI[eADNWQRHYaHYCjOa(·, ·) HYE,ObVLC6H«U_QSaHJ6NW[WNPC6XpADNWQRH0ªHIO L(·) V6C6H«U0QRaH[PNWC6XpADNWQRH0ª ?] HpE,MFA_GIH VC6U_ORXTM6QSXYGXYf4HYaaHYC0O
X2HpE>OV =
v ∈ C1(Ω), v = 0E,V6Q
∂Ω
.µ6W j
®2U_aaHBMLQRU4f4V6NPO9ESGAD[eADNWQSHBERV6QVC6U_VFEG&K6U_NeE,NeESE,U0CLE
〈w, v〉 =
∫
Ω
∇w(x) · ∇v(x) dx, rµL / E d0VLNvAMFU0V6QC6U_QSaHTA_ESE,U4GINWXH
‖v‖ =
(∫
Ω
|∇v(x)|2dx)1/2
.
;C´_XYQRNPLH¡A_NWERXaHCjOJdjV6HrµL / E Tf4XLC6NPOJV6CxM6QSUf6V6NOERGYAD[eADNWQRHERV6QV
H![PHE,HYV6[2M<U_NWCjOdjV6NaXYQRNPORHTf4H#EY] `¡AcOROSADQ&f4HYQHYE,O[eAM6QSU_M6QSNPXORX ‖v‖ = 0⇒ v = 06CHlCªHIOpLf4HT[^] XY0A_[PNPORX
∫
Ω
|∇v(x)|2dx = 0
Ecol
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repr
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G !"#$%
U0C f4XYf6V6NOTdjV6HvHYE,O;V6CLHJGIU0CLE,OSADCjOSHgf6ADCFE
ΩFHOBGU_aaH
v = 0ERV6Q
∂ΩU_C A\LNPHYC
v = 0 'A¡aU_ORNW¤ADORNWU_Cf4VG&K6U_NP³f4HrµL / E BGIU0aaHM6QSUf6V6NO#ESGA_[WA_NPQSHJHpE>OT6NWHCER6Qb[PH«rADNPO#djV6H[WA«U0QRaH6NW[PNWC6XYA_NPQSH
a(·, ·) HYE,OTm6kvnpq_3 m4nYoilk'|+3´|sªn¡¢0q|jy¤¢0oz<|MFU0V6Q\rµL / E ;C_XQSNPLHJMFADQA_NP[W[WHV6Q&ETADNeE,XYaHCjOgdjV6HaHpE>OgGU_CjORNWCV6H_°§U0V6Q#aU_CjORQSHQ#djV6H
LHYE,OgGIU_CjOSNPCV6H0NP[?«rADV4OT«rA_NPQSHA_M6M<H[vZ[^] NWC6XjAD[WNOSXTf4HT°§U_NWCLGA_QRXbf4V HYaaHTµ6 µ6 ¯ H4U_CAAD[WU_Q&E
∣
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∣
∣
∫
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f(x)v(x) dx
∣
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∣
∣
≤(∫
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Ω
|v(x)|2dx)1/2
≤ C‖v‖,U'&
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V 0'U_V4OSHYEg[PHpE#K`MFU_ORK6pE,HpE#f4V 0K6XYU_QSaHµ6 µ6Wf4H¡ 'Ac NP[W_Q&ADa ERHaJ6[WHCjOg_XYQRNPLXYHYEYHIOM<U_VLQ,O&ADCjONP[jHCTa\A_CLdjV6H-V6C6H?djV6N0HaM<¦YG&K6H?ERU_C#A_M6M6[WNWGYAcOSNPU0C HY[] HYERMLA0GIH
VC] HYE,OvMLA0EV6CTHpE,MFA_GIHf6HNW[P<HQROgGA_QTNP[Cv] HpE>O#MLA0ETGIU0aML[PHO#MFU0V6QT[WAC6U0QRaHNWCLf4V6NPORHMFADQrµL / E ª² ?] U0LE,ORQSVLGOSNPU0CCLHjNWHCjOMLA0EOSA_CjO¡f4V G&K6U_NPxf4VMLQRU4f4V6NPO¡ESGAD[eADNWQSHdjV6H³f6H[] HI4NW_HCFGIHf4H QSX0V6[WA_QRNPORX
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VESA«HYQRaHIOSV6QSHM<U_VLQT[PHM6QSU4f4V6NPOTESGA_[WA_NPQSHrµ60/ E
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f4U0CjO?[PHpEXY[PXYaHCjOSECLHgE,U0CjOM6[PVFEf4HYE«U_CLGIORNWU_CLEQRXY_V6[WNPYQRHpEa\ADNeEE,HYV6[WHaHCjOaHYERV6Q&AD6[WHYEYNCLHgADV4OSQRHf6NDWGIVL[OSXERHQ&ABf4H_U_NWQ!HCd0VLH[6ERHCLE![eA9°QSU_M<U0ERNPORNWU_CJµL /60GgrdjV6N4H4M6QRNWaH2[^] XYdjV6NW¤A_[PHYCLGIHHCjOSQRH[WHBM6QRU06[WaHBA_V4\[PNWaNOSHYETµ6Wp?HIOERA#«U_QSagV6[eAcOSNPU0CcADQSNWADORNWU_C6C6HY[P[WHJµ6 R?QSHYE,ORH;QSA_NF[WU_Q&ERdjV6HU0CQSHaM6[eA_GHb[] HYERMLA_GHVMFADQ
V9U_VLEHpE,M<XQSU_CFEA¤_U_NWQADNWCLERNvGIU0CjcA_NPCLGV[WHT[WHYGOSHV6Qf6V ¢_m4ycm6¢_nI:jyc|s!m4nYk'yc||0n\os'hrk!¢4=npm§r|w!|tT|jtY~§m6¢_|tTw!|q?q<r|jz³w!m6s!tTrmy¤htpqFkvnporq<sw'|t 1 qLy43k'm4nYoqFs!t;z6m4ycorm4nporq<s%=s!|r|jtfl] XYdjVLADORNWU_CLETA_V4f6XQSNP0XHYEBMLA_Q,OSNPHY[P[WHYEbHY[P[WNWM4ORNedjV6HYEY 0'HQSaNPC6U0CLETGIH\G&KLA_M6NPORQSHMFADQ#V6C[WHaaHORHYG&KLC6NWdjV6H0!A_M6M<H[WXNWC6XjAD[WNOSXf4H°§U_NWCLGA_QRX0!djV6HC6U_VLEA¤_U0CLE#V6ORNW[PNeE,XV6CxMFHYV M6[WVLEKFADV4Op
;2|+3 3´| \%] \%] `>bdcfeΩ g h b g i8jk em j RN b kuh m qdhpq4gEt b4c hNpg h>j m4c kvj- eKcBb h m jR jlp~|q'-jIw j - c p e j9g hj9 b hNp e q4h e j C > 0
e jfj gNj o | b g k eb g e jr b h e~cBb h v ∈ C1(Ω) g c p qdhNh_g j
pug ksj b k m ∂Ω o∫
Ω
|v(x)|2dx ≤ C∫
Ω
|∇v(x)|2dx.Yh+3´q<s'tYnycm4nporq<s#]§ ] K`jM<UDOSK6YERHgERV6Q9[WHgGYADQ&A_GOSQSHbFU0QRC6XJf4H
Ωf4NPOADM6QSYEVLC6H#X0HCjORVLH[W[PHQSUDO&AcOSNPU0CFTdjV6H¡M<U_VLQ#ORU0V4O
x ∈ Ω[eAM6QSHaNPYQRHGIU0aM<U0ESADCjOSH
x1HYE,OJ<U_QSC6XH0 −∞ < a ≤
x1 ≤ b < +∞ >(U_NPOvVLC6H#«U0CLGOSNPU0C f4H
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∂Ω ;CMFHYV4O;[^] XIORHYCLf4QSHMFADQ2GU_CjORNWCV6NPORXMLA_Q5JYXQSU#HC¡f6HK6U0QSEf6H
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C1MFADQaU_Q&GIHpADV4¡f6A_CLERN HIOXYGQRNWQSH_MFU0V6Q x ∈ Ω
v(x) =
∫ x1
a
∂v
∂x1(t, x2, ..., xN ) dt,f] U'&[^] U0Cf4XYf4VLNOMLA_Q[] NWC6XjAD[WNOSXbf6HJ®ADVLG&K`j (6G&KADQKJ
|v(x)|2 ≤ (x1 − a)∫ x1
a
∣
∣
∣
∣
∂v
∂x1(t, x2, ..., xN )
∣
∣
∣
∣
2
dt ≤ (b− a)∫ b
a
∣
∣
∣
∣
∂v
∂x1(t, x2, ..., xN )
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∣
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∫ b
a
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∣
∣
∣
∂v
∂x1(t, x2, ..., xN )
∣
∣
∣
∣
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∫
Ω
|v(x)|2dx ≤ (b− a)2∫
Ω
∣
∣
∣
∣
∂v
∂x1(x)
∣
∣
∣
∣
2
dx ≤ (b− a)2∫
Ω
|∇v(x)|2dx.
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9 " 9;3 7 2#
un(x) =
−x− 1 − 1 < x < −n−1,(n/2)x2 − 1 + 1/(2n) − n−1 ≤ x ≤ n−1,x− 1 n−1 < x < 1.
N = 2
9 <>"% 0 < α < 1/29 " 9;3 7 2#
un(x) = | log(|x|2 + n−1)|α/2 − | log(1 + n−1)|α/2.
N ≥ 3
9 <>"% 0 < β < (N − 2)/29 9); 3 7 2#
un(x) =1
(|x|2 + n−1)β/2− 1
(1 + n−1)β/2.
(% >%,* 7 2 ! un 9 2' 92 V & 2 *" >787 ! >% <'2 92 V7 @%K*" n >'9 >% 7 3 /
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L2 f6HYEb«U0CLGOSNPU0CLEgf4HGADQSQSXYEE,U0aa\A_6[PHpE4[eA9(HYGIORNWU_CIFL0/gf6U_C6C6HYCjOdjV6HY[WdjV6HpEQSA_M6M<H[eEZGHTE,V'A>HOY ;C`NWCjORQSUf6V6NOADVLESE,N<[eA#C6U_ORNWU_Cf6H;w!h0ycorz6m4nYoqFs 1 m6o §r|< vA(4HYGOSNPU0C FF µ#GU_CjORNWHCjOORU0V4ORHpE?[WHYEf4XLC6NPORNWU_CLEHIO[PHpETQRXpE,VL[O&AcOSETdjV] NW[-«rADV4OADLERU_[WV6aHCjO#GIU0C6CLA8VeORQSH\E,VLQ#[PHpEbHYERMLA_GHYETf4H(U_<U_[WH M<U_VLQgE,VLNPQSH[PHQSHYE,ORHf6V´GIU_VLQSEY vA (HpGOSNPU0C FL F f4U_C6CLHd0VLH[ed0VLHYE#QSXYERV6[POSADOSETGIU0aM6[PXYaHCjOSA_NPQSHYETM<U_VLQ#[PH[PHpGORHYV6QGIVLQRNWHV4l-C6LC?[eAy(HYGIORNWU_C FLMFHYQRaHIOA_V¥[WHYGIORHYV6QdjV6NGIU0C6CLA8VeO\[WAOSK6XU0QRNWHf4HYEf4NWE,ORQSNW6V4ORNWU_CFEBdjV6NlC] HYE,OMLA0EC6XYGHYESERA_NPQSH9NeGIN¬6f4H;«rADNWQSH;[WHb[PNWHCHCjORQSH;HpE,MFA_GIHpE2f6H (4U_<U_[WH\HIOf4NWE,ORQSNW6V4ORNWU_CFEv± [eALCf4V´G&KLA_M6NPORQSH[WH m!|m6k $]B^ych¢_m4~§onpk'|´npqFk!tr|tychtYk!npm4nYts'h¢_|tptYmLoyc|tJ~2qFk'yrmtYk!onp|]
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/ # +Q7 " % %
#?%'$g( 2 T!§ ( " % !$; b%'$ #c #$" 9*KJ*,+ 8UOvT8US´3NP 4 UORSx7r=R/21 3P!16=>5?/
0'U_VLEB[WHYEBQSXYERV6[O&AcO&E;f4HGIHO,OSHERU_VLE,E,HpGORNWU_C ERU_CjOTf4XIO&ADNW[P[WXYEBf6A_CLEB[PHGIU0V6QSE;f4Ha\AcORKLXa\AcOSNWdjV6HpEF ¯ H '(U_NPOΩV6CU_V60HQRO?f4H
RN aJV6C6NFf6H[WAbaHYERV6QSHf4H HYFHpE,0V6H_ ;Cf6XILC6NPO?[^] HpE,MFA_GIHL2(Ω)
f4HpE2«U_CLGIORNWU_CLEaHYERV6Q&AD6[WHYEf4HbGA_QRQSX;ERU_aa\AD6[WHbf6ADCLEΩ V6CLNf4VMLQRU4f4V6NPOERGYAD[eADNWQRH
〈f, g〉 =∫
Ω
f(x)g(x) dx,
L2(Ω)HpE>OV6CHpE,MLA0GIHTf4H 9NP[WFHYQ,Ogr_U0NPQ[WHbORK6XYU_QSaHTµ6 µ60/gf4H9F ¯ H ;CC6U_ORH
‖f‖L2(Ω) =
(∫
Ω
|f(x)|2dx)1/2
[eABC6U_QSaHGU_QSQRHpE,M<U_CFf6ADCjORH0 X9ADM6M<H[WU_CLE§djV6H[WHYE!«U_CFGORNWU_CFE'aHpE,V6Q&ADL[PHpE!fLADCLEΩE,U0C0O-f4XLC6NWHYE~!yc|tYilk!|~!m4y¤npqFk'n\f6ADCLE
ΩHªE,N§U0C³G&KLADCL_Hg[WHYE;cAD[WHV6Q&EBfl] VLC6Hg«U0CLGOSNPU0C aHYERV6Q&AD6[WH
fERV6QVLCERU_VFE> HCLERHaJ6[WHf6H
Ωf4H\aHpE,V6QSHCV6[P[WH_vU_CC6HG&KLA_C6_HMLA0ET[WA«U0CLGOSNPU0CaHpE,V6Q&ADL[PH
f±9V4ORQSHaHYC0Ogf6NOpvf4HV6³«U_CFGORNWU_CFEbaHpE,VLQSA_6[PHpE
fHOgERHQSU_CjO#f4NPORHpETXjAD[WHYEbE,N
f(x) = g(x)MLQRHpERdjV6HTMLA_Q,OSU_V4O9fLADCLEΩLGD] HYE,O,Zcf4NWQRH#EY] NW[HNeE,ORH
E ⊂ ΩORHY['djV6H#[eAaHYERV6QRHJf4H# HYFHpE,0V6Hf6H
EHpE>OCV6[W[PHbHO
f(x) = g(x)M<U_V6QOSU_V4O
x ∈ (Ω \E)
;C\C6U_ORHC∞
c (Ω)U0V D(Ω)
[] HYERMLA0GIHf4HpE-«U_CFGORNWU_CFEf4H9G[WA0ERERHC∞ ZTERV6M6M<U_QRO?GIU_aMLA0GOfLADCLE
ΩXHYa\ADQ&d0VLU_CLE#djV6H¡[] HYERMLA_GH
C∞c (Ω)
C] HYE,OgMLA_EgQRXpf4V6NPOZ[eA ERHV6[WH\«U0CLGOSNPU0C´CV6[P[WHMFADQRORU_V6OgrGIHJd0VLNvC] HYE,O9MLA0EXNef4HCjOv²-U_NWQ[PHJGIU_QSU_[W[eADNWQRHbµ60/4 ¯ f6ADCFE F ¯ H¬I<9UDORU0CLEA_VLERERNvdjV6H[WHYE«U_CLGIORNWU_CLE;f4HC∞
c (Ω)EY] A_C6CV6[WHCjOY<ADNWCLE,N!djV6H#OSU_V4OSHYE[WHVLQSE;f4XQSNP0XHpELERV6Q;[PHJFU0QSff4H
Ω9U_VLEQ&ADMLMFHY[PU0CLE[PHbQSXYERV6[POSAcOf4H#f6HCLERNOSX#E,VLNPcADCjO#0U_NWQ[PHBOSK6XU0QRYaHTµL FF µf4H9F ¯ H
£'hqFy8:+3´| $]%]B^ jlp~|q'@j C∞c (Ω) jup em jlhNplj m q4hNp L2(Ω) o _ jup e $ mdc kvj gNj| b g k e b g e
f ∈ L2(Ω)c $j- c p e jEg hjspug cfe j fn ∈ C∞
c (Ω)e jfj gNj
limn→+∞
‖f − fn‖L2(Ω) = 0.
'AM6QRU0M6QSNPXORXbE,V6NWcADCjORHb0XC6XYQSA_[PNeE,HB[WH# HaaHTµ60/4 6 qFycqFrm6ory¤| $]%] `>bdcfe
f ∈ L2(Ω) w `_c | b g k e b g e jr b h eKcBb h φ ∈ C∞c (Ω)
b hq∫
Ω
f(x)φ(x) dx = 0,
qd b k-p f(x) = 0 |kvjlp gNj|qdk eb g e2m q4hp Ω wYh+3´q<s'tYnycm4nporq<s#]$(4U_NPO
fn ∈ C∞c (Ω)
[eAE,V6NPORHTf4Hb«U0CLGOSNPU0CLEQSX_VL[PNWQSHYEdjV6NGIU0Cj0HQS_H;_HQ&Eff6ADCFE
L2(Ω)HC_HYQ,OSVf4V 0K6XYU_QSaH FL0/4W_ ;CA
0 = limn→+∞
∫
Ω
f(x)fn(x) dx =
∫
Ω
|f(x)|2dx,
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2 " n 4#8 µfl] U'&[^] U_Cf4XYf6V6NO9djV6H
f(x) = 0M6QSHYESdjV6HbMLADQRORU0V4Of6ADCFE
Ω
! "#$%&'()* +$-, Lp(Ω)
/. ,1 ≤ p ≤ +∞ 0 1 1 ≤ p <
+∞ Lp(Ω)
2354 +$-, 768,9%(:)+";" 3 "<"%+9,)p =?> @%9 -;'+ Ω 0"A "%
*(B-9‖f‖Lp(Ω) =
(∫
Ω
|f(x)|pdx
)1/p
, C5D"0E/FLp(Ω)
2G)+$,)7 IH7,JG,4 2 =LK/= %&9I"* +$-, .M ,)9-9%()M-9,)) 0 1 p = +∞ L∞(Ω)
2N54 )+$,): ")N6O,)9%1 +;"()f)++ 9%()( )P1;9) N+
Ω/,4 2 =LK-= %97QG4 %R"%29"*, 2-P9
C > 09)(SQ |f(x)| ≤ C
"9 9Q"*$/+9"< ""Ω 0$A "%N *(B-9
‖f‖L∞(Ω) = inf
C ∈ R+ 9 NQ" |f(x)| ≤ C
0 0 "" Ω
, C5D"0 TFL∞(Ω)
2SU +, H:-$-,J 0NV " !Q"W+% Ω)2X"Y.M+;9"1-9
Lp(Ω) ⊂Lq(Ω)
1 ≤ q ≤ p ≤ +∞ 0
*KJ*KJ Z 3F= P'16=,5?/\[-P-= nO!U;C f6XILC6NPO¡OSU_V4Ofl] A_FU0QSf¥[WH³GIU0CLGIHYM4O¡f4H³f6XQSNP0XH«rADNW6[WH³f6A_CLE
L2(Ω)®2HIORORH CLUDORNWU_C_XCLXQ&AD[WNWERHJ[eAf4XQSNW¤ADORNWU_CVLERV6H[W[PH rMLADQR«U_NeETADMLMFHY[PXYH_MLA_QTU_M6M<U0ERNPORNWU_Cf4XYQRNWcAcOSNPU0C³«U0Q,OSHpBHIOHYE,OV6CGA0EMLADQRORNeGIV6[WNWHQf4HB[WAf4XQSNW¤ADORNWU_CADVERHCFEf6HYEf4NeE>OSQRNW6V4OSNPU0CLEb0U_NWQ"F ¯ HU0V¡[eA (HpGOSNPU0C
FLgM<U_VLQV6C6QRH«'QSXYERV6aXpIYh[Z§s'ornYorq<s ] Q] \ `Qb4cfe
v g h>jr b h> e~cBb h m j L2(Ω) wxh m4cfe gNj v jlp e"m k c i8q'l j)q4gpjlhNprlq c j m q4hNp L2(Ω) p c $j- c p e j m jupr b h> e~cBb hNp wi ∈ L2(Ω) o | b g k i ∈ 1, ..., N o e jlf jup gNj o| b g k eb g e j#r b h e~cBb h φ ∈ C∞
c (Ω) o b hq∫
Ω
v(x)∂φ
∂xi(x) dx = −
∫
Ω
wi(x)φ(x) dx.
] q gNj wi jlp e q@|'|j jq i ,+ t.j m k c i j|Nq4k e~c jlfj2rlq c lj m j v j e h bde j m p b kut.q c p ∂v∂xi
w 'A@;XILC6NPORNWU_C FL0/4 µA6NPHYCTV6CgERHCLE?HHCTMFADQRORNeGIV6[WNPHYQYY[WACLUDOSADORNWU_Cwi = ∂v
∂xi
HYE,OvV6CLNP0UdjV6HGADQp_HYC_HYQ,OSVf4V®2U0QRU0[P[eADNWQRH FF /6 /6_[WHYE«U_CLGIORNWU_CLEwi
ERU_CjO?V6CLNWdjV6HpE;E,N<H[W[PHpE?H4NWE,ORHYC0O+*NPHYCE,6QpERNvHYE,O¡f4XYQRNWcAD6[WHADV E,HYCLEVLERV6H[HO¡djV6H E,HpE\f4XQSNW_XHpEMLADQRORNWH[W[PHpE\ADM6MFADQRORNWHC6C6HYCjO\Z
L2(Ω)lAD[WU_Q&E[WHYEbf4XYQRNW_XYHYEVFE,V6HY[P[WHHIOB«rADNW6[PHf4H
vGIU)L^CLGNWf4HYCjOYª@9U0C6C6U_CFE9OSU_V4OTf4HE,VLNOSHJV6CGIQSNOSQSHE,NWaM6[PH;HIO?MLQSADORNed0VLHMFU0V6Q2f6XIORHYQRaNWC6HQ2E,N<V6C6H«U0CLGOSNPU0C\HYE,O2f4XYQRNWcAD6[WH9ADVERHCFE-«rADNW6[WH_
;?|+3 3´| ] Q] `Qb4cfev g h>jr b h eKcBb h m j L2(Ω) w ` c j - c p e jg h>j) b hNp e q4h e j C > 0
e jlf j gNj o | b g k eb g e j#r b h> e~cBb h φ ∈ C∞c (Ω) j e | b g k eb g e#c h mdc -j i ∈ 1, ..., N o b hq
∣
∣
∣
∣
∫
Ω
v(x)∂φ
∂xi(x) dx
∣
∣
∣
∣
≤ C‖φ‖L2(Ω),FL µ0
q4 b k-p v jlp e2m k c i4qj9q4g pjhprlq c jIw
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F # +Q7 " % %
Yh+3´q<s'tYnycm4nporq<s#]$(4U_NPOL[eAJ«U_QSaHB[WNWC6XYA_NPQSHTf4XILCLNPHTMLA_Q
L(φ) =
∫
Ω
v(x)∂φ
∂xi(x) dx.
± M6QSNPU0QRNL(φ)
Cv] HpE>OJf6XILC6NWHdjV6HMFU0V6Qφ ∈ C∞
c (Ω)!a\ADNeE#_Q-,_GHZ [^] NWC6XY0AD[WNPORX³ FL µ0'U_CM<HV6O#XIORHYCLf4QSH
LMLA_QTGIU_CjOSNPCV6NPORX\Z¡ORU_V6ORHYEb[WHYEB«U0CLGOSNPU0CLETf4H
L2(Ω)GA_Q
C∞c (Ω)
HYE,O#f4HCLERHfLADCLEL2(Ω)
fl] A_M6QSYE;[PH 0K6XU0QRYaH FL0/4W_FC«rADNPOYª[] NWC6XjAD[WNOSX FL µ0MLQRU0V6_Hd0VLHJ[eA\«U0QRaH[WNWC6XYA_NPQSHLHYE,O2GU_CjORNWCV6HE,VLQ
L2(Ω)jC0HQRORV\f4V 0K6XYU_QSaHT /6P0P f6H9QSHM6QSXYERHCjOSADORNWU_Cf4H
XNPHpE6J04NP[H4NWE,ORHbV6C6Hb«U0CLGOSNPU0C(−wi) ∈ L2(Ω)
ORHY[P[WH#djV6HL(φ) = −
∫
Ω
wi(x)φ(x) dx,
GHTd0VLNM6QSU_V60HBdjV6HvHpE>Of4XYQRNWcAD6[WHTADVERHCLE«rA_NP6[WHTf6ADCFE
L2(Ω)
;u|jyc¢_or¢0| $]%]B^ Ω = (0, 1) / ="(% >% *" 7 2 ! " xα #69;>% 2 7 42 6)5 2 7 92 L2(Ω) # C>7 & ) α > 1/2 /;u|jyc¢_or¢0| $]%] Ω " )% >@% ;/ ="(%)% * ! "4#" !% Ω
9 ! C1
<'2)% & @% #2 #69;>% 2 7 :2 6) 2 7 :92 L2(Ω) /;u|jyc¢_or¢0| $]%] \ Ω >% >@% ;/ ="(%)%*" 5 ! " C1 < 2)% & @% !2 & 2 <'2 ! 4< 2 9;>% 2 7 26) 2 7 92 L2(Ω) /
;CQSHIORQSU_VL_HBV6CQRXpE,V6[POSADO6NPHYCGU_C6CVM<U_V6Q[eAf4XQSNP0XHBVLERV6H[W[PH0bycqF~?q<tYornYorq<s ] Q] `Qb4cfe
v g h>jEr b h eKcBb h m j L2(Ω)m k c i8q'l j q4g pjhp7rlq c jyj ee jlf j gNj e b g e juppjlp m k c i jlp|qdk eKc jfjlprlq c ljlp ∂v
∂xi
o | b g k 1 ≤ i ≤ N o p b h e hNg fjlpw %" b k-p o | b g k q gNjn b t| b pq4h e j b hNhj-+j m j Ω o c Nj- c p e jg h>j b hp e q4h e j C e jlf j gNj v(x) = C |k@jup gNj|Nq4k e b g em qdhpE-j e~e j b t| b pq4h e j b hNh>j -+j wYh+3´q<s'tYnycm4nporq<s#]°§U0V6QORU0V4O
ψ ∈ C∞c (Ω)
FU_CAf6U_CLG∫
Ω
v(x)∂ψ
∂xi(x) dx = 0.
FF F(4U_NPO
Q =] − `,+`[NV6C GIV6<HU_V60HQRONWCLG[PVLEfLADCLE
ΩrA¤_HpG
` > 0IHIO ERU_NPO
θ(t) ∈C∞
c (−`,+`) ORH[W[WH#d0VLH∫ +`
−`
θ(t) dt = 1.
°§U0V6QORU0V4ORHb«U0CLGOSNPU0Cφ ∈ C∞
c (Q)U_Cf4XLC6NPO
ψ(x′, xi) =
∫ xi
−`
(
θ(t)
∫ +`
−`
φ(x′, s) ds− φ(x′, t)
)
dt,
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2 " n 4#8 A¤_HYGb[eACLUDOSADORNWU_C
x = (x′, xi)M<U_V6Q
x′ ∈ RN−1 HO xi ∈ R ;C³_XQSNPLHT«rA_GNP[WHaHYC0ObdjV6H
ψADM6MLA_Q,OSNPHYCjOA_VLESE,NlZC∞
c (Q)HO9djV6H
∂ψ
∂xi(x′, xi) = θ(xi)
∫ +`
−`
φ(x′, s) ds− φ(x′, xi).
±v0HYGBV6C6HbOSH[W[PHb«U0CLGOSNPU0Cψ[^] XpdjVLAcOSNPU0C FL Fjf4HNWHCjO
∫
Q
v(x)φ(x) dx =
∫
Q
v(x)θ(xi)
(
∫ +`
−`
φ(x′, s) ds
)
dx′ dxi
=
∫
Q
φ(x′, s)
(
∫ +`
−`
v(x′, xi)θ(xi) dxi
)
dx′ ds
_Q-,_GIHA_V\ORKLXU_QSaHBf4HLV66NWC6NL®2U0aaHφHYE,OdjV6H[eGIU0CLdjV6H_0MLADQADM6ML[PNeGADORNWU_C¡f4V®2U_QSU_[W[WA_NPQSH
FL0/40/#U0CHYCf6XYf4V6NPOv(x) =
∫ +`
−`
v(x′, s)θ(s) ds,
GD] HYE,O,Zcf4NPQSHdjV6HvC6HBf4XM<HCFfMLA0E2f4H
xif6ADCFE
QjCQRXYMFXOSADCjOGIHIOADQS_V6aHCjO?M<U_VLQ?OSU_V4OSHYE[PHpE-GIU_aM<U0ESADCjORHpE
xicU0CJU04ORNWHCjO-ADNWCLERN6d0VLH
v(x)HYE,O-GU_CLE,OSA_CjO§f6A_CLE
Q0®2U_aaH2ORU_V6O-GU_V6ML[PHf4HMFU0NPCjOSE?f6ADCLE§VLC6Ha¦YaHGIU_aM<U0ESADCjORHGIU0C6C6HHf4H
ΩMFHYV6_HYCjO¦IORQSHQRHY[PNWXYE!MLA_Q-V6C6HG&KLA4VeCLHf4HgOSH[eETGIV6<HYEf4HOSADNW[W[PHcADQSNWA_6[PH¤9djV6N-ERHJQSHYGU_V6M<HCjOp<U0C³M<HV4OTHYCGU_CLG[PV6QSHJdjV6H
v(x)HpE>OGIU_CFE>O&ADCjOf6ADCLEG&KFA_djV6HTGIU0aM<U0ESADCjOSHBGU_C6C6H4Hbf6H
Ω
;CM<HV6O#«rA_GNP[WHaHCjOg0XC6XYQSA_[PNeERHQb[WA³@9XLC6NPORNWU_C FL0/4 µf4H[eAf4XQSNW_XH«rA_NPL[PH¡ZGHQROSA_NPCLEU_M<XQ&AcORHYV6Q&Ef4NC<XYQRHYC0OSNPHY[WE;djV6NvC6H#«U_CjO9NWC0OSHQS_HYC6NPQ;d0VLHgGIHYQ,O&ADNWC6HYEGU_aJ6NPCFADNeE,U0CLEf4Hgf6XQSNP0XHYEMLADQRORNWH[W[PHpEHO?C6U_CMLA0E!ORU0V4ORHpESI®;] HYE,OMLADQ-HHYaM6[PH[WHGA0E-f4H[eATf4NP0HQS_HYCLGIHfl] V6C6H«U0CLGOSNPU0CZJcA_[PHYV6QSE0HYGIORU_QSNWH[W[PHpEdjV6NlCLU_VLEERHQ&AV4ORNW[PHTMFADQ[eAERV6NPORH0Yh[Z§s'ornYorq<s ] Q] `Qb4cfe
σ g h>j r b h> e~cBb h m j Ωm q4hNp RN mb h eEe b g e jupjlp b t| b pq4h e jlp
q@||Nq4k e~c jhNhjh e $ L2(Ω) z b h h bde j σ ∈ L2(Ω)N wsxh mdcfe gNj σ q m t.j e g hj m4c i4jlk jh-jq4g pjhprlq c lj m qdhp L2(Ω) p c j - c p e j)g h>jsr b h> e~cBb h w ∈ L2(Ω)
e jlfj gNj o | b g k eb g e jr b h e~cBb h φ ∈ C∞
c (Ω) o b hq∫
Ω
σ(x) · ∇φ(x) dx = −∫
Ω
w(x)φ(x) dx.
qr b h e~cBb h w jup e q@||Nj jE q mdc i8jk jlh>@j2rlq c lj m j σ j e h b4e j m' p b kut9q c p divσ w 'AA>VFE>OSNFGYAcOSNPU0Cf4HB[WAJ@9XLC6NPORNWU_CIFL0/4 ¯ HYE,Od0VLH_4E,NσHpE>OV6C6H;«U_CFGORNWU_CQRXY_V6[WNWQSH_AD[WU_Q&EV6C6HE,NWaM6[WHNPCjORXY_Q&AcOSNPU0C´MLADQ\MFADQRORNWHr_U_NWQ[PH®2U0QRU0[P[eADNWQSH¡µ60/4 µ0gaU_CjORQSHd0VLH[^] U_C ALNPHYC
w = divσ$CLH_XYC6XQ&AD[WNWESAcOSNPU0C «rA0GINW[PH¡f6VGQRNPORYQRHf6H¡f4XQSNW¤ADORNWU_C«rADNW6[WH¡f4V´ HYaaH FF /6 FHYE,O9f4U_C6CLXHbMLADQ[WHTQSXYERV6[O&AcOERV6NPcA_C0Ogf4U_CjOC6U0VLE[eADNeERERU_CLE[eAf4XaU_CFE>OSQSADORNWU_CADV[PHpGORHYV6QHYC_V6NeE,HTfl] HI4HYQSGNWGHp
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0¯ # +Q7 " % %
;2|+3 3´| ] Q] `>bdcfeσ g hj#r b h e~cBb h m j L2(Ω)N w ` c j- c p e jEg hjE b hp e qdh e j C > 0
e jlf j gNj o | b g k eb g e j#r b h e~cBb h φ ∈ C∞c (Ω) o b hq
∣
∣
∣
∣
∫
Ω
σ(x) · ∇φ(x) dx
∣
∣
∣
∣
≤ C‖φ‖L2(Ω),
qd b k-p σ q m t9j e g hj m4c i4jlk jh-j.q4gpjlhNprlq c jIw;u|jyc¢_or¢0| $]%] Ω >% >@% ; #" ( ;I9 9)) >% Ω1 ! Ω2 C;><'2>% ;#<'2)%, #% 2'! Γ = ∂Ω1 ∩ ∂Ω2 / (% >%*" "'( (@% >787 9 >7 2 66 C1 !% 2'* & @% #2 Ω1 ! Ω2 2'9 & ! 9 )% >'! 2 7 92 L2(Ω) #C>7 & ) C2# & <>C2 @% & 27 #! (% 2 >% 7 2 #% 2'! Γ / N -9%( )9%./9%65%;"( )'9* '9)+&-9! X, +9 +" = + ,)9% 4 9 $ R +, )
Lp(Ω)"
1 ≤ p ≤ +∞ 0 !" p 6= 254 +$-,
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∣
∣
∣
∫
Ω
v(x)∂φ
∂xi(x) dx
∣
∣
∣
∣
≤ C‖φ‖Lp′(Ω)
.M , 1
p+
1
p′= 1
1 < p ≤ +∞,
) (:J" > E/T0(E0E " 9 9 + -9% " V % ! 2 9 " -, $/ 54 9%(%9/9%( " $-Lp(Ω)
"Lp′
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, ' ($&%'$# % !$I$ g" ( §$I%'(9*)*,+ ) S "P-:>U
H1(Ω)Yh[Z!s!onporq<s ] \%]K^a`>bdcfe
Ω g h b g i4jlk e2m j RN w jlp~|q'@j m j `>b b ji H1(Ω) jlp e2m ' h c |Nq4kH1(Ω) =
v ∈ L2(Ω)e j gNj ∀ i ∈ 1, ..., N ∂v
∂xi∈ L2(Ω)
, FL_
b*) ∂v∂xi
jlp e q m k c i j|Nq4k e~c jlfj2rlq c lj m j v q4g pjhp m jq ' h cfeKcBb h+*_w-,+w 'wC M6K`4ERNWdjV6HU_V HC aXpGA_C6NWdjV6H[^] HpE,MLA0GIHf4Hy(U_<U_[WHHpE>O ERU_V60HCjOA_M6MFHY[PX|tY~!m6¢0|w Phs!|jy8A<or| ADVE,HYCLEU'& NP[;HYE,O¡GIU_CFE>OSNOSV6X³f4HpE\«U_CFGORNWU_CFEfl] XC6HYQR0NPHLCLNPH GD] HYE,O,Zcf4NWQRHf4HCLU_QSaH ‖u‖H1(Ω)FC6NPH¤6 vHYE«U_CFGORNWU_CFEf] XYC6HQS_NWH9FC6NPHbM<HV60HCjOX0HCjORV6HY[P[WHaHYC0O¦IOSQRH¨RE,NWC40V6[WNPYQRHpE©GIH djV6N9AV6CERHCLEM6K`ERNed0VLHMFUjERERNPL[PHrGU_CLGHCjORQ&AcOSNPU0CxU0V HI4M6[WU0ERNPU0Cx[WU4GAD[WH f4HGHQROSA_NPC6HpE_Q&ADCLf6HV6Q&ESI ;CGIU_CFE,V6[PORHYQSAA¤_HpGBNPCjORXYQR¦O[PHpEHI4HYaML[PHpEHI4M6[WNWGNOSHYEf6HT[] -4HQ&GINeGIH
FF µL /gHIO9f6V vHaaHg40/4 µ_µLbycqF~?q<tYornYorq<s ] \%] g h cm g |k b m g cfe p@qdq c k@j
〈u, v〉 =∫
Ω
(
u(x)v(x) +∇u(x) · ∇v(x))
dx FL ¯
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0 4#! n #! Q # 2# n G
j em jq9h b kut9j‖u‖H1(Ω) =
(∫
Ω
(
|u(x)|2 + |∇u(x)|2)
dx
)1/2
R jup~|Nq-j m j `>b b ji H1(Ω) jup e g hjup~|Nq-j m j c -jk e wYh+3´q<s'tYnycm4nporq<s#] D[ªHpE>OXNef4HCjOdjV6HFF ¯ HpE>O6NWHCV6C¡M6QSU4f4V6NPOESGA_[WA_NPQSH;fLADCLE
H1(Ω) D[QRHpE>OSHf6U_CLGZBaU_CjORQSHQdjV6H
H1(Ω)HpE>OGIU0aM6[PHOMFU0V6Q[eABC6U_QSaHA0ERERU4GINWXH0(U_NPO
(un)n≥1V6C6HE,V6NPORHBf4Hb®A_VLG&K`fLADCLE
H1(Ω)°-ADQ2f6XILC6NPORNWU_Cf4H9[eATC6U0QRaHBf4H
H1(Ω)(un)n≥1
ADNWCLERN<djV6H(∂un
∂xi)n≥1
MFU0V6Qi ∈ 1, ..., N E,U0C0O#f4HpETE,VLNOSHYETf4H¡®ADVFG&Kj` fLADCLE L2(Ω)
!®2U0aaHL2(Ω)HYE,OGIU0aM6[PHOY0NP[<HI4NeE>OSH;f4HpE[WNPaNPORHYE
uHIOwi
ORHY[P[WHYE2d0VLHun
GIU_C0HQS_H_HYQSEuHIO ∂un
∂xi
GU_C_HYQR0H_HQ&Ewi
f6ADCLEL2(Ω)
;QYMLA_Qf6XILC6NPORNWU_Cf4Hb[eAJf6XQSNP0XH;«rADNW6[PHTf4Hun6M<U_V6QORU0V4ORHB«U0CLGOSNPU0C
φ ∈ C∞c (Ω)
6U_CA∫
Ω
un(x)∂φ
∂xi(x) dx = −
∫
Ω
∂un
∂xi(x)φ(x) dx.
FL GD°§A0ERESADCjOZ[WA[WNPaNPORH
n→ +∞ f6ADCFETFL GDI6U_CU_6ORNWHCjO∫
Ω
u(x)∂φ
∂xi(x) dx = −
∫
Ω
wi(x)φ(x) dx,GIH9djV6NFM6QRU0V6_HdjV6HuHYE,O2f4XYQRNWcAD6[WHADV\ERHCLE«rADNW6[PH9HIO2djV6H
wiHYE,O?[WA
i aH9f4XYQRNW_XYHMLADQRORNWH[W[PH«rADNW6[PH\f4H
u ∂u
∂xi
'@9U0CLGDuA_M6MLADQRORNWHCjOb6NWHCZ
H1(Ω)HO
(un)n≥1GIU_C0HQS_Hg0HQ&E
ufLADCLE
H1(Ω)
9u|jyc¢_o¢_| $] \]B^ (% >% *" 7 # "'( " ! 9 C1 < 2)% & @% !2 ! !<<>@%>@%. ;:92 Ω
9 2<<'2)% ) > H1(Ω) /?C f4NPaHYCLE,NWU_CN ≥ 2
4[WHYE«U_CLGIORNWU_CLEf4HH1(Ω)
C6H#ERU_CjOHC_XYC6XQ&AD[ls!o¢_q<sªnYorslk!|ts'o?qLycs!h|jt06GIU0aaHb[WHbaU0CjORQSHB[WH#GIU0CjORQSHI HI4HaM6[WHBERV6NWcADCjOY9u|jyc¢_o¢_| $] \] B 7 2 >"7 ;" )% 9 RN / N = 2
9& "(%)% * 7 2 ! "u(x) = | log(|x|)|α 2<< 2)% ) H1(B) <>"% 0 < α < 1/2
9& 2 #6 < 2 >@%. ;# 2 '2 9) 7 @% / N ≥ 39 & (% >% *" 7 2 "'( u(x) = |x|−β 2<< 2)% )
H1(B) <>"% 0 < β < (N − 2)/29& 2 4 # <'2 >@% ;#:2 '2 :9=7 @% @ / 'A f4NWaHCLERNPU0C´fl] HYERMLA_GH
N = 1«rA_NOT¨>H4GHM4OSNPU0C©Z[WACLU_C4GIU0C0OSNPCV6NPORXf6HYET«U0CLGOSNPU0CLEf4H
H1(Ω)GU_aaH[] A4W\QSaH\[PH[WHaaHERV6NWcADCjOgU&v'ESADCLETM<HQRORHf4H_XYC6XQ&AD[WNPORX_vU_CM6QSHCLf
Ω = (0, 1)
;?|+3 3´| ] \%] \b g k eb g e j#r b h> e~cBb h v ∈ H1(0, 1) j e | b g k e b g e x, y ∈ [0, 1] o b hqv(y) = v(x) +
∫ y
x
v′(s) ds. FL
g+p [ h kvqdjt.jh e o | b g k e b g e x ∈ [0, 1] o R q-|| c @q eKcBb h v → v(x) o m ' h c j m j H1(0, 1)m q4hNp R o jlp e g hjr b kut.j c h q c kvj6 b h e~c hNgNjpug k H1(0, 1) w2h|qdk eKc g c jk o eb g e jr b h e~cBb hv ∈ H1(0, 1) jup e b h e~c hNgNj p-g k [0, 1] w
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# +Q7 " % %
Yh+3´q<s'tYnycm4nporq<s#]$(4U_NPOv ∈ H1(0, 1)
;Cf4XILCLNOV6CLHb«U_CLGIORNWU_Cw(x)
E,VLQ[0, 1]
MLA_Qw(x) =
∫ x
0
v′(s) ds.®2HO,OSH#f4XILCLNOSNPU0CAV6CERHCLEGYADQp4MLADQ[^] NWC6XY0AD[WNPORX#f4Hg®ADVFG&Kj`j!(4G&K A_QCJ0∣
∣
∣
∣
∫ x
0
v′(s) ds
∣
∣
∣
∣
≤ √x√
∫ x
0
|v′(s)|2ds ≤√
∫ 1
0
|v′(s)|2ds < +∞.?C«rA_NOp6[WHTa¦YaHTQSA_NWERU_C6CLHaHCjOaU0CjORQSHbdjV6Hb[eAJ«U_CLGIORNWU_C
wHYE,O9GIU_CjOSNPCV6HTERV6Q
[0, 1]
|w(x) − w(y)| =∣
∣
∣
∣
∫ x
y
v′(s) ds
∣
∣
∣
∣
≤√
|x− y|√
∫ x
y
|v′(s)|2ds ≤√
|x− y|√
∫ 1
0
|v′(s)|2ds. U_CjORQSU_CLEbdjV6H
wHYE,O#f4XQSNW¤A_6[WHA_VERHCFEB«rADNW6[PH\HO#d0VLH
w′ = v′$(4U_NPO
φ ∈ C∞c (0, 1)
lCCLUDOSA_CjOT[WH;OSQRNeADCL_[WH
T = (x, s) ∈ R2, 0 ≤ s ≤ x ≤ 1 4U0CA∫ 1
0
w(x)φ′(x) dx =
∫ 1
0
(∫ x
0
v′(s) ds
)
φ′(x) dx =
∫
T
v′(s)φ′(x) ds dx.°-A_QADM6M6[WNeGAcOSNPU0Cf6VOSK6XU0QRYaHBf4H LV66NWC6N^LU_CA∫
T
v′(s)φ′(x) ds dx =
∫ 1
0
(∫ 1
s
φ′(x) dx
)
v′(s) ds = −∫ 1
0
φ(s)v′(s) ds,HOMLADQ;®A_VLG&K`j (4G&K ADQKJU_CHCf4Xpf4V6NPO∣
∣
∣
∣
∫ 1
0
w(x)φ′(x) dx
∣
∣
∣
∣
≤ ‖v′‖L2(0,1)‖φ‖L2(0,1).@;U_CFGDwHYE,O6NWHCf4XQSNW¤A_6[WHbA_VE,HYCLE«rADNW6[WHBHOMLADQf6XILC6NPORNWU_Cf4H
w′ U_CA−∫ 1
0
w′(x)φ(x) dx =
∫ 1
0
w(x)φ′(x) dx = −∫ 1
0
φ(s)v′(s) ds,M<U_VLQORU0V4Oφ ∈ C∞
c (0, 1)LGHgdjV6NvNWaM6[PNedjV6HgdjV6H
w′ = v′F vH# HaaHFF /6 C6U0VLEf4NPO;A_[PU0QSEdjV6H
w − v HYE,O9XjAD[WHbZV6C6HgGU_CLE,OSA_C0OSHbM6QRHpERdjV6HTMFADQRORU_V6O9f6ADCLE (0, 1)FGIH#djV6NvXOSA_6[PNPOJ FL ± MFADQRORNWQf6HFL 2HOHCV4ORNW[WNWESADCjO;®ADVLG&K`j!(4G&K A_QCJ;U_CU_4OSNPHYC0O
|v(x)| ≤ |v(y)|+√
|y − x|√
∫ y
x
|v′(s)|2ds ≤ |v(y)|+√
∫ 1
0
|v′(s)|2ds,HOHCNPCjOSX_Q&ADCjOMLA_QQSA_M6MFU0Q,OZ
y
|v(x)| ≤∫ 1
0
|v(y)| dy +
√
∫ 1
0
|v′(s)|2ds
≤√
∫ 1
0
|v(y)|2dy +
√
∫ 1
0
|v′(s)|2ds ≤√
2‖v‖H1(0,1),GHTd0VLNM6QSU_V60HBdjV6Hv → v(x)
HYE,OV6C6HB«U_QSaHT[PNWC6XpADNWQRHbGIU0C0OSNPCV6HTERV6QH1(0, 1)
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0 4#! n #! Q # 2# n
4 B9 /9% Q996O",)9% "H1(0, 1)
2X, 9%P X+) ;)K 9 % > 9 .P, P+ "%,)9%9 .M , 65%&BQP" ()S68,9%( H1(0, 1)
1,)B99 S()68",)9%< +; +P )#$"%() Q"*"9 9Q"$-+9" C +9 @ % )"<,J"),))+%)N. "9G",)9 N "
v9-N,J$-) 68,9%(S "
H1(0, 1) %W S ")+9"%9N(
,)9%"%&9 v F0 1"I9 + 9<, ):-9 I$/ "RP%6O"7+<- )3QPW4 :68",)9% "
H1(0, 1))2I) 65-%7":, (++ ":68,9%(I%9QG4 B "), %( "< W4 %( " 9%&#$7 "R!68,9%( M-
9 9Q $-+9" C -"+9 %&7 "R 9)"9 + -P9 4 ,) -++U " 68,9%( +)MR"9)9QP"@$-+9" F0 -, :,)$ %&9%( P@9 +&-3 "G) D0 0 %&3+<, 9 $ P9 "" (3+)QG4 %")R%293S9 "9)+ ":(',) -++< ":6O,)9%
v ∈ H1(0, 1)Q%P 2, 9%(P 0
•D[bHpE>OORQSYENPaM<U_QROSADCjO¡HYC M6Q&AcOSNWdjV6Hf4HESA¤_U0NPQERNbr|t 1^q<s!¢DnYoqFs!txy¤h+A<k'oB:jyc|txtpqFsªnw'|s!tY|t³w!m6s!t P|tY~!mL¢_|¥w'| vq 2qF|0z
H1(Ω)?®2HY[WA A>VFE>OSNLHHC MLADQRORNWH¡[eA C6U_ORNWU_C´fl] HYE,MLA_GHf4H9(U0FU0[PHY d0VLNADM6MLA_QSA4VWObADNWCLERN'OSQRpEbE,NWaM6[PHYaHCjO#GIU_aaH[] HCLERHaJ6[WHf4HpE;«U_CLGIORNWU_CLEQRXY_V6[WNPYQRHpEGIU0aML[PXORX#MLA_Q[WHYE[WNPaNPORHpE9f4HgERV6NPORHpE9f4HT«U0CLGOSNPU0CLEQSX0V6[PNWQSHYEfLADCLE[eA\C6U0QRaH#f4H[] XC6HYQR0NPH ‖u‖H1(Ω)
-®2HY[WA³MFHYQRaHOf4Hf4XaU0C0OSQRHYQ#«rA0GINW[PHYaHYCjOf4HC6U_aJ6QSHVLERHYEgM6QRU0M6QSNPXORXYEHC[PHpEbXIO&AD6[WNWESESADCjO#fl] A_FU0QSfERV6QT[PHpEb«U_CFGORNWU_CFEbQRXY_V6[WNPYQRHpEBM6V6NeETHCV4ORNW[WNWESADCjO#V6CADQS_VLaHYCjOf4H¨Rf4HCFE,NPORXI©´r_U0NPQMLADQHI4HaM6[WH³[WHYEf4XaU_CFE>OSQSADORNWU_CLEf6HYE0K6XU0QRYaHpE FF µLPpµHIO FF µLP¤GINP f4HpERERU_VFESI £'hqFy4:[3 | $] \] >w!| w!|s'tponph " `_c
Ω jlp e g h b g i4jlk e b kuh k # g c jk m j q4p@pj C1 o b g c jh p c Ω = RN
+o b g jh b k@jsp c Ω = RN o q4 b k-p C∞
c (Ω) jlp e2m jlhNplj m qdhp H1(Ω) w vAf4XYaU0CLE,ORQ&AcORNWU_Cf4V 0K6XYU_QSaH FL µ6djV] U0CMFHYV4OJA_f6aHO,OSQRH\HYCMLQRHYaNWQSH[PHpGOSV6QRH¤bE,HORQSU_V60HZ[WA (HpGORNWU_C FL FLQXA_M6M<H[WU_CLEbdjV6H[WAC6U_OSADORNWU_CRN
+
f6XYERNP0C6HJ[WH\f4HaNP^HpE,MFA_GIH x ∈RN ORHY[d0VLH
xN > 0 .¡|+3´m4ycilk!| ] \%] ] HYERMLA0GIH
C∞c (Ω)
djV6NªHYE,Of6HCLERHBfLADCLEH1(Ω)
HYE,OGIU_CFE>OSNOSV6XTf4HYE2«U_CFGORNWU_CLE\QSX0V6[WNPYQRHpEf6H GI[eA_ESE,HC∞ ZE,VLM6MFU0Q,O<U_QSC6XU0VGIU_aMLA0GOf6ADCLE[PH«HYQRaX
ΩCMLADQRORNeGIV6[WNWHQpjERN
ΩHYE,OFU0QRC6X0jORU_V6ORHYE[WHYE2«U0CLGOSNPU0CLEf4H
C∞(Ω)U_CjOCLXYGIHpERESADNWQSHaHCjO2VLCERV6M4MFU0Q,OFU0QRC6X02HOf4U0CLG
C∞c (Ω) = C∞(Ω)
°?QRXpGINeE,U0CLE¡djV6H[PHpE«U0CLGOSNPU0CLE¡f4HC∞
c (Ω)C6HE] ADC6CV6[WHCjOMLA0EC6XYGHYESERA_NPQSHaHYC0OERV6Q[WHTFU0QSff4Hg[] U_V60HQRO
ΩFGIH#djV6N'f4NDCªXQSHCFGINWH#GIHIO;HYERMLA_GHf4H
C∞c (Ω)
r_U0NPQ2[WA XHa\A_QSdjV6H9µL /6 /0j°-ADQGIU_CjOSQRH0jERNΩC] HpE>OMLA0E?<U_QSC6X_j[WHYE2«U_CLGIORNWU_CLEf4H
C∞c (Ω)
EY] A_C6CV6[PHYCjO¨RZ[] NWC4LC6N©& •.¡|+3´m4ycilk!| ] \%] vAC6U_ORNWU_Cf4H\QSX0V6[WA_QRNPORXfl] V6CU_V60HQRO#AXORX\NWCjORQSUf6V6NOSHf6ADCFEb[WA@;XILC6NPORNWU_C´µ60/44 D[?C] HYE,OgMLA0ETC6XYGHYESERA_NPQSH\f4H¡GU_C6CLA4VWOSQRH\MLQRXpGINeE,XYaHYCjOgGIHIORORH¡f6XILC6NPORNWU_C´f6H[WAQRXY_V6[eADQSNOSXgfl] V6C³U0V6_HYQ,Op D[§ERV WObf4HESA¤_U_NWQ k b p-p b t bIm'bJdjV6HJ[] U_Cf4Ha\A_CLf4HJdjV6H[WHg<U_Q&ff4H9[^] U0V6_HYQ,O?ERU_NPO?V6CLH;ERV6Q,«rA0GIHQSX0V6[PNWQSHHIOdjV6H[^] U0CHI6GI[WV4O2GHQROSA_NPC6HpE'¨,MLAcOSK6U_[WU_0NPHpE©T0U_NWQ[WAXHa\ADQ&djV6Hµ60/4 ¯ ID U_Q&ESd0VLHCLU_VLE§XYC6U_CGU_CLE§VLCQSXYERV6[POSADO?E,U0VLE§V6C6HK`MFU_ORK6pE,Hf4H9QRXY_V6[eADQSNOSXf4HT[] U_VL_HQROY6GHIORORHTQSX_VL[WA_QRNPORXBHYE,OORU0V'A>U_V6Q&EC6XpGIHYESESADNWQRH[WHTQRXpE,VL[O&AcOORU0ag<HTHCf4XI«rA_V4OM<U_V6QGIHQROSA_NPCFE-U_V60HQROSE§C6U0C\QRXY_V6[WNWHQ&Ec0U_NWQ[PH;GIU_CjOSQRH^HHYaM6[PHf4H9[] -4HQ&GINeGIHFF µL µjD9XYADCLaU0NPCFE[] K`jM<UDOSK6YERHTf4HTQSX_VL[WA_QRNPORXTERV6NPcA_C0O[eA\@9XLC6NPORNWU_Cµ60/4JMFHYV4O;ERU_V60HCjO¦IORQSH#A4ClADNW6[WNPH H6[^] ADM4MLADQRORHYCLADCLGHJZ¡[WAGI[eA_ESE,Hf4HYE;«U_CFGORNWU_CFE
C1 M<HV4Ob¦IOSQRHQRHYaM6[WA0GIXYH#MLADQB[^] A_M6MLADQRORHYCLADCFGIHJZ¡[WAGI[eA_ESE,H\f4HpEb«U_CLGIORNWU_CLE#[PNWMLESG&K6NOKJNWHC6CLHYE\0U_NWQ9FP4GIH$*NWHCdjV6HGIHpE#f4XOSADNW[eETf4XMLA0ERERHCjO#[eADQS_HaHYCjO[PH#GYA_f4QSHTf4H#GIHTGIU0V6Q&E4C6U0VLE«rA_NWERU_CLEGHIO,OSH#QSHa\ADQ&djV6HBADLCdjV6Hb[PHT[WHYGIORHVLQMFU0NPCjOSNP[W[PHYV4
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
E # +Q7 " % %
CLH9E] NWCLE,VLQR0HMFA_E-[WU_Q&ERdjV6HC6U0VLE-V4OSNP[WNeE,HYQRU0CLEf4HOSH[eE-QSXYERV6[O&AcO&EU'&\[] K`jM<UDOSK6YERHf4HQSX0V6[WA_QRNPORXHpE>OBC6XYGHYESERA_NPQSHpf6ADCLE;[WHJGA0E9fl] U_VL_HQROSE¨,A¤0HYG#f6HYEBGIU_NWCLE>©gd0VLN!A_M6MLADQ&ADNeESE,HYC0OCFAcORVLQRHY[P[WHaHCjOfLADCLE!OSU_VLE-[WHYE?GAD[eGIVL[WE§CV6aXQSNed0VLHYE0U_NWQ-[PHpE-f4NC<XYQRHYC0OSHYE-NWa\AD0HYE-f4Ha\ADNW[W[WA__HYE§djV6N6NW[W[PVLE,ORQSHCjOGHTGIU_VLQSE& •
9*)*KJ ) S "P-:>UH1
0(Ω)@;XILCLNWESE,U0CLEa\ADNWC0OSHCLA_CjOV6C ADV6ORQSHHpE,MLA0GIHf4H (4U_<U_[WH djV6NTHYE,OV6C ERU_VFE> HYERMLA_GH³f4H
H1(Ω)HIO#djV6N§C6U0VLEbE,HYQSAORQSYEBV4ORNW[WHM<U_VLQB[WHYEBM6QSU_L[PYaHpE;A¤0HYGJGIU_CFf4NOSNPU0CLEbADV4 [WNWaNPORHpETf4H@;NWQRNeG&K6[WHIOY
Yh[Z!s!onporq<s ] \%] `>bdcfeC∞
c (Ω) R jlp~|q'-j m jlpQr b h eKcBb hp m jnq4p@pj C∞ $ p-g |'| b k e b t|Nq em q4hNp Ω w jlpK|Nq-j m j `Qb b ji H10 (Ω) jlp e.m' ' h c b t t.jR q m kvjh-j m j C∞
c (Ω)m q4hp
H1(Ω) w;C0HQSQSAbVLC¡M<HVM6[WVLE2[PU0NPC_U0NPQ2[PHT®2U_QSU_[W[WA_NPQSH FL µ6W ¯ -djV6H
H10 (Ω)
HYE,O2HC¡«rADNPO2[WH;ERU_VLE,HpE,MFA_GIHTf4HH1(Ω)
GIU_CFE>OSNOSV6X#f4HYE1^q<s!¢DnYoqFs!tilk!ot Wm6s'slk!r|sªntpkvyr| ?qFycw∂Ω
M6VLNWESdjV6HOSH[6HpE>O[WHGA0E-f4HYE-«U_CLGIORNWU_CLEf6HC∞
c (Ω)0C\_XYC6XQ&AD[^
H10 (Ω)
HYE,O?tnYycor¢Dnp|+3´|sªn9~§rk!t;~2|0npondjV6HH1(Ω)
GADQC∞
c (Ω)HpE>O2V6CE,U0VLE,^HpE,MLA0GIHtYnycor¢_n9f4H
C∞c (Ω)
0U_NWQ[WH 0K6XU0QRYaH FL µ6BHIO[eAsXHa\A_QSdjV6HFL µ6 ¯ IdC6HH6GIHM6ORNWU_CNPaM<U_QROSA_C0OSHHYE,O[PHGYA_E§U&Ω = RN H_HCHlCªHIOp0f6A_CLE-GIHGYA_E
Ω = RN = ΩHIO9[WH 0K6XU0QRYaH FF µL AdWQSaH#djV6H
C∞c (RN )
HpE>O9f6HCLERH#f6ADCFEH1(RN )
f6U_CLG?U_C#AH1
0 (RN ) = H1(RN )D®2HO,OSH?HI6GHM4OSNPU0CgE,HGIU0aMLQRHYCLfTADNeE,XYaHCjO'M6VLNWESdjV6H?[^] HpE,MFA_GIHHYCjORNWHQ
RN C] AMLA_Ef6HTFU0QSflbycqF~?q<tYornYorq<s ] \%] g h c.m g |k b m g cfe pl-qdq c kvjz *w@ m j H1(Ω) o v jup~|Nq-j m j `>b b jiH1
0 (Ω) jup e g h jup~|Nq-j m j c -jk e wYh+3´q<s'tYnycm4nporq<s#]-°-ADQBf4XIFC6NOSNPU0C
H10 (Ω)
HYE,OBV6CE,U0VLE,^HpE,MLA0GIHb«HYQRaXJf6HH1(Ω)
rdjV6N§HYE,OVLCHpE,MFA_GIHTf4H 9NP[WFHYQ,O<f4U_CLGbG_] HpE>OADVFERERNlV6CHYERMLA_GHTf4H NW[P<HQROY9CQSXYERV6[POSAcOHYESE,HYCjORNWH[ªMFU0V6Q[PHpEA_M6M6[WNWGYAcOSNPU0CLE2f6V¡M6QSU4G&KLADNWC¡G&KLA_M6NOSQRH;HYE,O[^] NPC6XY0A_[PNPORXBE,V6NPcA_C0OSH_
bycqF~?q<tYornYorq<s ] \%]K^ Is!h+AFmLronphw'| ;qFos'¢0m4ych&" `>b4cfeΩ g h b g i4jk em j RN b kuh 7m q4hNp
qdgt b4c hp"g hj m4c kvj- eKcBb h m jnR jlp~|q'@j w Qj - c p e j"g h>jn b hp e qdh e j C > 0e jlfj gNj o | b g k e b g e j
r b h eKcBb h v ∈ H10 (Ω) o
∫
Ω
|v(x)|2dx ≤ C∫
Ω
|∇v(x)|2dx. FL 0Yh+3´q<s'tYnycm4nporq<s#]°§U_V6QJ[PHpET«U0CLGOSNPU0CLE
v ∈ C∞c (Ω)
U_C´A f4X6A,Z f4XYaU0CjORQSX[] NWC6XjAD[WNOSX¡f4H°§U0NPCLGYADQSXgFF jf6A_CLE[WH; vHaaHBµL µL ¯ D°-ADQV6C¡A_QR0V6aHCjO2f4HBf4HCLERNPORXB[PH;QRXpE,V6[POSADO?QSHYE,ORH9Q&ADNM<U_VLQORU0V4ORHg«U_CLGIORNWU_Cv ∈ H1
0 (Ω)ªC HlCªHIOp<GU_aaH
C∞c (Ω)
HpE>O;f6HCLERHJf6A_CLEH1
0 (Ω)MLA_QESA@9XLC6NPORNWU_C FL µ6 I4NP[lH4NWE,ORHTV6CLH#E,V6NPORH
vn ∈ C∞c (Ω)
ORHY[P[WH#djV6Hlim
n→+∞‖v − vn‖2H1(Ω) = lim
n→+∞
∫
Ω
(
|v − vn|2 + |∇(v − vn)|2)
dx = 0.
Ecol
e Po
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pas
repr
odui
re.
0 4#! n #! Q # 2# n LCMLADQRORNeGIV6[WNWHQp4U_CHCf4XYf6V6NO9djV6H
limn→+∞
∫
Ω
|vn|2 dx =
∫
Ω
|v|2 dx HIO limn→+∞
∫
Ω
|∇vn|2 dx =
∫
Ω
|∇v|2 dx.
°§A_QADM6ML[PNeGADORNWU_Cf4V HaaHTµ6 µ6 ¯ 4U0CA∫
Ω
|vn(x)|2dx ≤ C∫
Ω
|∇vn(x)|2dx. FLW E ;C¡MLA0ERERH;A_[PU0QSE2Zg[eAg[WNWaNPORH
n→ +∞ fLADCLEG&KLA0GIV6C¡f6HYEf4HV6\ORHQSaHYEf4HB[] NWC6XjAD[WNOSX FLW E MFU0V6Q?U04ORHYC6NPQ?[PHQSXYERV6[POSADOQRHpG&K6HQ&G&K6X_®2HO>`jM<H9fl] ADQS_VLaHYCjO'¨>MLA_Q?f4HYCLE,NPORXI©9E,HYQSABOSQRpEE,U0V6_HYCjOQRHYM6QRNeEMLADQ[eAE,VLNOSH_
.¡|+3´m4ycilk!| ] \%]K^_^¥ ] NPC6XY0A_[PNPORXf4H;°§U_NWCLGA_QRXTFF j§C] HYE,O?MLA0EQSA_NPHM<U_VLQ[PHpE-«U0CLGOSNPU0CLE?f4HH1(Ω)
DCHlCªHIOpc[PHpE!«U_CLGIORNWU_CLE-GU_CLE,OSA_CjORHYErC6U_CCV6[P[WHYE&!ADC6CV6[WHCjO-[WH2OSHQSaHf6Hf6QRU0NOSHf6A_CLEFL 0'a\ADNeE§MLA0E'[WHORHYQRaHf4H0A_VLG&K6H0¤ ?] K`M<UDORKLYERHERU_VFE>A,A_GHCjORHHYESE,HYCjORNWH[W[PHf6A_CLE![] NPCLX0A_[PNPORXf4HJ°§U0NPCLGYADQSXgHpE>O#djV6H[PHpE9«U0CLGOSNPU0CLEbf4HH1
0 (Ω)EY] A_C6CV6[WHCjOBERV6Qb[PH<U_Q&f
∂Ωf4H[] U_VL_HQRO
Ω_U0NPQ[eA9XHa\A_QSdjV6H FL µ6W MFU0V6Qf4HpEcADQSNWA_C0OSHYEf4H#GHIO,OSHbKj`M<UDOSK6YERHpI •9CGU_QSU_[W[WA_NPQSHNWaMFU0Q,O&ADCjO!f4H[^] NWC6XY0AD[WNPORX2f6H2°§U_NWCLGYADQSX?HYE,O§[PHQRXpE,VL[O&AcO-E,V6NWcADCjO-d0VLN0«U0V6QSC6NOV6C6HTC6U0QRaHbXYdjV6NWcAD[WHCjORHbM6[WVLEERNPaM6[WH#f6ADCFE
H10 (Ω)
qFycqFrm6oryc| $] \]B^& `>b4cfe
Ω g h b g i4jlk e7m j RN b kuh m q4hNp6q4g t b4c hNpg hj m4c kvj- eKcBb h m jR jup~|Nq-j w %" b k-pnqpjlt c h b kut.j
|v|H10 (Ω) =
(∫
Ω
|∇v(x)|2dx)1/2
jup e g h>j h b kut9j pug k H10 (Ω)
g c i8qdjlh e j $9 q9h b kut.jEg+pugNjlf j c h m g cfe j|Nq4kE-jfj m j H1(Ω) wYh+3´q<s'tYnycm4nporq<s#](U_NPO
v ∈ H10 (Ω)
L 'AJMLQRHYaNWQSHbNPCLX0A_[PNPORX|v|H1
0 (Ω) ≤ ‖v‖H1(Ω) =
(∫
Ω
(
|v|2 + |∇v|2)
dx
)1/2
HYE,OXNWf6HCjORH06@g] ADV4OSQRHbMLA_Q,Op6[^] NWC6XjAD[WNOSXTf4Hb°!U0NPCFGADQSXTf4V HYaaHTµ6 µ6 ¯ GIU_CFf4V6NPO9Z‖v‖2H1(Ω) ≤ (C + 1)
∫
Ω
|∇v|2 dx = (C + 1)|v|2H10 (Ω),
GIHTdjV6NM6QSU_V60HBdjV6H |v|H10 (Ω)
HYE,OV6C6HTC6U0QRaHbXYdjV6NWcAD[WHCjORHTZ ‖v‖H1(Ω)
Ecol
e Po
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Ne
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repr
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re.
[/ # +Q7 " % %
9*)*) 3P-:>US Ul1 [S5?3NM 8ORUS7U 3U%Uv/9U_VLE!A¤_U0CLEªVgdjV] HYC#f4NWaHYCLERNPU0C
N ≥ 2[WHYEl«U0CLGOSNPU0CLE!f4H
H1(Ω)CLH2ERU_CjOvMLA0EvGIU0CjORNWCjVLHYEr_U0NPQb[WHGU_CjORQSHI HI4HYaML[PHf4H[] -4HQ&GINeGIH FF µL /0v®2U_aaHMFU0V6QBORU_V6ORH«U_CFGORNWU_CaHYERV6Q&AD6[WH_U0C\C6HM<HV4Of4U_CLGMLADQS[PHYQ?f4H[eAbcAD[WHV6Q2MFU0CLGOSV6H[W[WH9fl] V6C6H«U0CLGOSNPU0C
v ∈ H1(Ω)d0VLH2¨>MLQRHpERdjV6HMFADQRORU_V6O>©fLADCLE
Ω0C\MLA_Q,OSNWGV6[PNWHQpDNW[LC] HYE,OMLA0EGI[eADNWQ?f4HESA¤_U0NPQ-ERNLU_CM<HV4Of4XILCLNPQ2[WA¨,cAD[WHV6QA_V<U_Q&fj©lU_V¨OSQSA0GIHI©f4H
vERV6QT[PH<U_Q&f
∂ΩGADQ
∂ΩHpE>O#V6CHCFE,HYag6[WH\C6X0[PNW_HpAD6[WHU_Vf4HaHpE,V6QSH9CV6[W[WH_ 6U_QRO2KLHV6QSHVLERHaHYC0OMFU0V6Q[PHpE?M6QSU_L[PYaHpE2A_V4[PNWaNOSHYEdjV6H9C6U0VLE2XIORVFf4NPU0CLEYjNW[`AORU_V6Obf4HJa¦aHJVLC aUc`_HC³MFU0V6QBf6XILC6NWQB[WAORQ&A_GIHv|∂Ω
fl] V6C6HJ«U_CLGIORNWU_C³f4HH1(Ω)
v®2HQSXYERV6[POSADOHpERERHCjOSNPHY[6A_M6MFHY[PXbOSK6XU0QRYaHBf4HbOSQSA0GIH04HYE,O[PHTERV6NPcA_C0Op £'hqFy8:+3´| $] \]B^ \ ,w'|nYycm6¢_|&" `>bdcfe
Ω g h b g i8jk e b kuh k # g c jk m j7q8p-pj C1 o b g c jhΩ = RN
+ w2xh m' ' h cfe v q@||> c -q e~cBb h e kvq-j γ0
H1(Ω) ∩ C(Ω) → L2(∂Ω) ∩ C(∂Ω)v → γ0(v) = v|∂Ω .
FLW_p] j eKe jq@||> c -q e~cBb h γ0 pjs|>k b b h j7|qdk) b h e~c hNg cfe jh g h>j)q-|| c @q eKcBb h c h q c k@j b h e~c hNgNjm j H1(Ω)
m q4hp L2(∂Ω) o h bde jsjh b kvj γ0 w 2h7|Nq4k e~c g c jlk o c j- c p e j"g hjn b hNp e q4h e j C > 0e jfj gNj o | b g k eb g e j#r b h e~cBb h v ∈ H1(Ω) o b h q‖v‖L2(∂Ω) ≤ C‖v‖H1(Ω).
FLW /_.|+3´m4ycilk'| $] \]B^ BQ-,_GHA_V 0KLXU_QSaHf4HORQ&A_GH FF µLPpµ U_C¥M<HV4O¡f4U0CLGMLADQS[PHYQ\f4H[WAcA_[PHYV6QJfl] V6C6H¡«U_CLGIORNWU_C´f4H
H1(Ω)ERV6Qg[WH¡<U_Q&f
∂Ω-®2H¡QSXYERV6[POSAcOJHYE,OgQRHYa\ADQ&d0VFAD6[WHGA_QgNW[Cv] HpE>OMLA0EQSA_N<M<U_VLQV6C6Hb«U_CFGORNWU_Cf4H
L2(Ω)r_U_NWQHCMLA_Q,OSNWGV6[PNWHQ[^] -4HQ&GINeGIH FL µ6 FjI •
Yh+3´q<s'tYnycm4nporq<s#]@9XYaU0CjORQSU_CLE[WHQSXYERV6[POSAcOM<U_V6Q[WHf4HYaN HYERMLA_GHΩ = RN
+ = x ∈RN , xN > 0 N(U0NO v ∈ C∞
c (RN+ )L±'_HpG;[eAC6UDO&AcOSNPU0C
x = (x′, xN )6U0CA
|v(x′, 0)|2 = −2
∫ +∞
0
v(x′, xN )∂v
∂xN(x′, xN ) dxN ,
HOY6HYCV6ORNW[PNeERA_CjO[] NPCLX0A_[PNPORX2ab ≤ a2 + b2
|v(x′, 0)|2 ≤
∫ +∞
0
(
|v(x′, xN )|2 +
∣
∣
∣
∣
∂v
∂xN(x′, xN )
∣
∣
∣
∣
2)
dxN .
°-A_QNPCjORXY_Q&AcOSNPU0C¡HCx′6U0CHYCf4XYf4VLNO
∫
RN−1
|v(x′, 0)|2 dx′ ≤∫
RN+
(
|v(x)|2 +
∣
∣
∣
∣
∂v
∂xN(x)
∣
∣
∣
∣
2)
dx,
G_] HpE>OR ZD f6NPQSH ‖v‖L2(∂RN+
) ≤ ‖v‖H1(RN+
)
l°-A_QBf4HYCLERNOSXf6HC∞
c (RN+ )
f6A_CLEH1(RN
+ )ªU_CU_4OSNPHYCjO9ADNWCLE,Nl[WHTQSXYERV6[O&AcOp
Ecol
e Po
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pas
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re.
0 4#! n #! Q # 2# n 0µ°§U0V6QV6C U0V6_HYQ,OFU0QRCLXQSX0V6[PNWHQf4HGI[eA_ESE,H C1 U_C V4ORNW[WNWERHV6C A_QR0V6aHCjOf6HGYADQRORHYE[PU4GA_[PHpEf4VFU0QSfdjV6NM<HQSaHIO;f6H#E,HTQ&ADaHCLHQADVGA0Ef4H
Ω = RN+
L9U_VLEC6H#f4XIO&ADNW[P[WU_CFEMLA_EGIHIOADQS_V6aHCjOf4H9GYADQRORHpE-[PU4GYAD[WHYErA0ERERH#JOSHYG&K6CLNWdjV6H¤-djV6NLHpE>O[WHa¦aH9djV6H9GH[WV6N6V4OSNP[WNeE,X9fLADCLE[WAf4XYaU0CLE,ORQ&AcORNWU_Cf4Hb[eAJ°?QRU0MFUjE,NPORNWU_CIFL FL0/GINP f4HpERERU_VFE vH 0K6XU0QRYaHf6HORQ&A_GHIFF µLPpµ³M<HQSaHIO¡f4H0XC6XYQSA_[PNeERHQ\ADV4x«U_CLGIORNWU_CLE\f6H
H1(Ω)[WA«U_QSagV6[WH9f4H BQSHHYC\M6QSXYGIXpf4HaaHCjO?XOSA_6[PNWHM<U_V6Qf4HYE«U_CLGIORNWU_CLE2f4H9G[WA0ERERH
C1 A_V®2U_QSU_[W[WA_NPQSHµ60/4 µ6 £'hqFy4:[3 | $] \]B^ ! vqFy43k!r|w!|5yc||js "y`>b4cfe
Ω g h b g i4jlk e b kuh k # g c jlk m jq8p-pjC1 w `_c u j e v p b h em jupr b h eKcBb hNp m j H1(Ω) o jfjlp7i k c ' jh e
∫
Ω
u(x)∂v
∂xi(x) dx = −
∫
Ω
v(x)∂u
∂xi(x) dx +
∫
∂Ω
u(x)v(x)ni(x) ds, FLWYµ0
b)n = (ni)1≤i≤N jlp e qh b kut9q4 j7g h cfe j - e k c jg kvj $ ∂Ω w
Yh+3´q<s'tYnycm4nporq<s#]XADMLMFHY[PU0CLEdjV6H[eAB«U_QSagV6[WH#FFPpµ0§AbXORXXOSADL[PNWHM<U_V6Q?f6HYE§«U0CLGOSNPU0CLE?f4HGI[eA_ESE,HC1 f6A_CLE[WHJ®2U_QSU_[W[WA_NPQSH;µL /6 µL ;CV4ORNW[PNeERHgZC6U_VL_HYA_V¡V6C A_QR0V6aHCjOf4H#f4HYCLE,NPORX0L°-ADQf4HCLERNPORXJf6H
C∞c (Ω)
f6A_CLEH1(Ω)
0U_NWQ9[WH 0K6XYU_QSaHFF µL 0<NP[!HI4NeE>OSHJf4HpE;ERV6NPORHYE(un)n≥1HIO
(vn)n≥1f6ADCFE
C∞c (Ω)
djV6N-GIU0Cj0HQS_HYC0O9fLADCLEH1(Ω)
0HQ&EuHIOvªQSHYERMFHpGOSNP0HaHCjOY<C_HQRORVf4V³®2U_QSU_[W[eADNWQRHBµ60/4 µgU0CA
∫
Ω
un∂vn
∂xidx = −
∫
Ω
vn∂un
∂xidx+
∫
∂Ω
unvnni ds. FLW#Fj
;C M<HV4OMLA0ERERHQZ[WA[WNPaNPORHn → +∞ fLADCLE[WHYE\f4HYV4´M6QSHaNPHYQSEJOSHQSaHpEf4HFLW#FgGYADQ
unHIO ∂un
∂xi
rQRHpE,M<HYGIORNW_HaHYC0Opvn
HO ∂vn
∂xi
gGIU0Cj0HQS_HYC0OT0HQ&EuHIO ∂u
∂xi
rQRHpE,M<HYGIORNW_HaHYC0OpvHIO ∂v
∂xi
bf6ADCLEL2(Ω)
v°§U_V6QbMLA0ERERHQ#Z[eA[PNWaNOSHf6ADCLEb[eAf4HQSC6NWQSHNPCjORXY_Q&AD[WHf6H FLW#FlU_CV4ORNW[PNeERH[WAgGIU_CjOSNPCV6NPORXf4H9[] ADM6ML[PNeGADORNWU_CORQ&A_GHγ00G_] HpE>OR ZD f4NWQSH[^] NPC6XY0A_[PNPORXg FLW /0_djV6NFMFHYQRaHIOfl] AdW\QRaHQTdjV6H
γ0(un)QSHYERMFHpGOSNP0HaHCjOY
γ0(vn)BGIU0C_HQS_HJ_HYQSE
γ0(u)rQRHpE,M<HYGIORNW_HYaHYCjOY
γ0(v)Bf6ADCLE
L2(∂Ω) ;C U04ORNWHCjObADNWCLE,N§[eA\«U0QRaJV6[PHFFPpµ0M<U_V6QTf4HpE«U0CLGOSNPU0CLE
uHOvf4H
H1(Ω)
®2U0aaHGIU0CLE,XpdjV6HCLGHgf4V 0KLXU_QSaH#f4HgORQ&A_GHFL µ6WYµU0CU_4OSNPHYC0OBV6C6HGADQ&A_GIORXYQRNeERADORNWU_CORQSYEERNPaM6[WH#f4Hb[] HYERMLA0GIHH1
0 (Ω)
qFycqFrm6oryc| $] \]B^ `>b4cfeΩ g h b g i8jk e b kuh k # g c jk m jyl q8p-pj C1 w jup~|Nq-j H1
0 (Ω) b h cBm j qdi8j-)j)p b g+p jup~|Nq-j m j H1(Ω) b hNp e~cfe g )m jlpr b h> e~cBb hNp g c p qdhNhNg jlh e pug k.j b k m ∂Ω wYh+3´q<s'tYnycm4nporq<s#]ª®2U_aaHORU_V6ORH«U_CFGORNWU_Cf4H
H10 (Ω)
HpE>O[WNPaNPORHfl] V6C6HERV6NPORHf4H«U0CLGOSNPU0CLEADM6MLA_Q,OSHCLA_CjOZC∞
c (Ω)djV6N'U_CjO6NWHC ER6QV6C6H#ORQ&A_GIHTCV6[W[WH_L[eA\GIU0CjORNWCjVLNOSXgf4H#[^] A_M6M6[WNWGYAcOSNPU0CORQ&A_GIH
γ0NWaM6[PNedjV6H;djV6H[eAbOSQSA0GIHf4H[eAT[WNPaNPORHHpE>OADVLESE,NLCjVL[P[WH_ ;C\HCf4Xpf4V6NPOdjV6H
H10 (Ω)
HpE>OGIU_CjOSHCVf6ADCFE[WH9E,U0VLE,^HpE,MLA0GIHf4HH1(Ω)
f4HYE«U_CLGIORNWU_CLE2djV6NFEY] A_C6CV6[WHCjO2ERV6Q?[WH<U_Q&f∂Ω
j vAQRXpGINWM6QRU4djV6HbHYE,OM6[WVLEOSHYG&K6C6NedjV6HTHIO9f6XYGIU0V6[WH#fl] VLCf6U_V66[WHTM6QSU4GIXYf6X#f4H#GA_Q,OSHYE[WUGYAD[WHYETr_U_NWQSH[WAJM6QSHV60HBf4Hb[eAg°?QRU0MFUjE,NPORNWU_C FL FL0/_?M6V6NeEf6HBQSX0V6[eADQSNWESAcOSNPU0C\HIOf4HBORQ&ADCLER[eAcORNWU_CrERNPaNW[WA_NPQSHBZ[WAM6QSHV60HTf4V 0K6XU0QRYaH FL FLWpI6U0VLEQRHYC_Uc`_U0CLEZ6F GIH %F µ0 HvM<U_VLQM6[PVFEf4HTf4XIO&ADNW[WEY
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
)F # +Q7 " % %
.|+3´m4ycilk'| $] \]B^ vH;®2U_QSU_[W[WA_NPQSH5FF µLP ¯ AdW\QRaHdjV6H[PHC6Uc`jADVf4H[^] A_M6M6[WNWGYAcORNWU_CJORQ&A_GHγ0HpE>OTM6QSXYGNWERXaHYC0O
H10 (Ω)
%9C6H\d0VLHYE,ORNWU_C³CFAcORVLQRHY[P[WH_ªa\ADNeEbFHpADVLGU_V6MM6[WVLETf4X[WNeGAcOSH_ªHYE,O#f4HGYADQ&A_GIORXQSNeE,HYQ?[^] NWa\AD_Hbf4Hγ0F®2U_CjOSHCjORU0CLE?CLU_VLEf4Hbf4NWQRHTdjV6HbGIHO,ORHbNWa\AD_H
Im(γ0)C] HYE,OMFA_E
L2(∂Ω)Fa\ADNeEVLCERU_VLE,^HpE,MFA_GIH#E,ORQSNeGOY<f4HCFE,Hf6ADCLE
L2(∂Ω)lGIU0CLE,ORNPORV6Xf4H#«U0CLGOSNPU0CLE2¨,M6[WVLEQSX0V6[WNPYQRHpE©0HOC6UDOSX
H1/2(∂Ω)6°§U0V6QM6[WVLEf4H#f4XOSADNW[eE4C6U0VLEQSHC_Uc`0U_CLE2ZF µ E H^ •
BQ-,_GH¡ADVx®2U0QRU0[P[eADNWQRH FL µ6W ¯ CLU_VLETM<U_VL_U_CFEgf4U0C6C6HQgV6C6H¡A_V4ORQSH¡f4XYaU0CLE,ORQ&AcORNWU_Cf4H¡[WA°?QRU0MFUjE,NPORNWU_C FL µ6W E ZM6QSU_M<U0E9f4HT[] NWC6XjAD[WNOSXgf4H#°§U_NWCLGYADQSX_F®2HIORORH#CLU_V60H[W[PH#f4XYaU_CLE,ORQ&AcOSNPU0CCv] HpE>O?ML[PVLE§¨RGIU_CFE>OSQRVLGIORNW_HI©a\ADNeEHpE>O?FA_ERXH9ERV6QV6CADQS_V6aHCjO?f4H;GIU_CjOSQSA0f4NWGIORNWU_C\djV6N<MFUjERERYf4H[WHTaXQSNOSHTf4H#E,Hb0XC6XYQSA_[PNeE,HYQORQSYE«rA_GNP[WHaHYC0Op6CHlCªHIOp6NW[HI4NeE>OSHbf6HTC6U_aJ6QRHYVLERHYEcADQSNWA_C0OSHYEf6H[^] NPC6XY0A_[PNPORX\f4H°§U_NWCLGA_QRX0<A0f6ADM6ORXHpEbADV4f6NDCªXQSHCjOSEbaU4f4Y[PHpEBf] Xpd0VFAcORNWU_CFETADV4³f4XYQRNW_XYHYEMFADQRORNWH[W[PHpE!±V´V f4H¡[] NPaM<U_QROSA_CLGIH¡f6HGIHO,OSH¡NWC6XjAD[WNOSXMFU0V6Qg[eA³ERV6NPORH0vNW[2C] HYE,Of6U_CLG\MFA_ENWCV4ORNW[WH#fl] HYCf4U_C6CLHQV6C6HTf4XYaU_CLE,ORQ&AcOSNPU0CA_NWERXaHCjOA_fLADM4O&AD6[WHTZgOSU_VLE[WHYEGYA_Ef4HbF_V6QSH_ kvnYyc|w!h+3´qFs!tYnycm4nYoqFsw!|rm bycqL~2qFtponporq<s ] \%]K^ ] ;C M6QSUGYf4HgMLADQ9GU_CjORQ&A_f6NWGIORNWU_C(l] NW[C] HI4NeE>OSHBMFA_Ef4H#GU_CLE,OSA_C0OSH
C > 0OSH[W[PH#djV6H04MFU0V6QORU0V4ORHb«U0CLGOSNPU0C
v ∈ H10 (Ω)
∫
Ω
|v(x)|2dx ≤ C∫
Ω
|∇v(x)|2dx,
GH[eA_HV6Of6NPQSH#djV] NW[lHI4NeE>OSHTV6C6HTERV6NOSHvn ∈ H1
0 (Ω)ORHY[P[WH#djV6H
1 =
∫
Ω
|vn(x)|2dx > n
∫
Ω
|∇vn(x)|2dx. FFP¤_?C\MLADQRORNeGIV6[WNWHQp6FLWp0'NPaM6[WNWdjV6HdjV6H[eAbE,VLNOSH
vnHpE>O<U_QSC6XYHfLADCLE
H10 (Ω)
D°-ADQ?A_M6M6[WNWGYAcORNWU_Cf6V 0K6XU0QRYaH#f6HXHY[P[WNWG&K FF µL /6_4NW['H4NWE,ORHTV6CLHJERU_VLE, ERV6NPORHvn′
d0VLN'GIU0C_HQS_HTf6A_CLEL2(Ω)
@;HbM6[WVLE FLWp0aU_CjOSQRHBd0VLHB[eAE,V6NPORH ∇vn′GIU0C_HQS_H9_HQ&EJXQSUf6ADCLE
L2(Ω)rGU_aM<U0ESADCjORHMFADQTGIU0aM<U0ESADCjOSHpª°-ADQ#GU_CLERXYdjV6HYCjOY
vn′HYE,OTV6C6H\E,VLNOSHf6H®ADVLG&K`f6ADCFE
H10 (Ω)
vdjV6N-HYE,OVLCHpE,MFA_GIHTf4H 9NP[WFHYQ,OpLf4U0CLGbH[W[PHTGU_C_HYQR0H;f6A_CLEH1
0 (Ω)_HYQSEV6C6HT[WNWaNPORH
v<®2U0aaHbU_CA
∫
Ω
|∇v(x)|2dx = limn→+∞
∫
Ω
|∇vn(x)|2dx ≤ limn→+∞
1
n= 0,
U0C¡HYCf4Xpf4V6NPOY4HYC0HQRORV¡f4V HYaaH FF /6 6jdjV6HvHYE,OV6C6HbGIU_CFE>O&ADCjORHbf6ADCFEG&KFA_djV6HbGIU_aM<UDESADCjOSHgGIU0C6C6HI4Hf4H
Ω ADNeEBGIU_aaH
vHYE,O;CV6[W[PHE,VLQ;[WHg<U_Q&f
∂ΩHC³_HQRORVf4V®2U_QSU_[W[eADNWQRH
FF µLP ¯ IvHYE,ONWf4HYCjORNed0VLHaHCjOCV6[P[WH#f6A_CLEORU0V4O
Ω6°-ADQADNW[P[WHVLQSEY
∫
Ω
|v(x)|2dx = limn→+∞
∫
Ω
|vn(x)|2dx = 1,
GHTd0VLNHYE,OV6C6HTGIU0CjORQ&A_f4NeGOSNPU0CA¤0HYG;[PHB«rADNPO9djV6Hv = 0
.|+3´m4ycilk'| $] \]B^ 'Af4XaU_CFE>OSQSADORNWU_CMLADQ#GU_CjORQ&A_f4NeGOSNPU0Cf6H\[WA°QSU_M<U0ERNPORNWU_C FL µ6W E E,H0XC6XYQSA_[PNeERHg«rA0GINW[PHYaHYCjOYl°QSHC6U0CLEbMLA_QTHI4HaM6[WH_l[WHGA0EBf] V6CU0V6_HYQ,OΩ<U_QSC6X\GIU_CLC6HI4HHIOQSX0V6[WNPHYQf4H?GI[eA_ESERH C1 Yf4U0C0O[WH<U_Q&f ∂Ω
E,H2f4XYGU_aMFUjE,H§HYC#f4HV4bMLA_Q,OSNPHpElf4NWE A>U_NWCjORHpEQSX_VL[PNWQSHYE
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
0 4#! n #! Q # 2# n j
∂ΩNHIO∂ΩD
f4U0C0OB[PHpEBaHpE,VLQRHpE;ERV6M<HQRFGINWH[W[PHpE;ERU_CjOBC6U_CCjVL[P[WHYE0U_NWQ;[eA !NW_V6QSH FFP¤ ;Cf4XILCLNOV6CHpE,MLA0GIHVMLADQV = v ∈ H1(Ω)
OSH[vdjV6Hv = 0
E,V6Q∂ΩD.°§A_Q9ADMLM6[PNeGADORNWU_C f4V 0KLXU_QSaH#f4H#ORQ&A_GIH:FF µLPpµ66NW[vHpE>O9«rA_GINW[WHgf4H#0U_NWQ9djV6H
VHpE>O9VLC E,U0VLE>HYERMLA_GH«HQSaXf6H
H1(Ω)f4U_CLGHYE,OV6C HpE,MLA0GIHf4H NW[P<HQRO\MFU0V6Q[PHM6QSUf6V6NO\ESGAD[eADNWQSHf4H
H1(Ω)®2U0aaHM<U_V6Q
H10 (Ω)
_[^] ADQS_V6aHCjOf4HGU_CjORQ&A_f4NeGOSNPU0CJM<HQSaHIO?f4H;f4XaU_CjOSQRHYQ-[] HI4NeE>ORHCFGIH³fl] V6C6HGIU_CFE>O&ADCjORHC > 0
OSH[W[PHd0VLHORU0V4ORH«U0CLGOSNPU0Cv ∈ V
_XYQRNPLH[^] NWC6XjAD[WNOSX f4H°!U0NPCFGADQSXFL 0I •°-A_QxADMLM6[PNeGADORNWU_C f4H [WA «U_QSagV6[WH f4H BQSHHYC f4V 0K6XYU_QSaH FL µ6Wp6#C6U0VLE´M<U_V60U_CLEGIU_CFE>OSQRV6NWQSHV6C6H¡«rADaNP[W[WHfl] HI4HYaML[PHpEJf4H¡«U_CLGIORNWU_CLEADMLMLADQRORHYCLADCjOJZH1(Ω)
®2HO,OSH¡«rA_aNW[W[PHfl] H4HaM6[WHYE#CLU_VLEJE,HYQSAOSQRpETV4ORNW[WH¡MLA_Q#[WA³E,V6NPORH¡MFU0V6QJGU_CLE,ORQSV6NWQRHf4HpEgERU_VLE, HYERMLA_GHYETf4H¡f6NaHYCLERNPU0C¡LCLNPH#f4HH1(Ω)
;?|+3 3´| ] \%]K^ `>bdcfe
Ω g h b g i8jk e b kuh k # g c jk m jyq8p-pj C1 w `>b4cfe (ωi)1≤i≤I g hj|qdk eKcfe~cBb h k # g c + k@j m j Ω o _ jlp e $ mdc kvj gNj q gNj ωi jup e g h b g i4jlk e k # g c jlk m j q8p-pjC1 o ωi ∩ ωj = ∅ p c i 6= j o j e Ω = ∪I
i=1ωi w `>bdcfe v g h>jr b h eKcBb h m'b h e q kvjlp e k c eKcBb h $ q gNj ωi
o vi = v|ωi
o q-||Nq4k e~c jlh e $ H1(ωi) w `c v jlp e b h e~c hNgNjsp-g k Ω o q4 b k-p v q-||Nq4k e~c jlh e$ H1(Ω) wYh+3´q<s'tYnycm4nporq<s#]v®A_[WGV6[PU0CLE[eAgf6XQSNP0XH«rA_NP6[WHbf4H
vHMFU0V6Q
φ ∈ C∞c (Ω)
MLA_QA_M6M6[WNWGYAcOSNPU0Cf4HT[WAJ«U0QRaJV6[PHTf4HBQSHHYCfLADCLEG&KLA0djV6Hωi
U_CA∫
Ω
v(x)∂φ
∂xj(x) dx =
I∑
i=1
∫
ωi
vi(x)∂φ
∂xj(x) dx
= −I∑
i=1
∫
ωi
∂vi
∂xj(x)φ(x) dx +
I∑
i=1
∫
∂ωi
vi(x)φ(x)nij (x) ds
= −I∑
i=1
∫
ωi
∂vi
∂xj(x)φ(x) dx,
GADQ-[WHYE§NWCjORXY_Q&AD[WHYE§f4H<U_Q&fJEY] A_C6CV6[WHCjOf4HV4Zbf4HV4lD?CHC<HOYjE,V6Q-[eABMLADQRORNWHΓ = ∂ωi∩∂ωkf4V FU0QSf GIU0aagV6C6H A_V4¥f6HV4¥U_V60HQROSE
ωiHOωk
U_CAni
j(x) = −nkj (x)
HIOf4U0CLGD?MLADQGIU_CjOSNPCV6NPORXTf4HvHOφ
∫
Γ
vi(x)φ(x)nij (x) ds +
∫
Γ
vk(x)φ(x)nkj (x) ds = 0.
;CHYCf6XYf4V6NPO9f4U0CLGbd0VLHvHYE,Of4XQSNPcA_6[PHTA_VERHCFE«rADNW6[PHbHO9d0VLH
∂v
∂xj
∣
∣
∣
∣
ωi
=∂vi
∂xj.
CMLADQRORNeGIV6[WNWHQp6GIHYGNlNPaM6[WNed0VLH#d0VLHvADM6MFADQRORNWHCjOZ
H1(Ω)
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
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re.
¯ # +Q7 " % %
∂ΩN
∂ΩD
Ω
FFP°§A_Q,OSNOSNPU0CHCf4HV4MLADQRORNWHYEf6NWE A>U_NWCjORHYEf6V<U_Q&ffl] VLCU0V6_HYQ,Op G4 2 $- 6O-9, 65-, % "U , +) .M)+ 9 %)
Ω
$/+9%9% W4 .M)+9X9 "(%)9(ωi)1≤i≤I 0 "9)+ W D"0 0E 9 29 .(;"(
+%() ".M)+9ωi
+ +) 9 "(%)9L- -9, -"R 0 "9%%(+9 -+68% , )+9 ")%9-9%W$+ > @, P "" " G D"0 0E0 •
x
y
y = x
Ω
r
FL0/\HYaM6[PH#f] V6CU0V6_HYQ,OC6U0CQSX0V6[PNWHQp;u|jyc¢_or¢0| $] \] \ ' 9:!! '>% #:#9) & (% >%*" 7 ;@% & :9) (% 2'! / - / - # <'2 % 2 7 " )% Ω < 2 :% ;7 )% /@ 7 " )% Ω ⊂ R2 9;3 < 2)% 0 < x < 1# 0 < y < xr 2 r > 2 + % 7 2 "% / 01#/ 7 2I"'( v(x) = xα / ="(% >%* v ∈ H1(Ω) ! 6>7 & >, 2α + r > 1 62'9 :* v ∈ L2(∂Ω) ! 6>7 & > 2α > −1 / >78% / + I<>>,2 6 & (% >%$2 # &)& & <7 =*" 7 ;@% & / - / 9 9> ; ! 7 2 %<>) / / 0 9) <)%7 " ) & > 6 < 2 % 2 <>"% )7 >%#/ 1;u|jyc¢_or¢0| $] \] '9!# '>% # #69 & "(%)% * 7 <>><'2 42 % 9 ) "9:(% 2'! <>%9)"'( 9 L2(Ω) #6"!$#%! 9 % * 7 <'2 9)=!" 62 C > 0)787 :*" & <>% ! " v ∈ L2(Ω) 2
‖ v|∂Ω ‖L2(∂Ω) ≤ C‖v‖L2(Ω).
Ecol
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lyte
chni
que.
Ne
pas
repr
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re.
0 4#! n #! Q # 2# n +G
""% & <7 )%$#" ! & & ?" >% Ω 7 2>"7 ;/ " (% % ?# ?9$ ! " %; "7 >% #92 Ω ;27 ## !% ∂Ω #9"?7 2: @% & 92 L2(Ω) ) 9 >% ;>% / "''78% /*)* / 3 $Sj8"O1 P!1 7U :ª5#M P-:l=>1
9U_VLEGU_CLESA_GQRU0CLEGIHO,OSHbERU_VFE>E,HpGOSNPU0CZ#[^] XORVLf4Hbfl] V6C6HbM6QSU_M6QSNWXIORX;f4HBGU_aMLA0GINPORXbGIU_CLCjVLHE,U0VLE[WHC6U0a f4H?ORK6XYU_QSaHf4H2XH[W[PNeG&K#djV6NA>U_V6HYQSAV6CgQCB0[PH-HpERERHCjORNWH[jf6ADCFE[eAORK6XYU_QSNPH2E,M<HYGIORQ&AD[WHf4HYEMLQRU06[PYaHYEADV6¡[PNWaNOSHYEJ_U0NPQ[WHJ®2KLA_M6NOSQRHEG_d0VLHTC6U_VLEV4ORNW[PNeERHQSU_CLEM<U_V6Q9QRXpE,U0VLf4QSH;[WHYEM6QRU06[WaHYEfl] X_U0[PV6ORNWU_C HYC ORHYaMLEXA_M6M<H[WU_CLE¡ORU0V4Of] A_FU0QSf d0VLH_f6ADCLEV6C HYERMLA_GHf4HNW[P<HQROf4Hf4NPaHYCLE,NWU_CNWC4LCLNPH0vNW[?C] HYE,OJMLA0ETQSA_N?djV6H_!f4H\OSU_V4OSHERV6NOSH<U_QSC6XH0U_CM6VLNWESE,HHIORQ&ADNWQRHV6CLHERU_VLE, ERV6NPORHGIU0C_HQS_HYCjORHrA_V GIU_CjOSQSA_NPQSHf4H³GIH djV6NBE,H³MLA_ESE,HHYC f4NWaHYCLERNPU0CNPC4FC6NPH06_U0NPQ[^] HYQSGNWGH FL µ6_ £'hqFy4:[3 | $] \]%^>w!| .¡|ror¢¤£"y`_c
Ω jlp e g h b g i4jlk e b kuh k # g c jk m jl q8p-pj C1 o q4 b k-pm j e b g e jpug cfe j b kuh j m j H1(Ω)b h"|jg e j- e kvq c kvj2g hjp b g+p p-g cfe j b h_i8jk jlh e j m q4hp L2(Ω)
z b h mdcfe gNj R c h j- e~cBb h -q4h b h c gNj m j H1(Ω)m qdhp L2(Ω) jlp e b t|Nq e j!w vH 0K6XYU_QSaH:FL µ60/4TM<HV6O;¦IOSQRH#«rADV4 ERNv[^] U0V6_HYQ,O
ΩC] HYE,O;MFA_E<U_QSC6X0L°-ADQ;HI4HaM6[WH_<E,N
Ω = RN ![] NWC'A>HYGIORNWU_C GYADC6U0C6NWdjV6Hf4HH1(RN )
f6A_CLEL2(RN )
C] HYE,OJMLA0EgGIU0aMLA_GIORH_!°§U_V6QE] HC GU_CcADNWCLGIQSHTNP[§ERV WOBf4HGIU_CFE,Nef4XQSHQ9[WAERV6NPORHun(x) = u(x + ne)
U&eHYE,O;V6C³_HYGIORHYV6QC6U_CCV6[lHIO
uV6C6HB«U0CLGOSNPU0Cf4H
H1(RN )U_CORQ&ADCFE,[eAcOSH
uf6A_CLE[eAJf4NWQSHYGOSNPU0C
e D[lHpE>OG[WA_NPQd0Vv] A_VLGIV6CLHbERU_VFE>E,VLNOSHbf6H
unC6HTGIU_C0HQS_Hbf6ADCLE
L2(RN )
.¡|+3´m4ycilk!| ] \%] <(NF[^] U_C\QSHaM6[eA_GIHH1(Ω)
MLADQH1
0 (Ω)AD[WU_Q&E-C6U0C¡E,HYV6[WHaHCjO?[WH0K6XYUDQRYaH FL µ60/4Bf4HsXH[W[PNeG&KQRHpE>OSH;Q&ADN^0a\ADNeEHCM6[WVLENP[ªC] HYE,OMLA_EC6XpGIHpERESADNWQRHBf4HbE,V6MLMFUjE,HYQdjV6H[] U_V60HQRO
ΩHYE,OQSX_VL[PNWHQp • 'AJf6XaU_CLE,ORQ&AcOSNPU0C¡f4V10K6XU0QRYaH,FL µ60/4;HYE,O[WU_CL_V6H;HIOf4X[WNeGAcOSHrH[W[PHB«rA_NOADM6M<H[lERU_NPOZ[WAbOSQSA_CLE,«U_QSaXYHf4H LU_V6QSNPHYQYjE,U0NOADVORKLXU_QSaHf4HBGIU0aMFA_GINPORXf] ±;ERGU_[WN 0U_NWQY_MLADQ2HI4HaM6[WH_F0G8H^F µ08H¬I0U0VLEA_[P[WU_CLE§CLU_VLE-GU_CjORHYC0OSHQ?f6Hf4U_C6CLHQ-V6C6H;f4XaU_CFE>OSQSADORNWU_CM6[PVFEE,NWaM6[WHHIO?M6[WVLE¨>MLA_QR[eADCjOSH©;HYCf4NWaHCLERNWU_C
N = 1
Yh+3´q<s'tYnycm4nporq<s#] ;C´ERHM6[eA_GH\HCV6C6Hf4NWaHYCLERNPU0C´fl] HpE,MFA_GIHN = 1
HIOp'ESADCLEgMFHYQ,OSH¡f4H_XCLXQ&AD[WNOSX_§U_C E,VLM6MFUjE,HdjV6HΩ = (0, 1)
#(U0NO(un)n≥1
V6C6H ERV6NOSHFU0QRC6XYHf4HH1(0, 1)
GD] HYE,O,Zcf4NPQSHbd0Vv] NW[HI4NeE>OSHK > 0
ORH['d0VLH‖un‖H1(0,1) ≤ K ∀n ≥ 1.@g] ADM6QSYE[WHT HYaaH:FF µL µLjM<U_VLQORU0V4O
x, y ∈ [0, 1]U_CA
|un(x)| ≤ CK HIO |un(x)− un(y)| ≤ CK√
|x− y|. FLW ¯ ;C A\ADMLM6[PNedjV6X#[^] NWC6XY0AD[WNPORX#f4H®A_VLG&K`j (4G&K ADQKJ;Z FL (U0NO;V6C6HgERV6NPORH¡f4XC6U0ag6Q&ADL[PH¤(xp)p≥1
f4HMFU0NPCjOSEbf4H[0, 1]
djV6N-HpE>O#f4HYCLERHf6A_CLEbGIHObNWCjORHYQRcAD[W[WHMFADQbHI4HaM6[WH_l[WHYEBM<U_NWCjOSEf4HQ ∩ [0, 1]
<°!U0V6Qp64X0F[eAERV6NPORH
un(xp)HYE,O;<U_QSC6XYHJf6A_CLE
RZGA_VLERHJf4HJ[WAM6QSHaNPYQRHNPC6XY0A_[PNPORXbf4H\FFP ¯ ;CMFHYV4O9f4U_CFG9HYCHjOSQSA_NPQSH9VLC6HTE,U0VLE>ERV6NOSHBdjV6NGU_C_HYQR0H9f6A_CLE
R ;C
Ecol
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repr
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re.
# +Q7 " % %
A_M6M6[WNWdjV6Hfl] A_FU0QSfGHM6QSUGXYf4XZ[WA ERV6NPORHun(x1)
HIOgU0CU_4OSNPHYCjOgV6C6H¡ERU_VFE>E,VLNOSHun1(x1)djV6NlGU_C_HYQR0H;f6A_CLE
Rj°?V6NWEU0C¡HjOSQSA_NOf4HTGIHO,OSHbERU_VFE>E,VLNOSH_jNPCLf6NWGXHbMLADQ
n14VLC6HbC6U_V60H[W[PHERU_VFE>E,VLNOSH_pNPCLf6NWGXH?MFADQ
n2pORHY[P[WHdjV6H
un2(x2)GIU0C_HQS_HfLADCLE
Ra\ADNeEYpLNPHYCJER6QYpU_CJA;A_VLERERN
un2(x1)djV6N!GIU_C0HQS_HpI6C HORQ&A¤`0ADCjOERVLGGHYESE,NW_HYaHYCjOV6C6HJERU_VFE>E,VLNOSH#f4H#[eA\M6QSXYGIXpf4HCjOSH_MFADQQRXpGIV6QSQRHYCLGIH0'U0CxGU_CLE,ORQSV6NPOADNWCLERNV6C6HE,U0VLE>ERV6NOSH_!NPCFf4NWGXHMLA_Q
np!ORH[W[WHdjV6H
unp(xp)GU_C_HYQR0H_FaA_NWEBADVFERERNunp(xk)
MFU0V6Q1 ≤ k ≤ p
NWf6HaaHCjOYl[WHYEbERU_VLE, ERV6NPORHpEunp
ERU_CjOf6H#M6[WVLEHYCM6[PVFE?¨>a\A_NP0QRHpE©TZaHpE,VLQRH#djV6HpHYE,O9_Q&ADCLf4°!U0V6QXYjNPORHYQ;djV] NP['C] HCQSHYE,ORHTM6[WVLEQSNWHC¡¨RZ[WA\[WNWaNPORHI©&<U_CV6ORNW[PNeE,HJV6C ADQS_VLaHYCjO;fl] HIORQ&A_GIORNWU_Cf4HgERV6NPORHJf6NWA__U_CFAD[WH_L±9V4ORQSHaHCjOf6NOp0U0C\HIORQ&ADNPO?V6C6HBf4HQSC6NWQSHE,U0VLE, ERV6NPORHgf4NOSH;f4NeAD0U_CLA_[PH¤§f4H;GHIO?HYCLERHagL[PHBf4H9ERU_VLE, ERV6NPORHpEG_] HpE>OR ZD f6NPQSHdjV] U_C G&K6U0NWERNO[WHM6QSHaNPHYQ\X[WXaHCjO¡f4H[eAM6QSHaNPYQRHERU_VLE,E,V6NPORH
un1
MLV6NWE[PHf6HV44NWaHTX[WXaHCjOf4HT[eA\f4HV6NWaHun2
4HO9ADNWCLE,N'f4H#E,VLNOSHB[WHp aHbX[WXaHYC0O9f6HT[WA
p aH
unp
' 'A³ERV6NPORHA_NPCFE,N2U_4OSHCV6HHpE>OC6UDOSXHum
HOJU0C_U_NPOdjV6H0vM<U_V6QJORU0V4Op ≥ 1
v[eA³ERV6NPORHum(xp)
GU_C_HYQR0H;f6A_CLER
(U0NO;aA_NPCjOSHCLA_C0Ox ∈ [0, 1]
HOε > 0
V6CM<HIOSNO9MFADQ&ADaIOSQRH¤F®2U_aaHT[eA\E,V6NPORH(xp)p≥1HpE>O#f4HYCLERHf6A_CLE
[0, 1]ªNP[-H4NWE,ORHV6CMFU0NPCjO
xpORH[djV6H |x − xp| ≤ ε
ª°-ADQTADNW[P[WHVLQSEYlGIU_aaHum(xp)
GU_C_HYQR0HJf6A_CLERdjVLA_CLf
mORHCFf0HQ&Eb[] NWC4LC6N^NW[HI4NeE>OSH
m0OSH[?djV6H0lMFU0V6QTOSU_V4O
m,m′ ≥ m04U_CA |um(xp)− um′(xp)| ≤ ε 4°-ADQGU_CLERXYdjV6HYC0Op4U_CU_4OSNPHYCjO
|um(x) − um′(x)| ≤ |um(xp)− um(x)| + |um(xp)− um′(xp)|+ |um′(xp)− um′(x)|
≤ ε+ 2CK√εGHJdjV6N'M6QRU0V6_HJd0VLHg[eAE,VLNOSH
um(x)HYE>OBf4H®A_VLG&K`f6A_CLE
R<HIObf4U_CFGgGIU0C_HQS_HbM<U_V6Q;ORU0V4O
x ∈ [0, 1]?°-ADQADM6M6[WNeGAcOSNPU0C f4VOSK6XU0QRYaHf4HGU_C_HYQR0HCLGHf6U_aNPCLXHf6H HYFHpE,0V6H_2U_CGU_CLG[PV6OdjV6HT[eAE,VLNOSH
umGU_C_HYQR0H;f6A_CLE
L2(0, 1)
;u|jyc¢_or¢0| $] \] @ Ω = (0, 1) ! un(x) = sin(2πnx) / ="(%)%* 7 2 # un #6 @% & ; & > >@% ; 92 L2(Ω) & 2 *" 7) 62'' C" ! ! ! >% ))/""%#>7 2 & (% >% 2 %"%'! # ;% 2 ,<'2>%3<'2>% &*)<>%"? ! " φ ∈ C∞
c (Ω) 2
limn→+∞
∫ 1
0
un(x)φ(x) dx = 0,
#?" >:9);9" % 2 #"(% 2'9 ( , 4C" ! ! 9) un #" )% 492 L2(Ω) / ;> ;>% 2%!7 C>% #!"(% & <7 # H1(Ω) > #" 9;>% 24 <>% &: :9 un /9*)* ) S "P-:>US
Hm(Ω);C\M<HV4OADNeE,XYaHYCjO_XCLXQ&AD[WNWERHQ-[eAT@9XLC6NPORNWU_C FF µLPf4H[^] HYERMLA_GHf4H"(U0FU0[PHY
H1(Ω)ADV6«U0CLGOSNPU0CLEdjV6NvERU_CjO
m ≥ 0«U0NWEf6XQSNPcADL[PHpEADVE,HYCLE«rA_NP6[WH_ª®2U_aaHCGU_CLEMFADQf4U0C6C6HQ9V6C6HGU_C_HYCjORNWU_C³fl] XYGQRNPORVLQRH6NWHCV4ORNW[WH_(U0NO
α = (α1, ..., αN )V6C 3k!npof=Sos'¢0orw!|<<G_] HpE>OR ZD f4NWQSHVLC´_HpGORHYV6QJZ
NGIU0aM<U0ESADCjOSHYE#HYCjORNWQSHYE#M<U0ERNPORNW_HYE
αi ≥ 0 ;CC6UDOSH |α| =
∑Ni=1 αi
HOYM<U_VLQV6C6Hb«U_CFGORNWU_Cv
∂αv(x) =∂|α|v
∂xα11 · · · ∂xαN
N
(x).
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0 4#! n #! Q # 2# n
± MFADQRORNWQf6HB[eA@9XLC6NPORNWU_C FL0/4 µgf4Hb[eAf4XQSNP0XHBM6QSHaNPYQRH;«rADNW6[WH_4U0Cf4XLC6NPOMLADQQSXYGIVLQRQSHCLGHE,V6Qm[eAf4XQSNW_XH2fl] U0QSf6QRH
m«rADNW6[PH,HU_C#f6NO§d0Vv] V6CLH-«U_CLGIORNWU_C
v ∈ L2(Ω)HpE>O
m«U0NWEf6XQSNPcADL[PHADV ERHCLE#«rADNW6[WHERN?OSU_V4OSHYEE,HpEgf4XYQRNW_XYHYEgMLADQRORNWH[W[PHpET«rA_NP6[WHYEfl] U_Q&f4QSH
m − 1E,U0CjOf4XYQRNWcAD6[WHYE«rADNW6[PHYaHCjO#ADVE,HYCLEbf4H\[WA@;XILC6NPORNWU_C FF /6 µL>XHa\ADQ&djV6U_CLEBdjV6H_fLADCLEb[eAf6XILC6NPORNWU_Cfl] V6C6Hf4XQSNP0XHGIQSU_NeE,XYH_§[^] U0QSf4QSHf4Hf4XQSNW¤ADORNWU_C¥C] HYE,OMLA_ENWaM<U_QROSA_CjOY?ZGA_VLERHf4V¥OSK6XU0QRYaHf4H
(4G&K A_QCJ ∂2v∂xi∂xj
= ∂2v∂xj∂xi
GH;djV6NA>VLE,ORNPLHB[WAgC6UDO&AcORNWU_C∂αv
U'&[^] U0QSf4QSH;f6H;f4XYQRNWcAcORNWU_C¡C] HpE>OMLA_ENWCLf4NedjV6X_Yh[Z§s'ornYorq<s ] \%] N\ b g k g hjlh eKc jk m ≥ 0 o R jup~|Nq-j m j `>b b ji Hm(Ω) jlp em ' h c |qdk
Hm(Ω) =
v ∈ L2(Ω)e j gNj o ∀α q4i4j- |α| ≤ m, ∂αv ∈ L2(Ω)
, FLW8G_
b) q m k c i j|qdk eKc jfj ∂αv jup e $n|>kvjlh m kvj q4g pjhprlq c j w9U_VLE[WA_NWESE,U0CLEADV[PHpGOSHV6Q[WH#E,U0NPCf4HT0XQSNLHYQ[PHTQSXYERV6[POSAcO«rA0GINW[PHTERV6NPcA_C0Opby¤qF~?q<tYornYoqFs ] \%] g h c#m g |>k b m g cfe pl-qdq c kvj
〈u, v〉 =∫
Ω
∑
|α|≤m
∂αu(x)∂αv(x) dx FLW
j em j qyh b kut9j ‖u‖Hm(Ω) =√
〈u, u〉 o R jlpK|Nq-j m j `>b b jli Hm(Ω) jlp e g h<jup~|Nq-j m j c -jk e w vHYE?«U_CLGIORNWU_CLEf4H
Hm(Ω)C6HBE,U0CjO2MLA0E-ORU0V'A>U_VLQSEGIU0C0OSNPCV6HpEU0VQRXY_V6[WNWQSHYEGIH[eAgf6XM<HCLff4H
mHOf4H9[WA#f6NPaHCFE,NWU_C
NI_a\ADNeE?ERN
mHYE,O2ERV WERA_aaHYC0O2_Q&ADCLfAD[WU_Q&E!ORU0V4ORH9«U_CLGIORNWU_Cf4H
Hm(Ω)HYE,O;GIU0CjORNWCjVLH_NXA_M6M<H[WU_CLE;d0Vv] HYC0HQRORV f4V vHaaHFL µ6 µ64HYC f4NWaHYCLERNPU0C fl] HpE,MFA_GIH
N = 1[PHpEB«U_CLGIORNWU_CLEgf4H
H1(Ω)ERU_CjO#GIU_CjOSNPCV6HpE9U_VLE#A0f4aHIORORQSU_CLEb[WH\QRXpE,VL[O&AcO#E,VLNPcADCjOd0VLNv_XYC6XQ&AD[WNeE,HB[PHJ HaaHFF µL µADV6f4NWaHCLERNPU0CLEERV6M<XQSNPHYV6QRHpE#0U_NWQ[WH# HaaHFF FF M<U_V6Q[WAf4XYaU0CLE,ORQ&AcORNWU_Cv4M6[PVFEE,NWaM6[PH0Lfl] V6CQRXpE,VL[O&AcOE,NWaNP[eADNWQSHp
£'hqFy4:[3 | $] \] `cΩ jup e g h b g i4jk e b kuh k g c jk m j q8p-pj C1 o j e p c m > N/2 o
q4 b k-p Hm(Ω) jlp e g h p b g+p jlpK|Nq-j m jER jhpjlt9j C(Ω)m jupr b h eKcBb hpE b h e~c hNgNjlpnp-g k Ω w
.¡|+3´m4ycilk!| ] \%] °§A_Q9ADMLM6[PNeGADORNWU_CQSXNPORXQSXH#f6V 0K6XU0QRYaH FF µL /0ZV6C6HT«U0CLGOSNPU0CHIO;ZE,HpEf4XQSNW_XHpE_U_CMFHYV4O2HYC«rADNPO2ADaXY[PNWU_QSHQESATGIU0CLGI[WVLERNPU0C'(ª] NP[FHI4NeE>OSHV6C\HCjOSNPHYQk ≥ 0
ORHY[FdjV6Hm −N/2 > k
ªA_[PU0QSEHm(Ω)
HYE,O;V6CE,U0VLE,^HpE,MLA0GIHgf6HJ[^] HYCLERHagL[PHCk(Ω)
f4HYE;«U0CLGOSNPU0CLEk«U_NeEf4NDCªXQSHCjOSNWA_6[PHpEE,VLQ
Ω • 'A¨,aU_Q&AD[WH©³f4V 0K6XU0QRYaH FL µ60/_HpE>OdjV6HM6[PVFE
mHpE>O_Q&ADCLf2M6[WVLE[WHYE¡«U_CLGIORNWU_CLEf4H
Hm(Ω)ERU_CjOTQRXY_V6[WNPYQRHpEG_] HpE>OR ZD f6NPQSHf6XQSNPcADL[PHpE#ADVE,HYCLETVLERV6HY[9NW[2ERV WOgfl] ADM6M6[WNed0VLHQE,VLGYGIHpERERNP0HaHCjO[PH 0K6XU0QRYaH4FL µ60/_TZJV6C6H;«U_CFGORNWU_C
v ∈ Hm(Ω)HOZJE,HpEf4XQSNP0XHpE
∂αv ∈Hm−|α|(Ω)
®2U0aaHM<U_VLQH1(Ω)
D[WHYE§«U0CLGOSNPU0CLE§QSX_VL[PNWQSHYE-ERU_CjOf4HCFE,HpE-f6ADCLEHm(Ω)
rERNLf4VaU_NWCLE[] U_V60HQROΩHYE,OTQRXY_V6[WNPHYQ _U0NPQb[eA@;XIFC6NOSNPU0CµL /6 0l vAf4XYaU_CLE,ORQ&AcOSNPU0Cf4V 0K6XU0QRYaHf4Hf4HCLERNPORXFF µL BERH_XCLXQ&AD[WNWERHORQSYE§«rA0GINW[PHYaHYCjO2Z
Hm(Ω)jU_VFE-C6H[eAbQRXYMFXORHYQRU0CLE-MLA0E-HIOC6U0VLEXC6U0CGIU0CLEE,HYV6[PHYaHCjO[PHTQSXYERV6[POSADOf6Hbf6HCLERNOSX#E,VLNPcADCjOp
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E)E # +Q7 " % %
£'hqFy8:+3´| $] \] `cΩ jlp e g h b g i8jk e b kuh k # g c jk m jq8p-pj Cm o b g c jhp c Ω = RN
+o
qd b k-p C∞c (Ω) jlp em jhpj m qdhp Hm(Ω) w
;CM<HV6OA_VLERERNLU04ORHYC6NWQf4HpE?QSXYERV6[POSADOSE?f6HORQ&A_GHHIOf4HYE?«U_QSagVL[PHpE?f4H BQRHYHC¡fl] U_Q&f4QSHM6[WVLEXY[PHY_XpEMFU0V6Q[] HYERMLA0GIHHm(Ω)
F°§A_QE,U0VLGINf6H#E,NWaM6[PNeGINPORX06C6U0VLEC6U0VLEGIU0C0OSHCjORU0CLEf4HbOSQSA_NOSHQ[WH#GA0Em = 2
rdjV6NvHYE,O[PHTERHV6[vdjV6HbC6U0VLEV4OSNP[WNWERHQSU_CFEMLADQ[eAE,V6NPORH¤ £'hqFy8:+3´| $] \] `Qb4cfe
Ω g h b g i4jk e b kuh k # g c jk m j6lq4p-plj C1 w"xh m' ' h cfe v q@||> c -q eKcBb h e kvq-j γ1
H2(Ω) ∩ C1(Ω) → L2(∂Ω) ∩ C(∂Ω)
v → γ1(v) =∂v
∂n
∣
∣
∣
∣
∂Ω
,
FLW 0qdi8j- ∂v
∂n = ∇u · n w ] j e~e jq-|| c @q eKcBb h γ1 pj|k b b h j|Nq4k b h e~c hNg cfe jhg h>jEq@|'| c -q eKcBb h c h q c kvjn b h eKc h_gNj m j H2(Ω)
m q4hp L2(∂Ω) w hs|qdk eKc g c jk o c _j- c p e jg h>j b hNp e qdh e j C > 0e jfj gNj o | b g k eb g e j#r b h e~cBb h v ∈ H2(Ω) o b h q∥
∥
∥
∥
∂v
∂n
∥
∥
∥
∥
L2(∂Ω)
≤ C‖v‖H2(Ω). FL0/ E
Yh+3´q<s'tYnycm4nporq<s#]# ?] HNeE,ORHCFGIHf6H [] ADMLM6[PNeGADORNWU_C OSQSA0GIHγ1
rHIOERHYE¡M6QSU_M6QSNPXORXpESHpE>OV6C6HERNWaML[PHGIU0CLE,XpdjV6HCLGHf4V M6QSXYGXYf4HYC0O10K6XU0QRYaHf4HORQ&A_GH FF µLPpµMFU0V6Q[PHpE«U_CLGIORNWU_CLEf4HH1(Ω)
<?C³HlCªHIOplE,Nv ∈ H2(Ω)
lAD[WU_Q&E ∇v ∈ H1(Ω)N HIObU_C M<HV4OTf4U0CLGgf4XLC6NWQb[WAORQ&A_GIHf6H ∇v E,V6Q ∂ΩGU_aaHTV6C6HB«U_CFGORNWU_Cf4H
L2(∂Ω)N F®2U_aaHT[WAC6U0QRa\AD[WHbHYE,OV6C6Hb«U_CFGORNWU_CGU_CjORNWCV6HbFU0QRCLXHTE,VLQ∂Ω
4U0CHYCf6XYf4V6NPO6NWHCdjV6H ∂v∂n ∈ L2(∂Ω)
%Ω
2 " ".M)+;-9 9 %) " ,) -++ C2 )(%-9) ("9 , " :J) > *+, D0 0EP0 9 )#$"%&@ 4 -%,/9%( +,
γ0
H2(Ω) ∩ C(Ω) → H1(∂Ω) ∩ C(∂Ω)v → γ0(v) = v|∂Ω ,
C5D"0 TPEFQP"%G+!9"!$/', 9%(P"%9) "!"%(, -9%%-%9S, P9%P!
H2(Ω) "
H1(∂Ω) 0 "+9) * %W( +-, γ0(v)
")* " ! )9%.M *- 9%()( 0 % Ω = RN+
W, X9 +&-S 2++ @65-, %( K , , .%&) K +9) 0 -3,I ")5M%6579.M%&: )#$"%"B +$-, < "; .+" "./9%)9 C %, % ∂Ω F ,):Q%P $-++I+ > 1(/9 I, P9: "I, 7,)"9 0 "%, , %PG4 2 9- %9) ) "%9QP"7, ) <)9) 4 )9$ %( " P > W4 Q-9%" R )9%.M $-+9% L.P%.-P" K (6O%(! ""'U %"S+"'+; C $/*)R "( ")*P > ' " "%+%.M%(Q)'+"+65-, %(Q;% P > X 4 (29%, %&9B.M("%(QP", "X/. ,!P > , PQ*++65,)% Q" F0 • vH0K6XU0QRYaH\f4HOSQSA0GIH FL µ60/ MFHYQRaHOgf4H\_XYC6XQ&AD[WNWERHQbADV6 «U_CLGIORNWU_CLEgf4H
H2(Ω)V6C6H«U0QRaJV6[WH9f4H BQSHHYC\M6QRXpGIXpf4HaaHCjO?XOSADL[PNWHM<U_V6Qf4HYE«U_CLGIORNWU_CLE2f4H9GI[eA_ESERH
C2 ADV®2U0QRU0[P[eADNWQSHµL /6 FF
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$# $ # # # $ 2!% E £'hqFy4:[3 | $] \] \ `Qb4cfe
Ω g h b g i8jk e b kuh k g c jk m j lq4p-plj C2 w `_c u ∈ H2(Ω) j ev ∈ H1(Ω) o b h q
∫
Ω
∆u(x)v(x) dx = −∫
Ω
∇u(x) · ∇v(x) dx +
∫
∂Ω
∂u
∂n(x)v(x) ds.
FF /'/_Yh+3´q<s'tYnycm4nporq<s#]#®2U_aaHFF /'/_JHYE,OQSA_NPHMFU0V6Q¡f4HpE«U_CLGIORNWU_CLE¡f4HGI[eA_ESE,H
C2 HIO¡djV6H[PHpE«U_CLGIORNWU_CLEQRXY_V6[WNPYQRHpEE,U0CjO9f4HCLERHYE9f6ADCLEH2(Ω)
HIOH1(Ω)
LU_CV4OSNP[WNeE,H#VLC ADQS_V6aHCjOf4Hf4HCLERNPORX_49U_VLE2QRHYC_Uc`_U0CLE-Z#[eAgf6XaU_CLE,ORQ&AcOSNPU0Cf4V 0K6XYU_QSaH,FF µLP¤BMFU0V6QM6[PVFEf6H;f4XOSADNW[eE H;ERHVL[<A_QR0V6aHCjO?C6U0V6_HpADVNWGNªHYE,OdjV] NP[F«rADV4OV4ORNW[WNWERHQ[WAgGIU_CjOSNPCV6NPORXBf4H9[^] A_M6M6[WNWGYAcOSNPU0C\ORQ&A_GHγ1LGD] HYE,O,Zcf4NWQRHB[] NWC6XjAD[WNOSX FL0/ E
" ( "I J%'( ¡%'$I" (®2HO,OSH#E,HpGORNWU_CM<HV4O¦ORQSHbU_aNWERHbHCM6QSHaNWQSHB[WHYGIORV6QSH_
* *,+ Z M 5?/"S013NP'16=,5?/ 7;8 53$M U *)* 7U 7Uv/"S=1 "@, ":$/<(, Ω = RN QP"% 2<("3+% 0
! "C∞
c (RN ) # %$'&)(* ,+-&.( # / '01.2' C∞ 3 24!&.5"+6 "&)78! ,+ # .(RN "+ # (9, # .(9 H1(RN ) :
1n
1n
nN
ρn 1
n 2n−2n −n
ζn
FL µLU_CLGIORNWU_CLEρn
f4HTQRXY_V6[eADQSNeERADORNWU_C³ZJjADVLG&KLHp4HOζn
f4HbORQSU_CLGYAcOSV6QRHrZf4QSU_NPORH¤; <>='? & ?A@B< N "2+-9% 4 ,)9"U- 9 (-9%9-9% +9,/9"9 0 % ρ ∈C∞
c (B) C .M), B Y; "%&9 F 9)( Q"
ρ ≥ 0) ∫
Bρ(x) dx = 1 0 ")#%&" +"%9
?9 (-9%9-P9 ρn(x) = nNρ(nx)
S( +"+S 2X, 9 P ""X( ;" BACM1/n
C .%&< ED1%("9 D0 F0 9 " /9%+S-<, ."9% 6O",)9%v ∈ H1(RN )
) "#$%+9-P
vn(x) = v ? ρn(x) =
∫
RN
ρn(x− y)v(y)dy,
Q"% 2 "',) -++C∞ )39 Q ∇vn = (∇v) ? ρn 0 .M9%# 6O, % <Q
vn C 9 + ,)9%.M =)P ∇vn F ,)P.M)93.M)9 v C 9)+ ,)9%.M ∇v F " L2(RN ) F ,4 27).%( I+% v C 9 + ,)9%.M =)P ∇v F 2I, 9%PG):-7 " "+%9< I68,9%(9 "(% > 9)3 - L2(RN ) C .%&7 :J - >
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E / # +Q7 " % %
D"0 TP0E/F !9)+&- 4 )9 $ K 9"9*68,9%(U "L2(RN ) 0 G*9 29!"'QG4 K +9$Q (+%&9
vn-# '%W "")3 +"+:, ,) 0 P%& ζ ∈ C∞
c (RN )9 Q
0 ≤ ζ ≤ 1)
ζ(x) = 1+% |x| ≤ 1ζ(x) = 0
+% |x| ≥ 2 0 +ζn(x) = ζ
(
xn
) C .%&'( D1%9 D0 F ) +9$Qvn "#$%+9-P
vn(x) = vn(x)ζn(x) 0 .M9%#*65-, %() *-++%1Q"vn C 9)+ ,)9%.M ∇vn F, P.M)9.M)9
v C 9 + ,9%(. ∇v F " L2(RN ) 0
G :J - > D"0 P0 " +%&9 " ".M)+!9 "%(B "X % = )+$,) 2 ", "+Q ", *% "%(/9* " 9 +"/<+%.-@, X;%'/. ,*#:J - > D0 D0E " +%&9 " 4 +, ! 9%)RN 0
=A@B?A@B< -Ω2+ 4(/&)4.5"+2&.5(5 .40 - 5 # % '01.2, C1 &.4,- ( 2-
Ω = RN+ 01&.5"
-0 .- 2+ 4( &5 )+ 45 # 5 & 01&.( 7 (9+ P # H1(Ω) # .(9 H1(RN ) 4- "+84(* ! 0 - 2.+-&.(0 - ( .- 5 2&.(+- (949+ ,0B0 49 &.45+ &)4+ v ∈ H1(Ω) : Pv|Ω = v : ‖Pv‖L2(RN ) ≤ C‖v‖L2(Ω) : ‖Pv‖H1(RN ) ≤ C‖v‖H1(Ω)& 01 2&.(92+ .(+
C > 0 # ( # ,40 7 (+ # Ω :; <>='? ?A@B< N W4 ;- W- " +9779 +"/"
Ω = RN+ 0 S-9 x = (x′, xN )/. ,
x′ = (x1, ..., xN−1) 0 %& v ∈ H1(RN+ ) 0 ")#%&
Pv(x) =
v(x′, xN )+%
xN > 0v(x′,−xN )
+%xN < 0.
.M)9%	 Q"1 ≤ i ≤ N − 1
∂Pv
∂xi(x) =
∂v∂xi
(x′, xN )+%
xN > 0
∂v∂xi
(x′,−xN )+%
xN < 0,
) QP"∂Pv
∂xN(x) =
∂v∂xN
(x′, xN )+%
xN > 0
− ∂v∂xN
(x′,−xN)+%
xN < 0.
)M%&9 +').%( 9 @+% v 2 *68",)9% 9 % > 9 -%(@" , ++%&9 29%&#$, -9%
(-99QvG4 2I "9%(.;"3Q:65-%; 0 @ )%$- %(,)%" I-9 9 C 65-, % F Q"%229%&# , %&9 C %( 6O"*- *9%(%+)!( C)+9%X$-9 $)R%Y "
Pv ! .M% " %" 7 ")-%( F0 " "%&I ",3 I"99%)9 I " +%&9 II54 -9 3 "39" )PP/. ,( , "29
C =√
2 C -@,- Ω = RN+ F0%
Ω 23" .M+3;9" 9 %)< " , (++ C1 P "9%%+* -9 3 L,/+9 3, $" +I ") X,
Ω = RN+ 0" X9 9 $-P1()"/9%( :( )#%&9% 0 TP0 W4 "X.M)+9 "(%)N%I)R%29U9 , .9 *#% "
Ω-X " .M+9
(ωi)0≤i≤I 0 %+9P "%&S-(-9 -+9%&9% *54 %&9 *++, % K , *9), ".P9) P,4 2 =LK-= "%&9! ")@68",)9% (θi)0≤i≤I
"C∞
c (RN )9 Q
θi ∈ C∞c (ωi), 0 ≤ θi(x) ≤ 1,
I∑
i=0
θi(x) = 1 "
Ω.
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$# $ # # # $ 2!% E µ
ωi Q
yN
y′ω0
φi
∂ΩΩ
Q+
FF F®ADQRORHpE[PU4GA_[PHpEfl] V6CU_V60HQROQSX0V6[PNWHQpC 4 )R%(29), 4 S9 $-+9%&9% "B54 "%&9B 2!, (++%(Q F .M%! 9J" > 0 T0 "" " &% 0 F .X ")#%& Pv
+" (X689Pv =
I∑
i=0
Pi(θiv),
,J$Q")/9 "Pi
2X ")#%(, "ωi 0 θ0v
2 K +-+S, ,) ""Ω "#$%&
P0(θ0v), *54 R9)+%
θ0v$/ )9 J-9'
Ω 0 1 ,J$Qi ∈ 1, ..., I - φi
54 -"(%,/9%YQP"%G+-26O-9ωi
" " %S "9)68)9 ", Q C .M%&( )#$"%9% 0 TP0 )<( D1%9 D"0 DF +
wi = (θiv) (φ−1i
∣
∣
Q+)/. ,
Q+ = Q ∩ RN+ .
+9 6O,)9%wi
--+9% K H1(Q+) 2 P -" .M%+%( "
∂Q+ ∩ RN+ 0 % 4 )9)$ $/ -
RN+ \ Q+ @ ;"9% "6O,)9%
wi ∈ H1(RN+ ) 0 " -9
9")'$/'9 )R%(wi
"*;"9 "%&'"6O,)9%Pwi ∈ H1(RN ) C "9%%+X54 )/9 "
*"9 PQG4 .% < *, "2+9%&9
RN+ F0 9).% < " ωi
)@ +Pi(θiv) = (Pwi) φi.
/:(*9 "(-9%&9C1 " φi
)7 "@+ %P.M)9+@;"9% I 79"9%)9): ")+%9) 7"Pi
)7 ",P 0
4 -9 7 " , -+9 :,- $'9%(%+ , % = ++7 27+ > 7, (++%(Q'3 W4 9% =%9-9% , 9 "" " "X;"9 "+ ")2+/9% 0 %@ 2B9" " ++ 9),J"%(Q354 %.PQP")9"@$-+68%<9< ")<@ "% 0 •
* *KJ ) S P-:>UH(div)
"N%+9P ""%(+G!+9I)+$,)%9)9 "%(-%9 +9 L2(Ω))
H1(Ω)N N68",)9%"
K .- "93.M),)99% 0 )@ +$-, * 23+ > 3"9% -:, +% <-%,/9%( C .M%:$-@R")( V /QP" 0 T0E F0; <>@B?A@ < ! "
H(div)"+ # (-)5
H(div) =
σ ∈ L2(Ω)N + ,0 49 divσ ∈ L2(Ω)
, C5D0 T MF
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E F # +Q7 " % %
&divσ
2+ 01 # - .5 (* " $')-'0 # σ)4/,(9 # 01 (- +-&.( : :!:
.M9%# 65-, %() Q" ,4 2 +$-, " %(;+ C ( "2+-9% 2 (%++ Y(),)9 "< "%(+S 4 R"9, %, F0 =A@B?A@B< 4(- # 4 5 & # 4- + 2 01.- 5
〈σ, τ 〉 =∫
Ω
(σ(x) · τ (x) + divσ(x)divτ (x))dx C5D"0 T/DF+ # 01 ( &.5"7 ‖σ‖H(div) =
√
〈σ, σ〉 0 ! " H(div)"+ 4( ! " # 8-0 "5+ :
"@ 3)+$,) < " P;(). @ " +9)39 +"&- "' " "+%9* " :68, =9%('9)% > 9 C ""'+9 ( "2+-9%UQ%1)2, " > 9 +%(%(%&9 K , )( ":J - > D"0 P0 F06 ) 9-
Ω2+ 4( &)4.5"+ 2&.5"(65 .40 - 5 # E '01)", C1 &.4 ,-( 2-
Ω = RN+ 01&.5"
C∞c (Ω)N "+ # (9, # )(9 H(div) : ' %(9) 9I 354 )+$,)
H(div) 27QG4 %")97 @ " +9) 9J" > @ ":+,)<*689X! " 9) .M , ", 9!%"@ "*9 (-9%&9XQP" -@54 +$-, ; .
H1(Ω) 0" )P+% σ
-$/+9% K H(div) L, +( 'QW4 "@+ "( , X;%$-%+ "<+ @ ")9%.M)
$-+9% C '" 9"9 <, ! "H1(Ω) F %< '""%< ") + "
K ( ? & < +-σ · n +
∂Ω 0 " $/ )(7Q" γ0
" +%354 "%(, -9%B+-, @ "H1(Ω)
""L2(∂Ω) C .M%&( :J) > S "+-, D0 0EF )*Q" Im(γ0) = H1/2(∂Ω)
QP"%N 2' + = +, " "+X ""L2(∂Ω) C .%&< V /Q D0 0E $ F0 @ "%&
H1/2(∂Ω) "( -9*+%.9
‖v‖H1/2(∂Ω) = inf
‖φ‖H1(Ω)
9 GQγ0(φ) = v
QP"%")B65%&IB)+$,)' @H:,J C I ):B)+$,)' '%;)+ F0 )#$"%&7-9 H−1/2(∂Ω), (! $-N H1/2(∂Ω) 0
6 ) - @ ! < #" &.- +Ω4( &.4 .5+2&.5(5 .40 - 5 # '01.2,
C1 :%$ ( # (- + 0 "!*0 - 2.+-&)('&+5 (*&.5"7 0 )( γn
H(div) ∩ C(Ω) → H−1/2(∂Ω) ∩ C(∂Ω)σ = (σi)1≤i≤N → γn(σ) = (σ · n)|∂Ω
&n = (ni)1≤i≤N
2+ 01 (*&)57 0 4(9- + .+ 5- 45 3 ∂Ω :* ++ /!*0 - 2.+-&.( γn, 5 & 01&.(
.5 2&.(+- (94- + ( 4(* !*0 - ")+-&.( 0 -B( .- 5 E 2&.(9+- (49 # H(div) # .( H−1/2(∂Ω) : 6 0 4 "-σ ∈ H(div)
+φ ∈ H1(Ω) &)( ∫
Ω
divσφdx +
∫
Ω
σ · ∇φdx = 〈σ · n, γ0(φ)〉H−1/2,H1/2(∂Ω). C5D"0 T F; <>='? ?A@B< P%
Ω)2:.M+I9 %)3 @, (++ C1 M 4 " R)9, %, P0 TP0E 689%&:(*6O-9 W4 %P9)-9%G %&9 "(B %(.)9 ", "$/ -+9%+%.9∫
Ω
divσφdx +
∫
Ω
σ · ∇φdx =
∫
∂Ω
σ · nφ ds, C5D"0 T F
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$# $ # # # $ 2!% E "B " S6O",)9%"B9 "(% > 9) σ
)φ 0 9 /Q -9;"() Q L.P% -%& 9)9 "
P9%&9 - C5D0 T F G4 2 -"+9 Q 9%( %P9) ; +)( " C5D0 T F0 .%&*-%+ QP"!()' "R 9)9 "!M-,JS " C5D0 T F <;% + "φ ∈ H1(Ω)
)σ ∈ H(div) 0 9 "$-< +%&9 :6O,)9%39 "% > 9 ' "" H1(Ω)
H(div)
()@99 3 "",J"@ " C5D"0 T F +7)9)$ "" $/7, 9%(P"%93I39)93 @ 9%9<-$/(I, :":689% %&9 , 9%(P"S+'54 % X54 "%,-9%+,)
Im(γ0)-9
H1/2(∂Ω) 0 9 S689% %&9 4 ,)9%&<)R",)9 )P:, ' "3(X6O-9 C5D0 T F )@(X+-, *"9 γn(σ) 2< ",
;"% )#$"%(,)'" )P3 "H−1/2(∂Ω) 0
@;% < ")#%&<"++%G54 Q%. "H1
0 (Ω)"@54 +,
H(div) 0 )#$"%&3+ = +, H0(div)
H(div)
, '54 "J")9 ", ! "C∞
c (Ω) -
H(div) 0 4 2'), -9)+$,)@ " %(;)+IQ%P 4 %9)9" > 9 C +%"54 ".M)+ 2 9 "%( F ,)73+ = +$-, @ ") 68",)9% H(div)
"@ +-, -9 * 2<P 0* *) ) S P-:>US
W m,p(Ω)' ") "* ")#%&* ')+$,)
W m,p(Ω)* )P9%)
m ≥ 0)'"
" 9 1 ≤ p ≤ +∞ 0 +, )*+, 2+9"%9'+"*54 +, H7$-,J Lp(Ω) C .M% C5D"0E/F C5D"0 TF+F0 '""<54 .M"< "%& K ( V ) -Q D"0 TP0 ( -9%( "! )9%.M '6O%;! 4 )9 K
Lp(Ω) 0 "<.M@ ,! ")@(B )#$"%&9%( +"%(.9 0; <>@B?A@ < &.45 + &.4+ (9+-5
m ≥ 0 0 ! " # *& 2& 0 W m,p(Ω)
2+ # (-.5
W m,p(Ω) = v ∈ Lp(Ω)+ ,0 49 ∀α
. |α| ≤ m, ∂αv ∈ Lp(Ω) , C5D"0 T $ F& 01 # 5- " .5+-,0 0 ∂αv
2+ 3 52( # 52 )4 ,(9 $')-'0 :A %G "' B"9
‖u‖W m,p(Ω) =
∑
|α|≤m
‖∂αu‖p
1/p
.M)9%&#$!Q"W m,p(Ω)
2'" +, X "!H7$-,J 0 +$-, '+ -+9%, "(% > 9) '% =-9<"< <"9;" > 3 = %-%9) C Q""<W4 ; "<<%, %5$.M%<-<)R " T % F -%3"++% ": 3"9; > I%("%&9 3) -%(+ :, > ;9 @< @B?,= - 0 I 9 I97 " 2+/9%( 0 % Ω 2:"".M)+ 9 "%(P ;% +%Ω = RN
Ω = RN+
-9
+%p < N W 1,p(Ω) ⊂ Lq(Ω) ∀ q ∈ [1, p∗]
.M ,1/p∗ = 1/p − 1/N+%
p = N W 1,p(Ω) ⊂ Lq(Ω) ∀ q ∈ [1, +∞[+%p > N W 1,p(Ω) ⊂ C(Ω),
C5D0 T MF
.M),*%2),)9% ,)9%",4 2 =LK-= "%&9Q" W 1,p(Ω) ⊂ E.M @ "%&9QG4 %)R%29*"', 29
C9)(!Q"39u ∈W 1,p(Ω)
‖u‖E ≤ C‖u‖W1,p(Ω).
G7, 1-+9%, %)p = 1
)m = N
)219 -Q; ,/ ! ")+9)W+ > 1+% "*%" M-(%&9S "* C ; . 0
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E_¯ # +Q7 " % %
! "W N,1(RN )
- ( ,+ 2&.(+- (7 (9+ # )(9 0 ! " # $'&.( ,+-&.(9 "&)(+- (949 2&.5"(2 245RN (*&.+ Cb(R
N ) +>&)45 + &)4+ u ∈W N,1(RN )&.(
‖u‖L∞(RN ) ≤ ‖u‖W N,1(RN ). C5D"0 T F; <>='? ?A@B< %&
u ∈ C∞c (RN ) 0 1 x = (x1, ..., xN )
u(x) =
∫ x1
−∞
· · ·∫ xN
−∞
∂Nu
∂x1 · · · ∂xN(y) dy1...dyN ,
W4 54 " %&
‖u‖L∞(RN ) ≤ ‖∂Nu
∂x1 · · · ∂xN‖L1(RN ) ≤ ‖u‖W N,1(RN ).
C∞c (RN )
2 +7 "W N,1(RN ) C , ( +3 +9 , I :J" > D"0 D0(E ": +%&9"
H1(RN ) F0 ,P-3 +%&9< ;9% 754 % %&9 C5D"0 T F 79 u ∈ W N,1(RN ) 0 /%()"9 ( 68)9)9"9: "C∞
c (RN )"(*-93 "
L∞(RN ) 2)R",)9 )P
Cb(RN ) C X6O%&, 3 "R +$-, :P7(S 3"9 F0 ", C5D"0 TMF %%(Q<Q@ 76O,)9%3 W N,1(RN )+<, 9%P 3)<;-9 0
9* * Z 8POR=1 V - !QX(B "$- V ′ W4 U +, '%;)+
V 2!54 +) ; " *689 %-%&9
, 9%P I+"V 0 -:"%,-9% ":J" > E/T0E0E <9)"9 + /9%( V % ! P* $- "
L2(Ω) 2<%( " 9%&#$ K L2(Ω)
% = 0 "<-++%N ")#%&@! "$-N 4 )+$,)X ; . 0" 4 , ,)"+9 ", !(! $-N H1
0 (Ω)2 !$/+9%, %) -@( +%&9 0
; <>@B?'@B< # 4 0 # 0 ! " # & 2& 0 H1
0 (Ω)"+ "!,0
H−1(Ω) : $ ( (*&)+ 〈L, φ〉H−1,H1
0(Ω) = L(φ)
0 95 & # 4- + # # 4 0 - + /(+52 H10 (Ω)
+%&.( # 4 0>&.45 + &.4+ $'&.5"7 0 -( .- 5 6 2&.(9+-B(949L ∈ H−1(Ω)
+ + &.4+ $,&.(* ,+-&.(φ ∈ H1
0 (Ω) :
@,--,)9)9%+)', ! "H−1(Ω) 0 ")++!*9 +&/'+"%(. C .M%& " $&% F0
=A@B?A@B< 9 ! "H−1(Ω)
2+ 2.5 ! ,+ 5"- .5
H−1(Ω) =
f = v0 +
N∑
i=1
∂vi
∂xi
. v0, v1, ..., vN ∈ L2(Ω)
.
4+5 7 (+ # - + + &.4+ $'&.5"7 0 - ( .- 52 2&.(9+- (49"45 H10 (Ω) (*&.+ 2 L ∈ H−1(Ω) ,5"- + &.45 + &.4+
φ ∈ H10 (Ω)
L(φ) =
∫
Ω
(
v0φ−N∑
i=1
vi∂φ
∂xi
)
dx
. v0, v1, ..., vN ∈ L2(Ω) :
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[ # $" % "6% 2 E G , K 4 +,
H−1(Ω) )" ")#%& "".M U"9% U ")9%.-9% " ()
68",)9%S L2(Ω) C "*65-%; ,)9 QX( ")9%.M) 65-%(;" "B( #$%&9% D"0 TP0 F0 .,))@B"R " -9%(3 * )9%./9%(G-++"9"3* ,)9 3%$Q%)3) %(9-@QG4 3+:9"9
*.-/9* "S( ")9%.-9%U+ "* * %(2+9%;""9%" C ,4 2229) '" " %(9) 9* "!(9J" 9%! < "%2+9%;"9%< W4 "%&#$)@, ) %.M)9: C @ ! ")9%.-9% F0 *&.-B+
v ∈ L2(Ω) : &.45 1 ≤ i ≤ N &.(84+ # (- 5 4(*$,&.57 0 - ( .-B52 2&.(9+-B(949∂v∂xi
# .(9 H−1(Ω).501 $,&.57640
〈 ∂v
∂xi, φ〉H−1,H1
0(Ω) = −
∫
Ω
v∂φ
∂xidx ∀φ ∈ H1
0 (Ω), C5D"0 F 4- 5"-
∥
∥
∥
∂v
∂xi
∥
∥
∥
H−1(Ω)
≤ ‖v‖L2(Ω) .
-v ∈ H1(Ω) 01&)5 01 $,&.57 0 - ( .- 52 "&)(9+- (4 ∂v
∂xi
2& (* ,- # 8. %01 # 5- 2 $')-'0 # .( L2(Ω)# v :; <>='? & ?A@B< .M)9%&#$ 65-, %() Q X;"9 " 9%&9 C5D0 F 2 " 689% %&9!, 9%P*+
H10 (Ω) 0 / ,)+ QP" %(W)R%29 Li ∈ H−1(Ω)
9 GQP"
Li(φ) = −∫
Ω
v∂φ
∂xidx.
.M)9%&# "++%365,)%( )P Q" 54 "%(, -9%v → Li
2% %&9 , 9%P "L2(Ω)
-H−1(Ω)
$*QG4 !"9(S( )9%./9%( +" S" <68",)9%v9 "(% > 9) C ( )9% =.-9% 65-%;
v ∈ H1(Ω) F0 -@,)+ QP" "39 $ P9'$/@, 9%P%&9'(B )9%./9%K 9"9 6O",)9%
L2(Ω))<"9)
Li = ∂v∂xi
0
, S- 9J) > *9 "9)+ -9% V %()! 9-% QG4 @% =9%&#) "I W4 +$-, "'%;)+X.M , % = 0 G -9%(Q W4 % 9%&#$2 -%(
H−1(Ω))
H10 (Ω) 0 " )1 ")#%
H10 (Ω)
, " +" = +$-, 2+9%(, C -%( + F " L2(Ω) 0 3 S " K , %( "' W4 %( " 9%&#$ L2(Ω) C "%$ + 9P ""%&:+,- -%&9+) F7+ $- C ,4 2 229<"@, P.M 9% %7 < 2:"%.M)9+ F P ",<@ 77)%( 9%#)H−1(Ω)
)H1
0 (Ω) C .M ,< -"+9'9P ""%&:+,- -%&9 F0 N!+%9-9% , -+9 ,)9*)7+" ()2' ",H1
0 (Ω) ⊂ L2(Ω) ≡(
L2(Ω))′
⊂ H−1(Ω),
@%, +%<+@2+9%,)9 0 •
$ " ( $I(l%'!$ ¡%'$#(N9 1 -9%N%(+9P %&9 1 ""1, 7,J$-%&+91"S% X.M ,7(39J -9%: ") "%2+9%;9%(
C .%&3*, "9" % F0 P+9%,)9S+ + :W4 .M:3; +% '(S9J" 9% 3 "%2+9%;9%3" )#$"%& +$-, " P;(). 9 + 9 "9; > -"R %%9) C J%299%(Q )P %2+9%(;""9%" P 3Q"<9J" 9%' /9J /9%(QP"M+3"$/9 3-" > 7(): +$-, 3 P; .354 -9,J"*./9%(-9%" :9 + 9* 3 QP-9%"3R ")9%.M) 3$/+9%()( F0 -P
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E # +Q7 " % %
@9J -9%3 " %2+9%(;""9%" 2 "X, 9:"%&#$, -9 19" , +, ) , )"RS ")1 ,)9 9QP"%I( , $-%++ " -$Q9!$- + ")SQ)(S+X%() )P+9,))+99J -9%(), QP" " .M W4 )R+) 0 -"R ,)9)"9 6O%%)9 .M , ,))+9 9J" 9%% .M)P@ %&9%() @ 4 %9)+9)'+"< %( < ++ 9% () K (B;$-+! " "%2+9%;9%( 0 4 )2":9-9%265-%9!, +9 "; , "9%+%&9!Q @.M": ,)9%&@, +9*+ ,)9% Q%+*.M)"3" C + > F;"9)6 @,-9%,/9"G9 +' "(X9J)9%! " < %(2+9%;""9%" 0%
Ω" .M+3 "
RN 0 -9 C∞c (Ω) C D(Ω) F 54 +$-, ! " I6O",)9%" ', (-++ C∞
K +-+ ,)$-,)X -Ω 0 "%&
C∞c (Ω)
W4 " "+ $ = 9%( 1,4 2 =LK-= "%&9 QG4 ")#%& " 9% " , P.M)9), "C∞
c (Ω) 0 "%& QG4 " +%&9(φn)n≥1
"C∞
c (Ω), P.M)9.M)9φ ∈ C∞
c (Ω)+%
E0 !+"+< " φn9 29! -@ , ,)
K
Ω
T0 "@9@ &9% = %$ "%, α∂αφn
, P.M9"%&6O9) 3 -K
.M)9∂αφ 0
= = @ ='? @ ?'@B<>= D′(Ω) 2*( "$- D(Ω)
W,4 2 =LK/= %&9 54 +$-, " <6O-9 (%"%&9 L, 9%(P" $ +" D(Ω) 0 G G"%( )9G+ 4 97,/1%MW4 C@N "9 )#$"%I+D(Ω)
-%(3+%"() :"*-9% *, .)9 ", 0 % "@ )#$"%&@"9 , %+ )PD′(Ω)
"@ @ 93+'- ) @ ='? @ ?'@B<>= 0; <>@B?'@B< ( # - 2+5"-,4+-&.( T ∈ D′(Ω)
"+ 4(* $'&)57 0 -B( .- 5 245 D(Ω) 4- 5"-
limn→+∞
T (φn) = T (φ)
&.45 + &.4+ 24-B+ (φn)n≥1 # C∞
c (Ω) 4- 2&.(.5 .5"φ ∈ C∞
c (Ω).4 ,(9 # (9- ,- # 2"4 :
-9 〈T, φ〉 = T (φ) 9P ""%X $-(%&9 +9" "%2+9%;9%(
T ∈ D′(Ω)X"
6O",)9%φ ∈ D(Ω) F , "9P %& "S "%&9 %+ !54 %9 - "+ ∫
ΩTφdx 0 " )
.M)9%&#$'QP"'+%f 2:<6O,)9% C , 3%9 -; "" Ω F "9@ @ ")#%&:" "%2+9%;"9%
Tf$/
〈Tf , φ〉 =∫
Ω
fφdx.
-<, "+Q % 9%&#$*(X68,9%( )@( "%2+9%;9%( -++, % Tf ≡ f 0
B 7-++%$X"% D′(Ω) 4 3"9% 3, P.M9 ", F "%&IQG4 "3+%&9 Tn ∈ D′(Ω) < = <>= = @ ='? @ ?A@B<>= .M)9
T ∈ D′(Ω)+%5$"39"
φ ∈ D(Ω)
limn→+∞
〈Tn, φ〉 = 〈T, φ〉.
)+9', P.M)9), + ": ")3 "%2+9%;9%(I 2:"<, P.M)9 ,) )RP+ ) 6O%; C )R%(9 F ,/< (!, -+9 + K "*, P.M)9), *%9 -! L CM " 0 )#%++"S -%P9)$' @ ?A@B< = <>= = @ ='? @ ?A@B<>= F +% T ∈ D′(Ω)
G ")#%& ∂T
∂xi∈ D′(Ω)
$/〈 ∂T
∂xi, φ〉 = −〈T,
∂φ
∂xi〉 ∀φ ∈ D(Ω).
.M)9%&#$SQG4 ,9%(. @( )9%.M ∂T∂xi
2'"! "%2+9%;"9%G,4 2 =LK/= %&9 Q ' "%2+9%; =9%(@+<%"#% @ "9%(.;" 4 2' K 54 " @"<- "@%P9 )9< " < "%2+9%;9% 0 .M)9%&#$<"++%Q@+%f 27"@68",)9% ")9%.;< + "7, (-++% Q"939! )9%.M + 7 "
"%2+9%;"9%@, ", %( ".M ,9 ")9%.M "+ 0
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[ # $" % "6% 2 E H7% P9 , $ (@ -:(S ")9%.-9% 65-%(;" + "3 " )#%&9% D"0 T0 ,7$- =9%, "%(: @ ")9%.-9%-B+ "7 "%2+9%;"9% 0 3 -9"I +$-, Lp(Ω)
B +, P; .
Hm(Ω)+@ :+ = +, @ '54 +$-, 3 "%2+9%;9% D′(Ω) 0 " $-+9%, "%( .M)9%&#$ Q9"9 S , P.M)9), -S, )+$,) %%(Q S( , P.M)9 ,) - +
%(2+9%;""9%" C %B(U9), %"9PQ 265-++ F0" "#$W B %&9 "B B6O-9 " /9%.-9%(/9%(" () C Q@:.MI%(9)99)9 I, @ 7)M%&9 39 9Q@$/+9" F %%(QP" ), -9@+% 7)<3 )- " @ %&9 '- + " @ "%2+9%;9% 0
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0 E # +Q7 " % %
HaaH:FL0/4 Fu ∈ L2(Ω)
HYE,Of4XQSNPcA_6[PHTA_VERHCLE«rADNW6[PHTERN ∀ i rf4XYQRNWcAcOSNPU0C¡«rA_NP6[WHp ∣
∣
∣
∣
∫
Ω
u∂φ
∂xidx
∣
∣
∣
∣
≤ C‖φ‖L2(Ω) ∀φ ∈ C∞c (Ω)
°QSU_M<U0ERNOSNPU0C FF µL /H1(Ω)
HYE,OV6CHYERMLA_GHbf6H NW[P<HQROMFU0V6Q[WHTM6QRU4f4VLNOESGAD[eADNWQSH 〈u, v〉 =
∫
Ω
(∇u · ∇v + uv) dx
0K6XYU_QSaH FL µ6C∞
c (Ω)HpE>O9f4HYCLERHbfLADCLE
H1(Ω)¬OSK6XU0QRYaHbf6Hbf6HCLERNOSXp°QSU_M<U0ERNOSNPU0C FF µLP E ∀u ∈ H1
0 (Ω)ΩFU0QRC6X¤NWC6XjAD[WNOSX#f4Hb°§U_NWCLGA_QRX¤ ‖u‖L2(Ω) ≤ C‖∇u‖L2(Ω)
0K6XU0QRYaH FF µLPpµu→ u|∂Ω
A_M6M6[WNWGYAcOSNPU0CGU_CjORNWCV6HORK6XYU_QSaHTf4HBORQ&A_GHp f4HH1(Ω)
f6A_CLEL2(∂Ω)
0K6XU0QRYaH FF µLP¤ ∀u, v ∈ H1(Ω)¬«U_QSagVL[PHTf4HBQSHHYCF ∫
Ω
u∂v
∂xidx = −
∫
Ω
v∂u
∂xidx+
∫
∂Ω
uv ni ds
®2U0QRU0[P[eADNWQSH,FL µ6W ¯H1
0 (Ω)HYE,O[PH#ERU_VLE, HYERMLA_GH;f4HpE«U_CLGIORNWU_CLEf6H
H1(Ω)rGA_QSA0GOSXQSNWESAcOSNPU0Cf4HH1
0 (Ω) djV6NE] ADCLCjVL[PHYC0OERV6Q
∂Ω
0K6XU0QRYaH FF µL /6 [^] NWC'A>HYGIORNWU_Cf4HH1(Ω)
f6A_CLEL2(Ω)
HYE,OGIU0aMFA_GOSH¬ORKLXU_QSaHTf4H7XHY[P[WNeG&KF Ω<U_QSC6XbQRXY_V6[WNPHYQ&
0K6XU0QRYaH FF µL µ E ∀u ∈ H2(Ω), v ∈ H1(Ω)¬«U_QSagVL[PHTf4HBQSHHYCF ∫
Ω
v∆u dx = −∫
Ω
∇u · ∇v dx+
∫
∂Ω
∂u
∂nv ds
FFP °QSNPCLGNPMFADV4QSXYERV6[O&AcO&EERV6Q¡[WHYEHYERMLA0GIHYEf4H (U_<U_[WHdjV] NP[;«rADV4OADFE,U0[PV6aHYC0OGU_C6CFA8VeORQSH_
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ªJ%v %!$#
@BADCFEgGIHG&KFADM6NPORQSH\C6U_VFEbORHYQRaNWC6U_CLE#[] ADCLA_[P`4ERH\a\AcORKLXa\AcOSNWdjV6Hf4HpETXYdjVLADORNWU_CLEgADV4f4XIQRNW_XYHYEbMLA_Q,OSNPHY[P[WHYETf4HO>`M<HH[W[WNPM4OSNWdjV6HGU_aaHCLGXHA_V´®2KFADM6NPORQSHµ6°§U0V6QTaU_CjORQSHQ#djV6H\[WHYEM6QRU06[WaHYEA_V4[WNPaNPORHpE9ERU_CjO6NWHCMFUjE,XpEM<U_V6Q9GHYEH0 f MFH[W[PNWM4OSNWdjV6HpELG_] HpE>OR ZD f6NPQSHTd0Vv] HY[P[WHYEA_f4aHIORORHYC0OV6C6H9ERU_[WV4OSNPU0C_V6CLNWdjV6H0_HIOf4XM<HCFf6ADCjO2GU_CjORNWC6aHCjO2f4HpE?f4U0C6C6XHpE0C6U_VFE?ERV6NP0U_CLE Pm6~'~!ycq¢¤£'|\z6m6y¤om4nYoqFs!s!|jr|TMLQRXpE,HYC0OSXHADV¡®2KLA_M6NPORQSHµbHIOC6U0VLE-V4OSNP[WNWERU_CFE-[PHpE|t~§m6¢0|jtTw!|q?q<r|jz NWC0OSQRU4f4VLNO&EADV ®2KLADMLNOSQRH FL
vHM6[eADC\f4H9GHG&KLADMLNOSQRHHpE>O?[WHERV6NPcA_C0Op0@BADCLE[WA7(HpGOSNPU0C¡40/9CLU_VLE-H4M6[PNedjV6U_CFE-HCf4XOSA_NP[[PH;«U_CLGIORNWU_C6CLHaHCjOf4HB[] ADM6MLQRU4G&K6HBcADQSNWADORNWU_C6C6HY[P[WH9M<U_VLQ[WHT vA_M6[WA0GINWHC¡A¤0HYG;f4NP0HQ&EO>`M<HYEf4HGIU_CFf4NOSNPU0CLEgA_V4[PNWaNOSHYEY!9U_VLEJf4XaU_CjOSQRU0CLE#f4HpE#ychtYk!npm4nYtw W|juortYnY|s'¢0| |jnw Pk!s!or¢_ornYhw'|ttYq<rk'nYoqFs!t09U_VLEaU0CjORQSU_CLEADVFERERNdjV6H#GIHpEE,U0[PV4OSNPU0CLE3´os'oK3´ortp|sªnbk!s!| hs'|jy4AFo|\HIOd0Vv] HY[P[WHYEJ_XYQRNPLHCjOV6CxGHQROSADNWC´C6U0ag6QSHf6H¡~!y¤qF~!y¤ohjnYhtilk'm6onpm4nYorzF|tORQSYEJCLAcOSV6QRHY[P[WHYEJHIONPaM<U_QROSADCjOSHYEf4VMFU0NPCjOf4HTV6HTf4HpEADM6M6[WNeGAcOSNPU0CLEbM6QSNWCLGINWMFH#f6Va\Ac4NWagV6a4QSX0V6[WA_QRNPORX¤4 vA(HYGIORNWU_C6 µgQRHYM6QRHYCLf[WH;a¦aH;M6QRU0_Q&ADaaHa\ADNeE?M<U_V6Qfl] ADV4OSQRHpE?aU4f4Y[PHpE?M6[WVLEGIU0aML[PNedjV6XYEGIU_aaH GIHY[PVLN9f4H Whrm6tYnYo¢_onph ros'hm4ycotYh|U_VGH[WV6N9f4HpEhilk'm4nporq<s't w'| lnpq<|t_(4N[WAORK6XYU_QSNPHbfl] HI4NWE,ORHYCLGIHbHO9fl] VLC6NWGNOSXbHYE,OORQSYEE,HYag6[eADL[PHTADVGA0EM6QRXpGIXpf4HCjOYNW[C] HCHYE,OMLA_Ef4Ha¦YaH#f4HBORU_V6ORHYE[PHpEM6QRU0M6QSNPXORXYEdjVLA_[PNPOSADORNW_HYEY
0_
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0 / " +# 2# % n)
\ % "I 2$ *KJ*,+ 5?/;79=16=,5?/S P-8 O,=vM =1 US7U Z =>3L=R: "O!Ul1
9U_VLEGU_CLERNef4XQSU_CLE[WHbM6QRU06[WaHTADV4¡[WNWaNPORHpEERV6NPcA_C0O
−∆u = ffLADCLE
Ωu = 0
ERV6Q∂Ω
^4WpU'&
ΩHpE>OVLCU_V60HQRO<U_QSC6X#f4HT[^] HYERMLA_GH
RN FHIO f HYE,O9V6C ERHYGIU0CLfaHYag6QSH#djV6N'A_M6MLA_Q,OSNPHYC0OZ[^] HYERMLA_GHL2(Ω)
F ?] A_M6M6QSU4G&K6HTcADQSNWADORNWU_C6C6HY[P[WHbMFU0V6Q9XORVLf6NPHYQg6P¤HYE,O9GIU0CLE>OSNOSV6XHJf4HTORQSU_NeEXOSA_MFHpEd0VLHTC6U_VLEf6XIOSA_NP[W[WU_CLEY nYm6~?| ^ OSA_6[PNeESE,HYaHYCjO9fl] VLC6Hb«U_QSagV6[eAcOSNPU0C¡cADQSNeAcORNWU_CLC6H[W[PH0
@BADCLEbVLC6HJMLQRHYaNWQSHgXOSA_MFHNW['«rA_V4ObM6QRU0MFUjE,HYQ;VLC6Hg«U0QRaJV6[eAcORNWU_C³¤A_QRNeAcOSNPU0C6C6HY[P[WHJf4VM6QSUDL[PYaHTA_V4\[PNWaNOSHYET6P¤4GD] HYE,O,Zcf4NWQRHBdjV] NW[ª«rADV6OOSQRU0V6_HYQ?V6CLH;«U0QRaHB6NW[PNWC6XYA_NPQSHa(·, ·) 4V6C6H«U0QRaHb[WNPC6XpADNWQRH
L(·) 6HIOV6CHpE,MLA0GIHTf4H 9NP[WFHYQ,O V OSH[eEdjV6H\4WpERU_NPOXYdjV6NWcAD[WHCjOZ H0'QRU0V6_HYQ
u ∈ V ORHY[vdjV6Ha(u, v) = L(v)
M<U_V6QOSU_V4Ov ∈ V. 40/_
vH;LV4Of6HBGHIO,OSH;MLQRHYaNWQSHBXIOSA_MFHBHYE,OERHVL[PHYaHYCjOf4H;ORQSU_VL_HQ[eAT«U_QSagVL[WADORNWU_CcADQSNeAcORNWU_CLC6H[W[PH^40/_ U0C _XYQRNPLHQ&A\[^] XpdjV6NPcA_[PHYCLGIHM6QSXYGINeERHJA¤_HpG¡^4Wp9M6[PVFE9O&ADQ&f ADVGIU0V6QSEBf4H[WA¡ORQSU_NeERNPYaHXOSA_MFH0°§U_VLQgOSQRU0V6_HYQg[eA«U_QSagV6[eAcOSNPU0CcADQSNWADORNWU_C6C6HY[P[WHU0C´agV6[PORNWM6[WNPH[^] Xpd0VFAcORNWU_C ^4Wp#MLADQV6C6H«U0CLGOSNPU0C¡OSHYE,OQRXY_V6[WNWQSHvHOU_CNPCjOS_QSHbMLADQMFADQRORNWH_F®2HTGA_[WGV6[HYE,OM6QSNPCLGNPMFAD[WHaHCjO«U_QSaH[A_V³ERHCLE;U& [^] U_CERV6M6M<U0ERHg[^] HI4NWE,ORHYCLGIHJHIOb[WA¡QRXY_V6[eADQSNPORXgf4H[eA¡ERU_[WV4ORNWU_C
uAcFC³djV6HgOSU_VLE;[WHYEGYAD[eGIV6[eEHC<HpGOSV6XYE2E,U0NPHYC0O2[PNeGINPORHpE± [^] A_NWf6H9f4H[eAT«U0QRaJV6[PHf6H BQSHHC FL0//00U_NWQ?ADVLESERN'rµL0G_,U0C¡OSQRU0V6_H
∫
Ω
fv dx = −∫
Ω
∆uv dx =
∫
Ω
∇u · ∇v dx−∫
∂Ω
∂u
∂nv ds.
^4 µ0®2U0aaH
uf4U_NPOESAcOSNWE,«rADNWQSHV6C6HGIU0CLf4NPORNWU_CxA_V4[PNWaNOSHYEf4H@9NWQSNWG&K6[WHIOp
u = 0E,VLQ
∂Ω!U_CG&KLU_NeE,NPOTV6CHYERMLA_GHf6HNW[P<HQRO
VORHY[?djV6HORU_V6ORH«U_CFGORNWU_C
v ∈ V 0XQSNLH\A_VLERERNv = 0
E,VLQ∂Ω
6@BADCLEGHTGA_EY4[^] XY0AD[WNPORX\6 µj2f6HNPHYCjO∫
Ω
∇u(x) · ∇v(x) dx =
∫
Ω
f(x)v(x) dx.6 F
°§U0V6Q2djV6H;[PH9ORHQSaH;f4H;0A_VLG&K6Hf4H4 Fj?ADNPO2V6CE,HYCLENW[<ERV WOd0VLH ∇u HO ∇v A_M6MLA_Q,OSNPHYC6C6HCjOZL2(Ω)
rGU_aMFUjERA_CjORH\MLA_QgGIU0aM<U0ESADCjOSHpvHIOgM<U_V6QdjV6H\[PHORHQSaH¡f4H¡f6QRU0NOSHf4H³6 FTADNPOA_VLESE,NV6C ERHCFE\NP[;E,V WOdjV6HvADM6MLA_Q,OSNPHYC6C6H Z
L2(Ω)rU_CAE,VLM6MFUjE,X djV6H
f ∈ L2(Ω)I°-A_QGU_CLERXYdjV6HYCjOY-V6C G&K6U_NP Q&ADNeE,U0C6CLA_6[PHMFU0V6Q\[^] HpE,MLA0GIHf4H NW[WFHYQ,OHpE>O
V = H10 (Ω)
[WHERU_VFE> HYERMLA_GH;f4HH1(Ω)
f4U0CjO[PHpEX[WXaHCjOSEEY] ADC6CV6[WHCjOE,VLQ[PHT<U_Q&f∂Ω
C GU_CLG[PVFE,NWU_C[eA«U_QSagV6[eAcOSNPU0C¡cADQSNeAcORNWU_CLC6H[W[PHbM6QSU_M<U0ERXHBM<U_V6Q#^4Wp2HYE,OH
OSQRU0V6_HYQu ∈ H1
0 (Ω)OSH[vdjV6H ∫
Ω
∇u · ∇v dx =
∫
Ω
fv dx ∀ v ∈ H10 (Ω).
^4_
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[ nn " ) $#$7 # _pµ NWf4HYaaHYCjOY¤C6U_VFE'A¤0U_CLE«rADNPO!V6CGIHYQ,O&ADNWCTC6U_aJ6QRHf4HG&K6U_NPBM<U_VLQ!A_QRQSNP0HQvZ6 0 fl] A_V4ORQSHYEG&K6U_NP¡C6U0VLEADV6Q&ADNWHCjOGIU0CLf4V6NPOZf] A_V4ORQSHYE«U0QRaJV6[WADORNWU_CLEcADQSNeAcORNWU_CLC6H[W[PHpEMFUjERERNPL[PHpE6 'A,A>VFE>ORNPFGADORNWU_Cf4H´6 0E] HlCªHYGIORV6HYQSAf4U0CLGAMFUjE>OSHQSNPU0QRN:HORU_V6O¡fl] A_FU0QSf-[WAf4HYV44NPYaHXIOSA_MFHGIU_CFE,NeE>OSHJZ_XYQRNPLHQbdjV6H4_9A_f4aHOB6NWHCV6C6HV6C6NedjV6HE,U0[PV4OSNPU0CªM6V6NeEB[WA¡ORQSU_NeE,NWaHJXIOSA_MFHd0VLH;[eAgERU_[WV4ORNWU_Cf4H^4_?HYE,OA_VLESE,N<V6C6HbE,U0[PV6ORNWU_C¡f4VM6QSU_6[WaH;A_V4\[PNWaNOSHYEb6P¤9f6ADCFE2VLCE,HYCLEZM6QRXpGINeE,HYQ&I npm4~?| QXXYERU_[WV4OSNPU0Cf4HT[eAJ«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WH_
@BADCFETGIHIORORH\f4HYV44NPYaHXOSADM<HC6U_VLEB0XQSNLU0CLEBd0VLH[eA¡«U_QSagV6[eAcOSNPU0C³cADQSNWADORNWU_C6CLH[W[PH4_A_f4aHIO\V6CLH E,U0[PV4OSNPU0C V6CLNWdjV6H0°§U_V6Q\GIHY[WAC6U0VLEV4OSNP[WNWERU_CFEJ[WH 0K6XYU_QSaHf6H 'AcNP[W_Q&ADaµ6 µ6WBf4U0CjOC6U_VLE0XQSNFU_CLE[WHYEK`MFU_ORK6pE,HpEA¤_HYG;[WHYEC6U_OSADORNWU_CLEa(u, v) =
∫
Ω
∇u(x) · ∇v(x) dx HIO L(v) =
∫
Ω
f(x)v(x) dx.
;C _U0NO«rA_GNP[WHaHYC0O\HYCxV4OSNP[WNeERA_C0O[^] NWC6XjAD[WNOSXf4H®A_VLG&K`j (4G&K ADQKJ¡djV6HaHYE,OVLC6H¡«U0QRaH6NP[WNWC6XYA_NPQSH¡GIU0CjORNWCjVLHERV6Q
H10 (Ω)
HIOdjV6HLHpE>OV6C6H«U0QRaH[PNWC6XYA_NPQSH¡GU_CjORNWCV6HERV6Q
H10 (Ω)
@9HML[PVLEYvHYC_HYQ,OSVf6H[] NWC6XjAD[WNOSX¡f4H°§U_NWCLGYADQSXr_U0NPQ#[WH®2U0QRU0[P[eADNWQRH FL µ6W / U_CV4OSNP[WNWERHNeGIN[PHGADQ&A_GIORYQRHb<U_QSC6Xgf4HJ[] U_V60HQROΩ<[eA«U_QSaH#6NW[PNWC6XYA_NPQSH
aHYE,OBGUjHYQSGNP0H_LG_] HpE>OR ZD f6NPQSH#djV] NW[HI4NWE,ORH
ν > 0OSH[vdjV6H
a(v, v) =
∫
Ω
|∇v(x)|2dx ≥ ν‖v‖2H10 (Ω) ∀ v ∈ H1
0 (Ω).
®2U_aaHH1
0 (Ω)HpE>OBV6C HYERMLA0GIHgf4H NW[P<HQRO\_U0NPQ;[WA°QRU0MFUjE,NPORNWU_C FF µL jLORU_V6ORHYE;[WHYE;Kj`M<UDORK6pE,HpEBf6V 0K6XYU_QSaHf4H 'AcNP[W_Q&ADaµL µLPE,U0C0OTESAcORNeE,«rADNPORHYEBHObU0C³M<HV4Ogf6U_CLGGIU0CLGI[WV6QSHd0Vv] NW[HI4NeE>OSHbV6C6HTV6C6NedjV6H#ERU_[WV4ORNWU_C
u ∈ H10 (Ω)
f6HB[eA«U_QSagVL[WADORNWU_C¡cA_QRNeAcOSNPU0C6C6H[W[WH^4_I.¡|+3´m4ycilk!| Q] Q]K^xU0VLE0HQSQRU0CLEM6[WVLE[WU_NWCADV ®2KLA_M6NOSQRH djV6H02f6A_CLE\[WH³GA0E\M6QSXYERHCjOYGIU_aaHB[WAg«U_QSaH;6NW[WNPC6XpADNWQRH
aHYE,OE,`aXIOSQRNedjV6H_NW[lHI4NWE,ORHBV6CADV4OSQRHbADQS_VLaHYCjOdjV6HB[PH 0K6XYUDQRYaHBf4H; 'AcNP[W_Q&ADa MFU0V6Q2GU_CLG[PVLQRH00C¡HlCªHIOY[eAgERU_[WV4ORNWU_Cf6H;[eAb«U_QSagVL[WADORNWU_C\cADQSNWADORNWU_C4C6H[W[PHTHpE>Of6A_CLEGIH#GYA_E[^] VLC6NWdjV6HbM<U_NWCjO9f4HTaNPC6NWagVLa f4HT[^] XYC6HQS_NWHbf4XILCLNPHTMLA_Q
J(v) =1
2a(v, v)− L(v) ∀ v ∈ H1
0 (Ω)
_U0NPQ9[WA\°?QRU0MFUjE,NPORNWU_C³6 /60G_L°-A_Q9GIU0CLE,XpdjV6HCjOY<E,N'U_C³f6XaU_CjORQSH#djV6HJA\V6C³V6C6Ned0VLHgM<U_NWC0Of4HTaNPC6NWagVLaLU_CAA_NPCFE,NlU04ORHYCjV[eAE,U0[PV4OSNPU0Cf6HT[WAJ«U_QSagVL[WADORNWU_C¡cA_QRNeAcOSNPU0C6C6H[W[WH_ •
npm4~?| \ djV6NW¤A_[PHYCLGIH#A¤0HYG;[] XYdjVLADORNWU_C 'AOSQRU0NWERNPYaHXIO&ADM<H9[eA9f4HYQRCLNPYQRHHO'[eAM6[WVLE!f4X[WNWGYAcOSHpGU_CLERNeE>OSH?Z_XYQRNPLHYQ'djV] HCTQSXYERU_[W¤A_CjO[WA¡«U_QSagV6[eAcOSNPU0C cADQSNWADORNWU_C6CLH[W[PH4_9U_CA¡LNPHYCQSXYERU_[WV [WHJM6QSU_L[PYaHADV6 [PNWaNOSHYE\6P¤ªHIOZ´M6QSXYGNWERHQf6ADCLEdjV6H[TERHCLE[eA ERU_[WV4ORNWU_C f6H¥^4_\HpE>OADVFERERN9V6CLHERU_[WV4ORNWU_C f4H¥^4WpI2Cfl] A_V4ORQSHYEORHQSaHYEYLNW['EY] A__NPO9fl] NPCjORHYQRMLQRXORHQ9[WA«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6CLH[W[PHTHO;f4H#QSHIOSU_V6QSC6HQZ[] XYdjVLADORNWU_Cj°§U_V6QGIHY[WATU0CM6QSU4GIYf6H;ADV6a¦YaHpE?NWCjORXY_Q&AcORNWU_CFEMLA_Q2MLA_Q,OSNPH;d0VLN<U0CjOGIU0CLf4V6NPOZ[WAJ«U_QSagVL[WADORNWU_CcADQSNeAcORNWU_CLC6H[W[PH0aA_NWEHYCERHCFE2NWC_HYQSERH_HIOHYC[WHYEA>VLE,ORNPFA_C0OERU_NW_C6HYVLERHaHCjOY
Ecol
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lyte
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pas
repr
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re.
0#F " +# 2# % n)
®2HO,ORH?A>VLE>OSN<GAcOSNPU0C\HYE,OORQSYE-«rA_GINW[WHE,NF[] U_C\ERV6M6M<U0ERHd0VLH[WATERU_[WV4OSNPU0Cuf4H[eAB«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WH6 0JHYE,OQSX0V6[PNWQSHrM6QSXYGINeERXaHCjO\E,N
u ∈ H2(Ω)JHIOdjV6H[^] U0V6_HYQ,O
ΩHYE,OA_VLESE,N4QSX0V6[PNWHQpDGIHdjV6HCLU_VLE§«rA_NWERU_CFE-f6ADCLEV6C\M6QSHaNPHYQ§ORHYaMFE0CHlCªHIOY0NP[<E,VWO?f] NWC_U4djV6HQ[eAJ«U_QSagV6[WHTf4HBQSHHCFF /'/_djV6NlC6U0VLEf4U0C6C6H_4M<U_VLQ
v ∈ H10 (Ω)
∫
Ω
∇u · ∇v dx = −∫
Ω
v∆u dx
MLV6NWESdjV6Hv = 0
ERV6Q[WHbFU0QSf∂Ω
/;CHYCf6XYf4V6NPO9AD[WU_Q&E∫
Ω
(∆u+ f) v dx = 0 ∀ v ∈ C∞c (Ω),
GHd0VLNNPaM6[WNed0VLH_HYC¥0HQRORV f4V ®2U_QSU_[W[WA_NPQSH FF /6 /6-d0VLH −∆u = ff6ADCFE
L2(Ω)?HIOU_C A[^] X0A_[PNPORX
−∆u = fM6QSHYESdjV6HbMLADQRORU0V4Of6ADCFE
Ω.6 ¯
@;H\M6[WVLEY'ERNΩHpE>O#VLCU0V6_HYQ,OT<U_QSC6X\QSX_VL[PNWHQ#f4HG[WA0ERERH C1 vA_[PU0QSEB[WH0K6XU0QRYaHf6HORQ&A_GIHFF µLPpµgrU_VM6[PVFE§M6QRXpGINeE,XYaHCjO§ERU_C®2U_QSU_[W[eADNWQRHFL µ6W ¯ vA4W\QSaHdjV6HORU0V4ORH«U_CFGORNWU_Cf4H
H10 (Ω)AV6CLHbORQ&A_GIHTERV6Q
∂ΩCV6[W[PH#f6A_CLE
L2(Ω) ;CHYCf4XYf4VLNOp6HCMLA_Q,OSNWGV6[PNWHQp4djV6H
u = 0M6QRHpERdjV6HbMLA_Q,OSU_V4OERV6Q
∂Ω.60G_
;CAf4U_CFGb6NPHYCQRHORQSU_V60X9[^] Xpd0VFAcORNWU_CHIO[eA\GIU0CLf4NPORNWU_CADV4[PNWaNOSHYEf4H\^4Wp(NL[] U_CJC6HE,V6MLMFUjE,HM6[WVLE-djV6H[WAbERU_[WV4ORNWU_C
uf4HT4_vHIO-[^] U0V6_HYQ,O
ΩERU_CjO§QRXY_V6[WNWHQ&EpNP[6«rADV4OOSQSA¤cA_NP[W[PHYQfLA¤¤A_CjOSAD0HgU0CCLH;M<HV4OM6[WVLEV4OSNP[WNeE,HYQ[eAg«U_QSagV6[WHBf6H BQSHHYCFF /'/_2djV6NªC6XYGHYESE,NPORHdjV6H
u ∈ H2(Ω)I ;C C6UDOSH
σ = ∇u djV6NHYE,OgV6C6H¡«U_CLGIORNWU_C ZcA_[PHYV6QSE#_HYGIORU0QRNWH[W[PHpE#f6A_CLEL2(Ω)N c°§A_Q§[] NPCLX0A_[PNPORXf4H9®A_VLG&K`0!(4G&K ADQKJ_IU0CJf4Xpf4V6NPO?f4H[WA9«U_QSagV6[eAcOSNPU0CgcADQSNeAcORNWU_CLC6H[W[PH^4_djV6H_LMFU0V6QORU0V4O
v ∈ H10 (Ω)
∣
∣
∣
∣
∫
Ω
σ · ∇v dx∣
∣
∣
∣
=
∣
∣
∣
∣
∫
Ω
fv dx
∣
∣
∣
∣
≤ C‖v‖L2(Ω).^4
®2U0aaHC∞
c (Ω) ⊂ H10 (Ω)
96 C] HpE>OQSNWHC fl] A_V4ORQSHdjV6H[PH GQRNPORYQRHfl] HI4NeE>OSHCLGH fl] V6C6Hf6NP0HQS_HCFGIH\«rA_NP6[WHf4Hσf6ADCFE
L2(Ω)0U_NWQJ[eA³@;XIFC6NOSNPU0C FF /6 ¯ HIO[PH HYaaHIFL0/4 GDgdjV6N0XQSNFH_4M<U_V6QOSU_V4O
v ∈ C∞c (Ω)
∫
Ω
σ · ∇v dx = −∫
Ω
divσv dx.
;CHCf4Xpf4V6NPO9f4U_CFGBdjV6H∫
Ω
(divσ + f) v dx = 0 ∀ v ∈ C∞c (Ω),
GH#djV6NlNPaM6[WNed0VLH_6HYC0HQRORVf4V³®2U_QSU_[W[WA_NPQSH FL0/40/44djV6H −divσ = ff6A_CLE
L2(Ω)6°-ADQGIU0CLERXIdjV6HYCjO
divσ = ∆uADM6MFADQRORNWHCjOZ
L2(Ω)Q&ADMLMFHY[PU0CLEdjV6H
div∇ = ∆?HO¡U_C QSHIOSQRU0V6_H
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[ nn " ) $#$7 # _¤[] XYdjVLADORNWU_C ^4 ¯ ;C QSHIORQSU_VL_H#[eA¡GU_CLf4NPORNWU_C A_V4[PNWaNOSHYE60G_GIU0aaHJM6QSXYGIXpf4HaaHCjObE,N[] U_V60HQRO
ΩHYE,OQRXY_V6[WNPHYQ¡f4H³G[WA0ERERH C1 2(4N Ω
C] HYE,O¡MLA_EQRXY_V6[WNPHYQYA_[PU0QSEU_C C6H M<HV4OMLA_ENPC_U4djV6HYQT[PH 0KLXU_QSaHf4HORQ&A_GIH FF µLPpµMFU0V6Q#U04ORHYC6NPQ60G_v9XYADCLaU0NPCFE'[PH¡ERNWaML[PH\«rA_NOfl] A_M6MLA_Q,OSHC6NWQ;ZH1
0 (Ω)HYE,O;V6CLHg_XYC6XQ&AD[WNeERADORNWU_Cf4Hg[eAGU_CLf4NPORNWU_CADV4[WNPaNPORHYEBf4H@9NWQRNeG&K6[WHIOMFU0V6Q9V6C³U_V60HQROC6U_CQSX0V6[PNWHQp6HO;U_C GU_CjORNWCV6HQ&A\Z\XpGIQSNPQSH71 qLy43´|r|+3´|sªnBdjV6H
u = 0ERV6Q
∂Ω ?C GIU_CFGI[WVLE,NWU_CC6U_VFEA¤_U0CLE2f6XaU_CjORQSXb[PHTQSXYERV6[POSADOERV6NWcADCjOY
£'hqFy4:[3 | %]%] `>bdcfeΩ g h b g i4jlk e b kuh m j RN w `>b4cfe f ∈ L2(Ω) w j- c p e jg h>jg h c gNj
p b g e~cBb h u ∈ H10 (Ω)
m j7 qr b kutg q eKcBb h i4q4k c q eKcBb hNhjfjz w w j2|Dg+p o u i k c ' j−∆u = f |>kvjup gNj2|Nq4k e b g em qdhp Ω, j e u ∈ H1
0 (Ω).^4 0
`_c$b h6pug |'| b plj jhE| g+p gNj Ω jlp e k g c jk m j q4p@pj C1 o q4 b k-p u jup e p b Dg eKcBb h m g |k b + t9jq4g -) c t cfe jlpz w-)q4g pjhp b)
−∆u = f |kvjlp gNj|Nq4k e b g e2m q4hNp Ω, u = 0 |>kvjlp gNj|qdk eb g e pug k ∂Ω.;C A_M6M<H[W[PH[eAE,U0[PV6ORNWU_C
u ∈ H10 (Ω)
f4H[eA«U_QSagV6[eAcOSNPU0C¥cA_QRNeAcOSNPU0C6C6H[W[WH^4_tpqFk%=nYorq<sz6m4ycom4nYoqFs!s'|r|gf6VM6QSU_6[WaH9A_V4[WNWaNPORHpE;^4WpI_°-ADQ2V6CQ&A_GGU_V6Q&GIN6f6H9[eADC6jAD_HLNPHYCGIU_aaU4f4H_ªq<s w!oycm´ilk!| Pk!s!orilk!|tpqFkvnporq<su ∈ H1
0 (Ω)w'|m 1 qLy43k'm4nYoqFsz6m6y¤om+=nYorq<s!s'|r| Q] " |jtYn Pk!s'oilk'|¥tpqFkvnporq<s w!k ~!ycq §B:+3´|xm6kvu roB3 onY|t Q]K^ "c-®2HO,ORHADM6M<H[W[WADORNWU_CHYE,O6NWHCE,LQA>VLE>OSNFXHTMLA_Q[WH 0K6XYU_QSaH#40/40/4 'AJERU_[WV4OSNPU0C¡f4HJ^4WpIdjV6H;C6U_VFE?0HC6U0CLE2f] U04ORHYC6NPQpjC6H;0XQSNFH q|k cBb k cL[^] XpdjVLAcOSNPU0CHIO[WAGIU_CFf4NOSNPU0C\ADV4g[WNWaNPORHpE?djV6Hf6A_CLE§V6CERHCLE¨>«rADNW6[WH©_GD] HYE,O,Zcf4NWQRHM6QSHYESdjV6HMLADQRORU0V4O9U0Va¦YaHM6NPQSH¡M<U_VLQJ[eA GIU0CLf4NPORNWU_C A_V4[PNWaNOSHYEERN2[^] U0V6_HYQ,OJC] HpE>OMLA_EJQSX_VL[PNWHQ ;C´MLADQS[WH¡AD[WU_Q&Egf4HtYq<rk'nYoqFs1^mLo §r|MLADQBU_MLMFUjE,NPORNWU_C³A_V4 E,U0[PV6ORNWU_CLE9«U_QRORHpE;djV] U_C³A_V6Q&ADNPO;M6VHYERMFXYQRHYQ9U_6ORHCLNPQf6ADCLE'[eA«U_QSagVL[WADORNWU_C#GI[eA_ESERNWdjV6H?f6HB^4Wpr_U0NPQ'[WAn(U_VLE,E,HpGORNWU_C#µLP0 /0¤@9H2a¦aH_¤U_CJADM6M<H[W[PHMLADQR«U_NeE[WA«U0QRaJV6[WADORNWU_C¤A_QRNeAcOSNPU0C6C6HY[P[WH 1^qFy43k!rm6nYorq<s 1^mLo §r|f4Hb[^] XpdjVLAcOSNPU0C
.¡|+3´m4ycilk!| Q] Q] \C«rA_NOp6[WA\ERU_[WV4OSNPU0C¡«rA_NP6[WHTM<HV4O9¦ORQSHTV6C6H#ERU_[WV4ORNWU_C«U_QRORH#ERN[WH#E,HpGIU0CLfaHYag6QSHfHpE>O9ML[PVLE;QSX_VL[PNWHQpF±9V4ORQSHaHCjObf4NOpF[^] XYdjVLAcOSNPU0CHIOb[WA¡GIU_CFf4NOSNPU0C ADV4[WNPaNPORHpE;f4H4Wp\MFHYV6_HYC0O¦IORQSH_XYQRNPLXYHYE\HYCV6C E,HYCLE¡G[WA0ERERNed0VLH_?G_] HpE>OR ZD f4NWQSHM<U_VLQ\ORU0V4O
x ∈ Ω2HIOORU_V6O
x ∈ ∂Ω4QRHpE,M<HYGIORNW_HYaHYCjOYF®;] HYE,OGHTd0Vv] U0CA_M6MFHY[P[WHbV6CQRXpE,VL[O&AcOf4HbQRXY_V6[eADQSNPORX9M<U_VLQ[WAE,U0[PV4OSNPU0Cr_U0NPQM6[WVLE[WU_NWC[PHg®2U_QSU_[W[eADNWQRHb40/40/'GDI •
.¡|+3´m4ycilk!| Q] Q] D[;«rADV4O6NPHYC GIU0aM6QRHYCLf4QSH[PHE,HYCLEMLQRXpGINeE¡f4H[] HI4M6QSHYESE,NWU_C∆u
fLADCLE[] XjAD[WNOSX^4 0f4V 0K6XU0QRYaH40/40/4§°§U_VLQ\V6C6H«U_CLGIORNWU_C djV6H[eGIU0CLdjV6Hvf4H
H10 (Ω)
C6U0VLEC] A¤0U_CLEgMLA_Ef4U0C6C6Xf4HE,HYCLEa¦aH«rADNW6[PH¤ZERU_C 'ADM6[eA_GNPHYC∆v
-°-A_QGU_CjORQSH_§M<U_V6Q[WAE,U0[PV4OSNPU0Cu ∈ H1
0 (Ω)f4HB[eAT«U_QSagVL[WADORNWU_CcADQSNeAcORNWU_CLC6H[W[PHJ4_IC6U_VLEA¤_U0CLEaU0C0OSQRXBdjV6H
∆uADM6MLA_Q,OSNPHYCjOZL2(Ω)
•°§U0V6Qd0VLH9[WH;M6QSU_L[PYaHBADV4\[WNPaNPORHpET4Wp?E,U0NO6NWHCMFUjE,XrADVE,HYCLEf4H ;A_f6A_a\ADQ&f _U0NPQ[WAB@9XLC6NPORNWU_C_4 µ0IpNW[j«rADV4OpcHCM6[WVLE-f4H[] HI4NeE>OSHCLGH2HO-f6H[^] V6C6NeGINPORXf4HERABE,U0[PV6ORNWU_C¤aU_CjORQSHQd0VLH[eAE,U0[PV6ORNWU_Cf4XM<HCLfGU_CjORNWC6aHCjO#f4HYETf4U0C6C6XYHYEY'®;] HpE>O#V6CLHGU_CLERXYdjV6HYCLGIHNWaaXpf4NeAcORHf4V 0KLXU_QSaHf4H 'AcNP[W_Q&ADa µ6 µ6Wga\A_NWEBC6U0VLEBHCf6U_C6C6U0CLE;V6CCLU_V60H[!XC6U0CLGIXHIOTV6C6HC6U_V60H[W[PHTf4XYaU_CLE,ORQ&AcOSNPU0C
Ecol
e Po
lyte
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pas
repr
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0 ¯ " +# 2# % n)
bycqF~?q<tYornYorq<s Q] Q] `Qb4cfeΩ g h b g i8jk e b kuh .m j RN o j e p bdcfe f ∈ L2(Ω) w q@|'| c -q e~cBb h g c $ f ∈ L2(Ω) rlq cfe b kukvjlpK| b h m kvj q p b Dg eKcBb h g h c gNj u ∈ H1
0 (Ω)m j6q r b kutg q eKcBb h
i4q4k c q eKcBb hNh>jlfj m jz w-6jlp e c h q c kvj.j e b h e~c hNgNj m j L2(Ω)m qdhp H1(Ω) w 2h.|Nq4k e~c lg c jk oc j - c p e jEg hj b hp e q4h e j C > 0
e jfj gNj o | b g k e b g e f ∈ L2(Ω) o b hq‖u‖H1(Ω) ≤ C‖f‖L2(Ω).
^4W E .|+3´m4ycilk'| %]%] ] NPCLX0A_[PNPORXT^4W E 'HpE>O?GHdjV] U_CADMLMFHY[P[WHV6C6H|tYnYoB3 m4nYoqFs w Phs!|jy8A<or|<?[P[WH0A_QSA_C0OSNOdjV6H[^] XYC6HQS_NWHf4H[eAERU_[WV4OSNPU0C¥HpE>O¡GU_CjORQKB_[WXHMLA_QGIH[W[WH f4H[WAf6U_C6C6XYH_ vHYEHpE>OSNPa\ADORNWU_CLEfl] XC6HYQR0NPH¡ERU_CjOJORQSYEJCLAcOSV6QRHY[P[WHYEfl] V6C M<U_NWCjOf4HV6H¡MLKj`4ERNWdjV6HHIOORQSYEJV4ORNW[WHYEf] V6CM<U_NWCjO9f4HbjVLHTaADORK6XYa\AcORNedjV6H_ •Yh+3´q<s'tYnycm4nporq<s#] 'A[WNPC6XpADQSNOSXf6H
f → uHpE>ObXNef4HCjOSH_ª°§U_V6QBU_4OSHC6NWQB[WAGIU0C0OSNPCV6NPORXJU_CMLQRHYCLf
v = uf6A_CLE[eAg«U0QRaJV6[eAcORNWU_CcADQSNWADORNWU_C6C6HY[P[WH\6 0
∫
Ω
|∇u|2dx =
∫
Ω
fu dx.
;Ca\A'A>U_QSHB[PH;ORHYQRaHTf4HTf4QSU_NPORHbZg[^] A_NWf6HBf4Hb[^] NWC6XY0AD[WNPORXbf4Hg®ADVLG&K`j (6G&KADQKJ__HIOU_CaNPC6U0QRHGH[WV6Nvf4HbjADVLG&K6HbMLA_Q[WA\GUjHYQSGNPNPORXbf4HT[eAg«U0QRaHb6NW[PNWC6XpADNWQRHν‖u‖2H1(Ω) ≤ ‖f‖L2(Ω)‖u‖L2(Ω) ≤ ‖f‖L2(Ω)‖u‖H1(Ω),f] U'&[^] U0Cf4XYf4VLNO[WHTQRXpE,VL[O&AcOY
9U_VLE!A¤_U0CLEvf4X6A,Z9f4NPO!djV6H2[WA«U_QSagV6[eAcOSNPU0CTcADQSNWADORNWU_C6C6HY[P[WHM<U0ESE,pf4H2ERU_VL_HCjOvVLC6H?NWCjORHQSM6QSXIO&AcOSNPU0C#M6K`4E,NedjV6HBGD] HYE,OY¤MLA_QvHI4HYaML[PH0p[WHM6QRNWCLGNPM<Hf6HYEvORQ&A¤¤A_V4TNPQRORV6HY[WE'HCJaXYGA_C6Ned0VLHppC«rA_NOpc[WAbERU_[WV4ORNWU_Cf4H[eA9«U0QRaJV6[eAcORNWU_CJcADQSNWADORNWU_C6C6HY[P[WHb4_'QSXYA_[PNeE,H[WHaNWC6NWagV6a fl] VLC6HXC6HYQR0NPHORQSYECLADORV6QSH[W[WH;HYCM6K`4ERNWdjV6HBU_VHCaXpGA_C6NWdjV6H¤6 vHBQSXYERV6[POSADOERV6NW¤A_CjOHpE>OV6CLHbA_M6M6[WNWGYAcOSNPU0CNWaaXYf4NeAcOSH#f4Hb[WA°QSU_M<U0ERNPORNWU_CµL µL FFbycqF~?q<tYornYorq<s Q] Q] `Qb4cfe
J(v) R h>jlk dc j m ' h c j| b g k v ∈ H10 (Ω) |qdk
J(v) =1
2
∫
Ω
|∇v|2dx−∫
Ω
fv dx.6P0p
`Qb4cfeu ∈ H1
0 (Ω) q9p b g eKcBb h g h c gNj m j7 qr b kut g q eKcBb hi4q4k c q eKcBb hNhjfj6z w dw %" b k-p u jlp eqdg+p-p c v g h c gNj| b4c h em j7t c h c t g t m j7v h>jlk c j o N jlp e $ m4c kvj gNj
J(u) = minv∈H1
0 (Ω)J(v).
" c |>k b gNjt.jh e o p c u ∈ H10 (Ω) jup e g h7| b4c h em jnt c h c t g t m jnR hjk dc j J(v) o qd b k-p u jlp e
q.p b g e~cBb hg h c gNj m jE qr b kut g q eKcBb hi4q4k c q eKcBb hNhjfjz w w
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[ nn " ) $#$7 # _4G.¡|+3´m4ycilk!| Q] Q] vA¡°QSU_M<U0ERNOSNPU0C³40/4 GQRHYMFUjE,HJf4Hga\ADCLNPYQRHgGQRVFGINeAD[WHgE,VLQ9[WH#«rADNPOBdjV6HJ[WA«U_QSaH\LNP[WNPCLXYADNWQSHf6H\[WA«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WHHYE,OgE,`aXIOSQRNedjV6H_(4N2GIHY[WAC] HYE,O#MLA0ET[PHGA_EY?[eA´ERU_[WV4ORNWU_C f6H [WA«U_QSagV6[eAcOSNPU0C¤A_QRNeAcOSNPU0C6C6HY[P[WHC6H aNWC6NPaNeE,HMLA_E¡[] XCLHQS_NWH 0U_NWQ[PHGIU_CjOSQRH^HHYaM6[PHbf4HT[^] HYQSGNWGH#40/4 µ0
(4U_V60HCjO[^] U_QSNP0NPC6H¡M6K`ERNed0VLHf4V¥ vA_M6[WA0GINWHC´HpE>OHC «rADNPO[eAQRHpG&K6HQ&G&K6H¡f4HpEaNWC6NWa\Af4H[] XC6HYQR0NPHJ(v)
>D[HYE,O-QRHYaA_QSdjVLA_6[WH?djV6HGIHM6QRU06[WaHf6H2aNWC6NPaNeERADORNWU_CC6XYGHYESE,NPORHf4H[WA;MLA_Q,Of4H#[eAE,U0[PV6ORNWU_CuaU_NWCLE;f4H#QSX_VL[WA_QRNPORX#djV6H#[^] XYdjVLAcOSNPU0C ADV4f4XQSNW_XHpEMLA_Q,OSNPHY[P[WHYE#rV6C6HE,HYV6[PHf4XQSNP0XHM<HQSaHIO2f4H;f4XILCLNPQ
J(u)O&ADCLf6NWE2d0Vv] NW[LHYC«rADV4O2f6HV4MFU0V6Q
∆u4®2HIO,OSH9GIU0CLE,OSAcO&AcOSNPU0CGIU_C6LQRaH [WHGADQ&A_GIORYQRH¡¨,CLAcOSV6QSH[© f6H³[eA«U_QSagVL[WADORNWU_CcADQSNWADORNWU_C6CLH[W[PH M<U_VLQA_CLAD[W`4E,HYQ¡V6C6HXYdjVLAcOSNPU0CA_V4f4XYQRNW_XYHYEMLA_Q,OSNPHY[P[WHYEY •
9u|jyc¢_o¢_| %]%]B^ 7 2 9 9 7 2<<>% 2>% 2 ")787 9); & "(% >%,7 ) # ! 7 ;9) 7 2=6"78 "9
−∆u+ u = f 92 Ωu = 0 #% ∂Ω
^4W /0 Ω #, " )% *" >7 #"'*" ,97 #!<'2'! RN ! f ∈ L2(Ω) / ="(%)% )=<'2>% '7 >% *7 2 "9 ,)% & ,9 @% 9"% ;)% 2 >2<7 2' ><>)% & !59:< 2 ?2 % >#6 9:7 <>> #C*" Ω >@% ;/9u|jyc¢_o¢_| %]%] @ Ω >% >@% ;I9 RN / 7 2 9I9 7 2<<>% @ 2)% 2 ")787 9); & "(%)%"7 >'!$#7 ;?9)7 2 C"78 "59<>% 7 & $# 2 95! ! "%! 9
V · ∇u−∆u = f 92 Ωu = 0 #% ∂Ω
^4WYµj f ∈ L2(Ω) # V #6 ="'( % ;7 )% # 27 >% !@% )787 #)787 * divV = 092 Ω /9u|jyc¢_o¢_| %]%] \ % ><>% >'9 7 )62 " # <>) 6?9) 7 )% ! / 0/ 0/ ="(% >% *" v ∈ H1
0 (Ω) ;)%
∫
Ω
vV · ∇v dx = 0.
(% >%=* 7 2 678 9) 7 2 @% & 7 2 2)% 2 >787 9 <>% 7 & 9 ! ! "%!9 " & &: 6 < 2 92 H1
0 (Ω) 7 ;> >% J(v) =
1
2
∫
Ω
(
|∇v|2 + vV · ∇v)
dx−∫
Ω
fv dx.
'A\MLA_Q,OSNPH9¨,V6C6NeGINPORXI©#f6V 0K6XU0QRYaHg6 /6 /HpE>O;V4ORNW[PHgMFU0V6Q;f4XYaU_CjORQSHQf4HpEM6QSU_MLQRNWXIOSXYEf4HE,`aXIOSQRNWHTGIU_aaHb[] NWCLf4Ned0VLHT[] HI4HQ&GINeGIHTERV6NW¤A_CjOY9u|jyc¢_o¢_| %]%] ! 9)% # " 2 7 =<)%7 & 42 7 & # + / 1!/ :#<<>)C *7 >% Ω # & ;#(% * <'2>% % 2<<>@% #7 <>>% <7 2 xN = 0 9 &>& * 47 249) ;# f + / /f(x′, xN ) = f(x′,−xN ) 1#/ ="(%)% * =7 26"78 ":9 + /1 2 7 2 &>& & ;#(% / (% >%*" + /1,#64;* 27 ) # <>% 7 & 2 7 & # <>)6; !% Ω+ = Ω ∩ xN > 0 2 :!"'9 "2 7 & # 9) & 2!% Ω ∩ xN = 0 /
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0 " +# 2# % n)
.|+3´m4ycilk'| %]%] _VFERdjV] NWGNLC6U0VLE?A¤0U_CLE?E,V6MLMFUjE,X9djV6H9[PHBE,HpGIU_CFfaHYag6QSHff4Hg^4Wp?ADM4MFADQRORHCFADNPOTZ
L2(Ω)va\ADNeEbV6C6H\0QSA_CLf4HMLADQRORNWH\f4HYETQSXYERV6[POSADOSEbQSHYE,ORHcAD[eAD6[WH\E,NU0CERV6M6M<U0ERHERHVL[PHYaHYCjOJdjV6H
f ∈ H−1(Ω)HpE,MFA_GIHf4HJ¨>«U_CLGIORNWU_CLE>©aU0NPCLEgQRXY_V6[WNWQSHYE&v vHYE#f4HYV4M6QSHIaNWQSHYEXOSADM<HYEQSHYE,ORHYCjONWf4HYCjORNed0VLHYEZGIU0CLf4NPORNWU_Cf4HTQSHaM6[eA_GIHYQ[] NPCjOSX_Q&AD[WHBVLE,VLH[W[PH ∫
Ωfv dxMFADQ[PHM6QSU4f4V6NPOf4Hf4VFAD[WNOSX 〈f, v〉H−1 ,H1
0 (Ω)
HCxMLADQRORNeGIVL[PNWHQL(v)
QSHYE,ORH¡LNPHYCxV6CLH¡«U0QRaH[WNWC6XYA_NPQSHGIU0C0OSNPCV6H;E,V6QH1
0 (Ω)@BADCLE?[WAbOSQRU0NWERNWaHXIO&ADM<H_0[] XjAD[WNOSX −∆u = f
C] ATM6[WVLE?[WNWHVdjV] ADVE,HYCLE;f6Hg[^] XY0AD[WNPORXgHYCjORQSH#X[WXaHCjO&E;f4HH−1(Ω)
HIObC6U0C M6[WVLE9MLQRHpERdjV6HJMLADQRORU0V4O;f6A_CLEΩI ®2HQ&A4W\C6HYaHYCjO§a\ADORK6XYaADORNedjV6HMFHYV4O?GIU0QRQSHYERMFU0CLf4QSH2ZBV6C6HaU4f4XY[PNeERADORNWU_CJM6K`ERNed0VLHM<HQROSNPCLHCjORH0°QSHC6U0CLE#GIU0aaH\H4HaM6[WH[WHGA0E#fl] V6CE,HpGIU0CLfaHYag6QSH\GIU0CLGIHYC0OSQRX\ERV6QTV6C6H¡Kj`jM<HQ&ERV6Q,«rA0GIHbM6[WV4OCB_O9djV6H#f4NeE>OSQRNW6V6X#f6A_CLEORU0V4O
Ω_(4U_NPO
ΓV6CLH#K`jM<HQ&ERV6Q,«rA0GIHV6CLHT¤A_QRNWXIOSX#f4Hf6NPaHCFE,NWU_C
N − 1§QSX_VL[PNWQSHNWCLGI[WVLERH9f6ADCFE
Ωj°§U_V6Q2aU4f4XY[PNeERHQ?VLC\ORHYQRaH;ERU_VLQSGHGIU_CFGIHCjOSQRXERV6Q
Γ6U_CM6QSHCLf
f ∈ L2(Γ)HOU_Cf4XIFC6NO
f ∈ H−1(Ω)MLA_Q
〈f, v〉H−1 ,H10 (Ω) =
∫
Γ
fv ds,
djV6NLHYE,O§LNPHYCVLC6H«U_QSaH[WNWC6XYA_NPQSHGIU_CjOSNPCV6HERV6QH1
0 (Ω)_Q-,_GIHADV0K6XYU_QSaHf4HORQ&A_GH FF µLPpµ6
•.|+3´m4ycilk'| %]%]B^ @BADCFE³6P¤C6U_VFE¡A¤_U0CLE\GIU0CLERNWf4XYQRXV6CLHGIU0CLf4NPORNWU_CADV6¥[WNPaNPORHpE¡f4H@;NWQRNeG&K6[WHIO;¨,K6U_aU_0C6HI©&lG_] HpE>OR ZD f4NWQSHJCV6[W[PH0a\ADNeETC6U0VLEbMFU0V6_U0CLETADVFERERN-6NWHCORQ&ADNPORHYQT[PHGYA_Ef6H#GIU_CFf4NOSNPU0CLEC6U0C4^KLU_aU_0C6HpE6®2U_CLERNef4XQSU_CLE[WHTM6QSU_6[WaHTADV4[PNWaNOSHYE
−∆u = ff6ADCLE
Ωu = u0
E,VLQ∂Ω,
^4W#FjU'&
u0HpE>O[eAORQ&A_GH#E,V6Q
∂Ωfl] VLC6HT«U_CFGORNWU_Cf4H
H1(Ω)6OSU_V'A>U0V6QSEC6UDOSXH
u06°§U0V6Q9ADCFAD[W`ERHQ^4W#F?U0CM<U0ERH
u = u0 + u6HOU_CG&K6HQ&G&K6HB[WA\ERU_[WV4ORNWU_Cf4H
−∆u = f = f + ∆u0fLADCLE
Ωu = 0
ERV6Q∂Ω.
^4Wp_(4V6NPcA_C0OB[WAXHYaA_QSdjV6H40/4 \U_C M<HV6OBQSXYERU_VLf6QRH¡^4Wp0MLA_Q9[^] A_M6M6QSUG&KLHgcADQSNWADORNWU_C6CLH[W[PHgGYADQfADM6MFADQRORNWHCjOZ
H−1(Ω)6CHC<HOY
〈f , v〉H−1 ,H10 (Ω) =
∫
Ω
fv dx−∫
Ω
∇u0 · ∇v dx
HpE>OLNPHYCV6CLHb«U_QSaHb[WNPCLXYADNWQSHbGU_CjORNWCV6HTE,VLQH1
0 (Ω) •
*KJ*KJ 5?/;79=16=,5?/S P-8 O,=vM =1 US7U<Uv8"M P§/;/9U_VLEGU_CLERNef4XQSU_CLE[WHbM6QRU06[WaHTADV4¡[WNWaNPORHpEERV6NPcA_C0O
−∆u+ u = ff6A_CLE
Ω∂u∂n = g
ERV6Q∂Ω
^4W ¯
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[ nn " ) $#$7 # _ U&
ΩHpE>OV6C U_V60HQRO¥CLU_C CLXYGIHpERESADNWQSHaHCjO³<U_QSC6Xpf4H¥[] HYERMLA0GIH
RN f ∈ L2(Ω)HIO
g ∈ L2(∂Ω) ?] XpdjVLAcOSNPU0C¡f4H4W ¯ -HpE>OV6C6HBcADQSNWA_C0OSH;f4V vA_M6[WA0GINWHCU'&C6U0VLEA¤_U_CFE?A'A>U0V4ORXV6CORHYQRaHgf] U0QSf4QSHJYXQSUAcLC³f] XYjNPORHYQJrHC M6QSHaNWHQBAD<U_Q&fLV6C6Hf4NW\GV6[PORXd0VLH#C6U_VFEQSX_[WHIQRU0CLEM6[WVLE9[PU0NPC³A_V 0K6XU0QRYaHJ40/4W L ] ADM6MLQRU4G&K6HTcADQSNeAcORNWU_CLC6H[W[PHgMFU0V6Q9XORVLf4NWHQ6P ¯ HpE>OE,HYCLE,NW6[WHaHCjO;f6NDCªXQSHCjORHf4HgGH[W[PHJM6QRXpE,HYCjORXH#Z[WAERU_VFE>E,HpGOSNPU0CMLQRXpGIXYf6HCjORHgfLADCLE9[PHgORQ&ADNPORHIaHYCjOgf4HYETGU_CLf4NPORNWU_CFETADV4³[WNWaNPORHpE§®;] HpE>O#M<U_V6Q&djV6U_N-CLU_VLETf4XOSA_NP[W[PU0CLETZC6U_VL_HYA_V³[WHYEbOSQRU0NWEXIOSA_MFHpEf4HT[^] A_M6M6QSUG&KLH_
npm4~?| ^ OSA_6[WNWESE,HYaHCjO9fl] V6C6Hb«U_QSagVL[WADORNWU_C¡cA_QRNeAcOSNPU0C6C6H[W[WH_°§U0V6QBORQSU_V60HQ9[eA¡«U_QSagV6[eAcOSNPU0C cADQSNWADORNWU_C6C6HY[P[WHgU0C agVL[OSNPM6[WNWH[^] XYdjVLAcOSNPU0C 6P ¯ 9MLADQBV6C6H«U_CLGIORNWU_C OSHYE,OBQSX0V6[WNPYQRH
vHIObU_C³NWCjORY_QSHgMLA_QBMFADQRORNWHgHCA_f4aHO,OSA_CjOTdjV6H[eA¡ERU_[WV4ORNWU_C
uHpE>OE,V WESADaaHCjOQSX_VL[PNWQSHTAcLC djV6HbOSU_VLE9[PHpEGA_[WGV6[WEHlCªHYGOSV6XYE;E,U0NPHYCjO[PNeGINPORHpE< vA«U_QSagVL[PH#f4H
BQRHYHCFL0//0B_U0NPQA_VLERERNrµL0G_,f4U0C6C6H∫
Ω
f(x)v(x) dx =
∫
Ω
(−∆u(x) + u(x)) v(x) dx
=
∫
Ω
(∇u(x) · ∇v(x) + u(x)v(x)) dx−∫
∂Ω
∂u
∂n(x)v(x) ds
=
∫
Ω
(∇u(x) · ∇v(x) + u(x)v(x)) dx−∫
∂Ω
g(x)v(x) ds.
^4W8G_
U_VFEgA¤_U0CLEbV4OSNP[WNWERX¡[eA GU_CLf4NPORNWU_C´A_V4[WNWaNPORHpEJf6H¡HYV6a\ADC6C fLADCLE4W8G_BHIONP[2C] HpE>OJMLA_EC6XYGHYESERA_NPQSH;f4Hb[^] NWCLESGIQSNPQSHTf6ADCFE2[WHTG&K6U_NP¡f4HT[^] HpE,MFA_GIHbf4H NW[WFHYQ,OV4°!U0V6QdjV6Hb[PHbM6QSHaNWHQHIO[PHpEBf6HV4 f4HYQRC6NWHQ&E9ORHQSaHYEBf4H4W8G_A_NPHYC0ObV6CE,HYCLE9NW[-E,VWOTf4HM6QSHCLf4QSH
V = H1(Ω)U_CV4ORNW[PNeERHT[PH 0K6XYU_QSaHTf4HbORQ&A_GH4FF µLPpµgM<U_V6QA>VFE>OSNLHYQ[^] NWCjORX0QSA_[PHbf4HT<U_Q&fL?C GIU0CLGI[WVLERNPU0C[eA«U_QSagV6[eAcOSNPU0C ¤A_QRNeAcOSNPU0C6C6HY[P[WHM6QRU0MFUjE,XYHMFU0V6Q³^4W ¯ \HYE,O H?OSQRU0V6_HYQ
u ∈ H1(Ω)OSH[vdjV6H∫
Ω
(∇u · ∇v + uv) dx =
∫
∂Ω
gv ds+
∫
Ω
fv dx ∀ v ∈ H1(Ω).^4W
HYEXOSA_MFHpEE,VLNPcADCjOSHYE A>VLE,ORNPLHQSU_CjO[WHbG&KLU_NPf4H6P I.¡|+3´m4ycilk!| Q] Q]K^_^¥ vAbM6QSNPCFGINWMLAD[WH9f4NC<XYQRHYCLGIHHYCjORQSH[WAB«U0QRaJV6[eAcORNWU_CcADQSNeAcORNWU_CLC6H[W[PH#^4W MFU0V6QV6C6HGIU_CFf4NOSNPU0CxA_V4´[PNWaNOSHYEf4HHYV6a\ADC6CxHIO\GIHY[P[WH^4_gMFU0V6QV6C6HGIU0CLf4NPORNWU_C A_V4[PNWaNOSHYEf4Hb@9NWQRNeG&K6[WHIONPHYC0Of4HbGIHBd0VLH;[eAJGU_CLf6NOSNPU0C¡f4Hb@9NWQRNeG&K6[WHIOHYE,ONWCLESGIQSNOSH;f6A_CLE[PHbG&K6U0Nf4HJ[^] HYERMLA_GHA_[PU0QSEBdjV6HJ[eAGIU_CFf4NOSNPU0Cf4H\9HV6a\ADCLCA_M6MLADQ&A8VeOBfLADCLEB[eA«U0QRaH[PNWC6XYA_NPQSHga\A_NWEMLA_EfLADCLE?[] HYERMLA_GH__ 'A#GIU0CLf4NPORNWU_C\f4HB@9NWQRNeG&K6[WHIO?HpE>Of4NOSH9|tYtp|sªnYo|jr|rU_V\HML[PNeGINPORH¤-GA_QH[W[PHHYE,O#«U_Q&GIXYHMFADQ#[^] A_M6MLA_Q,OSHCLA_CLGIH\ZV6CHpE,MLA0GIH0lOSA_CLf4NeE#djV6H[eAGIU_CFf4NOSNPU0Cf4H¡HYV6a\ADC6CHpE>Of4NOSHJs!m4nYk'yc|rr|U0V NWaML[PNeGINPORH¤9GYADQ9HY[P[WHgf4XpGIU0V6[PHf4H#[^] NWCjORXY_Q&AcORNWU_CMFADQ9MFADQRORNWHgdjV6N!GU_CLf6V6NOZJ[eAJ«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WH_ • npm4~?| QXXYERU_[WV4OSNPU0Cf4HT[eAJ«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WH_
Ecol
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8/ E " +# 2# % n)
@BADCLE9GIHIORORH#f4HYV44NPYaHBXOSA_MFHTC6U0VLE_XYQRNPLU0CLEd0VLHB[eAJ«U_QSagV6[eAcOSNPU0C¡cADQSNeAcORNWU_CLC6H[W[PH\^4W A0f4aHIO\V6C6H ERU_[WV4OSNPU0C V6C6NedjV6H_?°!U0V6Q\GIHY[WAC6U0VLEV4ORNW[WNWERU_CLE[WH 0K6XU0QRYaHf4H 'Ac NP[W_Q&ADaµL µLPbf4U0C0OCLU_VLE0XQSNLU0CLE[PHpEK`jM<UDOSK6YERHYEA¤0HYG;[PHpEC6UDO&AcOSNPU0CLEa(u, v) =
∫
Ω
(∇u · ∇v + uv) dxHIOL(v) =
∫
∂Ω
gv ds+
∫
Ω
fv dx.
?CV6ORNW[PNeERA_CjO#[] NPCLX0A_[PNPORX\f4H¡®A_VLG&K`0!(4G&K ADQKJgHIO#Z[^] A_NWf6Hf6V 0KLXU_QSaHf4HORQ&A_GH FL µ6WYµ6U0C\_U0NO2G[WA_NPQSHaHYC0Od0VLHaHpE>OV6C6H«U0QRaH6NW[WNPC6XpADNWQRH;GIU_CjOSNPCV6H9ERV6Q
H1(Ω)HIOdjV6H
LHpE>OV6C6H«U0QRaH[WNWC6XYA_NPQSH2GU_CjORNWCV6H?ERV6Q
H1(Ω)Y°-ADQ§ADNW[W[PHYV6QSEY[eA«U0QRaHLNP[WNPCLXYADNWQSH
aHYE,O'a\ADCLN«HpE>OSHaHCjOGUHQ&GINW_HrGD] HYE,OM<U_V6QGH[eAdjV] U_CAA'A>U0V4ORXbV6CORHYQRaH#fl] U_Q&f4QSH4JYXQSUA_V vADML[WA0GINWHCFGA_Q
a(v, v) = ‖v‖2H1(Ω) ∀ v ∈ H1(Ω).®2U0aaHH1(Ω)
HYE,O;VLC HYERMLA_GHgf4H NW[P<HQROr_U0NPQ;[WA\°?QRU0MFUjE,NPORNWU_C FF µL /06OSU_V4OSHYE;[PHpE9K`M<UDOSK6YERHYEbf4V 0K6XYU_QSaHf6H 'Ac NP[W_Q&ADaµ6 µ6WJERU_CjOTERADORNeE>«rA_NOSHYEBHIOTU0C³M<HV4Ogf4U0CLGGIU_CFGI[WV6QRHdjV] NW[HI4NeE>OSHBVLC6HTV6C6NedjV6H#E,U0[PV6ORNWU_Cu ∈ H1(Ω)
f4Hb[eA«U_QSagV6[eAcOSNPU0C¡cADQSNeAcORNWU_CLC6H[W[PH\^4W I.|+3´m4ycilk'| %]%]B^&°§U0V6Q-ADCLA_[P`4ERHQv[WHM6QRU06[WaHA_V4#[WNPaNPORHYE^4W ¯ 'f6A_CLEv[WHGA0EvU'&
g = 0U0CgADVLQSA_NO!M6Vg¦ORQSHORHYCjORXfl] NWCLESGIQSNPQSH?[eA9GU_CLf6NOSNPU0CgADV4T[WNPaNPORHpE!f6H9HV6a\ADCLCJf6A_CLEv[^] HYERMLA_GH2f4H
9NP[W<HQROV D[lC] HYE,OMLA0EMFUjERERNPL[PHTf4HbG&K6U_NeERNPQ
V = v ∈ H1(Ω), ∂v∂n = 0
ERV6Q∂Ω GYADQp0M<U_VLQVLC6H«U_CFGORNWU_C
v ∈ H1(Ω) ∂v∂n
C] A#MLA_Ef6H9E,HYCLE?ERV6Q∂Ω
D?CHlCªHIOp ∇v Cv] HpE>Od0Vv] V6CLH«U_CLGIORNWU_Cf6HL2(Ω)
rGIU0aMFUjERA_C0OSH?MLA_Q'GU_aMFUjERA_CjORHpHIO!U_CgESADNPO!djV] NP[C] `TAMLA0E'f6H?C6U_ORNWU_Cgf4H2ORQ&A_GH?ERV6Q∂Ω
M<U_VLQ-[PHpE!«U_CLGIORNWU_CLE-f4HL2(Ω)
;CMFU0V6QRQ&ADNPOG&K6U_NeERNPQV = v ∈ H2(Ω), ∂v
∂n = 0E,V6Q
∂ΩdjV6N-HpE>O#LNPHYCVLCERU_VLE, HYERMLA_GHg«HYQRaX\f4HH2(Ω)
HC_HYQ,OSVf4V0K6XYU_QSaHf4HORQ&A_GH FF µL / ³ADNeEBA¤0HYGJGIHf4HQSC6NWHQbG&K6U_NP V6C6HC6U_VL_H[W[WHf4NWGIV6[PORX\ERV6QS_NPO=Hl[eA«U0QRaH6NP[WNWC6XYA_NPQSH
aC] HYE,OMFA_EGIUHQ&GINW_HBE,V6Q
VHIOU_C¡CLH;M<HV4OMLA0EA_M6M6[WNWdjV6HYQ2[WH 0K6XU0QRYaHBf4Hb vAcNW[P0QSA_a)D[<Cv] `Af6U_CLGTMFA_EaUc`0HCf4H#M6QSHCFf4QRHgHC GIU0aM4ORH#[eAGU_CLf4NPORNWU_C A_V4[PNWaNOSHYE9f6HJHYV6a\ADC6C fLADCLE9[PHG&KLU_NPf4Hb[] HYERMLA0GIHTf4H NW[WFHYQ,Op •
nYm6~?| \ djV6NPcA_[PHYCLGIH#A¤0HYG;[] XYdjVLADORNWU_C9U_VLE¡NPCjORHYQRMLQRXORU_CFEa\ADNWCjORHCFADCjO[WA«U_QSagV6[eAcOSNPU0C¤A_QRNeAcOSNPU0C6C6HY[P[WH´6P M<U_V6Q_XQSNPLHQdjV] U_CA6NWHC QSXYERU_[WVx[WHM6QSU_6[WaH ADV6 [PNWaNOSHYE ^4W ¯ ?f6A_CLEV6CERHCLEZM6QSXYGNWERHQpU0VLEA_[P[WU_CFEE,V6MLMFUjE,HYQ9djV6HT[PHpEf4U0C6C6XHpEERU_CjOQRXY_V6[WNWQSHYET0U_NWQ[WA9XHa\ADQ&djV6H#40/4WpJ[PU0QSESdjV6Hb[] U_CCLH«rADNPO2MFA_E?GHIORORHBKj`M<UDOSK6YERHpI_°[WVLEM6QSXYGINeERXaHCjO?C6U0VLE?A_[P[WU_CFE?ERV6M6M<U0ERHQ2djV6H;C6U_VFE2ERU_aaHYEfLADCLE§[WHYEGU_CLf6NOSNPU0CLE-fl] ADM6M6[WNeGAcOSNPU0C\f4V[PHYaaHERV6NW¤A_CjO-f6HQRXY_V6[eADQSNOSXd0VLHCLU_VLEA0f4aHIORORQSU_CLEr_U0NPQ[eA\E,U0VLE>ERHYGOSNPU0C6 /6 FMFU0V6Qf4HpEQRXpE,V6[POSADOSEERNPaNW[WA_NPQSHYE&
;2|+3 3´| Q] Q]K^+\ `>b4cfeΩ g h b g i4jk e k # g c jk m j q4p@pj C1 m j RN w `Qb4cfe f ∈ L2(Ω) j e
g q e kvq-jpug k ∂Ωm g hj7r b h eKcBb h m j H1(Ω) w %" b k-p6 q p b g e~cBb h u
m j qr b kutg q eKcBb hi4q4k c q eKcBb hNh>jlfjz w u6q-||Nq4k e~c jlh e $ H2(Ω) w
BQ-,_GHbA_V vHaaHg40/4WYµJU_CMFHYV4OV4OSNP[WNWERHQ9[WAJ«U_QSagVL[PHTf4HBQSHHYCf4V 0K6XYU_QSaH FL µ6 µ E∫
Ω
∆u(x)v(x) dx = −∫
Ω
∇u(x) · ∇v(x) dx +
∫
∂Ω
∂u
∂n(x)v(x) ds.
6P 0
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[ nn " ) $#$7 # /6d0VLNHYE,OcAD[eADL[PHMFU0V6Q
u ∈ H2(Ω)HOv ∈ H1(Ω)
XADM6M<H[WU_CFEdjV6H[^] NWCjORXY_Q&AD[WHf4HFU0QSff6ADCLET^4W j?A6NWHCV6CE,HYCLEZGA_VLE,Hbf4V 0K6XYU_QSaHBf6H;OSQSA0GIH FL µ60/ djV6NvA4W\QSaHBdjV6HbM<U_V6Qu ∈ H2(Ω)
[eAf4XYQRNW_XYH#C6U0QRa\AD[WH ∂u∂n
AV6CERHCLE;f6ADCFEL2(∂Ω)
;C³f4Xpf4V6NPOBA_[PU0QSE;f4H^4W HIOg6P 0djV6H_4M<U_VLQORU_V6Ov ∈ H1(Ω)
∫
Ω
(∆u− u+ f)v dx =
∫
∂Ω
(
g − ∂u
∂n
)
v ds.^40/ E
(N2[^] U_CM6QSHCLfv ∈ C∞
c (Ω) ⊂ H1(Ω)fLADCLE40/ E '[PH¡ORHQSaH¡f4HFU0QSf´f6NWERMLADQ&A8VeO#HIO[] U_Cf4XYf4VLNOp<HYC³0HQRORV³f4V®2U_QSU_[W[WA_NPQSH:FL0/40/4<djV6H
∆u − u + f = 0fLADCLE
L2(Ω)f6U_CLGgM6QRHpERdjV6HMLADQRORU0V4Of6ADCLE
ΩL°§A_QGIU0CLE,XpdjV6HCjOY[WHTaHagLQRHTf4HTjADVLG&K6Hbf4H^40/ E 2HpE>OCV6[^Lf4U_CFG
∫
∂Ω
(
g − ∂u
∂n
)
v ds = 0 ∀ v ∈ H1(Ω).
;Q2[^] NWa\AD0H;f4HH1(Ω)
MFADQ[] ADM6ML[PNeGADORNWU_COSQSA0GIH;HYE,Of4HYCLERH;f6A_CLEL2(∂Ω)
r_U_NWQ[WAgQRHYaA_QSdjV6HFL µ6W8GDI_GIHBdjV6NFHYCjORQ&A8V¬C6H9djV6H
g− ∂u∂n = 0
f6A_CLEL2(∂Ω)
jHIOf6U_CLGMLQRHpERdjV6H;MLADQRORU0V4O2ERV6Q∂Ω
CGIU_CFGI[WVLE,NWU_CC6U0VLEA¤_U0CLEf4XaU_CjOSQRXb[WHBQSXYERV6[POSADOERV6NWcADCjOY £'hqFy4:[3 | %]%]B^ `Qb4cfe
Ω g h b g i8jk e k # g c jlk m j.lq4p-plj C1 m j RN w `>b4cfe f ∈ L2(Ω) j eg q e kvq-jEpug k ∂Ω
m g h>jr b h eKcBb h m j H1(Ω) w j - c p e jg h>jEg h c gNjEp b Dg eKcBb h u ∈ H1(Ω)m jsq2r b kutg q eKcBb h i4q4k c q e~cBb h_hjfj9z w uw j#| g+p o u q@||Nq4k e~c jh e $ H2(Ω) j e jlp e p b g e~cBb hm j)z w -q4gpjlhNp b)
−∆u+ u = f |>kvjlp gNj|qdk eb g e2m qdhp Ω,∂u
∂n= g |kvjlp gNj|Nq4k e b g e p-g k ∂Ω.
9u|jyc¢_o¢_| %]%] ; & (% >%*" 7 * 6"78 " u ∈ H1(Ω) 9 7 2I@% & 7 2 " 2>% 2%! ")787 =+ /1 ;>% 7 & 2 :9 ;> >% ! 2
‖u‖H1(Ω) ≤ C(
‖f‖L2(Ω) + ‖g‖L2(∂Ω)
)
,
C > 0 #64:#" 62 * 9;><>) 9I<'2 9) u, f ! g /®2U0aaHfLADCLEB[eAERU_VFE>E,HpGOSNPU0C³M6QSXYGXYf4HYC0OSH_lMLA_QbKLAD6NPORVFf4H HO#[PU0QSESdjV6HJ[WH\GIU0C0OSHIORHC6HM6QR¦ORH MFA_EZ´GU_C4«VFE,NWU_CF\U_C f4NPOdjV6H [^] VLC6NWdjV6HE,U0[PV6ORNWU_Cu ∈ H1(Ω)
f6H [WA«U_QSagV6[eAcOSNPU0C¤A_QRNeAcOSNPU0C6C6HY[P[WH^4W ?HpE>O[^] V6C6Ned0VLH#tYq<rk'nYoqFs 1 m6o §r|f6VMLQRU06[PYaHbA_V4¡[WNPaNPORHpE#6P ¯ @ ? " G-99Q'54 ".M)+
ΩW4 23$-:9 "%(@)@Q
g 2@+)
"S6O",)9% "L2(∂Ω)
W -++%I " +9X54 )R%29 , B W4 " "%(QP"X+"9%u ∈ H1(Ω) (U689X(-9% .-9%(-9%" C 0(E MF0 1+9)BQP" ,))+9 +9%( .)9%&#$ 54 Q$/9%
−∆u + u = f"9 9Q"$-+9"* "
Ω%N65-"*9%(%+)'/9)P<" <, %(QP" 0
" "-Pσ = ∇u ∈ L2(Ω)N " ""%&< " C 0(E MF Q@9" v ∈ C∞
c (Ω)
∣
∣
∣
∣
∫
Ω
σ · ∇v dx
∣
∣
∣
∣
≤∣
∣
∣
∣
∫
Ω
uv dx
∣
∣
∣
∣
+
∣
∣
∣
∣
∫
Ω
fv dx
∣
∣
∣
∣
≤ C‖v‖L2(Ω)
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8// " +# 2# % n)
QP"%W4 )279% W4 -"+9 QP"@(<,)9%& > 9 W4 )R%29 ,) 4 < %(.)9 ", @65%;"(' " σ "
L2(Ω) C .M% )#$"%&9%( D0 T0 @( W D0 TP0 $ F0 B ",divσ ∈ L2(Ω)
)∫
Ω
(divσ − u + f) v dx = 0 ∀ v ∈ C∞c (Ω),
, !Q%%%(QP" .)+9 9 -%&9 D0 T0 T Q −divσ = −∆u = f − u -
L2(Ω) 0 "'"++%N9)+9.M) (, "%&9%UR %%9)* " -G -< + "<+ > 65-%;+%I54 ".M)+! 2!9 %) C %$- g F0 1"S, 6O%&9 Y"9%%(+54 +,
H(div)%+9P ""% K (
" = + ,)9% D"0 D"0 T ")#%M- H(div) =
σ ∈ L2(Ω)N 9 GQdivσ ∈ L2(Ω)
.W :J) >
D"0 D"0 $B9SQ+% Ω 2 .M+<9)%)* ", (++ C1 $B(689X! W4 %9 /9%($-
$-+9%+"%(.9∫
Ω
divσv dx +
∫
Ω
σ · ∇v dx = 〈σ · n, v〉H−1/2,H1/2(∂Ω), C 0 TPEF"
v ∈ H1(Ω))
σ ∈ H(div) 0 GS9)9 B P9%&9B C 0 TE/F +%X "9P %&! $%&9 +9H1/2(∂Ω)
) + "57"9H−1/2(∂Ω) 0 P% σ
)v+ " 6O,)9%9 "% > 9 :, .P%(%& 9)9 W4 2 "+9Q" 54 %9 -U " ; +" ( ∫
∂Ωvσ · n ds 0 %"GI(Y6O-9 "(
C 0 TPEF "*" + " K σ · n +"∂Ω C , < < $
H−1/2(∂Ω) F " σ ∈ H(div) 0% -%(QI, I9 +"&- K <+"9%B " C 0EF7C .M , σ = ∇u F /X) ""%1Q <, "% =9%(R %%9)' - )2'.M9%# !, )M%&9S +9S)() 9' "H−1/2(∂Ω)
C ,- g ∈ L2(∂Ω) ⊂ H−1/2(∂Ω) F0 )!/9" ' 2'$-+9; )P', "% Q")' "-9%(Q+ , 9 9) %&9 QS( 689X(-9%U.-9%(-9%" ,)9% *"X %9-9% X(, "%&9% -"RU%(%&9 S " "G)!-SJ$-;%&9$ BY, 9%P) K ,)9%&9 + - <?QP" ∂u
∂n= g
+"∂Ω 0 •®2U0aaH9fLADCLE-[eA#ERU_VLE,E,HpGORNWU_CM6QSXYGXYf4HYC0OSH_DU0C\MFHYV4O2aU0C0OSQRHYQdjV6H[eATE,U0[PV4OSNPU0Cf4Hg^4W ¯ aNWC6NWaNeERH\V6C6HXC6HYQR0NPH0QXHa\ADQ&djV6U_CFEBdjV6H0vERN
g = 0AD[WU_Q&E;[^] XC6HYQR0NPH6 /'/_9HYE,O#[WAa¦aHdjV6HGH[W[PH6P0pf4XLC6NWHM<U_VLQ\[PH vA_M6[eA_GINWHC A¤_HpG¡GU_CLf4NPORNWU_CFE\ADV4 [WNWaNPORHpE\f4H @;NPQSNWG&KL[PHOY9XYA_C6aU_NWCLE[WHV6Q&EaNPC6NWa\A CLHERU_CjOHC _XCLXQ&AD[BMLA_E[WHYEa¦aHYEGA_QU_C aNWC6NWaNeERHERV6Qf6HV4HYERMLA0GIHYEJf4NC<XYQRHYC0O&E!Z³ESA¤_U0NPQ
H10 (Ω)
HIOH1(Ω)
- vA°QSU_M<U0ERNPORNWU_C´ERV6NPcA_C0OSH¡HpE>OV6C6HA_M6M6[WNWGYAcOSNPU0CNWaaXpf4NeAcORH#f6HB[eA°QSU_M<U0ERNOSNPU0CµL µL FFbycqF~?q<tYornYorq<s Q] Q]K^ `>b4cfe
J(v) R hjk c j m ' h c j| b g k v ∈ H1(Ω) |Nq4kJ(v) =
1
2
∫
Ω
(
|∇v|2 + |v|2)
dx−∫
Ω
fv dx−∫
∂Ω
gv ds.6 /'/_
`Qb4cfeu ∈ H1(Ω) qEp b Dg eKcBb h6g h c gNj m jnq2r b kutg q eKcBb h6i4q4k c q eKcBb hNh>jlfjz w lw %" b k-p u jlp e
qdg+p-p c v g h c gNj| b4c h em j7t c h c t g t m j7v h>jlk c j o N jlp e $ m4c kvj gNjJ(u) = min
v∈H1(Ω)J(v).
" c |>k b gNjt.jh e o p c u ∈ H1(Ω) jup e g h7| b4c h em jnt c h c t g t m jnR hjk dc j J(v) o qd b k-p u jlp e q.p b g e~cBb hg h c gNj m jE qr b kut g q eKcBb hi4q4k c q eKcBb hNhjfjz w uw
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[ nn " ) $#$7 # /_µ9u|jyc¢_o¢_| %]%] !<<>)6 * Ω = >%>@% ; % ;"7 >%59 '7 2 C6 C1 / 7 2 9)9) 7 2<<)% 2>% 2 ")787 9); & "(% >%?7 ) #:! 7 ;:9) 7 2 C"78 " 9 >2<7 2' >2 :!"'9 "2 7 &: # 9) '% >%
−∆u = f 92 Ω∂u∂n + u = g #% ∂Ω
^40/Dµj f ∈ L2(Ω) ! g #57 2 (% 2'! !% ∂Ω 9 "'( "9 H1(Ω) / ,9; & "(%)% 2 7 ;27 ;! 2 +.* ););)% 27 C #>787 :9) " 2>% ;1
‖v‖L2(Ω) ≤ C(
‖v‖L2(∂Ω) + ‖∇v‖L2(Ω)
)
∀ v ∈ H1(Ω).
9u|jyc¢_o¢_| %]%] !<<>)6 * Ω ," >%>@% ; ! '/ 7 2 9 9 7 2<<>% 2)% 2 >787 9; & (% >% 7 ) #! 7 ; 9) 7 24678 9 >2<7 2' ),2 9)#"'9 ! " 2 7 & & 7 ;#
−∆u = f 92 Ω∂u∂n = 0 !% ∂ΩN
u = 0 !% ∂ΩD
^40/'F f ∈ L2(Ω) ! (∂ΩN , ∂ΩD) #< 2)% 9 ∂Ω >787 ,* 7 # & #!% #?#<>)% " )787 #9) ∂ΩN ! ∂ΩD C", " 787 4+ %?7 2 "% / 1!/3+ 7 6)%?7 2 & 2>% *" / -/ / 1
9U_VLE'QSH_HYC6U_CFEla\ADNWC0OSHCLA_CjO'ADVT0XQSNO&AD6[WH-U_M<XQ&AcOSHV6Qv 'ADM6[eA_GNPHYCrESADCLE'A'A>U0V4O'fl] V6CTOSHQSaHfl] U0QSf6QRH JXYQRU\GIU0aaHTf6A_CLET6P ¯ R?HOC6U_VFEGIU_CFE,Nef4XQSU_CFE2[WHTM6QSU_6[WaHTADV6¡[PNWaNOSHYE −∆u = f
f6ADCLEΩ
∂u∂n = g
E,V6Q∂Ω
^40/_0U&
ΩHYE,OVLC U_VL_HQROFU0QRC6XGIU0C6C6HHf4H[] HYERMLA0GIH
RN f ∈ L2(Ω)HIOg ∈ L2(∂Ω)
- vAf4NDWGV6[OSXTC6U_VL_H[W[WHbfLADCLEb6 /0_2MLA_QQSA_M6MFU0Q,OZ6P ¯ ?HYE,O9d0Vv] NW[lC] HI4NWE,ORHTVLC6HTE,U0[PV4OSNPU0CdjV6HE,N<[PHpE2f6U_C6C6XYHYEfHIOg_XYQRNPLHCjO?VLC6H;¢0qFs!w'ornYoqFs´w!|¢_q3~!m6nYo §oronphFc?C¡HlCªHIOpNP[<HYE,O?«rA0GINW[PHf4H\_U0NPQTdjV6HEY] NW[-HI4NeE>OSH\V6C6H\E,U0[PV6ORNWU_C
u ∈ H2(Ω)'A_[PU0QSEBNWCjORX0QSA_CjOB[^] XpdjVLAcOSNPU0CERV6Q
ΩU_V6NPHYCV4ORNW[PNeESADCjO[WA«U0QRaJV6[PHTf4HBQSHHYC FL0//_R2U_CAC6XYGHYESERA_NPQSHaHCjO
∫
Ω
f(x) dx +
∫
∂Ω
g(x) ds = 0.6 / ¯
XHa\ADQ&djV6U_CFE;A_VLERERN§d0VLHERNuHpE>O#ERU_[WV4ORNWU_CAD[WU_Q&E
u + CA¤_HpG
C ∈ RlHpE>OTADVFERERN!ERU_[WV4OSNPU0CC «rA_NOp6 / ¯ gHYE,OV6CLHGU_CLf6NOSNPU0CxC6XpGIHpERESADNWQRH¡HIOE,VW\ESADCjOSHfl] HI4NeE>OSHCLGHfl] VLC6HERU_[WV4OSNPU0Cf6ADCLE
H1(Ω)vV6CLNWdjV6HZ[^] A0f6f4NPORNWU_Cfl] V6C6H¡GU_CLE,OSA_CjORHM6QSYEYXHa\A_QSdjV6U0CLETdjV6H_vERN[^] U0V6_HYQ,O
ΩC] HYE,O#MLA_ETGU_C6CLHI4H_lAD[WU_Q&EBNP[§«rA_V4O#XYGQRNWQSH^40/ ¯ ;M<U_V6Q#G&KFA_djV6HGIU_aM<U0ESADCjORHGIU0C6C6HH\f4H
ΩHIO[^] VLC6NWGNOSXTf4Hb[WAE,U0[PV6ORNWU_C¡cA_VLf4Q&AJZJ[] A_fLf4NOSNPU0C¡M6QSYEfl] V6C6H#GU_CLE,OSA_CjORH9MFADQGIU0aM<U0ESADCjOSHGIU_CLC6HI4H\rA¤0HYGBGIHYEaU4f4NPFGADORNWU_CLEOSU_VLE[PHpEQRXpE,V6[POSADOSEdjV6NERV6NW_HCjOQSHYE,ORHYC0OcAD[eADL[PHpESI
.¡|+3´m4ycilk!| Q] Q]K^ °K`ERNed0VLHaHCjOY![eAGU_CLf4NPORNWU_Cxf4HGU_aMLADORNW6NP[WNPORX^40/ ¯ #EY] NWCjORHQSM6QSIOSHGIU_aaH9VLC6H;¢0qFs!w'ornYoqFs´w Philk!oro !y¤| HfGU_QSQRHpE,M<U_CFfZ#V6C6H;ERU_VLQSGH_U0[PV6aNedjV6H_0HIO
gZ#V6C
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8/'F " +# 2# % n)
FV4HYCjORQ&ADCjO2ADV<U_Q&fl_°§U0V6Q2djV] NP[FHI4NeE>OSHV6CXOSAcOE>O&AcORNWU_CLCLADNWQRHU0Vfl] XpdjV6NP[WNW6QRHJrGD] HYE,O,Zcf4NWQRHVLC6HERU_[WV4OSNPU0C f4H40/_0,I'NW[«rADV4O\djV6HGHYEf4HV4OSHQSaHYEE,HLAD[eADCLGHCjOJMFADQR«rADNPORHaHYC0Op-@9Ha¦YaH0v[^] VLC6NWGNOSXJ¨RZV6C6H¡GU_CLE,OSA_C0OSHA_f6f6NOSNP0HM6QSYE>©GIU0QRQSHYERM<U_CLfZ[^] A_LE,HYCLGIHf] U0QRNW_NWC6Hf4HQSXI«XYQRHYCLGIHE,VLQB[^] XYG&K6HY[P[WHJdjV6N§aHYERV6QSHJ[WHYEBcAD[WHV6Q&EBf4HuGIU_aaHMFU0V6Q;[eA¡ORHaM<XQ&AcOSV6QRH0ªMLADQH4HaM6[WHp •
£'hqFy8:+3´| %]%]B^ `Qb4cfeΩ g h b g i8jk e b kuh b hNhj-+j"k # g c jk m j q8p-pj C1 m j RN w `Qb4cfe
f ∈ L2(Ω) j e g ∈ L2(∂Ω) g c i k c ' jh e q b h mdcfeKcBb h m j6 b t|Nq e~c c cfe z w-, @w j- c p e j
g h>jsp b Dg eKcBb hrlq c j u ∈ H1(Ω)m jz w-, o g h c gNj $9R q mmdcfeKcBb h m g hj b hp e q4h e j|k + pw
Yh+3´q<s'tYnycm4nporq<s#]<°§U_V6Q?ORQSU_V60HQ-[eAT«U_QSagVL[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WHU_C\MLQRU4GIpf4H9GU_aaHMFU0V6Q[^] XYdjVLAcOSNPU0C^4W ¯ CGYAD[eGIV6[ERNPaNW[WA_NPQSHTGIU0CLf4V6NPOZ∫
Ω
∇u · ∇v dx =
∫
∂Ω
gv ds+
∫
Ω
fv dx
M<U_VLQ9OSU_V4OSHg«U0CLGOSNPU0CORHpE>ObQRXY_V6[WNWQSHv<°§U0V6QBf6U_C6C6HYQ;VLC³E,HYCLEbZ\OSU_VLE;[PHpE;ORHQSaHYEBf4HGIHIORORHXY0A_[PNPORX0pU0CgM<U_V6QSQ&ADNPO!G&K6U0NWERNPQ
H1(Ω)GIU_aaHHYERMLA_GHf4H NW[WFHYQ,O
Vca\A_NWE!C6U0VLE!C6H2M<U_VLQRQSNPU0CLEaU0C0OSQRHYQ-[WAgGIUHQ&GINWNOSXf4H[WAT«U_QSaHLNP[WNPCLXYADNWQSH_j®2HO,ORH;f4NDWGV6[OSX9HpE>ONWCjORNWaHYaHCjO?[WNPXYH9ADV«rA_NOdjV6H0!ERN
uHpE>OE,U0[PV6ORNWU_C§AD[WU_Q&E
u + CHYE,O\ADVLESE,NE,U0[PV6ORNWU_C!°§U_V6QXNPORHQGIHONWCLGU_C_XYC6NPHYCjOYU0Cx[W0H[^] NPCLf6XIORHYQRaNWCLAcOSNPU0C¥f6HGHIORORH GIU0CLE,OSADCjOSHA0f6f4NPORNW_HHC¥C6HORQ&A¤cADNW[P[eADCjOdjV] A¤_HpGf6HYE«U0CLGOSNPU0CLEf4HTaUc`_HYC6C6HBCV6[P[WH_F±V4OSQRHYaHYCjO9f4NOp6U_CM<U0ERH
V =
v ∈ H1(Ω),
∫
Ω
v(x) dx = 0
HO[WAJ«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WHbf4H6 /0_?HYE,OH
OSQRU0V6_HYQu ∈ V OSH[vdjV6H ∫
Ω
∇u · ∇v dx =
∫
∂Ω
gv ds+
∫
Ω
fv dx ∀ v ∈ V. 6 /[GD;C M<HV4OX0A_[PHYaHCjO³G&K6U_NeERNPQ
V = H1(Ω)/R;GD] HYE,O,Zcf4NWQRHdjV6U_ORNWHCjORHYQ
H1(Ω)MLA_Q [PHpEGU_CLE,OSA_CjORHYETr[PHpEXY[PXYaHYCjOSEf4H
H1(Ω)/RE,U0CjO[PHpEGI[eA_ESE,HpEf4HT«U0CLGOSNPU0CLEf4H
H1(Ω)XjAD[WHYEZVLC6H#GIU0CLE>O&ADCjORHBM6QSYE&°§U_VLQBM<U_VL_U_NWQ;A_M6M6[WNWdjV6HYQ;[WH 0K6XU0QRYaHf4H vAcNW[P0QSA_a Z¡[WA¡«U_QSagV6[eAcOSNPU0CcADQSNWADORNWU_C4CLH[W[PHB40/'G_Y[WAERHVL[PH2Kj`M<UDOSK6YERH?f4XY[PNeGADORH2Z_XQSNPLHQ'HYE,Ov[WA;GIUHQ&GINWNOSX?f4H2[WA«U0QRaH26NP[WNWC6XYA_NPQSH_®2HY[P[WHIGINE] U_6ORNWHCjOg_Q-,_GH\ZV6C6H0XC6XYQSA_[PNeESAcORNWU_Cf4H[^] NPC6XY0A_[PNPORX¡f6H°§U_NWCLGA_QRX0vGIU0C6CV6H¡ERU_VLE[WH¡CLU_a fl] NPCLX0A_[PNPORXf4H¡°§U0NPCLGYADQSXINPQRORNWC60HQ H-ERN
ΩHpE>OFU0QRCLX¡HIOGIU0C6C6HI4H0'NW[HI4NWE,ORHV6C6HGU_CLE,OSA_CjORH
C > 0ORH[W[WH#d0VLH_4M<U_V6QOSU_V4O
v ∈ H1(Ω)
‖v −m(v)‖L2(Ω) ≤ C‖∇v‖L2(Ω)A¤_HYG
m(v) =
∫
Ω v dx∫
Ω dx.
^40/ ?] NWC6XY0AD[WNPORX³^40/ #E,Hf4XaU_CjOSQRHMFADQJGU_CjORQ&A_f6NWGIORNWU_CGIU0aaHf6ADCLEg[WAf4HV6NWaHf4XaU_CFE>OSQSADORNWU_Cf4H\[eA°QSU_M<U0ERNOSNPU0C FF µLP E rC6U_VFET[WA_NWESERU_CLETGH[eAADV[PHpGOSHV6Q#HYCHI4HQ&GINeGIH¤'®2U0aaHm(v) = 0
M<U_VLQJOSU_V4Ov ∈ V
6 / gM6QRU0V6_HdjV6H ‖∇v‖L2(Ω)
HpE>O\V6C6HC6U0QRaHf6ADCFEV
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[ nn " ) $#$7 # /0XYdjV6NW¤A_[PHYCjORHgZ¡[WA\CLU_QSaHgVLE,VLH[W[PH ‖v‖H1(Ω)
FHOBf4U0CLGgdjV6H#[eA«U0QRaHgLNP[WNPCLXYADNWQSH#HYE,O;GIUHYQSGNP0HE,V6QV
!NWCLAD[WHaHYC0OplMFU0V6QBaU0CjORQSHQbd0VLHJ[eAERU_[WV4OSNPU0C³V6C6NedjV6Hf4H^40/'G_HpE>Ob6NWHC³VLC6HERU_[WV4OSNPU0Cf4VMLQRU06[PYaH;ADV6[PNWaNOSHYET6 /0_IjU0C¡M6QSU4GIpf4HBGU_aaH;MLQRXpGIXYf6HaaHCjO[WU_Q&Ef4Hb[WAf4XaU0CLE>ORQ&AcORNWU_Cf4V 0K6XYU_QSaHb6 /6PFL ;CU_4OSNPHYC0O9A_NPCFE,NlM<U_V6QOSU_V4Ov ∈ V
∫
Ω
(∆u+ f)v dx =
∫
∂Ω
(
g − ∂u
∂n
)
v ds.^40/*0
;QY_djV6HY[WdjV6H;E,U0NOw ∈ H1(Ω)
j[WAT«U_CLGIORNWU_Cv = w−m(w)
ADMLMLADQRORNWHCjO?ZV_C¡G&KLU_NeE,NeERESADCjOV6C6HOSH[W[PH«U_CFGORNWU_Cxf6ADCLE6 / 0!HC QRHY_QSU_V6MLA_CjO#[PHpE#OSHQSaHpEgHYC«rA_GIORHYV6Qf6H¡[eA³GIU0CLE,OSADCjOSH
m(w)HIO-HCV4ORNW[WNWESADCjO-[WABGIU0CLf4NPORNWU_Cf4HGIU_aMLADORNW6NW[PNPORXT6 / ¯ 'ADNWCLERN6d0VLH2[^] XY0A_[PNPORX ∫
Ω∆u dx =
∫
∂Ω∂u∂n ds
6U0Cf4XYf4VLNO9f4U0CLGbf4H6 / 0∫
Ω
(∆u+ f)w dx =
∫
∂Ω
(
g − ∂u
∂n
)
w ds ∀w ∈ H1(Ω).
;CMFHYV4O;f4U0CLG#GIU0CLGI[WV6QSH#GIU0aaHgf] KLA_6NPORVLf4HJd0VLHu0XQSNLHTLNPHYC[PH#MLQRU06[PYaH#ADV4[PNWaNOSHYE40/_0
9u|jyc¢_o¢_| %]%] ; & (% >%$7 ;27 ; 9 " '#2)%;! % %>%$+ / 0 1!/9u|jyc¢_o¢_| %]%] #<<>)C?*" Ω #6 " )% >@% ;?#" %; "7 >% / @ f ∈ L2(Ω) / =! 9)%=7 2 @% & 7 2 " 2>% 2 " >787 ! 2@(% >% u ∈ H1(Ω) )7*
∫
Ω
∇u · ∇v dx+
(∫
Ω
u dx
)(∫
Ω
v dx
)
=
∫
Ω
fv dx ∀ v ∈ H1(Ω).
; & (% >%?7 ) #:! 7 ;9 7 2=C"78 "9 !# @% & 7 2 " 2)% 2 >787 /4)7<>%7 & 2I7 &: #,2%! "! 2 % ;#C"78 I< 2)% >7 )%$ #<<>)C *" ∫Ω f dx = 0 *" >7 <>%7 & :9); $# ;!('9 ; % !(% " !."! "*KJ*) 5U :l=!Uv/21S P§3L=!P"ORUS
@BADCFE;[WHYEBf4HYV4 E,U0VLE>ERHYGOSNPU0CLEM6QSXYGXYf4HYCjORHYE;C6U0VLE9A¤0U_CLE;GIU_CFE,Nef4XQSXgf4HpE9MLQRU06[PYaHYE9A_V4[PNWaNOSHYEM<U_VLQ[] U_M<XQ&AcOSHV6Q' vADML[WA0GINWHC ;CbM<HV4O'«rA_GNP[WHaHCjOv0XC6XYQSA_[PNeERHQª[PHpElQSXYERV6[O&AcO&ElU_4OSHCVLEZf6HYEU_M<XQ&AcOSHV6Q&EM6[PVFE_XCLXQ&ADV4l§f4NO&E\H[W[WNPM4OSNWdjV6HpE\f4V f4HV44NWaHU_Q&f4QSH ¢_qv| ´¢0or|sªnYtz6m4ycorm§r|t0<®2HTO>`MFHJf4HTM6QSU_6[WaHTHYE,O9NeERERVf4H#[eAaUf6X[WNWESAcOSNPU0Cf6HYEaNW[PNWHV6K6XORXQSU_0C6HpE(Nª[] U_CQRHYM6QRHYCLf\[^] H4HaM6[WHBf4H;[WAGIU0CLf4VLGIORNWU_C¡f6H;[eAJG&KFAD[WHV6QBrf4XOSADNW[W[PXbf6ADCFE2[WH#®2KLADM6NPORQSHg¤f6ADCLE2V6CaNW[WNPHYVK6XORXYQRU0_CLH[WAJGIU0CLf4VLGIORNWjNPORX
k(x)HYE,OV6CLH9«U0CLGOSNPU0CcADQSNWA_6[PHBfl] V6C¡M<U_NWCjOZ[] ADV4OSQRHTf4Vf4U0a\ADNWC6H_6@BADCLEGH#GA0E4U0CGU_CLERNWf6QSH;[WHTM6QSU_6[WaHTADV4[PNWaNOSHYE
−div(k∇u) = ff6ADCLE
Ωu = 0
ERV6Q∂Ω
^4 µ E U&
ΩHYE,OgV6CU_VL_HQRO#<U_QSC6Xf4H[^] HpE,MFA_GIH
RN vHO f ∈ L2(Ω)*NWHC´ER6QY'E,N
k(x) ≡ 1vU_CQRHORQSU_V60H;[WHT vADML[WA0GINWHC D[HpE>O«rA0GINW[PH#f4HT0XC6XYQSA_[PNeERHQ[PH 0K6XYU_QSaH#40/40/4
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8/ ¯ " +# 2# % n)
bycqF~?q<tYornYorq<s Q] Q]K^ `>b4cfeΩ g h b g i4jlk e b kuh sm j RN w `>b4cfe f ∈ L2(Ω) wxh)pug |'| b plj gNj
j) b j c jh e k(x) jup e g h>jr b h e~cBb h t9jlp-g k@qjj e g c #j- c p e j m jg - b hNp e q4h e jlp7p e k c e j t9jh e | b p cfe~c i8jlp 0 < k− ≤ k+ e jlfjlp gNj
0 < k− ≤ k(x) ≤ k+ |>kvjlp gNj|qdk eb g e x ∈ Ω.6 µLp
%" b k-p o c j- c p e j g h>j g h c gNj p b g e~cBb h z0rlq c lj! u ∈ H10 (Ω)
m jz w -wYh+3´q<s'tYnycm4nporq<s#]-°§U_VLQORQSU_VL_HQ;[WA«U_QSagV6[eAcOSNPU0CcADQSNWADORNWU_C6C6HY[P[WH#U0C³agVL[OSNPM6[WNWHg[^] Xpd0VFAcORNWU_C^4 µ j!MLADQVLC6H«U_CFGORNWU_COSHYE,O
vHIO?U0C\NPCjORY_QSHMLADQ?MLADQRORNWHHC\V4OSNP[WNeERA_C0O2[WAb«U0QRaJV6[WHf4H BQSHHCf6HT[] -4HQ&GINeGIHTµ60/4W
∫
Ω
divσ(x)v(x) dx = −∫
Ω
σ(x) · ∇v(x) dx +
∫
∂Ω
σ(x) · n(x) v(x) ds,
A¤0HYGσ = k∇u L°§U_VLQORHYC6NWQ;GIU0aM4ORHgf6Hg[eAGIU0CLf4NPORNWU_C ADV6[WNPaNPORHYEBf4Hg@;NWQRNeG&K6[WHIO;U0CG&K6U_NeE,NPO
H10 (Ω)
GU_aaHHYERMLA_GHf4H 9NP[WFHYQ,Op_HO?U_CORQSU_VL_H[WAT«U_QSagV6[eAcOSNPU0CJcADQSNeAcORNWU_CLC6H[W[PHf4HJ4 µ E HORQSU_V60HQ
u ∈ H10 (Ω)
OSH[vdjV6H ∫Ω
k∇u · ∇v dx =
∫
Ω
fv dx ∀ v ∈ H10 (Ω).
6 µ[/_BQ-,_GHTZJ[^] Kj`M<UDOSK6YERH\4 µ6¤2U_C_XQSNPLHTd0VLHB[eAJ«U_QSaHBLNP[WNPCLXYADNWQSHbf6H4 µ'/02HYE,OGU_CjORNWCV6H
∣
∣
∣
∣
∫
Ω
k∇u · ∇v dx∣
∣
∣
∣
≤ k+‖∇u‖L2(Ω)‖∇v‖L2(Ω),
HO9djV] HY[P[WHbHYE,O9GIUHQ&GINW_H∫
Ω
k∇u · ∇u dx ≥ k−∫
Ω
|∇u|2dx ≥ ν‖u‖H10 (Ω),
A¤0HYGν > 0
_Q-,_GIHZ¡[^] NPC6XY0A_[PNPORX\f4H°§U_NWCLGA_QRX0 ;C³M<HV4Ogf4U0CLGADM6ML[PNedjV6HQb[WH 0KLXU_QSaHf4H 'AcNP[W_Q&ADajGIHbd0VLNf4XaU_CjOSQRH;[] HI4NeE>OSHCLGHBHO[^] V6C6NeGINPORXTf4Hb[WAERU_[WV4ORNWU_C¡f6HB[eAg«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WH4 µ'/0F°!U0V6Q9aU0C0OSQRHYQ9djV6HJGHIORORHJERU_[WV4OSNPU0C¤A_QRNeAcOSNPU0C6C6HY[P[WHTHYE,O96NWHC VLC6HJERUD[WV4OSNPU0C´f4VM6QSU_6[WaHADV6[WNPaNPORHYE6 µ E vU_CM6QSU4GIYf6HGIU_aaH\M6QSXYGXYf4HYaaHYC0OJ[PU0QSETf4H¡[WAf6XaU_CLE,ORQ&AcOSNPU0Cf4V 0K6XYU_QSaH#40/40/4
.|+3´m4ycilk'| %]%] D[HpE>OJORQSYEJNPaM<U_QROSADCjOf4H[eADNeESE,HYQg[^] Xpd0VFAcORNWU_C 4 µ E TERU_VLEg«U_QSaHf4Hf6NP0HQS_HCFGIH H6U_CMFU0V6QSQSA_NO qs|>k cBb k cª[^] XpGIQSNPQSHTADVLESE,N−div(k∇u) = −k∆u−∇k · ∇u,a\A_NWE-GHIO,OSHf4HQSC6NWQSH«U_QSaHC] Abf4HERHCLE-djV6HERN6[PHGUjHWGINWHCjO
kHpE>O?f4XYQRNWcAD6[WH_D±9V\GIU_CjOSQSA_NPQSH_[^] XYdjVLAcOSNPU0C^4 µ E 2AV6CERHCLEa¦YaH#ERN
kHYE,Of4NWESGIU0CjORNWCjVv •C«rA_NOp6[WU_Q&ERdjV6HB[] XYdjVLADORNWU_C6 µ E 2HYE,OE,U0VLE«U_QSaH#f4H#f4NW_HYQR0HCLGH_4E,U0CNWCjORHYQRM6QSXIO&AcOSNPU0C¨RADVxE,HYCLET«rA_NP6[WH©GU_CjORNWHCjOM6[PVFEJfl] NWC4«U_QSa\AcORNWU_CFEJdjV6HE,U0CXYGIQSNPORV6QSHG[WA0ERERNWdjV6H0v°[WVLEM6QSXYGNERXaHYC0Op?[eAC6U_ORNWU_C f4Hf4NP0HQS_HYCLGIH«rA_NP6[WH³GU_CjORNWHCjO¡NWaM6[PNeGINPORHYaHCjOGHdjV] U_C ADMLMFHY[P[WH [WHYE
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[ nn " ) $#$7 # /[G¢_q<s'w!onporq<s't mLkvu roK3´onp|tw!|xnycm6s!tI3´otYtporq<sxHCjORQSH¡f4HYV4ERU_VLE,f4U_a\ADNWC6HpEbUGYGIV6M<XYEgMLADQf4HV4a\AcORXYQRNeADV6f4H¡GU_CLf4VFGORNWNOSXf4NC<XYQRHYC0OSH_!®2U0CLERNWf4XYQRU0CLEbV6CHHYaM6[PH\U'&
(Ω1,Ω2)HpE>OV6C6HTMLA_Q,OSNOSNPU0Cf4H
ΩE,VLQ[WA0d0VLH[W[PH
k(x)HYE,OGIU0CLE>O&ADCjOMLADQaU0QSGHYADV6
k(x) = ki > 0M<U_V6Q
x ∈ Ωi, i = 1, 2.6 µ0µ0
;C¡C6U_ORHΓ = ∂Ω1 ∩ ∂Ω2
[] NPCjOSHQR«rA_GIHrERV6M6M<U0ERXHbQSX0V6[PNWQSH9HONPCLG[PVFE,Hbf6ADCFEΩ?HYCjORQSH
Ω1HIO
Ω2_U0NPQ[eA!NP0V6QRHT4WpI6HIO
ui = u|Ωi
[eAJQSHYE,ORQSNeGORNWU_Cf4Hb[eA\E,U0[PV4OSNPU0CuZ
Ωi
1Ω
Ω
Γ
2
l4W DCjOSHQR«rA_GIHBHCjOSQRHTf4HYV4ERU_VLE,f4U_a\ADNWC6HpE?HO9GIU_CFf4NOSNPU0Cf4HBORQ&ADCLERaNWESE,NWU_Cv;?|+3 3´| Q] Q] Q^<`>b g+pnR | bde + pjz w u) j|>k b + t.j6z w -6jlp e g c i4q4jh e $
−ki∆ui = fm qdhp Ωi, i = 1, 2,
u1 = 0 pug k ∂Ω,u1 = u2 pug k Γ,k1∇u1 · n = k2∇u2 · n pug k Γ.
^4 µ>F jlp m jg - m jkuh c + kvjlp c hjlp m j z w *4p b h e q@|'|j jlp ¢_q<s'w!onporq<s'tTmLkvuroK3´onp|jt;w'|nycm6s!tl=3´ortptYoqFs pug ksR c h e jk rlq-j Γ wYh+3´q<s'tYnycm4nporq<s#](N
u ∈ H10 (Ω)
HpE>O9ERU_[WV4ORNWU_Cf4H^4 µ E 4MLA_QADM6M6[WNeGAcOSNPU0Cf4V 0K6XU0QRYaHf4HORQ&A_GHFL µ6WYµ6cU0Cf4U_NPO?A¤_U0NPQu1 = u2
ERV6QΓ'(NLU0C\MFUjE,H
σ = k∇u HIO σi = σ|Ωi= ki∇uiERA³QRHpE>OSQRNeGOSNPU0C´Z
Ωi!U0C ESADNPOdjV6H
σADNWCLE,NdjV6HESA³f6NP0HQS_HCFGIH_!ADM6MFADQRORNWHC6C6HYCjOZ
L2(Ω)±[WU_Q&E0HC¡0HQRORVf4V0K6XU0QRYaHFL FL G0[WAJGIU_aM<U0ESADCjORH9C6U_QSa\AD[WH
σ ·n A#VLC¡E,HYCLEE,V6Q ΓHO2U0Cf4U_NPO9A¤_U0NPQ
σ1 · n = σ2 · nERV6Q
Γ
XXYGINWM6QSU4d0VLHaHCjOYU0CGU_CLE,ORQSV6NPOV6CLH9«U0QRaJV6[WADORNWU_C¤A_QRNeAcOSNPU0C6C6HY[P[WH;f4H\^4 µ>FjMFU0V6QaU_C4ORQSHQbd0Vv] HY[P[WHJA0f4aHIObV6C6HV6C6Ned0VLHERU_[WV4ORNWU_Cd0VLN§GU)L^CLGINef4HA¤_HYG#[WAE,U0[PV4OSNPU0Cuf4H^4 µ E ;CG&K6HQ&G&K6H
ui ∈ H1(Ωi)HIObU_C³agV6[PORNWM6[WNPHG&KLA_djV6HJXYdjVLADORNWU_C³f6A_CLE
ΩiMFADQB[WA¡a¦YaHg«U0CLGOSNPU0CORHYE,O
v ∈ H10 (Ω)
6?CNWCjORXY_Q&ADCjOMFADQMLA_Q,OSNPHbHIOHYCE,U0aa\A_C0OU_CU04ORNWHCjO∫
Ω1
∇u1 ·∇v dx+
∫
Ω2
∇u2 ·∇v dx+
∫
Γ
(
k1∂u1
∂n1+ k2
∂u2
∂n2
)
v ds =
∫
Ω1
fv dx+
∫
Ω2
fv dx.^4 µ00®2U_aaHn1 = −n2
[] NPCjOSX_Q&AD[WHJERV6Qb[^] NWCjORHQR«rA_GHΓf6NWERMLADQ&A8VeOTZGYADVLERHf6H[eAGIU0CLf4NPORNWU_CA_V4[PNWaNOSHYE#f4HOSQSA_CLERaNeESE,NWU_C@J] ADV4OSQRHMLA_Q,OpvE,N
uHYE,Ogf4XILCLN?GIU0aaH
u1f6ADCLE
Ω1HIOu2
fLADCLE
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8/ " +# 2# % n)
Ω2L[eA\GIU0CLf4NPORNWU_C f4HbOSQSA_CLE,aNeERERNWU_C
u1 = u2ERV6Q
ΓNWaM6[WNWdjV6HgdjV6H
u ∈ H1(Ω)HC0HQRORVf4V vHaaH4FF µLP 60°§A_QGU_CLERXYdjV6HYCjOYª4 µ00§C] HYE,OQSNWHCf] A_V4ORQSH;djV6H;[WAg«U_QSagV6[eAcOSNPU0C\cADQSNWADORNWU_C4CLH[W[PH^4 µ'/_I
;u|jyc¢_or¢0| %]%]B^ @ Ω >%>@% ; # K # & <'2'( !" ' 9 RN >7892
Ω +." #<<>)C *" Ω \ K #6 % ;7 )% 1#/ f ∈ L2(Ω) / #" 9 >% <>% 7 & 9 #"'9"'( =92 Ω K ''78 < 2)% 2 & )! 9'((% !& #"!"#%! 9 % *" 7 '!" u +7 2 & <>;>% 2(% " 7 <>)> >7@;>7 !(% * & <'2)% ' & <7 1 #6!" 6292K +.#! !6624# 2C '!" 1!/ :!<<>6 *" 7 2 < 2 9,)% & C"% #492K / <)%7 & 4C & 9);)7 6=< 2)%
−∆u = f 92 Ω \Ku = C #% ∂K∫
∂K
∂u
∂nds = 0 #% ∂K
u = 0 #% ∂Ω,
C #, 4#" 62 '!"# 9);#>% & )% / % " )%$ @% & 7 2 " 2)% 2 >787 49)# <>%7 & :2 7 & # ! 9); & "(% >%$7 >'! #47 ; 9 :C"78 " (u,C) /
Y)"! ", 9B %+)S, Q% 9 , > K )/9 "9"))-"RU.M , , = , % 939 +-9% A(x) = (aij(x))1≤i,j≤N 0 ++SQP"*(B -+9%, A
2'"%&6O-9 ")#%:+%&9%.M3+
Ω C B, 9, %.M )(%"9%(Q F ,4 2 =LK/= %&9@QPW4 %(P)R%293"7, 2-P9 α > 09 (!Q"9 9Q*-+9"< -Ω
A(x)ξ · ξ =
N∑
i,j=1
aij(x)ξiξj ≥ α|ξ|2 39"ξ ∈ R
N , C 0 F)SQG4 2!"%689 )P';9" G,4 2 =LK-= "%&9QG4 %)R%(29 , "29 β > 0
9 ( Q""9 9Q'$-+9"< "
Ω
|A(x)ξ| ≤ β|ξ| "@9" ξ ∈ RN . C 0 $ F
"#$%&<-(-9 54 -9
−div(A∇·) = −N∑
i,j=1
∂
∂xi
(
aij(x)∂
∂xj·)
, C 0 F)< ,)+%( > 9!(9;" > R %%&9
−div(A∇u) = f -
Ω,u = 0
+∂Ω.
C 0 %F+% 9:9%.-9%"I"J C"+%(Q)I, $ %+ K 7P > : B C< " C 0 %F0 "I54 )R "< !, $ "",)9% "<(S,J$- "P" %% "%(+-+9 C "7Q (!, ",9%(.P%&9<G4 2::( < "799 : @ %9),)9% F )23,--,)9)9%+$/3" /+9%, ' C)+9%(Q< " , "",)9%.%&9A(x) C "9+9%" K 54 %( 9%9 F0 G, N /+9%, A(x)
S C)+9%(Q)G, -+9 + W$-')R K ( "9%+X)U,)"9X W4 " )! X, P.M ,)9% 0" Y +% U "), +
A =
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[ nn " ) $#$7 # /
As+Aa !9<-+9%7 C)+9%(Q As = (A+At)/2)N9<-+9%:-P9% C)+9%(Q
Aa = (A−At)/2
" ,-, W+%+9*Q"
−div(A∇u) = −div(As∇u) + V · ∇u.M ,
Vj(x) =
N∑
i=1
1
2
∂(aji − aij)
∂xi(x),
V 4 %9)9 > 9, "*.P%9)++! ", . ,)9% 0
! @ ' !#"$ %'&'()!#"$ %'*'()+,-!#"$ %'.'()/0,1234'5 +,6 75 8!:9;/<5>= (
u ∈ H10 (Ω)
25f ∈ L2(Ω)
$
*9 (-, )!( , $ %9% R(%%&9 ! %&9%,J"(S " C 0 %F $-!"X, "%&9%-"R %%9) ) " Q"% "<54 )-9)" C 0 F $ 4 ,)9%&
∂u
∂nA=(
A(x)∇u)
· n =
N∑
i,j=1
aij(x)∂u
∂xjni = 0
+∂Ω,
∂u∂nA
2 ( ")9%.M "%&9 , 9 - "u++,)%() K 54 )/9 " C 0 MF0 )+9 , "%&9% ) "Y)2S+ > S-99 ( 4 %PX B.PB"JC+%(QX%9QN+%Y%(+9P %&S( "R
σ = A∇uP ')R"9% Q< !, 99<-9 * "R 23P'+:*;
σ ·n = 0+
∂Ω 0 ! @ ' @?A
f ∈ L2(Ω) B g ∈ L2(∂Ω) B C>D 2E<5 6FGHH>D>'5 F5 -08 /I6 75 0
−div(A∇u) + u = f0/
Ω,∂u
∂nA= g
∂Ω.
*KJ* J³3L5? 93L= l1 SyT8PO,=>1 P'16= #US@BADCFEJGHIO,OSHERU_VLE,E,HpGORNWU_CC6U0VLE#XORVLf4NWU_CFEJdjV6HY[WdjV6HpE#M6QSU_M6QSNWXIORXpE#djVLAD[WNPOSAcOSNP0HYE#f6HYEE,U0[PV4ORNWU_CLEJf4V´ vA_M6[WA0GINWHC´A¤0HYGGIU0CLf4NPORNWU_CLEJADV4[PNWaNOSHYEf4H¡@;NPQSNWG&KL[PHOY'@BADCFETORU0V4ORHGIHO,ORHE,U0VLE>E,HpGORNWU_Cv
ΩHpE>ObV6CU_V60HQRO9<U_QSC6Xf6H
RN HIOf ∈ L2(Ω)
l H 0K6XU0QRYaH40/40/«U0V6QRCLNObV6C6HV6C6Ned0VLH#E,U0[PV4OSNPU0Cu ∈ H1
0 (Ω)f4H
−∆u = ff6ADCFE
Ωu = 0
E,VLQ∂Ω.
^4 F E by¤os'¢0o~?|w!k3´m4uoK3k3
9U_VLEBGIU0aaHYCGIU0CLEMLA_QQSHIORQSU_VL_HQ;GIHIORORH#MLQRU0M6QRNWXIOSX#f4XYGU_V60HQRORH#A_V®2KLADMLNOSQRH\¡r_Q-,_GHZf4HYE«U_QSagV6[WHYEHI4M6[WNWGNOSHYEHYC f4NWaHYCLERNPU0CN = 1 0U_NWQ[WA)XHa\ADQ&djV6H\_0/4W E HIO9H4M6[PU0NOSXHCjVLaXYQRNedjV6HaHYC0OA_V®2KLA_M6NPORQSH /6
£'hqFy4:[3 | %]%] bycors!¢_or~?|w'k3 m4uoB3k$3 " `_cf ≥ 0 |k@jup gNj9|Nq4k e b g em q4hp Ω o
q4 b k-p u ≥ 0 |>kvjup gNj|Nq4k e b g em qdhp Ω w
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pµ E " +# 2# % n)
.|+3´m4ycilk'| %]%]_\ vH;MLQRNWCLGINWM<Hbf6V¡a\Ac4NWagV6a C6HB«rA_NOdjV6HBORQ&A_f4V6NWQSH9V6C6HbM6QSU_MLQRNWXIOSX9MLA_Q,«rA_NOSHaHCjO-CLADORV6QSH[W[PHf4VM<U_NWCjO-f6HV6HM6K`4ERNWdjV6H HcMLA_Q-HI4HaM6[WHf6ADCLE§[WHGIU0C0OSHIORHf4H[^] XpdjVLAcOSNPU0C´E>O&AcOSNPU0C6CLADNWQSHf6H[WAG&KFAD[WHV6QpE,N2U_CG&KLA_V C<H f ≥ 0
Iv[eAORHaM<XQ&AcOSV6QRH\NWCjORXYQRNWHV6QSH\HYE,OOSU_V'A>U0V6Q&EM6[WVLE;_Q&ADCLf4H#djV6HJ[WA\ORHaM<XQ&AcOSV6QRHADV<U_Q&f´u ≥ 0
< HgMLQRNWCLGINWM<HJf4V a\ADNWagVLaQSHYE,ORHg¤A_[WA_6[WHgE,N'U_C QSHaM6[eA_GHT[PH vA_M6[WA0GINWHCMLA_Q[^] U0MFXYQSADORHVLQ9M6[WVLE0XC6XYQSA_[?6 µ Z¡GIUHlWGNPHYCjOSE!cADQSNWA_6[PHpEvf6A_CLEL∞(Ω)
U_VJ6NWHCJERN[] U_CGIU0CLE,Nef4YQRH[WHM6QRU06[WaHB^4W ¯ A¤_HpGGIU0CLf4NPORNWU_CA_V4\[PNWaNOSHYEf4HBHVLaA_C6C4 vAgcAD[WNWf4NPORXbf4VMLQRNWCLGINWM<HBf4Va\Ac4NPaJV6a HYE,O?«U0CLf6ADaHYC0O&AD[WHaHCjO[WNWXHADVGYADQ&A_GOSQSHb¨RESGAD[eADNWQSH©f4H\[^] XYdjVLAcOSNPU0C¥GD] HYE,O,Zcf4NWQRH\djV6H\[^] NWCLGIU0C6CV6HuHpE>OgZcAD[WHVLQSEfLADCLE
RI<®2HTM6QSNPCLGNPM<H#f4Va\Ac4NWagV6a OSU_aJFHT0XC6XYQSA_[PHYaHYCjOHYCf6XI«rADV6O9E,Nl[^] NWCLGU_C6CV6H
uHpE>OZcAD[WHVLQSE0HYGOSU_QSNPHY[P[WHYEbrMLADQHHYaM6[PHbM<U_V6Q[WH#ER`E,ORYaH^4 ¯ 2f6HT[] X[eA_E,ORNeGINPORX¤ •
Yh+3´q<s'tYnycm4nporq<s#];C´V6ORNW[PNeE,H[eA«U_QSagVL[WADORNWU_CcADQSNWADORNWU_C6CLH[W[PH³^4_gf4H^4 F E TA¤_HpGv =
u− = min(u, 0)djV6NADMLMLADQRORNWHCjO6NPHYC¥Z
H10 (Ω)
HYCx0HQRORVxf6V¥ vHaaH6 /6 />FxGADQu =
u+ + u−/;CA
∫
Ω
fu− dx =
∫
Ω
∇u · ∇u− dx =
∫
Ω
1u<0∇u · ∇u dx =
∫
Ω
|∇u−|2dx ≥ 0.6 FFp
³ADNeEu− ≤ 0
HOf ≥ 0
M6QRHpERdjV6HMLA_Q,OSU_V4O#f6A_CLEΩ°-ADQTGIU0CLERXYdjV6HCjOpªORU_VFEb[PHpEbORHYQRaHYETf6H^4 FL¤bE,U0C0O#CV6[eEYHIOJGU_aaH
u− ∈ H10 (Ω)
U_CHC´f6XYf4V6NPOJdjV6Hu− = 0
vGD] HYE,O,Zcf4NWQRH\djV6Hu ≥ 0
M6QRHpERdjV6HbMLA_Q,OSU_V4Of6A_CLEΩ
;2|+3 3´| Q] Q] `_cv ∈ H1
0 (Ω) o qd b k-p v+ = max(v, 0) q@|'|qdk eKc jh e $ H10 (Ω) j e
∇v+ = 1v>0∇v |k@jup gNj|Nq4k e b g em q4hNp Ω,
b*)1v>0(x) jlp e qr b h e~cBb h g c i4q4g e 9 $ b)
v(x) > 0 j e 6q c fjg k-pw.|+3´m4ycilk'| %]%] 'ATf4XaU0CLE>OSQSADORNWU_Cf4V\ HYaaH96 /6 />FBHpE>O?A0ERERH#J[WU_C60V6HHIO-OSHYG&K6CLNWdjV6HrH[W[PHgMFHYV4O;¦ORQSH#U_aNeE,H#HYCM6QSHaNPYQRHT[WHYGIORV6QSHpIL-4M6[WNed0VLU_CLE9C6XYA_C6aU_NWCLEM<U_V6Q&djV6U_N!GIHIORORHgf6XIaU0CLE>OSQSADORNWU_C´Cv] HpE>O\MLA0EJERNPaM6[WH_-HOHYCxMFADQRORNeGIV6[WNPHYQJM<U_VLQSdjV6U0NU0C C6HM<HV4O\MLA0EJV6ORNW[PNeE,HYQ[PH vHaaHFL µ6W djV6N!A4W\QRaHgdjV] V6C6H#«U_CLGIORNWU_C f4XLC6NWH;¨>MFADQaU_Q&GIHpADV4j©6ADM6MFADQRORHCFADCjO9Z
H1f6H9G&KLA_djV6HERU_VFE>f4U_a\A_NPC6HHIOGIU_CjOSNPCV6H9A_V4JNWCjORHYQ,«rA0GIHYE§HYCjORQSH[PHpEERU_VLE, f6U_a\ADNWC6HYE-A_M6MLA_Q,OSNPHYC0OHYC«rADNPOZH1 f4V f4U0a\ADNWC6HHCjOSNPHYQY?C´HlCªHIOp'U0C´ADM6M6[WNed0VLHQ&ADNPOg[PH HaaH FF µLP Z
vERV6Q[WH¡ERU_VLE,f4U_a\ADNWC6H
v > 0HOZ E E,V6Qg[PHE,U0VLE>f4U0aA_NPCLH
v < 0vHOg[PHORU_VLQJERHQ&ADNPOA>U_V6X0' vHMLQRU06[PYaH;HpE>OdjV] HYC¡0XC6XYQSA_[F[eA#«QSU_CjOSNPYQRHbf4HbGIHYEf4HYV4¡ERU_VLE, f6U_a\ADNWC6HYEBrf6XILC6NWHbMLADQ
v = 0Cv] HpE>O-MLA0E'QSX0V6[WNPYQRHBrADVE,HYCLE-f4H[WAB@;XILC6NPORNWU_C\µL /6 0la¦aHE,N
vHpE>O-V6C6H«U0CLGOSNPU0CgQSX_VL[PNWQSH_
•; <>='? ?A@B< A +9'9" 4 ; UQ$+%
v ∈ H10 (Ω)
*+%G(t)
2*!6O,)9% R ""
R$ "S, (++
C1 9 Q"G(0) = 0
G′(t)
)2*;9"!+"R-9
G(v) ∈ H10 (Ω)
)∇(G(v)) = G′(v)∇v 0 - ")#%&9% "
H10 (Ω)
%1)R%29X"S+%&9X "!6O,)9%"vn ∈ C∞
c (Ω)QP"%, P.M)9*.M)9v ""3 X9* "
H1(Ω) C ) $/+9%, %)"@'+ = +"%9 vn) ∇vn, P.M)9@"9 9Q@-+9: ""
Ω.M)9
v) ∇v
9 + ,)9%.M )P ! .%&:<, 9((-%9 0 P0 "#" % F0
|G(vn)−G(v)| ≤(
supt∈R
|G′(t)|)
|vn − v| , C 0 DTF
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[ nn " ) $#$7 # YµL ",
G(vn), P.M)9*.M)9
G(v) ""
L2(Ω) 0 4 -"+9*-+" 1 ≤ i ≤ N
∣
∣
∣
∣
∂G(vn)
∂xi−G′(v)
∂v
∂xi
∣
∣
∣
∣
≤∣
∣G′(vn)−G′(v)∣
∣
∣
∣
∣
∂v
∂xi
∣
∣
∣+
(
supt∈R
|G′(t)|)
∣
∣
∣
∂vn
∂xi− ∂v
∂xi
∣
∣
∣. C 0 D MF
|G′(vn)−G′(v)|∣
∣
∂v∂xi
∣
∣
, .)9S"9 9Q"S-+9"*.)9 C "*"!+" = +%&9 F * 2 2-9 -2 (sup |G′(t)|)
∣
∣
∂v∂xi
∣
∣
Q%--+9% K L2(Ω)@--"(%,/9% " 9J) > Y "
,)P.M)9 , "%(" W ; +" C .M%B( 9J - > T0 P0 $ "+" % F , )+9 +"%9 " 68",)9%,)P.M)9 .)9 ""
L2(Ω) 0 W )9%)!9)9B - C 0 D F , .)9 "++%:.M)9 ""L2(Ω)
7 ",G(vn)
2 " +"%&9 " -,JC ""H1
0 (Ω)QP"%@ 2 +$-, " '% =;+ F !, P.M9*.M9<"*%%&9 w ∈ H1
0 (Ω) 0 -<%( " 9%&#$,/9% @%(%&9 " +9".MQw = G(v)
"L2(Ω)
) Q ∂w∂xi
= G′(v) ∂v∂xi
"L2(Ω)
W4 *9 +&- 0 "" %9 $-*-"9,J( 68,9%( t → max(t, 0)
$/*"X+"%&9B "S6O,)9%"Gn(t)
"B C<9 , :"7 " P+9:Qv+ -$/+9% K H1
0 (Ω) 0 %& G(t)"36O,)9%
C1(R)
9)(SQ
G(t) = 0+%
t ≤ 1
2, 0 ≤ G′(t) ≤ 1
+% 1
2≤ t ≤ 1, G′(t) = 1
+%1 ≤ t.
U )#$"%Gn(t) = G(nt)/n
"n ≥ 1
U9%&-54 -9 *9 , QGn(v) ∈
H10 (Ω)
∂Gn(v)∂xi
= G′n(v) ∂v
∂xi0 4 +9$-+" .M)9%&#$!Q"
|Gn(v)− v+| ≤ supt∈R
|Gn(t)− t+| ≤ 1
n,
",Gn(v)
,)P.M)9.M)9v+ ""
L2(Ω) 0 B-++%5$"@9"1 ≤ i ≤ N
∣
∣
∣
∣
∂Gn(v)
∂xi− 1v>0
∂v
∂xi
∣
∣
∣
∣
= |G′n(v)− 1v>0|
∣
∣
∣
∂v
∂xi
∣
∣
∣≤ 10<v<1/n
∣
∣
∣
∂v
∂xi
∣
∣
∣,
X,)10<v<1/n
, P.M9B.M9 "9 9Q $-+9"GB9J - > " , P.M)9 ,) "%" G ;)+9".MQ ∂Gn(v)∂xi
, P.M)9.)91v>0
∂v∂xi
"L2(Ω) 0 Y ""%W,
9 , PQGn(v)
, P.M)9.)9v+ "
H10 (Ω)
'Q" ∇v+ = 1v>0∇v 0
9u|jyc¢_o¢_| %]%]B^ \ ="(%)%*7 2<<7 2 ",+6 !K7 ;2 % 1 v → v+ #6 !" ?9 L2(Ω)92 78 ! &>& 2 *" 9 H1(Ω) 92=78 ! &)& I+ 7 6>%7 = 2 4* ∇u = 0 <)%C*< 2)% !%?7 > 6 & 7 u−1(0) 1#/.¡h+AFk!rm6y¤ornYh
9U_VLEgaU_CjORQSU_CLEba\A_NPCjORHYCLADCjOgdjV6H\[eA ERU_[WV4ORNWU_Cfl] V6CM6QSU_6[WaHA_V4[WNPaNPORHpE#H[W[WNPM4OSNWdjV6HHYE,OM6[PVFEQRXY_V6[WNPYQRHbdjV6HTM6QSXV6HTERNl[PHpEf4U_CLC6XHpEE,U0C0OM6[WVLEQSX0V6[PNWQSHYEdjV6HTC6XpGIHpERESADNWQRH0 £'hqFy4:[3 | %]%] >w!|y¤h+A<k'm4yconph " `Qb4cfe g h jh e~c jk m ≥ 0 w `Qb4cfe Ω g h b g i8jk e b kuh m j RN m jslq4p-plj Cm+2 w `Qb4cfe f ∈ Hm(Ω) w %" b k-p o R g h c gNjp b g e~cBb h u ∈ H1
0 (Ω)m jEz w * -
q@||Nq4k e~c jh e $ Hm+2(Ω) w j|> g+p o R q-||> c -q e~cBb h f → u jlp e c h q c kvj9 b h eKc h_gNj m j Hm(Ω)m q4hNp Hm+2(Ω) o _ jup e $ m4c k@j g c j- c p e j g h>jE b hNp e qdh e j C > 0e jfj gNj
‖u‖Hm+2(Ω) ≤ C‖f‖Hm(Ω).
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pµ'/ " +# 2# % n)
°-ADQ!ADM6ML[PNeGADORNWU_CbNPaaXYf6NWADORH2f6V 0K6XU0QRYaHf6H?QSX_VL[WA_QRNPORX40/40/ ¯ HO'f4V 0KLXU_QSaH$FF µL /0E,VLQT[WAGU_CjORNWCV6NOSX\f4HYEb«U0CLGOSNPU0CLETf4HHm(Ω)
IvU0CU04ORNWHCjOgGIU0aaHGU_QSU_[W[WA_NPQSHg[WHQRXpE,VL[O&AcOA_C6C6U0CLGIX9A_V6MLA_QSA¤cADCjOf6A_CLE[eA7XHYaA_QSdjV6HB40/4 µ6_Z#ESA¤_U0NPQdjV6H9[PHpE2ERU_[WV4ORNWU_CFE§«rADNW6[WHYEfl] XpdjVLAcOSNPU0CLE?A_V4f6XQSNP0XHYEMLADQRORNWH[W[PHpE-H[W[WNPM4OSNWdjV6HpE?ERU_CjOHC«rA_NOf4HYE2E,U0[PV6ORNWU_CLE§«U0Q,OSHYErU_VG[WA0ERERNWdjV6HpESERNl[WHYEf4U0C6C6XYHYEERU_CjOQRXY_V6[WNPYQRHpE qFycqFrm6ory¤| %]%] `c
Ω jlp e g h b g i8jk e b kuh 9m j RN m j9l q8p-pj Cm+2 o p c f ∈ Hm(Ω) oj e p c m > N/2 o q4 b k-p7q9p b g eKcBb hi4q4k c q eKcBb hNh>jlfj u ∈ H1
0 (Ω)m jz w * - jup e g hj7p b g e~cBb h
r b k e j -q4kEjfj.q@|'|qdk eKc jh e $ C2(Ω) w2h |qdk eKc g c jk o p c Ω jup e g h b g i8jk e b kuh m j RN m j)q8p-pj C∞ o j e p c f ∈ C∞(Ω) o
qd b k-p7 q.p b g e~cBb h u ∈ H10 (Ω)
m j)z w * ujlp e q4g+p-p c#m qdhp C∞(Ω) w.|+3´m4ycilk'| %]%] D[<HYE,O?NWaMFU0Q,O&ADCjO2f4H;6NPHYC¡GIU0aM6QRHYCLf4QSH[WA#MFU0Q,OSXH9f4HBGIHpEQSXYERV6[O&AcO&E?f4HQSX0V6[eADQSNOSX_ªC´ERV6M6M<U0ESADCjOgdjV6H
∆uvdjV6N?HYE,O#V6C6HGIU0ag6NWCLA_NWERU_CMLADQRORNeGIVL[PNWQSH\f4HGIHYQ,O&ADNWC6HYEf6XQSNP0XHYE9E,HpGIU_CFf4HYEf6H
u<ADM6MFADQRORNWHCjO;ZV6C³GHQROSA_NPCHYERMLA0GIHT«U0CLGOSNPU0C6C6H[^LU0CHC f4Xpf4V6NPOBdjV6HnpqFk'nY|t\[WHYEJf4XQSNW_XHpE#ERHYGIU0CLf4HpE#f4H
uA_M6MLA_Q,OSNPHYC6C6HCjOgZ GHa¦aHHpE,MLA0GIH²$*2NWHC´ER6QplORU0VLEGHYEJQRXpE,VL[O&AcOSEf4HQRXY_V6[eADQSNPORX¡ERU_CjOJXYNWf4HYCjOSEHC¥f4NPaHYCLE,NWU_C
N = 1M6V6NeERdjV6H[PH vA_M6[eA_GINWHCGU)L^CLGINef4HA¤_HYGg[WAf4XYQRNW_XYHJERHYGIU0CLf4HJHIO#f4U_CFGg[^] XpdjVLAcOSNPU0C«U_V6QSC6NPO#f4NPQSHYGIORHYaHYCjOB[eAQSX0V6[WA_QRNPORXf6H#GIHIORORHTf4XYQRNW_XYHbERHYGU_CLf6H_ •
.|+3´m4ycilk'| %]%] vH0K6XYU_QSaH;f6H;QSX0V6[WA_QRNPORXB40/40/ ¯ HIOERU_C®2U0QRU0[P[eADNWQRHB40/40/'G;QSHYE,ORHCjOcA_[WA_6[PHpEMFU0V6Q¡f4HpEGIU0CLf4NPORNWU_CLE¡ADV4¥[PNWaNOSHYE¡f6H HVLaA_C6CD[WEERH_XYC6XQ&AD[WNeE,HYC0OA_VLERERNADVGYA_Ef6HYE\U0MFXYQSADORHVLQSEHY[P[WNPM6ORNed0VLHYEZGUjHWGINWHCjOSE\cA_QRNeAD6[WHYEGIU0aaH f6A_CLE\[eA (U_VFE>E,HpGOSNPU0C6 /6 µj<@;A_CLEbGIHJf6HQSC6NPHYQBGYA_EYLNW[!«rADV4OTA'A>U0V4ORHYQ;Z[^] Kj`M<UDOSK6YERHgKLA_6NPORV6HY[P[WHf6HJGIUHYQSGNPNPORXJf6HYEGUHlWGINWHCjOSEYª[] K`MFU_ORK6pE,HdjV6HJ[WHYEbGUjHWGINWHCjOSEbERU_CjObf6HG[WA0ERERHCm+1 f6A_CLE Ω
¬O&ADCLf6NWETdjV6HΩHYE,OFU0QRCLXBQSX0V6[WNPHYQf6H#GI[eA_ESE,H Cm+2 HIO9djV6H f ∈ Hm(Ω)
I • <! ")+9"3$@ :J - > ! "*9)(/9%9 0 T0 T "39"99B %&9X -%(+ * "*,-$/+9%(,)%)!'+%(" C *R""% Q")9"* -*( V /Q 0 TP0 MT, ! $++ ,X$/+9%(,)%)- ,S ") F0 + " -, "
Ω = RN N)X"f ∈ L2(RN )
, "+% > 9"9; > −∆u + u = f
"R
N . C 0 DDMF <G4 C $- " , $ "%&9% "R %%&9 A @ @B? - C 0 DDMF "%9QG4 %@G4 C $- ; 0 -%" "!, +9 K 54 %#$%G "( +"9% 2<"*+-+9! ", $ "%&9% "R %%&9 0" Y"9
f "
L2(RN )!.M,J%+%I , "%&9% -"RU%%9) @ @ @B? Q% 2
"S,J")9,J")u -
H1(RN ),4 )2 = K/= "%&9BQ -'" , )+-%(U+ "
u(x)L9)$ .M)9 )9 K 4 %"#%-#$ Q"3 4 %9 - ∫
RN |u|2dx, P.M9 0 :689 "(-9% .-9%(-9%" " C 0 DDMF 2 F+9.)
u ∈ H1(RN )9 GQ∫
RN
(∇u · ∇v + uv) dx =
∫
RN
fv dx ∀ v ∈ H1(RN ). C 0 D F 'X"%,-9% "%&9 ,9 :J" > B " N/R =2A %- 0 P0(E "9.MXQG4 % )R%(29X"X"%(Q+"9%
u ∈ H1(RN ) " C 0 D F0 " "#G-S%+" )P %( 9% Q" K 9" , "RXQ7"/. " K 6O%&3 ""7, @,J$"%&+9<+9 Q"3, )+9<+"9% <(*6O-9 " /9% ./9%(-9%" )23"++%
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[ nn " ) $#$7 # Yµ0µ+"9% " 54 Q$/9%( "R ")9%.M) X-+9% C 0 DDF0 S".M"X -%P9)$S ",)) 9)+&- " 9 " /9%&9 0 *=A@ ?A@B< -
f ∈ L2(RN ) 01&.5" 01 & 0 4+-&.( u ∈ H1(RN ) # : !)5+- (+ ($,.- + 3 H2(RN ) : 6 77 2- f ∈ Hm(RN ) .
m ≥ 0 01&.5" u
!.5"+- (9+ 3 Hm+2(RN ) :; <>='? & ?A@B< 4 %9 "% ++ 9% < " (Y "2+-9% 2B L)9JP " S++( =9%$ 0 1" h ∈ RN
h 6= 0" )#$"%@" Q-9% < "%)9 9%
Dhv(x) =v(x + h)− v(x)
|h|Q"% "$-+9% ';% K H1(RN )
+%v ∈ H1(RN ) 0 .M)9%&#$X%+ )PQP" ∇(Dhv) = Dh(∇v)'Q""
v, φ ∈ L2(RN )" " ?689 " W4 %9 /9%( $/<$-+9%! %(+, > 9 ∫
RN
(Dhv)φ dx =
∫
RN
v(D−hφ) dx.
4 -"+9 99%(9 "SQ-9%()P Dhv+ " ""1 G 0 T0 PE0 "1( 6O-9 (/9%(.-9%(/9%(" ( C 0 D F "9 v = D−h(Dhu)
") "% Q@ 3 > <, % = " ++"< ;9% ∫
RN
(
|∇(Dhu)|2 + |Dhu|2)
dx =
∫
RN
fD−h(Dhu) dx.
""%&@ B 2--9%
‖Dhu‖2H1(RN ) ≤ ‖f‖L2(RN )‖D−h(Dhu)‖L2(RN ).
"$/<"%,-9% C 0 D MF $ B-++%
‖D−h(Dhu)‖L2(RN ) ≤ ‖∇(Dhu)‖L2(RN ) ≤ ‖Dhu‖H1(RN ).
,' ‖Dhu‖H1(RN ) ≤ ‖f‖L2(RN )
)< -+9%, %)"1 ≤ i ≤ N
∥
∥
∥Dh
∂u
∂xi
∥
∥
∥
L2(RN )
≤ ‖f‖L2(RN ),
Q"%$%%(QP"P 4 -" > 3 G 0 T0 PE , % = " ++ "Q ∂u∂xi
-+9% K H1(RN )P,4 2 =LK-= "%&9Q"
u ∈ H2(RN ) 0"+! %9 $-SQP"f ∈ H1(RN ) 0 A P+9SQ ∂u
∂xi
2S 4 "%(QB+(9% -H1(RN )
−∆ui + ui =
∂f
∂xi
""R
N . C 0 D FP%,4 2<.%5$-3"%(, -9% '(X-+9%*9 , 9* " ( ")2+/9%GP " "%&SQ∂u∂xi
-+9% K H2(RN )),4 2 =LK-= "%&9IQP" u ∈ H3(RN )
,) "9). 0 ,)9%.M"G(:6O-9 " /9%.-9%(/9%(" ( C 0 D F .M ,*(X68,9%( 9 2 ∂φ∂xi
"φ ∈ C∞
c (RN )
∫
RN
(
∇u · ∇ ∂φ
∂xi+ u
∂φ
∂xi
)
dx =
∫
RN
f∂φ
∂xidx.
Ecol
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re.
pµ>F " +# 2# % n)
u ∈ H2(RN )
f ∈ H1(RN ) @%(9 -9) $-@-+9%<;"9)%&
∫
RN
(
∇ ∂u
∂xi· ∇φ +
∂u
∂xiφ)
dx =
∫
RN
∂f
∂xiφ dx C 0 D $ F
QP"%5 $/ ")+%&9 9 29 .-(; "X9φ ∈ H1(RN ) 0 , 2-9 Q C 0 D $ F 2( 6O- = (/9%( ./9%(-9%" B C 0 D F0 /*, +Q)P ∂u
∂xi= ui
2*;"% 4 "%(Q!+"9% "H1(RN )
C 0 D F0G ,-f ∈ Hm(RN ) ⇒ u ∈ Hm+2(RN )
+ +9B-!9 , +9 , +m
, .% < **65%&9!
m = 1 0
&.45v ∈ L2(RN ) h ∈ RN h 6= 0 &.( # (- + 4( 4&.+- (+ # - 52(+- ,0
Dhv(x) =v(x + h)− v(x)
|h| ∈ L2(RN ).
9-v ∈ H1(RN ) &.( 0 "+-B7 .+-&)(
‖Dhv‖L2(RN ) ≤ ‖∇v‖L2(RN ). C 0 D F ,- 5 & 497 (9+ &.- + v ∈ L2(RN )
-0 .-B"+ 4( 2&.(92+ .(+ C + ,0 0 49 &.45 + &.4+ h 6= 0 &.(
‖Dhv‖L2(RN ) ≤ C, C 0 D F 01&.5"
v ∈ H1(RN )+ ‖e · ∇v‖L2(RN ) ≤ C
&.45 + &.4+. ,+ 454(9- + e ∈ RN :
; <>='? ?A@B< +9@ W4 -; C 0 D MF0 1" v ∈ C∞c (RN )
" ,)9%&
Dhv(x) =
∫ 1
0
h
|h| · ∇v(x + th) dt,
QPW4 2-9$/|Dhv(x)|2 ≤
∫ 1
0
|∇v(x + th)|2 dt.
9 -@ x%G.%
‖Dhv‖2L2(RN ) ≤∫ 1
0
∫
RN
|∇v(x + th)|2 dxdt ≤∫ 1
0
‖∇v‖2L2(RN )dt = ‖∇v‖2L2(RN ).
- " "+%&954 29% -9% C 0 D F )2 .-%("@99'6O,)9% "H1(RN ) 0%< -%(9
v ∈ L2(RN )Q%.M)9%&#'54 29% -9% C 0 D MF0 ) ""%<Q"P39"
φ ∈ C∞c (RN )
∣
∣
∣
∣
∫
RN
Dhvφ dx
∣
∣
∣
∣
≤ C‖φ‖L2(RN ).
-@%9 /9%( $/@$-+9%! %(+, > 9$ ∫
RN
(Dhv)φ dx =
∫
RN
v(D−hφ) dx,
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[ nn " ) $#$7 # Yµj,
φ 2@9 % > 9$+%G + h = te
.M ,e ∈ RN
e 6= 0"
limt→0
D−hφ(x) = −e · ∇φ(x).
/<, "+Q ) " "%&<Q""1 ≤ i ≤ N
)@9"φ ∈ C∞c (RN )
∣
∣
∣
∣
∫
RN
v∂φ
∂xidx
∣
∣
∣
∣
≤ C‖φ‖L2(RN ),
,)@QP"%PG4 2 9% W4 -"+9@Q:( )#$"%9%B "354 "$/+9 $-, @ v K H1(RN ) C .%& )#$"%&9%(
D0 0E/F0
@ ? " " R""% Q"@, 3 $++S ,-<+-%9S "@(B 9 =+%&9% 0 T0 C .M , Ω = RN F ,X ")< :J - > 0 TP0 T 0 9%(%+ " -9 *,/+9 @(, '@'$-+9%&9% "*54 "%&9, ' "3(B " 2+/9%( *(B 9+%&9%D0 D"0 TP0 "%(.'()*--9%"! " ( )#%&9% 0 TP0 4 U.)+'9 %) C .M%++% ( D1%("9D0 DF P%$)R%29 "9 , ".9 #$% " Ω
-3 7.M+9(ωi)0≤i≤I
)I" -+9%&9% "<54 "%&9(θi)0≤i≤I
9 SQ
θi ∈ C∞c (ωi), 0 ≤ θi(x) ≤ 1,
I∑
i=0
θi(x) = 1 ""
Ω.
4 .)+ω0
2 K 54 %(9)9% B Ω C 65-%&
ω0 ⊂ Ω F -$ "%SQB X-"+9 S".M)+9 ωiN"
i ≥ 19 ,).9 <!;
∂Ω 0 % u ∈ H10 (Ω)
(B+(9%"% Q" " C 0 D F0 1" +9)@(9)(/9%9B "u =
∑I
i=0θiu
U. +9) 9 (-9%&9 ,J$,)U " <9)9 θiu 0 1"*99
θ0uU$-9 X( 9 " /9%&9B%P99%()"9B
u 0 1 2*%( "%(-*QP" θ0u 2( +(9%
"%(Q ""H1(RN )
−∆(θ0u) + θ0u = f0
"R
N ,.M),f0 = θ0(f − u) − 2∇θ0 · ∇u − u∆θ0
Q"%*"$/+9%()P K L2(RN ) 0 / -%,/9%( "( 9+%&9%( 0 TP0 " %& ",Q
θ0u ∈ H2(RN ) 0 , % )9) 4 %9) (9)(/9%9 f0G!$-S-"(%,/9% +", , ++%.M B( 9+%9% 0 TP0 , , SQP"
f ∈Hm(Ω)
%"(%(Q<Qθ0u ∈ Hm+2(Ω) 0 17P+9I(*9 (-9%&9' " I+9 I9)9) θiu
"i ≥ 1
%365-" 4 -; L9 9 ++)" ;- + - )B ,Ω = RN
+ 0 L:65 ", +9)79 +"/< " 3 C QP"'( 9+%&9%( 0 T0 %:" Ω = RN+ 0 4 2<")U" %(, -,/ 9 9)+9 ;- -, -+9 ,-()Y ,J$-X *, B, % 9! "
54 -9 3 %(9%(QP" C G" -, % " .P% 7" )/9 " K , , % 93./9%(; 7, < -( = + ,)9% 0 TP0 MFF " 9 P.MAC K " $% ' ' ")-%( 0 " 9 +$%N65-" 9)9 %&'Q(9 "(-9%&9* "9:54 +$-, '): -7" " % = +, <+ B7:"9.M7(9 " /9%&9* -" .M)+@;-9*9 %) 0 •
9u|+3~§r|w'|³tYos$A<k'm4yconph-Uc`0U_CLEga\ADNWC0OSHCLA_CjOJV6CxHI4HaM6[WHf4HE,U0[PV6ORNWU_CLEtYos$A<k!roB:jyc|t_G_] HpE>OR ZD f6NPQSHC6U0C´QRXY_V4[PNWQSHYEYD[§E] AD0NOBfl] VLC M6QSU_6[WaH#M<U0ERXgf6A_CLEV6C³U_V60HQRO9C6U_CQSX0V6[WNPHYQMFU0V6Q9[WHYdjV6HY['[WH 0KLXUDQRYaHTf4HbQSX0V6[WA_QRNPORXT40/40/ ¯ HIO9ERU_C ®2U_QSU_[W[WA_NPQSH;6 /6 /[G#ERU_CjO«rADV4l4CMLA_Q,OSNWGV6[WNPHYQY6NWHCdjV] NP[HI4NWE,ORHf4HYEBERU_[WV4ORNWU_CLE«rADNW6[WHYErGD] HYE,O,Zcf4NWQRHJADM6MLA_Q,OSHCLA_CjO;Z[^] HpE,MFA_GIHJf4H (U0FU0[PHY
H1(Ω)NW[C] `ABMLA_E-f4HERU_[WV4OSNPU0CLE!«U_QRORHpErGD] HYE,O,Zcf4NWQRHf4HYV4#«U0NWEf6XQSNPcADL[PHpES'M<U_V6QGHM6QRU06[WaH_D9U_VLE
Ecol
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repr
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re.
pµ ¯ " +# 2# % n)
Γ0
Ω
ΘΓ2
Γ1
Γ2
Ω
Γ0
Γ1
Θ
l40/ (4HYGOSHV6QA_C6_VL[WA_NPQSHΩfl] ADC60[PH
ΘrM6[WVLEM<HIORNPOU_VM6[WVLE0QSA_CLfdjV6H
πI
CLU_VLEM6[eA)GU_CLEHYCf4NPaHYCLE,NWU_CN = 2
fl] HYERMLA0GIH_4HOC6U_VFEGIU_CFE,Nef4XQSU_CFE2VLCERHYGIORHYV6Q9ADCL_V6[eADNWQRHΩf4XLC6NHYCGUjU0QSf6U_C6C6XYHYEQ&A_f4NeAD[WHYEMLA_Q#_U0NPQ[eA 'NW_VLQRH#40/_
Ω = (r, θ) OSH[vdjV6H 0 ≤ r < RHO
0 < θ < ΘA¤0HYG0 < R < +∞ HIO
0 < Θ ≤ 2πQ&ADM6M<H[WU_CFEBdjV6H
x1 = r cos θHIOx2 = r sin θ
;CCLUDORHΓ0
[WAMLA_Q,OSNPHbf4VFU0QSff4HΩU&
r = RΓ1
GIHY[P[WHbU&θ = 0
HIOΓ2
GH[W[PHTU'&θ = Θ
F®2HIOU0V6_HYQ,OΩM6QSXYERHCjORHbOSQRU0NWE2¨,GU_NWCLE©6a\ADNeEERHV6[WH#[^] U0QRNW_NWC6H\[WHgGU_NWCHCjORQSHT[PHpE<U_Q&f6E
Γ1HIO
Γ2M<HV6OBM<U0ERHQBM6QSU_6[WaHgM<U_VLQ;[eA¡QSX0V6[eADQSNOSXgf6ADCFE9[WHYE;HI4HaM6[WHYEbGNf4HYESERU_VLEYL°K`4ERNWdjV6HYaHYCjOY[WH;GYA_E2fl] VLCADCL_[WH
Θ < πHYE,O2QSHM6QSXYERHCjOSADORNP«fl] VLC¡HlCªHIOf4H9M<U_NWCjORH0_O&ADCLf4NeEd0VLH9[WH;GA0E
Θ > πGU_QSQRHpE,M<U_CFf¡ZV6C6HTHYCLGIU4G&K6H\U0Va¦YaH#ZV6CLH Z!tptYk'yc|ERNΘ = 2π
°§U_VLQV6CHCjORNWHQk ≥ 1
4U_CXIOSVLf4NWHT[PHpEf4HV6¡M6QSU_6[WaHYEA_V4¡[WNPaNPORHpEE,VLNPcADCjO&E
−∆u = 0f6ADCLE
Ωu = cos
(
kπθΘ
) ERV6QΓ0
∂u∂n = 0
E,VLQΓ1 ∪ Γ2
^4 E HO
−∆u = 0f6ADCLE
Ωu = sin
(
kπθΘ
) ERV6QΓ0
u = 0E,VLQ
Γ1 ∪ Γ2
^44p9U_VLEM<U_VLQRQSNPU0CLEXIOSVLf4NWHQ[eAQRXY_V6[eADQSNOSXBf6HYEERU_[WV4ORNWU_CFEf6H4 E 2HIOg6 6p?HCORHYQRaHpEfl] A_M4MFADQRORHCFADCLGHJU0VC6U0CA_V4 HYERMLA0GIHYEbf4H(U0FU0[PHY
Hm(Ω)la\A_NWEYlMLADQTERU_VLGNf4H\E,NWaM6[WNWGNOSX\HIOM<U_VLQ#0A_QSf4HYQTV6CE,HYCLETM6K`4ERNWdjV6H\XYjNef4HYC0OZC6U0EbQSXYERV6[POSAcO&EvC6U_VLEgAD[W[PU0CLE#ERNPaM6[WHaHCjO#C6U0VLENWCjORXYQRHpERERHQ9A_V GIU0aM<U_QRORHYaHCjO9f4V 0QSA0f4NPHYCjO ∇u ADV0U_NeE,NWCLAD0HTf4H#[^] U0QRNW_NWC6H_F@g] V6C M<U_NWCjOBf4HV6HM6K`4E,NedjV6HU_V³aXpGADCLNWdjV6H0ªGHJ0QSA0f4NWHCjOBGU_QSQRHpE,M<U_CLf³Z¡V6C LV4f6HG&KLA_[PHYV6Qp<A_VG&KLA_aMXY[PHpGOSQRNedjV6H_ªU_VZV6CG&KLADaMf4H\GIU_CjOSQSA_NPCjORHpE HlNW[-NWaM<U_QRORH\f4H\ESA¤_U0NPQbE,NGHIORORH\djVLADCjORNPORXHpE>O<U_QSC6XYHbGU_CjORNWCV6HbU_VC6U_CZ[^] U0QRNW_NWC6H_
;2|+3 3´| Q] Q] \_\ j - c p e jg hj6g h c gNj6p b g e~cBb h rlq c lj m j z w - m qdhp H1(Ω) o m'b hNh j|Nq4k qr b kutg j
u(r, θ) =( r
R
)kπΘ
cos
(
kπθ
Θ
)
.^4/_
Ecol
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lyte
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Ne
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repr
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re.
[ nn " ) $#$7 # Yµ+G
jtt.j o c j- c p e j)g hj9g h c gNj9p b g e~cBb h)rlq c l j m j z w u m q4hNp H1(Ω) o mb hNh j |qdk9qr b kut g j
u(r, θ) =( r
R
)kπΘ
sin
(
kπθ
Θ
)
.^4Dµj
qdhpjlp m jlg --q4p o p c k = 1 j e π < Θ o qd b k-psj kvq mdc jlh e ∇u hQ jlp e |q4pn b kuh $Ev b k c c hj oe qdh m4c p gNj p c k ≥ 2b g π ≥ Θ o q4 b k-p7j kvq m4c jh e ∇u jup e b h e~c hNg $R b k c c hj w
.¡|+3´m4ycilk!| Q] Q] \ U_Q&ESd0VLHT[] U_CNWaMFUjE,H#VLC6HJf4U0C6C6XYHgf4Hg@;NPQSNeG&K6[PHO9M6[WVLE;_XCLXQ&AD[WHbfLADCLE[PHpE?M6QSU_L[PYaHpE9^4 E HIOb44¤jU_CM<HV6O[eAgf4XpGIU0aM<U0ERHQ2HCVLC6HBERXQSNWH9f4HLU_V6QSNWHQ?HYCθERV6Q
Γ0HIOgA_M6M6[WNWdjV6HYQgZG&KLA_djV6HOSHQSaHf6H\[WAERXQSNPH[WH HYaaH¡6 /6 µ0µrNP[-«rA_V4OgC6XpADC6aU0NPCLETdjV6HGIHIORORH9f6U_C6C6XYHf4H@;NPQSNWG&KL[PHO2E,VLQ
Γ0ERU_NPO?GIU0aMFAcORNW6[WHA¤_HpG2[WHYE?GIU_CFf4NOSNPU0CLEA_V4J[WNPaNPORHpE?ERV6Q
Γ1HIOΓ2 ;CHYC\f4XYf4VLNO2djV6H0_ERN
Θ ≤ π 0AD[WU_Q&E![PHpEERU_[WV4ORNWU_CLE-f6H#4 E !HIO9^44p!ERU_CjO-ORU0V'A>U_VLQSEQRXY_V6[WNPYQRHpEbd0VLH[W[PH\djV6HERU_NPOb[eAf4U_CLC6XH\f4H@;NPQSNeG&K6[PHO#E,V6QΓ0±9VGIU_CjOSQSA_NPQSH_lERN
Θ > π[WHYEE,U0[PV4OSNPU0CLEf4H\6 E 2HIOg^44¤?M<HVL_HCjO¦IOSQRH#ERNWC6_V6[WNWQSHYEY°?Kj`4ERNWdjV6HYaHCjOY-U0C NWC0OSHQSM6QSIORHGIHQSXYERV6[POSAcO\f4HQSX_VL[WA_QRNPORXHC f4NWESADCjO\djV] VLC6HHCLGUG&KLH
Θ > πHYC6_HYCLf4QSHTV6C6H#ERNPCL_V6[eADQSNOSX_6A_VGU_CjORQ&ADNWQSHBfl] V6C6HgMFU0NPCjORH
Θ ≤ π <@;A_CLEGH#GA_EY6NW[«rADV4OMLADQR«U_NeE2QRHY_U_NWQ?[eAJaU4f4XY[PNeESAcORNWU_CGADQV6C LV4¡f4HbG&KLAD[WHVLQYjV6CG&KFADaM¡XY[PHpGORQSNed0VLH_jU_V¡VLCG&KLADaM\f4H9GU_CjORQ&ADNWCjORHYE§NWC4LCLNFZT[^] U0QRNW_NWC6HC] ATMLA0Ef4H9ERHCLEM6K`ERNed0VLH__@;HV4\ADMLM6QRU¤4NWa\AcORNWU_CFEf4V aU4f4Y[PHM<HVL_HCjO¦ORQSHZ[^] U0QRNW_NWC6Hf4H GHIO,OSH E,NWC60V6[WA_QRNPORXC6U0C4 M6K`ERNed0VLH H?f] V6CLHMLADQROYV6CA_C6_[WHC] HpE>OA,A_aA_NWE¨,MLADQR«rADNPO>©aA_NWEbERU_V60HCjObV6CM<HV ¨RADQSQRU0CLf4N©&<fl] ADV4OSQRHMLA_Q,Op<[WU_Q&ESd0VLH∇u HYE,O9OSQRpE90QSA_CLflFU_C³djV6NPO,OSHg[WHf4U0a\ADNWC6Hgf4HcAD[WNWf4NPORXf4HYE;XYdjVLADORNWU_CLE9[PNWC6XpADNWQRHpE9djV6HgCLU_VLEXIORVFf4NPU0CLEO>`jMLNWdjV6HYaHYCjOYDV6CLH[PU0N6GIU_CFE>OSNOSV4ORNW_HGU_aaH[eAB[PU0NLf4H LU_V6QSNPHYQ>_ µ0§f6HNPHYCjO§CLU_C4[PNWC6XYA_NPQSHGADQ§[eA;GU_CLf6VLGOSNPNPORXORK6HYQRaNedjV6HHYE,O!HY[P[WHI a¦aHV6C6H«U_CLGIORNWU_Cf4H ∇u c?CJOSU_V4O-XOSAcOf4HGA_VLE,H0lV6CQRXpE,VL[O&AcOgf4H\QSX0V6[WA_QRNPORXU0VC6U_CFbf6U_C6C6Hf6H\M6QRXpGINWHVLERHYEBNWCLf4NeGADORNWU_CLE#ERV6QT[WHYE[PNWaNOSHYEf4HTM<HQRORNWC6HYCLGIHTf4VaU4f4[WHTV4ORNW[WNWERX_@BADCFE;[WHJGYA_E;U&
Θ = 2πF[^] U0V6_HYQ,O
ΩHYE,O;V6Cf4U_a\ADNWC6HZ!tptYk'ychHIOb[PH HaaH\40/4 µ_µ\AV6C6HbNPCjOSHQSM6QRXOSADORNWU_CaXYGYADC6NedjV6HBORQSYENWaM<U_QROSA_CjORHbE,Nª[WHYEM6QSU_6[WaHpEB^4 E -HIO#^44p?aU4f4XI[PNeE,HYCjOb[WHGINeESADNW[P[WHaHCjO#ADCjOSN M6[eADCfl] VLCG`[PNWCLf4QSH\f4HLA_ERH
Ωr_U_NWQb[] -4HQ&GINeGIH6 µL0G_@BADCLEGIHGA_EY![PHGIUHlWGNPHYC0O\f4V ORHQSaHf] U0QSf4QSH
k = 1f6ADCLE[eA³ERXQSNPHf4H16U_VLQRNWHQrdjV6NGU_CLf4VLNOADV ERHV6[GIU0aMFU0Q,OSHaHCjOJERNPCL_V6[WNPHYQZ [] U_QSNP0NPCLHpbHpE>O\ADM6M<H[WX¡[WH61 m6¢_nY|k'y w Worsªnp|js!tponYhxw'|t¢_q<sªnycm6orsªnp|tdjV6N-HYE,O#E,U0V6_HYC0ObV4OSNP[WNWERXf6A_CLETf4HYEbaU4f4Y[PHpETf4HM6QRU0MLADjAcOSNPU0C³f4HFESE,V6QSHYEBU0Vf4HTQRVLM4ORV6QSH\0U_NWQMFADQHI4HYaML[PH9F0/ H •
.¡|+3´m4ycilk!| Q] Q] \ HYE§ERU_[WV4ORNWU_CFE!ERNWC6_V6[WNWQSHYEv«U_V6QSC6NWHYE!MLADQ§[WH vHaaH40/4 µ_µCLHERU_CjO§MLA_Ed0VLHBf4HpEGIU_CjOSQRH^HHYaM6[PHpE-OSK6XU0QRNedjV6HYE HH[W[PHpEMFHYV6_HYC0O¦IOSQRHBaNWERHYEHYC¡XNef4HCFGIHBCjVLaXYQRNedjV6HIaHYCjOJ0U_NWQ[WA!NW_V6QSH ¯ W L±?A>U_V4OSU_CLEdjV6HJGIHYE9E,U0[PV4OSNPU0CLEERNPCL_V6[WNPYQRHpEE,U0CjOV6C6HgERU_V6Q&GIHTf4Hf4NDWGV6[OSXYETM<U_VLQ#[PHpETaXIOSK6U4f4HYETCV6aXYQRNedjV6HYE\r_U_NWQT[eA XHa\ADQ&djV6H ¯ µ6Wp0;GHdjV6N?GIU0C4LQSaH[] NPCjOSXQS¦IOf4HT[WHV6QXORVLf4H0 •.¡|+3´m4ycilk!| Q] Q] \ ;C´MFHYV4OA_VLESE,NGIU_CFE,Nef4XQSHQg[PHGA_EJfl] V6C6HGIU_CFf4NOSNPU0C´ADV4[WNPaNPORHpEJf4H@9NWQRNeG&K6[WHIOERV6Q
Γ1HO9f4HTHVLaA_C6CE,VLQ
Γ2rf6A_CLEGIH#GYA_EYNP[l«rA_V4OM6QSHCLf6QRH
u = sin(kπθ2Θ )
ERV6QΓ0IL vAERHV6[WHTf4NDCªXQSHCFGIHbMFU0Q,OSHTE,V6Q[WH#GA0E
kπ = ΘM<U_V6Q[WHYdjV6H[l[WHb_Q&A_f4NWHCjO ∇u HpE>O<U_QSC6XaA_NWEbMFA_ETGIU0CjORNWCjVZ[] U_QSNW_NWC6H_lCMLA_Q,OSNWGV6[WNPHYQYlU0C³OSQRU0V6_HYQSAdjV6HMFU0V6Q
k = 1HOπ = Θ
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
pµ " +# 2# % n)
§or|silk Wor's P m6onB~!mLtBw!|¢_q<ors2c[WAbERU_[WV4ORNWU_CHYE,OE,NWC60V6[PNWQSH__®2HY[WABHYE,Of4\ADVG&KLADCL_HaHYC0Of6H#GIU_CFf4NOSNPU0CA_V4¡[WNPaNPORHpEE,VLQV6C6HTM<U_QRORNWU_CQRXY_V6[WNPYQRHbf4Hb[eA«QRU0C0OSNPYQRH0 •Yh+3´q<s'tYnycm4nporq<s#]L9U_VLE§CLU_VLE§<U_QSC6U_CFE!ZBOSQSA_NOSHQ§[WHM6QRU06[WaHT4 E vA¤0HYGGIU0CLf4NPORNWU_CLE-A_V4[WNWaNPORHpE;f6HJHYV6a\ADC6Cf6ADCLE9[PHGIU0NPC´r[] ADV4OSQRHJM6QSU_6[WaH¡^44pE,HTOSQSA_NOSH#HI6A0GORHYaHCjO;f4HJ[WAa¦YaHa\ADCLNPYQRH¤ ;CGU_aaHCLGHJMLA_Q¨>QSH[WH0HQ,©T[eAGIU_CFf4NOSNPU0CA_V4 [WNPaNPORHYEbCLU_C4 K6U_aU0_XCLHHYC f4XIFC6NWESESADCjO9V6C6Hg«U_CLGIORNWU_C
u0 ∈ H1(Ω)f4U_CjO9[eAOSQSA0GIH#ERV6Q9[WH#<U_Q&fGIU)L^CLGNWf4HJA¤_HYGTGHIORORHGU_CLf6NOSNPU0CA_V4¡[WNPaNPORHYEY ;C_XYQRNPLHbADNeE,XYaHCjOd0VLH
u0(r, θ) =r2
R2cos
(
kπθ
Θ
)
QSXM<U_CFfZ[eAdjV6HYE,ORNWU_CªGD] HYE,O,Zcf4NWQRH#djV6Hu = u0 + v
U&vHpE>OB[WAERU_[WV4OSNPU0C fl] VLC³M6QSU_6[WaHKLU_aU_0C6H
−∆v = ∆u0f6ADCFE
Ωv = 0
E,V6QΓ0
∂v∂n = 0
E,V6QΓ1 ∪ Γ2.
^4'FjXHa\ADQ&djV6U_CLEdjV6H#GHBQSH[W0HaHCjOHYE,OMFUjERERNW6[PHTGYADQ[eAf6U_C6C6XYH#E,VLQ
Γ0HYE,O9GIU0aMFAcORNW6[WHTA¤_HYG[WHYEGU_CLf6NOSNPU0CLEADV4[WNPaNPORHpEE,V6Q
Γ1HIO
Γ26GD] HYE,O,Zcf4NPQSH;fLADCLE[WHTGA_EM6QSXYERHCjOdjV6HTERAf4XQSNW_XHHYC
θN^ H0ERAf4XYQRNW_XYH¡C6U0QRa\AD[WHpJEY] A_C6CV6[PHHC
θ = 0§U_V
Θ?®2U_aaH
∆u0ADM6MFADQRORNWHCjOZ
L2(Ω)ªNP[§HI4NeE>OSHgV6CLHJV6CLNWdjV6HE,U0[PV6ORNWU_C³f4H6 >Fj9f6ADCLE
H1(Ω)ªHOBMLA_QBGU_CLERXYdjV6HYCjO6 E A0f4aHIO\ADVLESERN2V6CLHV6CLNWdjV6HE,U0[PV6ORNWU_C f6A_CLE
H1(Ω) 9XYQRNPLU_CFEJdjV6H6 '/_THpE>OM6QSXYGNWERXaHCjOGHIORORHTV6C6NedjV6H#ERU_[WV4ORNWU_CX9ADM6M<H[WU_CLEdjV6H
∆φ(r, θ) =∂2φ
∂r2+
1
r
∂φ
∂r+
1
r2∂2φ
∂θ2.
9CGAD[eGIVL[E,NWaM6[PHTaU0C0OSQRHTdjV6H(
∂2
∂r2+
1
r
∂
∂r
)
( r
R
)kπΘ
=
(
kπ
Θ
)21
r2
( r
R
)kπΘ
,
f6U_CLG^4/_<HpE>OLNPHYC#E,U0[PV4OSNPU0CTf4H96 E I ;Cb_XQSNPLH-ADVLESE,ND«rA_GINW[WHaHCjO'djV6H^4/_ªA_M6MLA_Q,OSNPHYC0OZH1(Ω)
'NWCLA_[PHYaHYCjOY§[WA«U_QSagV6[WHf4V _Q&A_f4NWHCjOgHYC GIUU_Q&f4U0C6C6XHpE#Q&A_f4NeAD[WHYEgf6ADCLEJ[eA LA0E,H(er, eθ)
∇φ(r, θ) =∂φ
∂rer +
1
r
∂φ
∂θeθCLU_VLEf4U0C6C6H
∇u =( r
R
)kπΘ −1 kπ
Θ
(
cos
(
kπθ
Θ
)
er − sin
(
kπθ
Θ
)
eθ
)
djV6N^vG[WA_NPQSHaHCjOp<Cv] HpE>O#MLA0EbFU0QRCLXZ[^] U0QRNW_NWC6HE,N0 < kπ < Θ
HIOTHpE>OgGU_CjORNWCVZ[^] U0QRNW_NWC6HERNkπ > Θ
ª HGA0E9[WNPaNPORHkπ = Θ
GU_QSQRHpE,M<U_CLfA_VLESE,N§Z¡VLC³_Q&A_f6NPHYC0OBGIU_CjOSNPCVGYADQBHC³«rADNPOu = x1
²
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 % .2 % Yµ, ( g" %!$# > %v ( " (*)*,+ S01*M U 7U O! ORPS_16=R:l=1 aO,=R/ %P§3L=vS U
9U_VLE;ADM6ML[PNedjV6U_CLE[] ADM6MLQRU4G&K6HTcADQSNeAcORNWU_CLC6H[W[PH#Z[eA\QSXYERU_[WV4ORNWU_Cf4V ER`4E>OSaHgfl] XYdjVLADORNWU_CLEf4H[] X[eA_E,ORNeGINPORX[WNWC6XYA_QRNeE,XYH_2®2U0aaHYCGIU0CLEMLADQf4XYGQRNWQRHGHaU4f4Y[PHaXpGADCLNWdjV6Hf4U0C0OCLU_VLEA¤_U_CFEVV6CGA_E2MLA_Q,OSNWGV6[PNWHQADV®2KLADM6NPORQSH_6®2HYE2XYdjVLAcOSNPU0CLE2aU4f4XY[PNeERHCjO[WHYEf4X«U_QSaADORNWU_CLEfl] VLC¡E,U0[PNef4H9ERU_VFE[^] K`MFU_ORK6pE,H;f4H9M<HIOSNOSHYEf4XI«U0QRa\ADORNWU_CLE-HOf4H;MFHORNPOSEf4XML[WA0GIHaHYC0O&E9rKj`M<UDORK6pE,Hd0VLNjM<HQSaHIO§fl] U04ORHYC6NWQ§f6HYE'XYdjVLAcOSNPU0CLE[WNWC6XYA_NPQSHYE fl] U&J[PH2C6U_a fl] X[eA_E,ORNeGINPORXros'hm4ycortph|<_U_NWQ9MFADQ9H4HaM6[WH)F µ ¯ H ;C³GU_CLERNWf6QSH#[PHpEXpd0VFAcORNWU_CFE;E,OSADORNWU_C6CLA_NPQSHYE9f4Hg[^] XY[WA0E>OSNWGNOSX_ªGD] HYE,O,Zcf4NPQSH#NWCLf4XYMFHYCLf6ADCjOSHYEBf4V OSHaMLEY>(U0NOΩV6C³U0V6_HYQ,O;FU0QRC6Xf4H
RN Q(U_NPO;V6CLHg«U0QSGH f(x)V6C6Hb«U_CFGORNWU_Cf4H
Ωf6A_CLE
RN L ] NPCFGIU_CLCjVLH u [WH#f4XM6[eA_GHaHCjOHYE,O9ADVLESE,NvV6C6Hb«U_CFGORNWU_Cf4HΩf6A_CLE
RN ' vAaU4f4X[WNeERADORNWU_CaXpGA_C6NWdjV6H«rA_NO#NWCjORHYQR0HC6NWQT[PHOSHCLERHVLQ#f4HYE#f6XI«U_QSa\AcOSNPU0CLEC6UDOSXe(u)
Fd0VLNHYE,OV6C6Hb«U_CFGORNWU_CZcAD[WHV6Q&Ef6ADCFE[] HCLERHaJ6[WH#f4HYEa\AcOSQRNeGIHpEE,`aXIOSQRNedjV6HYEe(u) =
1
2
(
∇u+ (∇u)t)
=1
2
(
∂ui
∂xj+∂uj
∂xi
)
1≤i,j≤N
,
ADNWCLE,N-djV6H[WHJOSHCLERHVLQ#f4HYEbGIU0CjORQ&ADNWC0OSHYEσrV6C6H\ADV6ORQSHJ«U0CLGOSNPU0CZcAD[WHV6Q&Ebf6ADCFE;[^] HCLERHaJ6[PHf4HYEa\AcOSQRNeGIHpEE,`aXIOSQRNedjV6HYE&djV6NHYE,OQRHY[PNWXTZe(u)
MLA_Q[WA[WU_Nf4H 9UU _Hσ = 2µe(u) + λ tr(e(u)) Id,U&
λHOµERU_CjO[PHpEGUHlWGINWHCjOSEf4Hb vA_aXBf4VaADORXYQRNeADVKLU_aU_0C6HNeERUDORQSU_M<H;djV6NªU4GGV6M<H
Ω°!U0V6Qf4HpEQSA_NWERU_CLEf4HbOSK6HQSaU4f4`CLADaNed0VLHB[WHYEGUHlWGINWHCjOSEf6H# vADaXb_XYQRNPLHYC0O
µ > 0HIO
2µ+Nλ > 0.;CA>A>U_V4OSHbZGHIO,OSHT[PU0NvGIU0CLE,ORNPORV4OSNP0HB[WHT6NW[WA_Cf6HYE«U_Q&GIHpEf6ADCFE[PH#ERU_[WNWf6H
−divσ = ff6ADCLE
ΩU&4MLA_Qf4XIFC6NOSNPU0C6[eAf4NP0HQS_HYCLGIHbf4HσHYE,O[PHb_HpGOSHV6Qf4H#GU_aM<U0ESADCjORHpE
divσ =
N∑
j=1
∂σij
∂xj
1≤i≤N
.
ORNW[WNWESADCjO[WH;«rA_NO9djV6Htr(e(u)) = divu
6U0CHYCf6XYf4V6NPO[WHYEXYdjVLADORNWU_CLEM<U_VLQ1 ≤ i ≤ N
−N∑
j=1
∂
∂xj
(
µ
(
∂ui
∂xj+∂uj
∂xi
)
+ λ(divu)δij
)
= fif6A_CLE
Ω^4_0
A¤_HYGfiHIOuiFM<U_VLQ
1 ≤ i ≤ NF[WHYE;GIU0aM<U0ESADCjOSHYE9f6H
fHIOuf6A_CLE[eA\LA0E,HgGYADC6U0C6Ned0VLHgf4H
RN 4CA>A>U_V4O&ADCjOVLC6HTGIU_CFf4NOSNPU0CA_V4[WNPaNPORHYEf6H#@9NWQRNeG&K6[WHIOp4HIOHCV4ORNW[WNWESADCjOf4HYECLUDOSADORNWU_CLE_HYGIORU0QRNWH[W[PHpE[WHTM6QSU_6[WaHbADV4[PNWaNOSHYEGIU0CLERNWf4XYQRXbHpE>O
−div (2µe(u) + λ tr(e(u)) Id) = ff6A_CLE
Ωu = 0
ERV6Q∂Ω.
^4 ¯
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
F E " +# 2# % n)
9U_VLEbM<U_VL_U_CFE;XYC6U_CFGIHQbHOgf4XaU_CjOSQRHYQBVLCMLQRHYaNWHQbQSXYERV6[POSAcO#fl] HI4NeE>OSHCLGHHIOgfl] V6C6NeGINPORXf4VER`4E>OSaHTf4HT[^] XY[WA0E>OSNWGNOSXB[WNWC6XYA_QRNeE,X0 £'hqFy8:+3´| %] \]B^`>bdcfe
Ω g h b g i8jk e b kuh m j RN w `>b4cfe f ∈ L2(Ω)N w j - c p e j g hjg h c gNj p b g e~cBb h z rlq c l j u ∈ H1
0 (Ω)N m jz w @wYh+3´q<s'tYnycm4nporq<s#]\°§U_VLQORQSU_V60HQ[eA¥«U_QSagV6[eAcOSNPU0C ¤A_QRNeAcOSNPU0C6C6HY[P[WHU_C agVL[OSNPM6[WNWH G&KLA_djV6HXpdjVLAcOSNPU0C³6 0_-MFADQV6C6H;«U_CLGIORNWU_COSHYE,O
virdjV6NlEY] A_C6CV6[WHBERV6Q[PHBFU0QSf
∂ΩM<U_V6QM6QSHCFf4QRH;HCGU_aM4OSHB[eA\GIU0CLf4NPORNWU_CADV4[PNWaNOSHYEf4H#@;NWQRNeG&K6[WHIO&HOU_CNPCjOS_QSHBMLADQMLA_Q,OSNPHbM<U_V6QU04ORHYC6NWQ
∫
Ω
µ
N∑
j=1
(
∂ui
∂xj+∂uj
∂xi
)
∂vi
∂xjdx+
∫
Ω
λdivu∂vi
∂xidx =
∫
Ω
fivi dx.
;C¥E,U0aaHAD[WU_Q&EGIHYEXpd0VFAcORNWU_CFE!MFU0V6QiAD[W[eADCjOf4H
1ZN-ADLCxf4H«rA_NPQSHA_M6MLA_QSA4VWOSQRH[eAf6NP0HQS_HCFGIHbf4HT[WAJ«U0CLGOSNPU0C
v = (v1, ..., vN )HO9f4H#E,NWaM6[WNLHYQ[WAM6QSHaNWQSHbNPCjORXY_Q&AD[WHBGYADQ
N∑
i,j=1
(
∂ui
∂xj+∂uj
∂xi
)
∂vi
∂xj=
1
2
N∑
i,j=1
(
∂ui
∂xj+∂uj
∂xi
)(
∂vi
∂xj+∂vj
∂xi
)
= 2e(u) · e(v).
®2KLU_NeE,NeERESADCjOH1
0 (Ω)N GIU_aaHbHYERMLA_GHTf4H NW[P<HQROYFU_CU_4OSNPHYCjO9AD[WU_Q&E2[eA«U_QSagVL[WADORNWU_C¡cA_QRNeAcOSNPU0C6C6HY[P[WH HORQSU_V60HQu ∈ H1
0 (Ω)N OSH[vdjV6H∫
Ω
2µe(u) · e(v) dx+
∫
Ω
λ divu divv dx =
∫
Ω
f · v dx ∀ v ∈ H10 (Ω)N .
^4'GD;C_XYQRNPLHTADNeERXaHCjOdjV6HTG&KLA_djV6HBORHYQRaHTf4H^4'GDA6NWHCV6CERHCLEY°§U_VLQgM<U_V60U_NWQgADM6ML[PNedjV6HQJ[PH 0KLXU_QSaH¡f6H¡ vADjNW[P0QSA_a µ6 µ6WZ[eA«U_QSagVL[WADORNWU_CcAcQSNeAcORNWU_CLC6H[W[PH4'G_<[WAE,HYV6[WHgK`MFU_ORK6pE,Hgf6X[WNWGYAcORHZ0XQSNLHYQ9HpE>OB[WAGIUHQ&GINWNOSXgf4H[WA«U0QRaHLNP[WNPCLXYADNWQSH_ ;CM6QRU4GYf4HbHCOSQRU0NWEXOSADM<HYEY4°QSHaNWQSHaHCjOY4aU0C0OSQRU0CLEdjV6H
∫
Ω
2µ|e(v)|2dx+
∫
Ω
λ|divv|2dx ≥ ν∫
Ω
|e(v)|2dx,
A¤0HYGν = min(2µ, (2µ +Nλ)) > 0
<°§U0V6QTGIH[eA6ªU0C³V4OSNP[WNWERHV6C6HNPCLX0A_[PNPORXAD[W_XY6QRNedjV6H HERNU0C C6U_ORHA · B =
∑Ni,j=1 aijbij
[PHM6QSUf6V6NOTESGAD[eADNWQSHTVLE,VLH[-f4HYEBa\AcOSQRNeGIHpE9ER`aXORQSNWdjV6HpE<U_CM<HV6O9f4XYGU_aMFUjE,HYQORU0V4ORHba\AcOSQRNeGIHbQSXH[W[WHbER`aXIORQSNed0VLHAERU_VFE[WAJ«U_QSaH
A = Ad + Ah A¤_HpGAd = A− 1
NtrA Id
HIOAh =
1
NtrA Id,
f6HbORH[W[WHBa\A_C6NPYQRHTdjV6HAd · Ah = 0
HO |A|2 = |Ad|2 + |Ah|2 ;CA\AD[WU_Q&E2µ|A|2 + λ( trA)2 = 2µ|Ad|2 + (2µ+Nλ)|Ah|2 ≥ ν|A|2A¤0HYG
ν = min(2µ, (2µ+Nλ))cGIHd0VLNf6U_C6C6H[WHQSXYERV6[POSADO'M<U_V6Q
A = e(u)¤ vH«rADNPO§djV6H
ν > 0Cv] HpE>OMLA_E2V6CKLA_ESADQ&f H[PHpEA_QR0V6aHCjOSE?aXYGA_C6Ned0VLHYE2HIOORK6HYQRaU4f4`CLADaNedjV6HYEdjV6N<GU_CLf6V6NWERHCjO
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 % .2 % #FFADV4gNWC6XY0AD[WNPORXYE
µ > 0HO
(2µ+Nλ) > 0E,U0C0O-M6QSXYGNWERXaHYC0O-[WHYE-a¦YaHpE0@9HYV44NPYaHYaHCjOYDU0CV4ORNW[PNeERH[] NWC6XjAD[WNOSXf4H;U_QSCrU_VM6[WV4OCB_O?V6CGA0E-MLA_Q,OSNWGV6[PNWQSHaHYC0OERNPaM6[WH9f4H;GIHIORORHNWC6XjAD[WNOSX__U_NWQ[PH# vHaaH#4 µ60/JGNf4HpERERU_VLE&2d0VLNvf4U_CLC6HbV6C6H#GIU0CLE,OSADCjOSH
C > 0OSH[W[PH#djV6H
∫
Ω
|e(v)|2dx ≥ C∫
Ω
|∇v|2dx
MFU0V6QOSU_V4Ov ∈ H1
0 (Ω)N 0'QRU0NWERNPYaHaHCjOY0U_C\V4OSNP[WNeE,H[^] NPC6XY0A_[PNPORX9f6H9°§U_NWCLGA_QRX#GIU0aM<U0ESADCjOSHMLADQGIU0aM<U0ESADCjOSH_j_U_NWQ[WAJ°QSU_M<U0ERNOSNPU0C FL µ6W E -djV6Nf4U_CLC6H;VLC6HBGU_CLE,OSA_CjORHC > 0
ORHY[P[WHbd0VLH_MFU0V6QORU0V4Ov ∈ H1
0 (Ω)N ∫
Ω
|v|2dx ≤ C∫
Ω
|∇v|2dx.±VORU_OSAD[^6GIHpEORQSU_NeENPCLX0A_[PNPORXpEGIU0CLf4V6NeE,HYCjOZ[eA\GIUHQ&GINWNOSX∫
Ω
2µ|e(v)|2dx+
∫
Ω
λ|divv|2dx ≥ C‖v‖2H1(Ω). H 0KLXU_QSaH?f6H2 vAD NW[W_Q&ADa µL µLPf6U_C6C6Hf4U_CFG[^] HNeE,ORHCFGIH?HO'[^] VLC6NWGNOSXf6H?[eAE,U0[PV6ORNWU_Cgf4H[WA#«U_QSagV6[eAcOSNPU0CcADQSNWADORNWU_C6C6HY[P[WHg^4'GDI !NWCLAD[WHaHYC0OpMFU0V6Q2aU0C0OSQRHYQdjV6H;[WAERU_[WV4ORNWU_CVLC6NWdjV6Hf4H#6 [GD'HYE,O?6NWHCV6C6HERU_[WV4OSNPU0Cf6VMLQRU06[PYaHADV4g[WNWaNPORHpE9^4 ¯ IcU_CM6QSU4GIYf6HGU_aaH[PU0QSEf4HT[WAf4XYaU_CLE,ORQ&AcOSNPU0Cf4V 0K6XYU_QSaH#40/40/#M<U_V6Q[WH# vA_M6[eA_GINWHCv
;?|+3 3´| Q] \%] `Qb4cfeΩ g h b g i4jlk em j RN w b g k e b g e jr b h eKcBb h v ∈ H1
0 (Ω)N o b hq‖∇v‖L2(Ω) ≤
√2‖e(v)‖L2(Ω).
^4 Yh+3´q<s'tYnycm4nporq<s#](U_NPO
v ∈ C∞c (Ω)N L°§A_QNPCjORXY_Q&AcOSNPU0CMLA_QMLADQRORNWHbU_CU_4OSNPHYCjO
2
∫
Ω
|e(v)|2dx =
∫
Ω
|∇v|2dx+
∫
Ω
∇v · (∇v)tdx =
∫
Ω
|∇v|2dx +
∫
Ω
|divv|2dx.
°§A_Qf4HCFE,NPORX#f4HC∞
c (Ω)f6ADCLE
H10 (Ω)
6U_CHYCf4Xpf4V6NPOJ6 I
9u|jyc¢_o¢_| %] \]B^ (% >%3*" :7 2<<7 #2 49 L2(Ω)N 92 H10 (Ω)N * # f 2 5!@% % # !
<>"'9"% u 37 * 678 2 7 9 + / 1 @,7 ;2 % :! / ?] A_CLAD[W`4E,Hf6VM6QRU06[WaH¡ADV6[WNPaNPORHYE6 ¯ TA¤0HYGV6C6H¡GU_CLf6NOSNPU0C´ADV4[PNWaNOSHYEf4H¡@;NQRNeG&K6[WHIO#tYk'y npqFk'nr| ?qFycw∂Ω
HYE,O#V6CMFHYVOSQRU0aMFHYVLE,HMLA_Q#ERAERNPaM6[WNWGNOSX_?CHC<HOYvf6YEd0VLH¡[^] U0CNPCjORQSU4f4V6NPOJV6CLHADV6ORQSH¡GIU0CLf4NPORNWU_C´ADV6[WNWaNPORHpErMLADQJHI4HaM6[WH_!f4H9HV6a\A_C6CF#ERV6QV6C6HBMLADQRORNWH;f4V<U_Q&fl0[eAf4XaU_CFE>OSQSADORNWU_Cf4HB[WAGIUHYQSGNPNPORX;f6H;[eAT«U0QRaJV6[WADORNWU_CcA_QRNeAcOSNPU0C6C6H[W[WHf4HNWHCjO-FHpADVLGU_V6MJM6[PVFE-f4NDWGNP[WHGADQ§U0Cf4U0NO-QSHaM6[eA_GHQ![] NPCLX0A_[PNPORXXY[PXYaHYCjOSADNWQSHf4V HYaaH4 µ60/MLADQBERA_XCLXQ&AD[WNWESAcOSNPU0C6C6HO,ORHYaHCjOBML[PVLE9ORHpG&K6C6NedjV6H_ªf4NPORHNPC6XY0A_[PNPORXgf6H;U_QSC´_U0NPQ;[PH HaaH;6 µL µbGINPf4HYESE,U0VLE&XA_M6M<H[WU_CLEdjV] U_C\CLHMFHYV4O2MFA_EYDHC\0XC6XYQSA_[DERH9GIU0CjORHCjOSHQ?fl] V6C6HGIU_CFf4NOSNPU0C ADV6´[PNWaNOSHYEf4H@9NWQRNeG&K6[WHIO\E,VLQJ[^] NWCjORXY_Q&AD[WNOSXf4VxFU0QSf∂Ω
GYADQH[W[PHE] NWC0OSHQSM6QSIORHGIU_aaH[WH\«rADNPOdjV6H¡[PHE,U0[PNef4HHpE>O#LX¡HIOJNWaaU_6NW[PHE,VLQgE,U0CFU0QSfvC´MLQSADORNed0VLH_OSU_V4O[PHFU0QSfC] HYE,OMLA_E6[PU4djV6XTHO9E,U0V6_HYC0OVLC6HTMLADQRORNWH#f4V<U_Q&fHpE>O[WNW6QRH#f6HTFU0V6_HYQY6U0V6NPHYCf4HYE
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
F[/ " +# 2# % n)
«U0QSGHYE;E,V6QR«rA_GNWdjV6HpE9ERU_CjO;A_M6M6[WNWdjV6XYHYEBE,V6QBV6C6HADV4OSQRHgMLADQRORNWH_®2HYEBf4HV6GYA_E;f4H#L0V6QSHYE9ERU_CjOaU4f4XY[PNeE,XpEMLADQf6HYEGIU0CLf4NPORNWU_CLEA_V4¡[WNPaNPORHpEf4H#HYV6a\ADC6CdjV6NEY] XpGIQSNP0HCjONWGNσn = g
ERV6Q∂Ω,
6 0U'&
gHYE,O9V6CLHT«U_CLGIORNWU_C Z\cA_[PHYV6QSE_HYGIORU0QRNWH[W[PHpE6 'AGIU0CLf4NPORNWU_Cf4Hg9HV6a\A_C6C 6 0EY] NWCjORHYQ,MLQRORHHCf6NWESADCjOBdjV6H
gHYE,OBV6CLHg«U0QSGHA_M6M6[WNWdjV6XYHJERV6Q;[WHFU0QSfl%(N
g = 0lADVLGV6C6HJ«U_Q&GIHJC6HEY] ADM6M6[WNed0VLHBHO[PHb<U_Q&fMFHYV4OFU0V6_HYQERA_CLEQRHpE>OSQRNeGOSNPU0C HLU_Cf4NO9djV6Hb[WHTFU0QSfHYE,O[WNP6QSH_9U_VLEAD[W[PU0CLE#a\A_NPCjORHYCLADCjOJGU_CLERNef4XQSHQ#[WHER`4E>OSaH¡f4H[] X[eA_E,ORNeGINPORXA¤_HpG\f4HYEJGIU_CFf4NOSNPU0CLEA_V4¡[WNPaNPORHpEa¦[WXHYETrV6CaX[eADC60Hbf6H#@9NWQRNeG&K6[WHIOHIO9f6H#HVLaA_C6CFILGD] HYE,O,Zcf4NWQRH
−div (2µe(u) + λ tr(e(u)) Id) = ff6ADCFE
Ωu = 0
E,VLQ∂ΩD
σn = gE,VLQ
∂ΩN ,
^4 ¯)E U'&
(∂ΩN , ∂ΩD)HYE,O#V6C6H\MLA_Q,OSNOSNPU0Cf6H
∂ΩORHY[P[WHdjV6H\[PHpETaHYERV6QSHYETERV6MFHYQ,<GINWH[W[PHpE#f4H
∂ΩNHO∂ΩD
E,U0CjObCLU_CCjVL[P[WHYE\r_U_NWQb[WA1!NW_V6QSH=FFP¤ ?] A_CLAD[W`4E,Hf4H\GIHC6U0V6_HpADV³MLQRU06[PYaHA_V4[WNWaNPORHpEHpE>O9ML[PVLE;GIU0aML[PNedjV6XHgdjV6HJf6ADCLE[PHgGYA_Ef6Hg[eA\GIU_CFf4NOSNPU0C ADV4[PNWaNOSHYEBf4Hg@;NPQSNeG&K6[PHO HNW[l«rA_VLf4Q&AgV4OSNP[WNeE,HYQ[] NWC6XjAD[WNOSXbf6H 9U0QRCGINP f4HpERERU_VFE;2|+3 3´| Q] \%] \ Is'h+A<m6rornYhw!|¡qFycs" `>bdcfe
Ω g h b g i8jk e b kuh k # g c jk m jnq8p-pj C1 m jRN w j - c p e j7g hj b hp e q4h e j C > 0
e jfj gNj o | b g k e b g e jr b h> e~cBb h v ∈ H1(Ω)N o b hq‖v‖H1(Ω) ≤ C
(
‖v‖2L2(Ω) + ‖e(v)‖2L2(Ω)
)1/2
.^4 ¯ p
?] NWC6XjAD[WNOSXJ^4 ¯ pC] HYE,O2MLA0E?V6CLH;LA_CLAD[WNPORX HjHCHlCªHIOpERU_C¡aHYag6QSH;f6H90A_VLG&K6H9GU_CjORNWHCjOOSU_V4OSHYE?[PHpE?f4XYQRNW_XYHYE-MLA_Q,OSNPHY[P[WHYE?f4HvAD[WU_Q&E-d0VLH9E,U0C\aHagLQRHf4HBf4QSU_NPORH«rADNPO2NWCjORHYQR0HC6NWQE,HYV6[WHIaHYC0O!GIHQROSA_NPCLHYElGIU0ag6NWCLA_NWERU_CLEª[WNWC6XYA_NPQSHYEf6HYEf4XYQRNW_XYHYEªMLA_Q,OSNPHY[P[WHYEYp®2U0aaH-[^] NWC6XY0AD[WNPORX-NWCj0HQ&E,Hf6H4 ¯ ¤2HYE,OXYNWf4HYCjORH_LU_CHCf4XYf6V6NO9djV6HT[WHYEf6HV4aHYag6QSHYEf6H^4 ¯ ¤2ERU_CjOf4HpEC6U_QSaHYEXpdjV6NPcA_[PHYC0OSHYEYv vAf6XaU_CLE,ORQ&AcOSNPU0Cf6V vHaaH4 µ6 µHYE,OgGIU0aML[PNedjV6XH\HOJf4XMFA_ESE,H[WH¡GA0f4QRHf6HBGH;GIU0V6Q&E;r_U_NWQp0MLA_Q2H4HaM6[WH_FP 8H¬I6U0VLE?[^] A_f4aHIORORU0CLEf4U_CLG9HCQRHYaA_QSdjVLA_CjO2djV6H9C6U0VLEA¤0U_CLEbLNPHYC´f4XaU_CjOSQRX\[^] NWC6XY0AD[WNPORXf4H ;U_QSC6 ¯ pB[WU_Q&ERdjV6H
vA_M6MLA_Q,OSNPHYC0OgZ
H10 (Ω)N vCHC<HOYf6ADCFE?GIH;GA_EV6C6H;GU_aJ6NPCFADNeE,U0Cf6V HaaHB6 µL /BHO2f4H;[] NPCLX0A_[PNPORX;f4H°§U_NWCLGA_QRX#rMFU0V6QVLCU0V6_HYQ,O<U_QSC6Xpf4U0C6C6Hb6NWHC[^] NWC6XjAD[WNOSX\4 ¯ ¤ ?] NWCjORHQSM6QSXIO&AcORNWU_CaXYGA_C6Ned0VLHf4H[^] NWC6XjAD[WNOSXf6H ;U_QSC HpE>Ob[WAERV6NPcA_C0OSH_ ?] XYC6HQS_NWHgXY[WA0E>OSNWdjV6H09M6QSU_M<U_QRORNWU_CLC6H[W[PHZ[eA C6U0QRaH´f4V OSHCLERHVLQ³f4HYEf4XI«U0QRa\ADORNWU_CLE
e(u)f6A_CLE
L2(Ω)GU_CjORQKB_[WH[WACLU_QSaH f6V f4XM6[eA_GHaHCjO
uf6ADCLE
H1(Ω)N 2Z[^] A0f6f4NPORNWU_CM6QSYEf6H[WACLU_QSaHf6HufLADCLE
L2(Ω)!®2U_aaHC6U_VLETA_[P[WU_CFE;[WH\_U0NPQTf6A_CLEb[^] -4HYQSGNWGH\6 µL /6lGIH\f4HQSC6NWHQTA'A>U0V4OHpE>O9f6HYE,ORNWC6XgZM6QSHCLf6QRHTHYC GIU_aM4OSHT[PHpE3´q<kvz<|+3´|sªnYtw'|¢_qFyp~§t\ycoBA<orw!|tTGD] HYE,O,Zcf4NWQRHb[WHYEf6XM6[eA_GHaHCjOSE
uCLU_CCV6[WEa\A_NWEfl] XCLHQS_NWHBXY[WA0E>OSNWdjV6HbCV6[W[PH0
;u|jyc¢_or¢0| %] \] @ Ω " >% #" :9 RN / @ 7 )C & 7 R 9# & & )% 9#49 Ω 9;3 <'2>%R =
v(x) = b+Mx 2 b ∈ RN ,M = −M t & 2(% !2 & ;#(% * . ^4 ¯ /_ ="(% >% * v ∈ H1(Ω)N
;)% e(v) = 0 92 Ω # C>7 & ) v ∈ R /
Ecol
e Po
lyte
chni
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Ne
pas
repr
odui
re.
[0 % .2 % #Fjµ9U_VLE#MFU0V6_U0CLEBaA_NPCjOSHCLA_C0OTXYC6U_CLGHQbV6Cf4HYV44NPYaHQSXYERV6[POSAcO#fl] HI4NeE>OSHCLGH\HIO#fl] V6C6NeGINPORXMFU0V6Q[PH#ER`4E>OSaHTf4HT[^] XY[WA0E>OSNWGNOSXB[WNWC6XYA_QRNeE,XTA¤0HYGBGIU_CFf4NOSNPU0CLEADV6¡[PNWaNOSHYEa¦Y[PXYHYEY
£'hqFy4:[3 | %] \] `>bdcfeΩ g h b g i8jk e b kuh b h_hj-[jEk # g c jlk m j.q8p-pj C1 m j RN w `Qb4cfe
f ∈ L2(Ω)N j e g ∈ L2(∂ΩN )N wx2h m ' h cfe R jlpK|Nq-jV =
v ∈ H1(Ω)N e j gNj v = 0 pug k ∂ΩD
.^4 ¯ µj
nj- c p e j g hj g h c gNj p b g eKcBb h z0rlq c j! u ∈ Vm j z w - g cm' |Njh m c h q c k@jlt9jlh e j e
b h e~c hN t9jlh em jlp mb hNh jlp f j e g wYh+3´q<s'tYnycm4nporq<s#]; vA³«U_QSagV6[eAcOSNPU0C´cADQSNWADORNWU_C6C6HY[P[WHf6H^4 ¯)E gE] U_6ORNWHCjOGU_aaHfLADCLEJ[WAf4XaU_CFE>OSQSADORNWU_C\f4V0K6XYU_QSaH4 µ6W__ ?] HpE,MLA0GIH
V_f4XIFC6NFMFADQ6 ¯ µ0I_GIU0C0OSNPHYCjO?[WATGU_CLf6NOSNPU0CADV4[WNPaNPORHpEf6HB@;NPQSNeG&K6[PHOE,V6Q
∂ΩDHIOHpE>O6NWHC¡VLCHpE,MLA0GIHBf4H 9NP[W<HQROGU_aaHBERU_VFE> HYERMLA_GH«HQSaXTf4H
H1(Ω)N MLA_QA_M6M6[WNWGYAcOSNPU0Cf4V10K6XU0QRYaH;f4HBORQ&A_GH4FF µLPpµ0I ;CU_4OSNPHYCjOA_[PU0QSE2[WA«U_QSagV6[eAcOSNPU0C¡cADQSNWADORNWU_C6CLH[W[PH H4OSQRU0V6_HYQu ∈ V ORHY[vdjV6H
∫
Ω
2µe(u) · e(v) dx+
∫
Ω
λ divu divv dx =
∫
Ω
f · v dx+
∫
∂ΩN
g · v ds ∀ v ∈ V. 6 ¯ Fj°§U0V6Q2M<U_VL_U_NWQADM6M6[WNed0VLHQ?[WH 0K6XU0QRYaH;f4HB 'AcNP[W_Q&ADa µ6 µ6W;Zg[WA#«U_QSagV6[eAcOSNPU0C\¤A_QRNeAcORNWU_C6C6HY[P[WH6 ¯ FjI4[WAERHV6[WHbKj`M<UDOSK6YERH#f4X[WNeGAcOSHTZJ0XQSNFHQHYE,OHCLGU_QSHbV6C6Hb«U_NeE[eA\GIUHQ&GINWNOSXf4H[eAB«U_QSaH96NP[WNWC6XYA_NPQSH__±9V4ORQSHaHYC0Of4NOp_NW[6«rA_V4O?aU_CjORQSHQ2d0Vv] NW[LHNeE,ORHV6C6H;GIU_CFE>O&ADCjORH
C > 0ORH[W[WH#d0VLH_4M<U_V6QOSU_V4OSHB«U_CLGIORNWU_Cv ∈ V 6U_CA‖v‖H1(Ω) ≤ C‖e(v)‖L2(Ω).
^4 ¯ 00vU0V4OBfl] AD<U_Q&flFU_CC6U_ORHJdjV6H ‖e(v)‖L2(Ω)
HYE,O;V6CLHgC6U0QRaHgERV6QV< vHgERHV6[!MFU0NPCjObZ\_XYQRNPLHQd0VLNaXQSNOSHd0VLH\[] U_CEY] `AcO,O&ADQ&f4HHYE,OgdjV6H ‖e(v)‖L2(Ω) = 0
NWaM6[WNWdjV6Hv = 0
(V6MLMFUjE,U0CLEf4U_CLG\djV6H ‖e(v)‖L2(Ω) = 0H'A_[PU0QSEb[^] HYQSGNWGH4 µ60/aU_CjORQSHdjV6H
vHpE>O#VLC´f4XM6[eA_GHaHCjOQRNW_Nef4H_<GD] HYE,O,Zcf4NWQRH#djV6H
v(x) = b +MxA¤_HpG
M = −M t ;C³_XQSNPLHb«rA_GNP[WHaHCjObd0VLH_ªE,NM 6= 0
A_[PU0QSE9[PHpEbMFU0NPCjO&ExE,U0[PV6ORNWU_CLEbf4H
b + Mx = 0F«U0QRaHYC0ObV6C6H\f4QSU_NPORHf6A_CLE
R3 HIOV6CM<U_NWC0Of6A_CLER2 &;Q v(x) = 0
ERV6Q∂ΩD
djV6N<HYE,Of4HBaHYERV6QRH;E,V6QR«rA_GNWdjV6H9C6U_CCV6[W[WH_f4U_CLGC6XYGHYESERA_NPQSHaHCjOM = 0
HIOb = 0
ª@9XYaU0CjORQSU_CLEa\A_NPCjORHYCLADCjOJ^4 ¯ 0MLADQ;V6C ADQS_VLaHYCjO;f4HGIU_CjOSQSA0f4NWGIORNWU_C_U0NPQ[eAXHa\ADQ&djV6H FL µ6W I+(4N?6 ¯ _?HYE,O«rA_V4l4NP[vHI4NWE,ORHBV6C6H#ERV6NPORHvn ∈ VORH[W[WH#d0VLH
‖vn‖H1(Ω) = 1 > n‖e(vn)‖L2(Ω).C MLA_Q,OSNWGV6[PNWHQp[eAE,VLNOSHe(vn)
OSHCLf 0HQ&E E f6A_CLEL2(Ω)N2 @J] ADV4OSQRHMLA_Q,Op2GU_aaH
vnHYE,O<U_QSC6XYHf6A_CLEH1(Ω)N MLA_Q\ADM6M6[WNeGAcOSNPU0Cxf4V 0K6XYU_QSaHf4H XH[W[PNeG&K FL µ60/4_!NW[HI4NWE,ORHV6C6HERU_VFE>E,VLNOSH
vn′djV6NGIU_C0HQS_H\f6ADCFE
L2(Ω)N ! ] NPCLX0A_[PNPORXf4H ;U_QSCf4V HYaaH4 µ6 µNPaM6[WNWdjV6H#djV6H‖vn′ − vp′‖2H1(Ω) ≤ C‖vn′ − vp′‖2L2(Ω) + ‖e(vn′)− e(vp′)‖2L2(Ω),fl] U'&[] U_Cf4Xpf4V6NPO;djV6HT[WA\ERV6NPORH
vn′HpE>OBf4HJ®ADVLG&K`¡fLADCLE
H1(Ω)N <f4U_CFGBGU_C_HYQR0HCjORHB_HYQSEV6C6H[PNWaNOSHv∞
djV6NB_XYQRNPLH ‖e(v∞)‖L2(Ω) = 0®2U_aaH³NW[bEY] AD_NPOfl] V6C6HC6U0QRaH³U0C HC
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
F)F " +# 2# % n)
f6XYf4V6NPOJdjV6H\[eA[PNWaNOSHHYE,OgCV6[W[PHv∞ = 0
vGIHdjV6N2HYE,O#V6C6H¡GU_CjORQ&A_f6NWGIORNWU_CA¤_HYG[WH«rADNPOJdjV6H‖vn′‖H1(Ω) = 1
?] NWCjORHQSM6QSXIO&AcORNWU_Cf4HJ[eA\«U0QRaJV6[eAcORNWU_CcA_QRNeAcOSNPU0C6C6H[W[WH6 ¯ FjMFU0V6Q;QSHIOSQRU0V6_HYQ[^] Xpd0VFAcORNWU_C^4 ¯)E !HYE,O2E,HYag6[eADL[PH9ZTGH[W[PH;d0VLHC6U_VFE?A¤_U0CLE!«rADNPO?[WU_Q&Ef4H[eA#f4XYaU_CLE,ORQ&AcOSNPU0Cf6V0K6XU0QRYaH6 /6PFTE,V6Q?[PH9 'ADM6[eA_GNPHYCA¤0HYGGU_CLf4NPORNWU_C\A_V4J[WNPaNPORHpE?f4H99HV6a\A_C6C!NWCLAD[WHaHYC0Op_[^] A_M6M6[WNWGYAcOSNPU0C(f, g)→ u
HYE,O[PNWC6XYA_NPQSH_4°§U0V6QaU_CjORQSHQdjV] H[W[PHTHpE>O9GU_CjORNWCV6HTf4HL2(Ω)N × L2(∂Ω)NfLADCLE
H1(Ω)N LC6U0VLEM6QSHCLU_CLE v = ufLADCLE9[WA«U0QRaJV6[WADORNWU_CcADQSNeAcORNWU_CLC6H[W[PH\^4 ¯ FjILC³V4ORNP[WNeERA_C0O[eAGIUHQ&GINWNOSX#f4HT[eAg«U0QRaHT6NW[WNPC6XpADNWQRHbHO9HCa\A'A>U0QSA_C0O[eA«U_QSaHT[WNWC6XYA_NPQSH_4U0CU04ORNWHCjO P|tYnYoB3 m4nYoqFs´w Phs!|jy8A<or|
C‖u‖2H1(Ω) ≤ ‖f‖L2(Ω)‖u‖L2(Ω) + ‖g‖L2(∂ΩN )‖u‖L2(∂Ω).^4 ¯_¯
BQ-,_GHBZ[^] NWC6XY0AD[WNPORXTf4Hb°§U_NWCLGA_QRXBHIOADVORKLXU_QSaHTf4HBORQ&A_GH_U_CM<HV4Oa\A'A>U0QRHYQ[PHBORHYQRaHTf4Hf6QRU0NOSH;f4H6 ¯0¯ -MFADQC(
‖f‖L2(Ω) + ‖g‖L2(∂ΩN )
)
‖u‖H1(Ω)
4GH;djV6NªM6QSU_V60H9[eAgGIU0CjORNWCjVLNOSX_
.|+3´m4ycilk'| %] \] °§U_V6Qb[WHER`4E>OSaHf6HJ[^] XY[WA0E>OSNWGNOSXA¤0HYGJf4HYEbGIU0CLf4NPORNWU_CLEbA_V4[PNWaNOSHYETf4H@;NWQRNeG&K6[WHIOU0Vf6H9HV6a\A_C6CF!U_CAB[PHpE§a¦aHYE-QSXYERV6[POSADOSE§f4HQSX_VL[WA_QRNPORXd0Vv] A¤0HYG2[PH 'ADM6[eA_GNPHYCr_U0NPQJ[PH 0KLXU_QSaH6 /6 / ¯ I°-ADQJGU_CjORQSH_!Z[eA f4NC<XYQRHYCLGIHf4V vA_M6[eA_GINWHCvvNW[2C] `A MLA0Egf4HMLQRNWCLGINWM<H³f4V a\Ac4NPaJV6a MFU0V6Q[WH E,`4E,ORaH f4H³[] X[eA_E,ORNeGINPORX´GIU_aaH M<U_V6Q¡[WAM6[WV6MLA_Q,O¡f6HYEER`4E>OSaHYEJf4H¡ML[PVLERNWHV6Q&E#Xpd0VFAcORNWU_CFESI'U0VLEf4U_C6CLU_CLEgV6C GIU0CjORQSHI HI4HaM6[WH\CV6aXQSNWdjV6H¡Z [WA!NW_V6QSH;6 µ HjHC¡[] ADLERHCFGIH9f4H9«U_Q&GIHpE
f = 00[WHYEGIU0CLf4NPORNWU_CLE2ADV4[WNPaNPORHYEERU_CjOf4HBHYV6a\ADC6CERV6Q[WHYE«rA_GHYE?ERV6M<XQSNPHYV6QSHHIO?NWC4«XQSNWHV6QSH9f4Vf4U0a\ADNWC6H_jf4H9@;NPQSNWG&KL[PHO
u = 0Zgf4QRU0NOSHHIO
u = e1ZJ0A_VLG&K6H06®2HY[WAQSHNWHCjOZJXIOSNPQSHQ[WHbf4U_a\ADNWC6HTdjV6N^6f4VGU_V6Mv6E] ADaNWCLGINPOY6GU_CLf4VLNWESADCjOADNWCLE,NlZVLCf6XM6[eA_GHaHCjO_HYQ,OSNWGYAD[ldjV6NvG&KLA_C6_Hbf4H#ERNP0C6H_ •
Xd3d 8.0.3 (1/10/2003)
09/10/03 allaire
elasticite.def.amdba
elasticite.bb
Triangles 2D P1
noeuds : 70
éléments: 108
NAN
4 µ ®2U0CjORQSHI HI4HaM6[WHCV6aXYQRNedjV6HMFU0V6Q\[PHMLQRNWCLGINWM<H f4V aAD4NPaJV6a HC XY[WA0E>OSNWGNOSX_± jADVLG&KLHJ[WHf4U_a\A_NPC6Ha\ADNW[W[PX\ADVQSHM<U0EYlHIO#Zf4QSU_NPORH[PH\f4U0a\ADNWC6H\f4XI«U0QRaXU&[PHpE LYG&KLHYEQSHMLQRXpE,HYC0OSHCjO[PH#f4XYM6[eA_GIHYaHCjO#rESAGU_aM<U0ESADCjORHb_HYQ,OSNWGYAD[WHBG&KFADC60HBf4H#ERNW_C6H¤9U_VLEA¤0U_CLE\f6XA,Zf4NPO¡djV6H[WA«U0QRaJV6[eAcORNWU_C¥cADQSNWADORNWU_C6CLH[W[PHC] HYE,OQRNWHCfl] ADV6ORQSH d0VLH[PH~!ycors!¢_or~?|³w!|t\nycm0z6m6kvu zoy¤nYk!|rtTHYC aXYGA_C6Ned0VLH_6°§U_VLQSERV6NW¤A_CjO;GIHO,ORHJADCLA_[PU0_NWH_6[^] HpE,MFA_GIH
VHpE>O[^] HYERMLA_GHf6HYEw!hj~!m6¢_|+3´|sªnYt
v¢_os'h+3´m4nporilk!|+3´|sªn9m6w3´ortptYo !|t_YHIO?[^] HpE,MFA_GIHf4HpEOSHCLERHVLQSE;E,`aXIOSQRNedjV6HYE
σ ∈ L2(Ω)N2 LOSH[eEBdjV6H −divσ = ff6A_CLE
ΩHOσn = g
E,V6Q∂ΩN
HpE>ObGIHY[PV6N§f4HpEOSHCLERHV6Q&E9f6HJ¢_q<sªnycm6osªnY|ttnpm4nYoilk'|+3´|sªnm6w3´otYtpo §r|t06®2U0aaHJMFU0V6QB[PH 'ADM6[eA_GNPHYCc[eATE,U0[PV4OSNPU0C\f4H[WA;«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6CLH[W[PHb6 ¯ Fj!QSXYAD[WNeE,H[WHaNWC6NWagV6a fl] V6C6HXYC6HQS_NWHBaXpGA_C6NWdjV6H#f6XILC6NWHbMFU0V6Qv ∈ V MLADQ
J(v) =1
2
∫
Ω
(
2µ|e(v)|2 + λ|divv|2)
dx −∫
Ω
f · v dx−∫
∂ΩN
g · v ds. 6 ¯ GD
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 % .2 % #FCORHQSaHYEaXYGYADC6NedjV6HYE
J(v)HpE>O[eAERU_aaH#f4HT Whs'|jy4AFo|w'| w!h+1 qLy43´m4nporq<s1
2
∫
Ω
(
2µ|e(v)|2 + λ|divv|2)
dx
HIO!f6H2 Whs'|jy4AFo|b~?qFnY|sªnYo|jr|Tw!|t1 qLyc¢_|t|julnph0yco|jk'yc|t9U0VTORQ&A¤cADNW[0f4HpE«U0QSGHYEvHIORXYQRNWHV6QSHYEADVE,NW_C6HbM6QSYE&−∫
Ω
f · v dx−∫
∂ΩN
g · v ds.
9u|jyc¢_o¢_| %] \] \ (% >%* u ∈ V #657 *678 9 7 2 @% & 7 2 " 2)% 2 ")787 + / @1? # C>7 & )5 u % ;27 C=7 & & & #% V 9 7 ;))% J(v) 9;3 <'2>%+ / 1!/+ 9 #2 @" <>"% % 2= #< % >%9) 7 2 % "<>) "-/ -/ 1#/9u|jyc¢_o¢_| %] \] @ Ω " )% >@% ;I! ' 9 RN / #" 9 >% 7 6 & I9)7 ;)7 2 ; 2 7 2 #"'9 9 ) & 2 + / 1 !%" 7 >@% 9 ∂Ω / ="(%)%*" 7 2!"'9 "9 ;* 7 >%∫
Ω
f · (Mx+ b) dx+
∫
∂Ω
g · (Mx+ b) ds = 0 ∀b ∈ RN , ∀M = −M t ∈ RN×N
# ! 9 " ;!6C2 % # ! K2 9 >'! ! 9 ; 9 C"78 " 92H1(Ω)N +7 ;;#62?) #=7 2'99 " 9 & " & >?9) !@% < % 9 <>% #% %+ / )0161#/.¡|+3´m4ycilk!| Q] \%] U0QSESdjV6H[PHpEBGUHlWGINWHCjOSETf6H 'ADaXE,U0C0O#GU_CLE,OSA_C0O&EBHIO#djV6H[PHpEBGU_CLf6NORNWU_CLEA_V4 [WNPaNPORHpEE,U0CjO¡f4V¥O>`jM<H f4H @;NPQSNeG&K6[PHOK6U_aU0_CLH_-[WHYE\Xpd0VFAcORNWU_CFE\f4H[] X[eA_E,ORNeGINPORXMFHYV6_HYC0O9ERHbQRXpADQSQSA_C6_HYQ?M<U_V6Qf6U_C6C6HYQ[PH#ER`4E>OSaHTf4HT vA_aXrM6QSXYERHCjORXTADV ®2KLA_M6NOSQRH¤
−µ∆u− (µ+ λ)∇(divu) = ffLADCLE
Ωu = 0
ERV6Q∂Ω.
^4 ¯ ] A¤¤A_CjOSAD0H#f4H^4 ¯ MLA_Q9Q&ADM6M<U_QRO;Z³^4 ¯ HYE,OBdjV6H[PHgORHYCLE,HYV6Q
e(u)Af4NeE,MLA_QRV³HIOBdjV] U_CMFHYV4O2f4U0CLGE,HMFA_ESE,HYQf4H[] NWC6XjAD[WNOSXf4H;U_QSC D[6«rADV6O?GIHM<HCFf6ADCjO«rA_NPQSHAcO,OSHCjORNWU_CdjV6H#^4 ¯ C] HpE>O2M6[PVFE?Xpd0VLNPcAD[WHCjO2Z\^4 ¯ -E,NF[PHpE?GUjHWGINWHCjOSE2f4H9 vA_aX9f4XM<HCFf4HCjO2f4HxU_VERNLU0C\M6[eA_GIHf4HYE#GU_CLf6NOSNPU0CLE#ADV6[WNPaNPORHYEJf4HHYV6a\ADC6C´ERV6Q#VLC6H\MLADQRORNWHf4V<U_Q&flv@J] VLCMFU0NPCjOJf4H\V6HaXpGA_C6NWdjV6H04[PH#ERHVL[laU4f4Y[PH#GU_QSQRHpGOf6ADCFEGIHTGA0EHYE,Og4 ¯ •
9u|jyc¢_o¢_| %] \] !<<>6 * Ω #6 " )% >@% ; 9 RN ! * f ∈ L2(Ω)N / (% >% 7 ) #! 7 ;9:7 2 678 ,9)+ / 192 H1
0 (Ω)N C2 7 C>% 7 ;2%!7 ; 9 @% /?;)% >% *" <>) 2 2 7 %7 # <>> #6 9 <> ) ;=#%7 4!@ >9)>2 & ;:)!<<>C26)7 & >* µ > 0 # 2µ+ λ > 0 /9u|jyc¢_o¢_| %] \] ?;>% >%7 ;*" 27 ) #,9+ / 1#+ / 1 λ ! µ 6"?!662/ !(%)%3* + / 1!+ / 1 C" <78 ?;* 27 >$ λ ! µ 6?9#$ ! " +% ;"7 >% # 1 &>& % & <7 2'# 7 ;*"'2 " (@% >787 :9 + / 1,< 2)%
−div(µ∇u)−∇((µ+ λ)divu) = f 92 Ω.
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
F ¯ " +# 2# % n)
@BADCLEJV6C GYA_EbOSQRpE#MLA_Q,OSNWGV6[WNPHYQY!ADM6M<H[WX~!y¤q§B:+3´|´w!k ¢0ortpm6orr|+3´|sªnm6sªnYo =R~§rm6s2![PHER`4E>OSaHf4H[^] X[eA_E,ORNeGINPORX[PNWC6XYA_QRNeERXHERHE,NWaM6[PNPLHGIU0CLE,Nef4XYQSA_6[PHYaHCjOM6VLNWESdjV] NW[TE,HQ&ADaC6HZ[eAQSXYERU_[WV4ORNWU_Cfl] V6CM6QSU_L[PYaH A_V4x[WNWaNPORHpEM<U_V6Q[PH vADML[WA0GINWHC®2HO¡HI4HYaML[PH M<HQSaHIOf6U_CLG\f4H«rA_NPQSH\V6C[WNPHYC´f4NPQSHYGIO#HCjORQSH[PHpETXYdjVLADORNWU_CLEgf4H\[] X[eA_E,ORNeGINPORX\HO#[PH 'ADM6[eA_GNPHYCGIHdjV6NH4M6[PNedjV6Hfl] V6C6HGIHYQ,O&ADNWC6HaA_C6NWQSH2M<U_V6Q&djV6U_N4[WHYE§QSXYERV6[POSADOSE!MFU0V6Q-[WHYE-f4HYV4gaU4f4[WHYE-ERU_CjO!ORQSYEERHaJ6[eAD6[WHYEfLADCLEv[] HCLERHaJ6[WH_¤®2H2GA_EvMLADQRORNeGIVL[PNWHQvf6V#GINeERA_NP[W[PHYaHCjO'ADCjOSN M6[eADCbHYE,O'XORVLf4NWX?f6A_CLE[^] HI4HQ&GINeGIHTERV6NPcA_C0Op;u|jyc¢_or¢0| %] \] ' 9#! '>% #5 9?(% " )% 678 ,<'2>% '7 >% 59 6 & 9 7 ;>7 2 6 ; 7 ;2)% 6;#I92 7 I#2 9 @% ! 9 K2 787 & )2 !C<7 2 / ! 9)%I9 & 2 7 '9% * & > Ω 9 7 %)% L > 0 # 9) C! " ω ω >%>@% ; #" =%; "7 >% 9 RN−1 +67 # !@ >9 >2 & ; λ # µ C"!662 1!/ (% & )9 Ω = ω × (0, L) "!:<>"% x ∈ Ω " > x = (x′, xN ) 2 0 < xN < L# x′ ∈ ω / =#" 9 >% 7 <>% 7 & 2 7 & # 2
−div (2µe(u) + λ tr(e(u)) Id) = 0 92 Ωσn = g !% ∂ω × (0, L)u′ = 0 !% ω × 0, L(σn) · n = 0 !% ω × 0, L
^4 ¯ 0
"42 7 6; 7 2 >62 <>% !>% v = (v1, ..., vN ) v = (v′, vN ) 2 v′ ∈ RN−1
# vN ∈ R / !<<>)6 *" 7 2I@% ! !% 2' *" g #6 9"I <> K2 787 & ):2 !C<7 2 "!"#%! 9 % * g′ = (g1, ..., gN−1) = 0 ! gN 9;><>) 9 *" :9) x′ /
="(%)% *" 7 26"78 " * 9) + / 15#69) ;# < 2)% u = (0, ..., 0, uN) uN(x′)47 2 C"78 "9">2<7 2' )! 2
−∆′uN = 0 92 ωµ∂uN
∂n = gN !% ∂ω ∆′ #6,7 >2<7 2' >92 7 2 2)% 2 7 x′ ∈ RN−1 /;u|jyc¢_or¢0| %] \] ;> ;>% 27 6>%7 )% ! / -/ 2 #2 9 ! 9 " 2 7 & 7 2;)% 27 9: <>
u′ = 0 # (σn) · eN = gN !% ∂ω × (0, L).
;u|jyc¢_or¢0| %] \] 7 2 9 9 7 2<<>% 2)% 2 ")787 9); & "(%)%,7 ) # ! 7 ;9 7 2 C"78 "9=7 ;*'2 "9) <7 2'* #
∆ (∆u) = f 92 Ωu = 0 !% ∂Ω∂u∂n = 0 !% ∂Ω
^4 G E f ∈ L2(Ω) / <>"% % 2 % & 2)% *)% *" &" u ∈ H2
0 (Ω) "27 @% ∂u∂xi∈ H1
0 (Ω) #∫
Ω
|∆u|2dx =N∑
i,j=1
∫
Ω
∣
∣
∣
∣
∂2u
∂xi∂xj
∣
∣
∣
∣
2
dx.
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 % .2 % #F G =2'9 & ! (% 2 7 % ;##7 629) % ;7 2)% ; ! 2 w ∈ L2(Ω) ! f ∈ L2(Ω)
;>% @) <>"%" v ∈ C∞
c (Ω)
−∫
Ω
w∆v dx =
∫
Ω
fv dx,
27 @% (θw) ∈ H2(Ω) *" >787 :* 6 ,7 2"'( " θ ∈ C∞c (Ω) /
*)*KJ T8P!16=>5?/"S´7U 2145#US9U_VLE;ADM6ML[PNedjV6U_CLE[] ADM6MLQRU4G&K6HTcADQSNeAcORNWU_CLC6H[W[PH#Z[eA\QSXYERU_[WV4ORNWU_Cf4V ER`4E>OSaHgfl] XYdjVLADORNWU_CLEf4H(jOSU _HpE(U_NPO
ΩV6CU_VL_HQROFU0QRCLXgGIU0C6C6HI4H#f4H
RN >(U_NPOV6C6HT«U0QSGH f(x)FVLC6HT«U_CFGORNWU_Cf4H
Ωf6A_CLE
RN D[-`³Af4HV6NWCLGIU0C6CV6HYE Hl[eAjNPORHpERERHudjV6NHYE,OTV6C6H«U0CLGOSNPU0C0HYGOSU_QSNPHY[P[WH_HIO[WAgM6QRHpERERNWU_C
pdjV6NªHpE>OV6C6H;«U_CLGIORNWU_C¡ESGA_[WA_NPQSH_0C¡C6U_OSADORNWU_C0HYGOSU_QSNPHY[P[WH_0[PHBM6QSU_6[WaH9A_V4[PNWaNOSHYEGU_CLERNWf6XQSX;HpE>O
∇p− µ∆u = ff6A_CLE
Ωdivu = 0
f6A_CLEΩ
u = 0ERV6Q
∂Ω
^4 G¤U&
µ > 0HYE,O;[eA\NeERGU0ERNOSXgf4V FV6NWf6H_ª vAf4HYV44NPYaHgXYdjVLADORNWU_C³f6H60G4pHpE>OB[WA¡GIU_CjOSQSA_NPCjORHfl] NWCLGU_aM6QSHYESE,NW6NW[PNPORXTf4V LV6Nef4H_6OSADCFf4NWEdjV6Hb[eAM6QRHYaNPYQRHBHYE,O[PHb6NW[WA_Cf4HYE«U0QSGHYEY 'AJGU_CLf6NORNWU_CADV4[PNWaNOSHYEf4HT@;NPQSNeG&K6[PHOaUf6X[WNWERHb[] A_f4KLXQSHCLGHbf6V FV6NWf6H#E,V6Q[WHYEMFADQSU_NeE
.¡|+3´m4ycilk!| Q] \%] HM6QSU_6[WaHf4Hy(jORU _HYE60G4p¡HYE,OVLC aU4f4[WHtYoK3~!o Z§h fl] XYGU_V4[PHYaHYCjOfl] V6C LV6Nef4HNWESdjV6HV4NWCLGIU0aM6QRHpERERNW6[PH¥HYC QSX0NPaHE,OSADORNWU_C6CFADNWQRH¤C HC<HOY[WHYE¨>Q&ADNWHYE>©BXYdjVLADORNWU_CLE;f4V aU_VL_HaHYC0OBfl] VLCORHY[ LVLNWf4HE,U0CjO9[PHpEXpd0VFAcORNWU_CFE9f4Hg;A¤NPHYQ,!(jORU _HpEE>O&AcORNWU_CLCLADNWQRHpEb_U0NPQMLA_QHI4HaM6[WH9F µ H
(u · ∇)u+∇p− µ∆u = ffLADCLE
Ωdivu = 0
f6ADCLEΩ
u = 0E,VLQ
∂Ω.
^4 Gd/0 U_Q&ESd0VLH-[WANPORHpERERH?f4V FV6NWf6H
uHpE>Ov«rADNW6[PH0Y[WH-OSHQSaH2C6U_C6^[WNPCLXYADNWQSH
(u·∇)uXIOSA_CjO'djVLA_f4Q&AcOSNWdjV6HHC
uf4HYNPHYC0OCLX_[WNW_HYA_6[WH_ ;CU_4OSNPHYCjO;AD[WU_Q&E[PHpEXYdjVLADORNWU_CLEf6H(jORU _HpE"D[«rADV4O;f6U_CLGTA¤0U_NWQZ¡[^] HpE,MLQRNPO#d0VLHJ[WH\f4U_a\ADNWC6Hf4HcAD[WNWf4NPORX\f4H\GHaU4f4Y[PHHYE,OT[PNWaNOSX\MLADQbGIHO,ORH\K`M<UDORKLYERHf4HMFHORNPORHTNOSHYESE,H0 •
£'hqFy4:[3 | %] \] `>bdcfeΩ g h b g i8jk e b kuh b h_hj-[jEk # g c jlk m j.q8p-pj C1 m j RN w `Qb4cfe
f ∈ L2(Ω)N w #j - c p e jg hjg h c gNj p b g e~cBb hz rlq c l j u ∈ H10 (Ω)N j e p ∈ L2(Ω)/R
m jz w u z~ qs|>kvjup-p cBb h jup e g h c gNj $g hjE b hp e qdh e jEq mm4cfe~c i4j|k + p m q4hNp Ω dwYh+3´q<s'tYnycm4nporq<s#]\°§U0V6QORQSU_VL_HQ[eA¥«U_QSagVL[WADORNWU_C cADQSNWADORNWU_C6C6HY[P[WHU_C aJV6[OSNPML[PNWH G&KLA0d0VLHXYdjVLAcOSNPU0Cf4V E,`4E,ORYaH´^4 G¤gMFADQV6CLH«U_CLGIORNWU_C¥ORHpE>O
virdjV6NBE] ADC6CV6[WH ERV6Q[WHFU0QSf
∂ΩMFU0V6Q9M6QSHCFf4QRHgHC³GU_aM4ORHg[WAGIU_CFf4NOSNPU0C ADV4[WNPaNPORHYEBf4H@9NWQRNeG&K6[WHIO<U_CNWC0OS0QRH#MFADQMLA_Q,OSNPHHIO?U0CERU_aaHM<U_V6QiAD[W[eADCjO?f4H
1ZN
M<U_V6Q?«rADNWQRHA_M6MLA_QSA4VWOSQRH[eATf4NP0HQS_HYCLGIHf4H[eAB«U_CFGORNWU_Cv = (v1, ..., vN )
∫
Ω
µ∇u · ∇v dx−∫
Ω
pdivv dx =
∫
Ω
f · v dx,
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
F " +# 2# % n)
A¤0HYG[eATC6U_OSAcOSNPU0C ∇u ·∇v =∑N
i=1∇ui ·∇vi±LC¡f6HORHYC6NWQGU_aM4ORH;f4H9[eA#GIU0CLf4NPORNWU_C¡f] NWC4GU_aM6QSHYESE,NW6NW[PNPORX
divu = 0ªU0C³G&K6U_NeERNOTGIU0aaHJHYERMLA_GHJf4HNW[P<HQROb[PHE,U0VLE,^HpE,MLA0GIHgERV6NWcADCjOf6H
H10 (Ω)N
V =
v ∈ H10 (Ω)N ORH['d0VLH
divv = 0M MvLf6ADCFE
Ω
,^4 Gcµ0djV6N?HYE,ObLNPHYCVLCHpE,MFA_GIH\f4H9NP[WFHYQ,OgGU_aaHE,U0VLE,^HpE,MLA0GIHg«HQSaXf6H
H10 (Ω)N ;C³OSQRU0V6_HA_[PU0QSE[eAJ«U_QSagV6[eAcOSNPU0CcADQSNeAcORNWU_CLC6H[W[PH H
ORQSU_VL_HQu ∈ V ORHY[vdjV6H ∫
Ω
µ∇u · ∇v dx =
∫
Ω
f · v dx ∀ v ∈ V, 60G'FjfLADCLE2[WA0djV6H[W[PHB[eA#M6QSHYESE,NWU_CAgf4NeE,MFADQSVT² ;C_XQSNPLH;A_NWERXaHCjOd0VLHBG&KLA0djV6HOSHQSaHBf4H4 G FALNPHYCVLCERHCFE ?] A_M6M6[WNWGYAcORNWU_Cf4V10K6XU0QRYaHBf4HT vADjNW[P0QSA_a ZJ[eA#«U0QRaJV6[WADORNWU_CcA_QRNeAcOSNPU0C6C6H[W[WH^4 G FjCLHMFUjE,H?MLA_E'f4H?M6QSU_L[PYaH0ICJMLADQRORNeGIV6[WNWHQpI[WA9GIUHQ&GINWNOSX?f4H[eA«U_QSaH6NW[PNWC6XpADNWQRHHpE>O'XNWf6HCjORHfLADCLE
H10 (Ω)N 0Q@,0GIH9Zg[] NWC6XjAD[WNOSX;f4H;°!U0NPCFGADQSXp?f4U_CFG9f6A_CLE
VdjV6NFHpE>OV6CE,U0VLE> HYERMLA0GIHf4H
H10 (Ω)N vH9M<U_NWC0O[PH9M6[PVFE2f4XY[PNeGADO2HpE>O?NeGINªf4H;aU0CjORQSHQ?djV6H[eAgERU_[WV4OSNPU0C\V6C6Ned0VLH;f4Hg^4 G F§HYE,O26NWHCVLC6HERU_[WV4OSNPU0C f4V MLQRU06[PYaHADV6´[PNWaNOSHYE60G4p!-4M6[WNWdjV6U0CLE[WAf4NW\GV6[PORXHC¥E,V6MLMFUjERA_C0Oa¦YaHaU0aHYCjOSADCLXaHCjO¡djV6H
uHpE>O¡QSX0V6[WNPYQRH0GD] HYE,O,Zcf4NPQSHADM6MLA_Q,OSNPHYCjO¡Z
H2(Ω)N ?°-ADQNWCjORXY_Q&AcORNWU_CMLADQMFADQRORNWH_60G'Fj2NWaML[PNedjV6H#djV6H∫
Ω
(µ∆u+ f) · v dx = 0 ∀ v ∈ V,a\A_NWE§U0CJC6HM<HV6OMLA_E§HYCf4Xpf4V6NWQRHdjV6H
(µ∆u+f) = 0GA_Q§[] U_QRORK6U0_U0CLAD[4f4H
Vf6A_CLE
L2(Ω)NCv] HpE>O9MFA_ENef4H²L?C HlCªHIOpLU_C0U_NPO«rA_GNP[WHaHCjO;djV6HJE,NφHpE>O9VLC6HT«U_CFGORNWU_CQSX0V6[WNPYQRH06AD[WU_Q&EM<U_VLQORU_V6O
v ∈ V 4U_CA∫
Ω
∇φ · v dx = −∫
Ω
φdivv dx = 0,
G_] HpE>OR ZD f6NPQSHdjV6H[^] U0Q,OSK6U_0U_CLA_[f4HV
GIU_CjOSNPHYC0OA_VxaU_NWCLEOSU_VLE[WHYE\0QSA0f4NWHCjOSEY§C¥«rADNPO[PH0KLXU_QSaHf4HBf4HXKFADa 4 µ6 bC6U_VFE?f4NPOdjV6H[^] U_QRORK6U0_U_CFAD[6f4H
VGIULCLGNWf6HHI6A_GIORHaHYC0OA¤_HYG[^] HYERMLA_GHbf6HYE_Q&A_f6NPHYC0O&E
BQ-,_GHA_V 0K6XYU_QSaHf4Hf4HXKLADa 4 µ6 C6U0VLE§MFU0V6_U0CLE§GIU0CLGI[WV6QSHGU_aaHERV6NPOY ;CJMFUjE,HL(v) =
∫
Ω
µ∇u · ∇v dx−∫
Ω
f · v dx,djV6NvHYE,O96NWHCV6C6HT«U0QRaHT[WNPCLXYADNWQSHbGU_CjORNWCV6H#ERV6Q
H10 (Ω)N HIOCV6[W[PHJE,V6Q
V6°-ADQ;GIU_CFE,Xpd0VLHCjOYNW[HNeE,ORH
p ∈ L2(Ω)6V6C6NedjV6H#ZV6C6H#GU_CLE,OSA_C0OSHbM6QRpEOSH[vdjV6H
L(v) =
∫
Ω
p divv dx ∀ v ∈ H10 (Ω)N .
(4NU_CMFUjE,Hσ = µ∇u− p Id
d0VLNvADM6MFADQRORNWHCjOZL2(Ω)N2 LU_CAf6U_CLG
∣
∣
∣
∣
∫
Ω
σ · ∇v dx∣
∣
∣
∣
=
∣
∣
∣
∣
∫
Ω
f · v dx∣
∣
∣
∣
≤ C‖v‖L2(Ω),
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[0 % .2 % #F GIHd0VLN!MLQRU0V6_Hd0VLH
σA_f4aHIOTV6CLHf4NW_HYQR0HCLGH#«rA_NP6[WHf6ADCLE
L2(Ω)N lHOBU0CA −divσ = f°§A_QGIU0CLE,XpdjV6HCjOY4U0CHYCf4XYf4VLNO9djV6H
∇p− µ∆u = fM6QSHYESdjV6HbMLADQRORU0V4Of6ADCLE
Ω.60G__
@g] ADV4OSQRHbMLA_Q,Op6GIU0aaHu ∈ V divu
HpE>OCV6[vf6A_CLEL2(Ω)
LGIH#djV6NlNWaM6[PNedjV6Hdivu = 0
MLQRHpERdjV6HbMLA_Q,OSU_V4Of6A_CLEΩ.@9HBa¦aH_4[eAgGU_CLf4NPORNWU_CADV4[WNWaNPORHpEE] NPCjOSHQSM6QRORH;MLADQ[PH;ORKLXU_QSaHbf4H;ORQ&A_GIH;HIOU_CU_4OSNPHYC0Od0VLH
u = 0M6QSHYESd0VLHBMFADQRORU_V6OERV6Q
∂Ω
9U_VLEbXYC6U_C GIU_CFE;a\A_NPCjORHYCLADCjOb[PHQSXYERV6[POSAcOBORQSYEbMLQRU_«U_CLf³HIO#f4NW\GNP[WH\djV6N-C6U_VFETA¡M<HQSaNWEf4H9QSHIOSQRU0V6_HYQ-[WAgM6QRHpERERNWU_C\Z#MLA_Q,OSNPQf4H9[eAb«U0QRaJV6[WADORNWU_C\cADQSNeAcORNWU_CLC6H[W[PH#^4 G F-f4V¡ER`4E>OSaH;f4H(jORU _HYEU'&H[W[PH#A¤cADNPO9f4NeE,MFADQSVT²N(4A\f4XaU0CLE>OSQSADORNWU_Cf4XMFA_ESE,HBORQSYE[WA_QR0HaHCjO[PHgGYA_f4QSHTf4H#GIHGIU_VLQSEE,NW_CFAD[WU_CLE-djV] NP[FC] HYE,O?MLA_E-«rA_GINW[WHf4HORQSU_VL_HQfLADCLE?[WAb[WNORORXYQSADORV6QSHV6C6H9f6XaU_CLE,ORQ&AcOSNPU0C¨>XY[PXYaHYCjOSADNWQSH©;HO9ADV4OSUDGIU_CjOSHCV6H _U0NPQC6XpADC6aU0NPCLEsF0/'FdH¬I £'hqFy4:[3 | %] \] >w!| w!| .¡£!m3 "y`>bdcfe
Ω g h b g i8jk e b kuh b hNh>j -+j"k # g c jlk m j q8p-pjC1 m j RN w `>bdcfe L g h>jr b kut9j. c h q c kvj b h eKc h_gNjEp-g k H1
0 (Ω)N w %" b k-p L p q4h_hNg jpug k Vp c j e pjg jt.jh e p cc $j - c p e j7g h>jr b h> e~cBb h p ∈ L2(Ω)
e jfj gNjL(v) =
∫
Ω
p divv dx ∀ v ∈ H10 (Ω)N .
^4 G ¯ j| g+p p jlp e g h c gNj $g h>jE b hNp e q4h e jq m'm4cfe~c i8j|>k + pw.¡|+3´m4ycilk!| Q] \%]K^ °!U0V6Q(jORU _HpE!GU_aaHMFU0V6Q§[PH vA_M6[eA_GINWHCJU_VJMFU0V6Q-[] X[eA_E,ORNeGINPORXU0CgM<HV4OADVLESE,Nf6XILC6NWQf4HpEGIU0CLf4NPORNWU_CLEA_V4¡[WNPaNPORHYE9f4HTHVLaA_C6CdjV6NvEY] XpGIQSNP0HCjO
µ∂u
∂n− pn = g
ERV6Q∂Ω,
60G'GDU&
gHYE,O2V6CLH«U_CFGORNWU_CrZ#cA_[PHYV6QSE?_HpGORU0QRNWH[W[WHYE&-f4H
L2(∂Ω)N L®2U0aaHM<U_VLQ2[^] XY[WA0E>OSNWGNOSX_j[WAGIU_CFf4NOSNPU0Cf6Hb9HV6a\A_C6C^4 GG_?EY] NWCjORHYQRM6QSIOSH;HYCf6NWESADCjOd0VLHgHpE>OV6CLH;«U0QSGHBA_M6M6[WNWdjV6XYHBERV6Q[PHT<U_Q&fl •
.¡|+3´m4ycilk!| Q] \%]K^_^x°!U0V6Q[PHTER`4E>OSaHTf4H7(jORU _HYEb^4 Gp?U_CAJ[PHpEa¦YaHYEQSXYERV6[O&AcO&Ef6HBQSXI_V6[eADQSNOSXdjV6HM<U_V6Q-[WH vA_M6[WA0GINWHCJU_V[PHER`E,ORYaHf6H[^] X[eA_E,ORNeGINPORX[PNWC6XpADQSNWERXHb0U_NWQ![WH 0K6XU0QRYaH40/40/ ¯ I4°-ADQ9GU_CjORQSH_6[WHYEXYdjVLAcOSNPU0CLEf4H (jORU _HYEXOSA_C0O;V6C E,`4E,ORYaH#f6H#M6[WVLE,NWHVLQSEXpdjVLAcOSNPU0CLENP[Cv] `AMFA_ETf4H\M6QSNWCLGINWMFHf6Va\ADNWagVLa rG_] HpE>O#[eAa¦aHE,NPORVFAcORNWU_CdjV6H\M<U_V6Q#[WH\E,`4E,ORYaHf4H¡[^] XY[WA0E>OSNWGNOSXpI'°K`4ERNWdjV6HYaHYCjOY§Z³GA_VLERH¡f4H¡[eA³GU_CLf6NOSNPU0C fl] NPCLGU_aM6QSHYESE,NW6NW[PNPORXf4V LVLNWf4HGIH[eAE,HbGIU0aMLQRHYCLfOSQRpE6NPHYC4CHlCªHIOp6E,Nl[^] U0CXORVLf6NPHbV6CXYGU_V6[WHaHCjOf4H7(jORU _HpE2f6A_CLEV6C6HORV6`0QSHTM6QRXpE,HYCjOSADCjOV6CQSHYESE,HYQRQSHaHYC0O#rU_VGIU0[eMLQRU0C6U_CLGX_4[WHgf4XY6NO;XIOSA_CjO9GIU_CFE>O&ADCjO9f6ADCFE[WAE,HpGORNWU_Cv0[eA#NPORHYESERH;f4V LV6Nef4H;HpE>OC6XpGIHYESESADNWQRHYaHYCjO?M6[WVLEX[WH_XYH;A_VC6NW_HYA_V¡f4HBGIHbGIU_[ªdjV6H;ERV6Q[] HCjORQSXH9U_V[eA#ERU_QRORNWH9f4H9[eATOSV6`_YQRHJrGIHBdjV6NFNWU_[WH[WH9M6QSNPCFGINWMFHBf4Va\Ac4NWagV6a HC¡[] ADLERHCFGIH9f4H«U_Q&GIHbHIORXYQRNWHVLQRH¤ •
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¤ E " +# 2# % n)
®2U0aaHMFU0V6Q[PHER`E,ORYaHf4H[^] XY[WA0E>OSNWGNOSXNW[HI4NeE>OSHV6CxM6QSNPCLGNPM<Hf4HaNWC6NWaNWESAcOSNPU0Cxf4H[^] XC6HYQR0NPH\rf6NOSH#f4H#w'otYtpo~§m4nporq<s´zortpilk!|jk!tp|M<U_VLQ[PHpEXYdjVLAcOSNPU0CLEf4H7(jOSU _HpE;u|jyc¢_or¢0| %] \]B^ @ V 7 ##< 2'# 9# '2 & < 9) > 6C # 9 )% >'! 787 9;3 < 2)%+ / )-1!/ @ J(v) 7 ;))% 9;3 <>"% v ∈ V < 2)%
J(v) =1
2
∫
Ω
µ|∇v|2dx−∫
Ω
f · v dx. 60G u ∈ V 7 2 C"78 " *" 9) 7 2 @% & 7 2 2)% 2 >787 + / 1!/ ="(% >%,* u,2 6 7 * <> 9 &: & & 9 7 ;> >% & #6"!$#%! 9 % * J(u) = minv∈V J(v) / ; <>% *" & ) & "(%)%3* & u ∈ V <> 59 & & & 9) 7 ;))% J(v) @27 @%u 47 2 C"78 " *:9=7 2@% & 7 2 2)% 2 >787 + / 1#/@BADCLE9[] HI4HYQSGNWGHbERV6NWcADCjOC6U_VFEAD[W[PU0CLE_U0NPQdjV6H#fLADCLEGHQROSA_NPCLE9GA_EOSQRpEMLA_Q,OSNWGV6[PNWHQ&E[WHYEXpdjVLAcOSNPU0CLEf4H7(jORU _HpEE,HbQSXYf4V6NeERHCjOADV vA_M6[WA0GINWHC;u|jyc¢_or¢0| %] \]B^^ ' 9 #! '>% # 9?(% " >%3 6"78 "4< 2)% '7 >% 9#;* 2%! 9(92 4=2 27 %( 7 " :9):C! " @% & & 2<<>>7 ; <>% 37@9 " 6> 787 / Ω = ω × (0, L) L > 0 #6 7 2 7 " " >% 9=#2'27 ! ω C2=6! " 3=" >%>@% ;#" %; "7 >%9 RN−1 / ""% x ∈ Ω ) x = (x′, xN ) 2 0 < xN < L !x′ ∈ ω / #" 9 >% =7 <>% 7 & 2 7 & # 2
∇p− µ∆u = 0 92 Ωdivu = 0 92 Ωu = 0 !% ∂ω × (0, L)pn− µ ∂u
∂n = p0n !% ω × 0pn− µ ∂u
∂n = pLn !% ω × L
^4 G0
p0 ! pL 659> <>% #C " #" 62#/ ="(%)% *" 7 26"78 " * 9) + / 1?#6p(x) = p0 + xN
L (pL−p0) ! u = (0, ..., 0, uN) uN #7 2 678 ,9" >2<7 2' >,# 2
−µ∆′uN = − (pL−p0)L
92 ωuN = 0 !% ∂ω
∆′ #6,7 >2<7 2' >92 7 2 2)% 2 7 x′ ∈ RN−1 /;u|jyc¢_or¢0| %] \]B^& ;> ;>% 27 6)%7 )% ! / -/ 2 #2 9# ;*'2 " 9 2 > )% ! (+ / )01!/
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0'HQSaNPC6U0CLE\GIHO,OSHNPCjORQSU4f4VLGIORNWU_C¥HCf4NeERA_C0OdjV] HC M6[WVLEf4HpEaXIORKLUf6HYEf4Hf4NCªXQSHCLGHYEFC6NPHpE\HIO¡f] XY[PXYaHYCjOSELCLNWEY?djV6N9ERU_CjO[WHYEERHV6[WHYEH4MFUjE,XYHYE\f6A_CLEGIH GU_V6Q&EY-NP[H4NWE,ORH6NWHCf] A_V4ORQSHYEaXIOSK6U4f4HYECjVLaXYQRNedjV6HYEf6HTQRXpE,U0[PV4OSNPU0Cf] Xpd0VFAcORNWU_CFEADV4f6XQSNP0XHYEMFADQRORNWH[W[PHpEGIU_aaH[WHYE§aXIOSK6U4f4HYEf6H0U_[WV6aHYE'LC6NeEY_fl] X[WXaHCjOSE!FC6NWE?f4H«QRU0C0OSNPYQRHbU0VJaXORK6U4f4HNWCjORX0QSA_[PH¤ERM<HYGOSQSA_[PH0Ff4H 6U0V6QRNWHQpLHOSG_!r_U_NWQ9[WHYE;HCFGI`4GI[WU_M<XYf4NWHYEEFP0uH $FW8GIHª°§U_VLQ9M6[WVLEBf4HJf4XOSA_NP[eE9ERV6Q[eATaXIOSK6U4f4H9f4HpEXY[PXYaHYCjOSE§LCLNWE?C6U_VLE?QRHYC_Uc`_U0CLE§Z.F FH^_FP E H NF0/4uH NF0/'F4H NF µ>FdH^F µj H!r_U0NPQ?A_VLESE,N
FWp8H^%FW ¯ H^%F µ6lHMFU0V6Q9f4HpEA_ERMFHpGOSEMLQSADORNed0VLHYEf4HbM6QSU_0QSA_aa\ADORNWU_C¡NWC4«U0QRa\AcOSNWdjV6H¤*,+-*KJ , ;365=vM P!16=>52/ =R/?1 U3F/U v/ 3NPORU
9U_VLEGIU0CLERNWf4XYQRU0CLE?ZgC6U_VL_HYA_V\[PHbGA0f4QSH9_XYC6XQ&AD[<f4V«U0QRa\A_[PNeE,aHcA_QRNeAcOSNPU0C6C6H[<NWC0OSQRU4f4VLNOA_Vx®2KLADMLNOSQRH¡µL OSADCjOf4U_C6CLXV6C´HpE,MLA0GIH¡f6H NW[WFHYQ,OV!V6C6H\«U0QRaH6NW[WNPC6XpADNWQRHGIU_CjOSNPCV6HHOgGIUHQ&GINW_H
a(u, v)lHIO#VLC6H«U_QSaH[WNWC6XYA_NPQSH\GIU_CjOSNPCV6H
L(v)U0CGIU_CFE,Nef4QSH[WA«U_QSagVL[WADORNWU_CcA_QRNeAcOSNPU0C6C6H[W[WH=H
ORQSU_VL_HQu ∈ V ORHY[vdjV6H
a(u, v) = L(v) ∀ v ∈ V, ¯ Wpf6U_CjO-U_CJESADNPOdjV] HY[P[WHA_f4aHIO-V6CLHVLC6NWdjV6HERU_[WV4ORNWU_CMLA_Q![WH 0K6XU0QRYaHµL µLPf4H vAcNW[P0QSA_a; Pm4~!~!y¤quoK3´m4nporq<s osªnY|jycs'|f4H ¯ Wp;GIU_CFE,NeE>OSHZ¡QRHYaM6[WA0GIHYQ;[^] HpE,MFA_GIHf4HNW[P<HQRO
VMLADQVLCERU_VFE> HYERMLA_GH;f4H#f6NPaHCFE,NWU_CLC6NWH
Vh6G_] HpE>OR ZD f4NWQSHBZG&K6HYQSG&K6HYQ[WA\ERU_[WV4ORNWU_Cf4H H
OSQRU0V6_HYQuh ∈ Vh
ORH['d0VLHa(uh, vh) = L(vh) ∀ vh ∈ Vh.
¯ 0/_ 'AQRXpE,U0[PV4OSNPU0Cf4H\[^] ADM6M6QSU¤4NPa\ADORNWU_C NWCjORHQSC6H ¯ 0/_;HYE,Ob«rA_GINW[WHGIU_aaH[PHaU_CjOSQRH[WH\[PHYaaHERV6NWcADCjOY;2|+3 3´| %]K^_]K^a`>bdcfe
V g h jup~|Nq-j m j c -jk e k j o j e Vh g hp b g+p jup~|Nq-j m j m4c t9jlhNp cBb h' h c j w `>bdcfe a(u, v) g h>jr b kut9j. c c h q c kvj9 b h e~c hNgNjEj e b jkv c i4j7p-g k V o j e L(v) g h>jr b kut.j c h q c kvj b h e~c hNgNj)pug k V w %" b k-p6v q@||>k b - c t9q e~cBb h c h e jlkuh>jz w-,l q m t.j e g hjg h c gNj6p b Dg eKcBb hw q4kq c fjlg k-p)-j eKe j p b g e~cBb h |jg e p b e jlh c k9jh k p b i4q4h e g h p p e + t9j. c h q c kvj m jt9q e k c -j m' ' h c j| b p cfeKc i4j6z j e p t e k c gNj p c a(u, v) jlp e p t e k c gNj!wYh+3´q<s'tYnycm4nporq<s#] ] HI4NWE,ORHYCLGIH\HO#[] V6C6NeGINPORXf6H
uh ∈ VhvE,U0[PV6ORNWU_Cf4H ¯ /0f4XpGIU_VL[PHf4V
0KLXU_QSaHµ6 µ6Wf4H vADjNW[P0QSA_a A_M6M6[WNWdjV6XZVh§°!U0V6QaHIO,OSQRH[WHM6QSU_L[PYaHE,U0VLEV6C6H
Ecol
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chni
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[ # ! !"#$% p_µ«U_QSaHM6[WVLEbE,NWaM6[WH_lU_C³NPCjORQSU4f4V6NPOTV6C6HLA_ERH
(φj)1≤j≤Nh
f4HVh%(N
uh =∑Nh
j=1 ujφjªU0CMFUjE,H
Uh = (u1, ..., uNh)[PH0HYGIORHVLQf6ADCLE
RNhf4HpE-GIUU_Q&f4U0C6C6XHpE-f4H
uh_ vHM6QRU06[WaHT ¯ 0/_HYE,OXYdjV6NW¤A_[PHYCjOZIH
OSQRU0V6_HYQUh ∈ RNh
OSH[vdjV6Ha
Nh∑
j=1
ujφj , φi
= L(φi) ∀ 1 ≤ i ≤ Nh,
GIHTdjV6NvEY] XpGIQSNOERU_VLE[eAJ«U_QSaHTfl] V6CER`E,ORYaHB[WNWC6XYA_NPQSHKhUh = bh,
¯ µ0A¤_HYG_MFU0V6Q
1 ≤ i, j ≤ Nh(Kh)ij = a(φj , φi), (bh)i = L(φi). vA GUjHYQSGNPNPORXf4H¡[eA«U0QRaH6NW[WNPC6XpADNWQRH
a(u, v)HCjORQ&A8V¬C6H[WHGYADQ&A_GIORQSHf4XLC6NMFUjE,NPORNP«9f4H[WAaADORQSNWGH Kh
6HIO9f6U_CLGbERU_CNPC_HYQSERNW6NP[WNPORX_4?CHC<HOY6M<U_VLQORU_V6O_HYGIORHYV6QUh ∈ RNh
6U0CA
KhUh · Uh ≥ ν
∥
∥
∥
∥
∥
∥
Nh∑
j=1
ujφj
∥
∥
∥
∥
∥
∥
2
≥ C|Uh|2A¤_HpG
C > 0,
GADQORU0V4ORHpE¡[WHYEC6U0QRaHpEERU_CjOXpdjV6NPcA_[PHYC0OSHYEHC f4NPaHYCLE,NWU_C LC6NWH¥ | · | f4XYERNP0C6H³[eAxC6U_QSaHHVLG[PNef4NWHC6C6Hf6A_CLERNh
l@9Ha¦aH_ª[eAER`aXIORQSNWHJf4Ha(u, v)
NWaML[PNedjV6HGH[W[PHf4H Khl@BADCLE[PHpEADM6M6[WNeGAcOSNPU0CLEaXYGYADC6NedjV6HYE[eAaADORQSNWGH Kh
HpE>OA_M6MFHY[PXYH73´m4nYy¤o¢_|w!|ycoBA<orw!onYh<
9U_VLE;AD[W[PU0CLEa\ADNWCjORHCFADCjO9GIU0aMFADQSHQ[] HQSQSHV6QGU_aaNWERHTHCQSHaM6[eA)GYADCjO[] HYERMLA_GHVMLADQE,U0C³E,U0VLE,^HpE,MLA0GIH
Vhª°[WVLEBM6QSXYGNWERXaHCjOpLC6U0VLE;A_[P[WU_CFE9a\A'A>U0QRHYQ[eA¡f4NC<XYQRHYCLGIH ‖u− uh‖
U'&uHpE>Ob[WAE,U0[PV6ORNWU_C³f6A_CLE
Vf4H ¯ P¤HIO
uhGH[W[PHf6ADCFE
Vhf4H ¯ 0/_ª°QSXYGNWERU_CFE9ADVLMLADQ&A¤cADCjOd0VLH[ed0VLHYECLUDOSADORNWU_CLE HFU0CC6UDOSH
ν > 0[eAGIU0CLE>O&ADCjORHJf4HJGUjHYQSGNPNPORX#HO
M > 0[eA¡GU_CLE,OSA_CjORHf4H#GIU0C0OSNPCV6NPORXTf4HT[eAJ«U_QSaHBLNP[WNPCLXYADNWQSH
a(u, v)djV6Nl_XYQRNPLHYC0O
a(u, u) ≥ ν‖u‖2 ∀u ∈ V,|a(u, v)| ≤M‖u‖ ‖v‖ ∀u, v ∈ V H[WHaaH9ERV6NPcA_C0Op_f4Z 0HYA_C¡®2XYA6_aU0CjORQSHd0VLH[WATf6NWE,OSA_CLGIHHYC0OSQRH9[WATERU_[WV4ORNWU_CH4A0GOSH
uHIO[WA#ERU_[WV4OSNPU0CADM6M6QSU4G&K6XH
uhHpE>Oa\A'A>U_QSXHk!s'oK1 qLy43´h+3´|sªnB~!m4yycm4~!~?qLy¤n#m6kxtYq<k!tl=&|jtY~§m6¢_|
VhMFADQ[eAf6NWE,OSA_CLGIHbHCjOSQRH
uHOVh
;?|+3 3´| %]K^_] ,w'| hm " xh pj2| q-jsp b g+psjlp | bde + pjlp m g jtt9j www `>bdcfe u qp b g e~cBb h m jz w-6j e uh -jfj m j6z w-,lwxhq
‖u− uh‖ ≤M
νinf
vh∈Vh
‖u− vh‖. ¯ Fj
Ecol
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repr
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¤'F "+ n " I # $ !6
Yh+3´q<s'tYnycm4nporq<s#]J°V6NeERdjV6HVh ⊂ V
U_C f6XYf4V6NPOYMFADQE,U0VLE>OSQSA0GOSNPU0C f4HYE¡«U_QSagV6[eAcOSNPU0CLEcA_QRNeAcOSNPU0C6C6H[W[WHYEb ¯ P¤?HOg ¯ /0LdjV6Ha(u− uh, wh) = 0 ∀wh ∈ Vh.?CG&KLU_NeE,NeERESADCjO
wh = uh − vhU_CU_6ORNWHCjO
ν‖u− uh‖2 ≤ a(u− uh, u− uh) = a(u− uh, u− vh) ≤M‖u− uh‖‖u− vh‖,f] U'&[^] U0Cf4XYf4VLNOg ¯ Fj
;u|jyc¢_or¢0| ]B^]B^ 2 7 #2'9% 9 ' && 9) ;2 // 0 9); & "(% >%:*" & 7 2 @% & 7 ;2 % a(u, v) #6 & ;#(% * & 27 @% 2 & ;>7 @% + / @15)
‖u− uh‖ ≤√
M
νinf
vh∈Vh
‖u− vh‖.
'9 2 " " 7 6)% 2 7 2 * 7 2 6"78 " uh 9) + / 01,%;27 6:2 6 7 & & & 9 ;))% /
!NWCLAD[WHaHCjOp_M<U_V6Q?f6XaU_CjORQSHQ-[eAbGIU0Cj0HQS_HYCLGIHf4HGIHO,OSH9ADM6MLQRU¤4NWaADORNWU_CJ¤A_QRNeAcOSNPU0C6C6HY[P[WH_CLU_VLEf4U0C6C6U0CLEV6Cf4HQSC6NWHQ[WHaaHT_XYC6XQ&AD[^ XADM6M<H[WU_CFEd0VLH#f6ADCLE[eAC6UDO&AcOSNPU0CVh
[WHTMLADQ&AcaORQSHh > 0
C] AMLA_EBHYCLGIU0QRHf4H\E,NW_CLNFGYAcOSNPU0C³M6Q&AcOSNWdjV6H0ª9XYA_C6aU_NWCLEªC6U0VLEbE,VLM6MFUjE,HYQRU0CLEdjV6HG_] HpE>Of6ADCLE#[WA[WNPaNPORHh → 0
d0VLH\[] ADM6MLQRU¤4NWaADORNWU_CNPCjORHYQRCLH³ ¯ /0¨,GU_C_HYQR0H©g_HYQSEb[eA«U0QRaJV6[eAcORNWU_CcADQSNWADORNWU_C6C6HY[P[WH\ ¯ P¤;2|+3 3´| %]K^_] \ xh pj | q-j p b g+pjlp | bde + pjlp m g jtt9j dw'ww.xh pug || b pj g c j- c p e jg h p b g+p jlpK|Nq-j V ⊂ V
m jhpj m qdhp V j e g hjyq-||> c -q e~cBb h rhm j V m qdhp Vhz!q-||Njl j qF~?hjy¤m6nY|kvyw WorsªnY|jy¤~?q<rm4nporq<s e j0p gNj
limh→0‖v − rh(v)‖ = 0 ∀ v ∈ V . ¯ 0
%" b k-p q9t e bIm j m q@|'|k b - c t9q e~cBb h i4q4k c q eKcBb hNhjfj c h e jkuhj b hNi4jlk j o _ jup e $ m4c k@j gNjlimh→0‖u− uh‖ = 0.
¯ ¯ Yh+3´q<s'tYnycm4nporq<s#]#(U0NO
ε > 0ª°-ADQbf4HCLERNPORXJf6H V ªNP[!HI4NeE>OSH v ∈ V ORH[-djV6H ‖u− v‖ ≤ ε
°-A_QADNW[P[WHV6Q&EYjNW[lHI4NeE>OSHTV6Ch0 > 0
f4XM<HCFf6ADCjO9f4Hε2ORHY[d0VLH_4M<U_V6QGHIOX[WXaHYC0O
v ∈ V 4U_CA‖v − rh(v)‖ ≤ ε ∀h ≤ h0.?C0HQRORVf4V HYaaH ¯ P0 /6jU0CA
‖u− uh‖ ≤ C‖u− rh(v)‖ ≤ C (‖u− v‖+ ‖v − rh(v)‖) ≤ 2Cε,f] U'&[^] U0Cf4XYf4VLNO[WHTQRXpE,VL[O&AcOY
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[ # ! !"#$% p0 'AgE,ORQ&AcOSX_NWHNWCLf4NedjV6XH;MLADQ2[PHpE2 vHaaHYE ¯ P0P0 ¯ P0 /bHO ¯ P0 µTGNf4HYESERVLEHpE>OaA_NPCjOSHCLA_C0OGI[eADNWQRH0 9q<k'yq!nY|s'oryk!s!|m4~!~!y¤quoK3´m4nporq<s´slk$3 h0ycoilk'|w'|rmtpqFkvnporq<s´|jum6¢Dnp|w'k~'ycq!K:[3 | z6m4ycorm6nYorq<s!s'| %]K^ "?or1 m6k'nosªnycqw!k'oryc|k!s |jtY~§m6¢_|
Vhw!|w'oK3´|s'tporq<s
Z!s!or|¥~§k'otychtpqFk!wvyc| k!s tpoB3´~!|xtYtn :+3´|xors!hm6oyc|¥m6tYtpq¢0orh Pm4~!~!y¤quoK3´m4nporq<sz6m4ycorm4nporq<s's!|rr|orsªnp|jy¤s!| ] "¤<XYA_C6aU_NWCLEYF[WHJG&K6U0N f4HVh
C] HYE,ObMLA_E;XNef4HCjOY D[!«rADV4Od0Vv] NW[QSHYERMFHpGORHTf4HYV4GQRNPORYQRHpE H_U0Cf6U_NPO;M<U_V60U_NWQ9GIU0CLE>OSQRVLNPQSHTV6CU_M<XQ&AcOSHV6Q;fl] NWCjORHYQRM<U_[eAcOSNPU0Crh
f4H V fLADCLEVh
ERADORNeE>«rA_NWESADCjOT ¯ _brU&O>`M6NWdjV6HYaHCjO V HYE,OV6CHYERMLA_GHBf6Hb«U_CLGIORNWU_CLEQSX0V6[PNWQSHYE&/4NW[«rADV4O\djV6H[eA QSXYERU_[WV4OSNPU0C f4VxER`4E>OSaH¡[WNWC6XYA_NPQSH KhUh = bh
ERU_NPOXpGIU_CLU_aNWdjV6HHYCM6Q&AcOSNWdjV6HbGIHpEER`E,ORYaHYE[WNPC6XpADNWQRHpEE,U0C0Of4HbOSQRpE_Q&ADCLf6HYE2OSA_NP[W[PHpESI vAaXIORKLUf6Hf4HYEbXY[PXYaHYCjOSEBLC6NeE#GU_CLERNWE,ORHM6QSXYGNWERXaHCjO#Z«U_V6QSC6NWQ#f4HORHY[WE9¨,<U_CLE>©JHYERMLA0GIHYEVh<±'cADCjO;fl] HCjOSQRHYQ;f6A_CLE[WHYE;f4XIO&ADNW[WEYFf4NeERU_CLE;d0VLH[ed0VLHYEaU_OSE9f6H#[WA¡aXIORKLUf6Hgf4H bA_[PHYQ NWCd0VLNQRHYCjORQSHBA_VLERERNf6A_CLEGIH#GYA_f4QSH_
*,+-*) l1957U 7U PORU3 =,/ 'A³aXORK6U4f4Hf4H bAD[WHQ NPC AXIOSXVLC M6QSXYGV6QSERHVLQf4H[eA³aXIOSK6U4f4Hf4HYEX[WXaHYC0O&EJLC6NeE
*2NWHCdjV] HY[P[WHC] A¡MLA0Ebfl] NWCjORXYQR¦OBCV6aXQSNed0VLHJHYC0XC6XYQSA_[<H[W[PHHYE,OBORQSYEBV4ORNW[WHf] V6CM<U_NWC0O#f4HjVLHORK6XYU_QSNWdjV6HTCLUDOSA_aaHYCjO?MFU0V6Q[^] XORVLf6Hf4HYE-MLQRU06[PYaHYE-C6U0C4 [PNWC6XYA_NPQSHYE&¤[W[WHQRHYCjORQSHfLADCLE[PH#GYA_f4QSHBf6HT[] ADM6MLQRU¤4NWaADORNWU_CcA_QRNeAcOSNPU0C6C6H[W[WHBNWCjORHYQRC6HTf4XpGIQSNOGNf4HYESERVLE;CERV6M6M<U0ERHdjV6H[^] HpE,MFA_GIHf4H NW[WFHYQ,O
VHYE,OE,XYMLADQ&ADL[PHf4Hf4NWaHYCLERNPU0CJNWC4LC6NWH_DGHd0VLN4HC4ORQ&A8V¬C6H_MLA_Q[eAJ°QRU0MFUjE,NPORNWU_C³ /4W_Wp6jdjV] NW[<H4NWE,ORHbV6C6HBLA0E,HBK6NW[P<HQRORNWHC6CLH
(ei)i≥1f6HV ;CG&K6U_NeE,NPO9AD[WU_Q&E V GIU_aaH#[WHgERU_VLE,^HpE,MFA_GIHbHYC6_HYCLf4QSXTMLADQ9GHIORORHgFA_ERHTK6NP[W<HQRORNWHC6C6HHYC6_HYCLf4QSXMLADQBGIU_aJ6NWCLADNeE,U0C[PNWC6XpADNWQRHpELC6NWHYE&djV6N§HpE>OB6NPHYC³ER6QBf6HCLERHJf6A_CLE
VªC M<U0ESADCjO
h = 1/nU_Cf4XIFC6NO
VhGU_aaH\[PHERU_VFE> HYERMLA_GHf6Hf4NPaHYCLE,NWU_CLCLNPH\HYC6_HYCLf4QSXMFADQ
(e1, ..., en) !NCLAD[WHaHCjOp0[^] U0MFXYQSADORHYV6Q2fl] NWC0OSHQSMFU0[WADORNWU_C
rhHpE>OE,NWaM6[PHYaHCjO2[eA#M6QSU>A>HpGOSNPU0C\U_QRORK6U0_U_CFAD[WHERV6Q
VhdjV6NHYE,ONeGINf4XLC6NWH#f6ADCLEOSU_V4O
VHOMLA_EERHV6[WHaHYC0O9fLADCLE V 0'U_V6ORHYE[PHpEK`jM<UDOSK6YERHYE9f4HYE vHaaHYE ¯ P0P0 ¯ W_0/JHIO ¯ W_ µE,U0CjOf6U_CLGTESAcOSNWE,«rADNPORHpEHIOU0CHCf6XYf4V6NPOdjV6H;[eAE,U0[PV4OSNPU0CA_M6M6QSUG&KLXHuh
GIU0Cj0HQS_H0HQ&E?[eAJERU_[WV4OSNPU0C¡HI6A0GORHu XA_M6M<H[WU_CLEd0VLH
uhHpE>OgGYAD[eGIV6[WX\HCQSXYERU_[WcADCjOT[PHER`4E>OSaH\[PNWC6XpADNWQRH KhUh = bh
U&Uh
HpE>O#[WH0HYGOSHV6Qf6ADCLERn f4HpEGIUU_Q&f4U_CLC6XYEf4H
uhf6ADCFE[WALA0E,H
(e1, ..., en)
³AD[W_QSX³E,U0C A_f6XYdjVLAcOSNPU0CADV GYA_f4QSHORKLXU_QSNed0VLH³f4X0H[WU_M6M<XGNf4HpERERVLEY?[eAaXIORKLUf6Hf4HbAD[WHQ jNWCHpE>OM<HVGIU_aaU4f4Hbfl] VLC¡M<U_NWC0Of4HBV6HbCV6aXQSNWdjV6H0j?CHC<HOY4[eAga\AcOSQRNeGIH Kh
djV6H[] U_CU04ORNWHCjO;A_NPCLERNHpE>O90XC6XYQSA_[PHYaHCjO?¨>M6[WHNWC6HI©&<GD] HYE,O,Zcf4NPQSHTdjV6HTORU0VLE9GHYEGUHlWGINWHCjOSE9E,U0C0OC6U_CCV6[eETHC0XC6XYQSA_[lHOB¨,aA_[GIU0CLf4NPORNWU_C6CLXH©vG_] HpE>OR ZD f4NWQSHdjV6H\[eAQSXYERU_[WV4ORNWU_CCV6aXQSNWdjV6Hf4VE,`4E,ORaHb[WNPC6XpADNWQRHbERHQ&AgNWCLE,OSADL[PH#GYADQORQSYEERHCLERNW6[PHTA_V4¡HQSQSHV6Q&Efl] A_QRQSU_CFf4Nlf4VGYAD[eGIV6[ERV6QU_Q&f4NPCFAcORHYV6QpY@;H2GIH?MFU0NPCjO!f6H?V6H_p[eAaXIOSK6U4f4H2f4HpEvXY[PXYaHCjOSELC6NeEvHpE>O!6NPHYC#M6[WVLE'MFHYQ,«U0QRa\ADCjOSHHIO9f4Hb[WU_NWCMLQRX«XQ&AD6[WHBZ[eAaXIORKLUf6H#f4HbAD[WHQ jNWC*,+-* l1957U 7US O M Uv/21Sb/9=RS# ;3L=,/;:l= US v/ 3NP§8
vH\M6QSNPCLGNPM<H\f4H[WAaXIOSK6U4f4Hf4HpEbX[WXaHCjOSEBFC6NWE#HYE,OTf6HGU_CLE,ORQSV6NWQRHf4HpEbHYERMLA_GHYEbfl] ADM4M6QRU¤4NWa\AcORNWU_C\NWCjORHYQRC6HVh
f4HpE?HpE,MFA_GIHpE«U0CLGOSNPU0C6C6HY[WE2VLERV6H[eEH1(Ω), H1
0 (Ω), H2(Ω), ...f4U_CjO
Ecol
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re.
¤ ¯ "+ n " I # $ !6
[eA´f4XLC6NPORNWU_CHpE>O¡FA_ERXH ERV6Q[eAC6UDOSNPU0C_XYU_aXIOSQRNedjV6Hf4H 3´m6ormAF| f4V f4U_a\A_NPC6HΩ2Ca\A_NP[W[WA__HHYE,O#V6CMLA¤cA__Hf4H\[] HYERMLA0GIH\HC0U_[WV6aHYEbX[WXaHYC0O&ADNWQRHpEBORQSYE#ERNWaML[PHpE HlORQSNeADC60[PHpEOSXIOSQSA_Yf4QSHYEYªMLADQ&AD[W[PXY[PXYM6NWMFpf4HYE0U_NWQYªMLA_QbHI4HaM6[WH_l[eA1'NW_VLQRH ¯ GDIlU_VFETf4U_C6CLHQSU_CLEBM6[WVLE[WU_NWC´V6CLHf4XLC6NPORNWU_C´M6QSXYGNWERH¡fl] V6C´a\A_NP[W[WA__H¡fLADCLEg[PHGA_f6QRH¡f6H¡[eAaXIORKLUf6Hf4HpEgXY[PXYaHYCjOSEFC6NWEY@BADCLE\GHGIU_CjOSHIORH[PHMLA_QSA_aIORQSH
hf6H
VhGIU0QRQSHYERM<U_CLfxZ³[eAnYm6orr|3 m4uoB3´mLr|¥w'|t
3 m6orr|tU_V GH[W[PV6[WHYEdjV6N;GU_aMFUjE,HYCjO¡[PHa\ADNW[P[eAD0H_ 02`M6NWdjV6HYaHCjO¡V6C6HLA_ERH³f4HVh
E,HYQSAGU_CLE,ORNPORVLXH;f6H«U_CFGORNWU_CFE?f4U0C0O[PHBE,VLM6MFU0Q,OHYE,Oq¢_m6ortphgERV6Q?VLC6H9U0V¡djV6H[edjV6HYE?aA_NP[W[WHYEY4®2HpGINA_V6Q&A¡f4HYV4 GIU_CFE,Xpd0VLHCLGHYE;NPaM<U_QROSADCjOSHYE=Hªf] V6CLHMFADQROYªf6ADCLE;[eA¡[WNPaNPORHh → 0
F[^] HYERMLA_GHVhERHQ&Af4HJM6[WVLEBHC M6[WVLE¨>0QRUjE©THIOTADMLM6QRU4G&K6HYQSA\f6HgaNPHYV4HCaNWHV6[^] HYERMLA_GH
VOSU_V4OBHCjORNWHQpHOf] A_V4ORQSH9MLA_Q,Op_[eA#a\AcOSQRNeGIHBf4HBQRNW_Nef4NPORX Kh
f4VER`4E>OSaH9[WNPCLXYADNWQSHJ ¯ µjERHQ&A#¢Dyc|k!tY|<GD] HYE,O,ZD f6NPQSH djV6H³[eAM6[WV6MLA_Q,Of4H³ERHYE¡GUHlWGINWHCjOSEERHQSU_CjOCV6[WErGH³djV6NB[PNWaNOSHQ&A[PHGIU04Of4H³[WAQSXYERU_[WV4OSNPU0C¡CV6aXQSNWdjV6H¤ 'AaXORK6U4f4Hf4HYETXY[PXYaHCjOSEbFC6NWE#HYE,O#V6C6H¡f4HpETaXIOSK6U4f4HYET[WHYE#M6[PVFE#HlWGYA_GIHHOg[PHpETM6[WVLEM<U_MLV6[WA_NPQSHBMFU0V6QQRXpE,U0VLf4QSHCV6aXQSNWdjV6HYaHYCjOf4HpEM6QRU06[WaHYEADV6[PNWaNOSHYEYj?[P[WHbHYE,OZJ[eAgLA0E,Hf] NWC6C6U0ag6Q&AD6[WHYE[WU_0NWGNPHY[WEf4HTGYAD[eGIV6[eENPCFf4VLE,ORQSNPHY[WEY¡ " g%!( ($I( $ ('$#
N = 1°§U_VLQgE,NWaM6[WNLHYQ#[^] HM<U0ERNPORNWU_CC6U_VFE#GIU0aaHYCGIU0CLEbMLADQgM6QRXpE,HYCjORHQT[eAaXIOSK6U4f4Hf4HYE#X[WXIaHYC0O&ETLCLNWEgHCV6C6Hf4NPaHYCLE,NWU_Cfl] HYERMLA0GIH_(4ADCFETMFHYQ,OSH¡f4H0XC6XYQSA_[PNPORX\CLU_VLE#G&KLU_NeE,NeERERU_CFEb[PHf6U_a\ADNWC6HΩ =]0, 1[
<?Cf6NPaHCFE,NWU_CxVLCa\ADNW[W[WA__HHYE,OTERNWaML[PHYaHYCjOTGU_CLE,ORNPORVLXfl] V6C6HGU_[P[WHYGIORNWU_C f4HJMFU0NPCjO&E(xj)0≤j≤n+1
rGIU0aaHJMFU0V6Q9[eA\aXIOSK6U4f4Hgf4HpE9f4NCªXQSHCLGHYEFC6NPHpE<_U0NPQ9[PH®2KFADM6NPORQSHp?ORHY[WEdjV6Hx0 = 0 < x1 < ... < xn < xn+1 = 1. vHbaA_NP[W[eAD_HTERHQ&AJf6NO9k!s'oK1 qLy43´|\ERNl[PHpEM<U_NWC0O&E
xjE,U0CjOXpdjV6NWf6NWE,OSA_C0O&E6G_] HpE>OR ZD f6NPQSHBdjV6H
xj = jhA¤0HYG
h =1
n+ 1, 0 ≤ j ≤ n+ 1. vHYEM<U_NWCjOSE
xjERU_CjOADVLESERNªA_M6M<H[WXYE[WHYEtpq_3 3´|jnYtgrU_VC6UHVLfLESf4Va\ADNW[W[WA__H_°-ADQERU_VFGINf4HERNWaML[PNeGINPORXTC6U0VLEGIU0CLERNWf4XYQRU0CLEYjM<U_VLQ[] NPCFE>O&ADCjOY6[WHbM6QSU_6[WaHbaUf6[WH#E,V6NWcADCjO
−u′′ = ffLADCLE
]0, 1[u(0) = u(1) = 0,
¯ GDf6U_CjO;C6U0VLEBERA¤0U_CLEdjV] NW[§A0f4aHIOBV6C6HJERU_[WV4OSNPU0C V6C6NedjV6HfLADCLE
H10 (Ω)
ERNf ∈ L2(Ω)
r_U_NWQ9[WH®2KFADM6NPORQSH0¤@;A_CLEOSU_V4O§GIHd0VLNERV6NPO'U0C#C6UDOSHQ&APk
[^] HCLERHaJ6[PHf4HYE'M<U_[W`jCB_aHYE!ZGIUHW\GNPHYCjOSEQSXHY[WEf] V6CLHT¤A_QRNeADL[PHBQRXYH[W[PHTf4H#f6H_QSXBNPC4«XYQRNWHVLQU_VXjAD[lZk
*KJ*,+ O M Uv/21S B/;=RSP1 'AaXIOSK6U4f4HJf4HpEXY[PXYaHYCjOSE9LC6NeE
P1QRHYMFUjE,HgERV6Q;[] HYERMLA_GHgf4NeERGQRHOBf4HpE«U0CLGOSNPU0CLE0[PU0LAc[WHaHYC0O9GU_CjORNWCV6HYEHO9A4W\C6HYEERV6QG&KLA0djV6HbaA_NP[W[WH
Vh =
v ∈ C([0, 1])ORHY[vdjV6H
v∣
∣
[xj ,xj+1] ∈ P1MFU0V6QORU0V4O
0 ≤ j ≤ n
, ¯
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[ # $ !66# "! # +N = 1
p'GHIO9ERV6QE,U0CERU_VFE> HYERMLA_GH
V0h = v ∈ VhOSH[djV6H
v(0) = v(1) = 0 . ¯ j vAaXORK6U4f4Hf4HpEbX[WXaHCjOSE;LCLNWE
P1HYE,OTAD[WU_Q&E;ERNPaM6[WHaHYC0Ob[eA¡aXIOSK6U4f4H\fl] ADM6M6QSU¤4NPa\ADORNWU_C¤A_QRNeAcOSNPU0C6C6HY[P[WHNPCjOSHQSC6H;f4H;[WA(U_VLE,E,HpGORNWU_C ¯ P0 /TA_M6M6[WNWdjV6XYH9ADV6HpE,MLA0GIHpE
VhU0V
V0hf4XILCLNWEMLADQ# ¯ 2U_V ¯ j
x0 xj xn xn+1=1=0 x x1 2
vh
φ j
¯ W ADNW[P[eAD0HBf6HΩ =]0, 1[
HO«U_CLGIORNWU_Cf4HTLA0E,HbHCXY[PXYaHYCjOSELC6NeEP1
;CM<HV6O;QSHM6QSXYERHCjORHYQ[PHpE«U0CLGOSNPU0CLE9f4HVh
U_VV0h
FAdW\C6HYE9MLADQaU0QSGHYADV6ª6Z[] ADNef4Hgf4H«U_CLGIORNWU_CLEf4HTFA_ERH;OSQRpEE,NWaM6[WHYEY DC0OSQRU4f4VLNWERU_CLE[eAT¨«U0CLGOSNPU0CG&KFADM<HYADV4©φf6XILC6NWHbMLADQ
φ(x) =
1− |x| ERN |x| ≤ 1,0
E,N |x| > 1.
(Nv[WH#a\ADNW[P[eAD0HTHYE,OV6C6NP«U_QSaH_LM<U_VLQ0 ≤ j ≤ n+ 1
U_Cf4XLC6NPO;[WHYE«U0CLGOSNPU0CLE9f6H#LA_ERH0U_NWQ[WA !NP0V6QSH ¯ P¤φj(x) = φ
(
x− xj
h
)
. ¯ W E
;?|+3 3´| %] Q]K^ jup~|Nq-j Vho m ' h c |Nq4k z w u o jup e g h p b g+p jlp~|q'-j m j H1(0, 1)
m j m4c t.jhp cBb h n+2 o j e$e b g e jr b h eKcBb h vh ∈ Vh jlp em ' h c j m j"t.qdh c + kvj"g h c gNj|qdkpjupi4q4jg k-pq4g -p b tt9j e p (xj)0≤j≤n+1
vh(x) =n+1∑
j=0
vh(xj)φj(x) ∀x ∈ [0, 1].
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
¤ "+ n " I # $ !6
j)tt.j o V0ho m' ' h c |Nq4k z w o jlp e g h p b g+p jup~|Nq-j m j H1
0 (0, 1)m j mdc t.jhp cBb h n o j ee b g e jr b h eKcBb h vh ∈ V0h jlp enm ' h c j m jt9q4h c + k@jg h c gNjs|qdk.pljlp.i8qdjlg k-p)qdg - p b t t9j e p
(xj)1≤j≤n
vh(x) =
n∑
j=1
vh(xj)φj(x) ∀x ∈ [0, 1].
Yh+3´q<s'tYnycm4nporq<s#]XA_M6MFHY[PU0CLEgdjV] HC0HQRORVf4V vHaaH FL µ6W Ll[PHpET«U0CLGOSNPU0CLEgGU_CjORNWCV6HYEHOf4HGI[eA_ESE,HC1 MLA_Q¡aU_Q&GIHpADV4 ADM6MFADQRORNWHC6C6HYCjOZ
H1(Ω)@;U_CLG
VhHO
V0hERU_CjO6NWHCf6HYE9ERU_VFE> HYERMLA_GHYEf6H
H1(0, 1)ª H#QSHYE,ORHJf4Hg[eA\M6QSHV60H#HYE,O;NWaaXYf4NeAcO9HYC QSHa\ADQ&djVLADCjO9djV6H
φj(xi) = δijU'&
δijHpE>O[WHBER`ag<U_[WHBf4H bQSU_C6HpG_HQ2d0VLNl¤A_V4ObBERN
i = jHO E E,NWC6U0C0U_NWQ2[eA
!NW_V6QSH ¯ WpI
.|+3´m4ycilk'| ]%] vA\FA_ERH(φj)
<f4XLC6NWHgMLA_QJ ¯ W E ILM<HQSaHIOBf6HJGA_QSA0GOSXQSNWERHQV6CLH#«U_CFGOSNPU0Cf6HVh
MLA_QERHYEcAD[WHVLQSEADV6C6UHVLf6Ef6Va\ADNW[W[WA__H_@BADCLEGHbGYA_E2U_CMLADQS[WHBw Phrh+3´|sªnptZ§s!ort\w!|;2mALycm6sA<|F<U0VLE0HQSQRU0CLEM6[WVLE[WU_NWCZ\[WA6(U_VLE,E,HpGORNWU_C ¯ 0/4djV] U_CMFHYV4O;NWCjORQSUDf6V6NPQSHbfl] A_V4ORQSHYEHpE,MLA0GIHpE
VhM<U_V6Q[WHYESdjV6H[eEV6C6HB«U_CLGIORNWU_CE,HYQSAGA_QSA0GORXYQRNeERXH_0C6U_CERHV6[WHaHCjOMFADQbE,HpE9cAD[WHVLQSEYLa\ADNeEBADVLESE,N!MLA_Q;[WHYE;cAD[WHV6Q&E9f4HESA¡f4XYQRNW_XYH_/;C³MLADQS[PHAD[WU_Q&E9w Phrh+3´|sªnpt
Z§s!ortw'||jy43´onp|FD>GIN^ªGU_aaHg[WHYE;«U_CFGORNWU_CFE;ERU_CjO;[WU4GA_[PHYaHYCjOP1FU0Cf4NPOTd0VLHJ[^] HpE,MFA_GIH
VhLf4XIFC6NMLA_Q# ¯ I4HYE,O[] HYERMLA0GIHTf4HYEXY[PXYaHCjOSELCLNWEf6H# vAD0QSA_C6_HBfl] U_Q&f4QRH_®2HOHHYaM6[PH;f4HYE2X[WXaHCjO&ELCLNWE
P1M<HQSaHIOZTC6U_VL_HYA_Vf4H;GU_aM6QSHCLf6QRH[^] NWCjORXYQR¦Of4H;[WA«U0QRaJV6[eAcORNWU_CTcA_QRNeAcOSNPU0C6C6H[W[WH_?C#HlCªHIOpp[WHYEv«U_CFGORNWU_CFEvf4H
VhC6HE,U0C0O!MLA_E'f4HV6B«U0NWE!f4XYQRNWcAD6[WHYEERV6Q[PHbERH_aHYC0O
[0, 1]HOGH[eA#C] AgMFA_Ef4HBERHCFEf6H;QSXYERU_VLf6QRH0_a¦aHbf4H;a\A_C6NPYQRHBADM6MLQRU4G&K6XYH_[^] XYdjVLAcOSNPU0C ¯ 0G_\HYC«rADNPOJ[eA f4XQSNW_XHERHYGU_CLf6Hfl] VLC6H«U0CLGOSNPU0C´f4H
VhHpE>OV6C6H¡ERU_aaH¡f4Ha\A0ERERHYEf4H#@;NPQ&A_GBADV6¡C6UHVLfLEf4VaA_NP[W[eAD_H?² I6±VGIU0C0OSQSA_NPQSH_jNW[HYE,OMLADQR«rADNPORHYaHYCjO[WX0NOSNPaHf] V4OSNP[WNeE,HYQ;f4HpE«U_CFGORNWU_CFEf4H
VhfLADCLE[eA«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WH ¯ /0djV6NvCLH#QRHpdjV6NPHYQ,OdjV] V6C6H#ERHVL[PHTf4XYQRNW_XYH_ •@;XYGQRNW_U0CLE[eATychtYq<rk'nYoqFs~!ycm4nYoilk'|f4VMLQRU06[PYaH9f4H;@;NWQRNeG&K6[WHIOB ¯ GD§MFADQ?[eA#aXORK6U4f4Hf6HYE;X[WXaHCjOSE;LCLNWE
P1< 'A\«U_QSagVL[WADORNWU_C cADQSNeAcORNWU_CLC6H[W[PH ¯ 0/_;f4H[] ADMLM6QRU¤4NWa\AcORNWU_CNWCjORHQSC6Hf6HNPHYCjONeGIN5H
OSQRU0V6_HYQuh ∈ V0h
OSH[vdjV6H ∫ 1
0
u′h(x)v′h(x) dx =
∫ 1
0
f(x)vh(x) dx ∀ vh ∈ V0h. ¯ W_p
;Cf4XpGIU_aM<U0ERHuh
ERV6Q[eALA_ERHBf6HYE(φj)1≤j≤n
HIOU0CM6QRHYCLfvh = φi
GHTd0VLNvf4U_CLC6Hn∑
j=1
uh(xj)
∫ 1
0
φ′j(x)φ′i(x) dx =
∫ 1
0
f(x)φi(x) dx.
?CC6U_OSA_C0OUh = (uh(xj))1≤j≤n
bh =
(
∫ 1
0f(x)φi(x) dx
)
1≤i≤n
lHOBHYCNWCjORQSUf6V6NWESADCjOB[WA3 m4nycor¢0|w'|ycoKAFow'ornYh
Kh =
(∫ 1
0
φ′j(x)φ′i(x) dx
)
1≤i,j≤n
,
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[ # $ !66# "! # +N = 1
p*[WAJ«U_QSagVL[WADORNWU_C¤A_QRNeAcOSNPU0C6C6HY[P[WHBfLADCLE
V0hQRHYjNWHCjOZQRXpE,U0VLf4QSHBfLADCLE
Rn [PH#ER`4E>OSaHb[PNWC6XpADNWQRHKhUh = bh.®2U_aaH[PHpET«U0CLGOSNPU0CLEgf4H¡LA_ERH
φjU_CjOgV6C¨,M<HIORNPO>©¡ERV6M6M<U_QROY'[] NPCjOSHQ&E,HpGORNWU_Cf4HpEgERV6M6M<U_QROSEf4H
φjHIOφi
HYE,O;ERU_V60HCjONWf4HgHIO9[eA\M6[WV6MLA_Q,O;f6HYE9GUHlWGINWHCjOSE9f4H KhERU_CjO9CV6[eEC GYAD[eGIV6[E,NWaM6[PHTaU0C0OSQRHbdjV6H
∫ 1
0
φ′j(x)φ′i(x) dx =
−h−1 ERNj = i− 1
2h−1 ERNj = i
−h−1 ERNj = i+ 1
0ERNWC6U_C
HIO[eAJa\ADORQSNWGH KhHpE>OORQSNef4NWA__U0CLAD[WH
Kh = h−1
2 −1 0−1 2 −1
−1 2 −10 −1 2
. ¯ W /0
°!U0V6QU_6ORHCLNPQ[WH#E,HpGIU0CLf¡aHYag6QSHbh
NP[«rADV4O9GYAD[eGIV6[WHQ[WHYENWCjORX0QSA_[PHpE(bh)i =
∫ xi+1
xi−1
f(x)φi(x) dxERN
1 ≤ i ≤ n.
] XcAD[WVLAcOSNPU0CHI6A_GIORH#f4V ERHYGU_CLfaHaJ6QRHbh
M<HV4O9¦ORQSH#f4NW\GNP[WH#U0VNPaM<U0ESE,NW6[WH#E,N'[WA«U0CLGORNWU_CfHYE,OGU_aM6[WNWdjV6XYH_§C M6Q&AcOSNWdjV6HU_C¥A³QSHYGU_V6Q&EJZf4HpE 1 qFy43k!r|tw'| ilk!m6w'ycm4nYk'yc|U_V«U0QRaJV6[PHpEfl] NPCjORXY_Q&AcOSNPU0C¡CV6aXQSNWdjV6H¤djV6Nvf4U0C6C6HCjOV6C6H#A_M6M6QSU¤4NPa\AcOSNPU0Cf4HpENWC0OSX0QSA_[PHpEf4XILCLNWESERA_CjO
bh4°-ADQH4HaM6[WH_6U0CM<HV4OV6ORNW[PNeE,HYQ[eAg«U0QRaJV6[WHbf6V\¨,M<U_NWC0OaNW[PNWHV©
1
xi+1 − xi
∫ xi+1
xi
ψ(x) dx ≈ ψ(
xi+1 + xi
2
)
,
U_V[WAJ«U_QSagVL[PHTf4HpE?¨OSQSA_MFJHpE©1
xi+1 − xi
∫ xi+1
xi
ψ(x) dx ≈ 1
2(ψ(xi+1) + ψ(xi)) ,
U_V6NPHYCHCLGU_QSH1
xi+1 − xi
∫ xi+1
xi
ψ(x) dx ≈(
1
6ψ(xi+1) +
1
6ψ(xi) +
2
3ψ
(
xi+1 + xi
2
))
.
HYEf6HV4¡MLQRHYaNWQSHYE«U0QRaJV6[WHYEE,U0CjOH6A_GOSHYEM<U_V6Q[WHYE«U0CLGOSNPU0CLEψA4W\C6HpE4HO[WAJOSQRU0NWERNPYaHHYE,O'HI6A0GORH2MFU0V6Q
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
¯)E "+ n " I # $ !6
f ∈ Vh(Nª[eAT«U_CFGORNWU_C
ψHpE>OQRXY_V6[WNWQSH9djV6H[eGIU0CLdjV6H_AD[WU_Q&E?GHYE?«U_QSagV6[WHYEE,U0CjOERNPaM6[WHaHCjOA_M6M6QSU4G&K6XHpEA¤0HYG9V6CQRHpE>OSHbf6HB[^] U_Q&f4QRHbf4H O(h2)
Lf4H O(h2)4HO9f4H O(h3)
QSHYERMFHpGORNW_HYaHCjOY 'A¡QSXYERU_[WV4ORNWU_Cf4VER`4E>OSaHg[WNPCLXYADNWQSH KhUh = bhHYE,O;[eA¡MLA_Q,OSNPHJ[WAM6[WVLEbGIU04ORHYVLE,Hf4H[WAaXORK6U4f4HHYCJOSHQSaHf4HORHYaMFEf4HGA_[WGV6[^0®;] HpE>OM<U_V6Q&djV6U_N4CLU_VLE-MLQRXpE,HYC0OSU_CLE!HYCA_C6C6HHf6A_CLE[eA (HYGIORNWU_C YµLPf4HpEbaXORK6U4f4HpETM<HQR«U_QSaA_CjORHYEbf4H\QSXYERU_[WV4OSNPU0C%XADMLMFHY[PU0CLETdjV6H[WAa\AcOSQRNeGIH
KhHpE>OC6XpGIHYESESADNWQRHYaHYCjONWC_HQ&ERNP6[WHbMLADQA_M6M6[WNWGYAcOSNPU0Cf4V HYaaH ¯ P0P0
.|+3´m4ycilk'| ]%] \ vAa\AcOSQRNeGIHf4HQRNW_Nef4NPORX KhHpE>OOSQRpE\E,NWaNP[eADNWQRHZf4HYEa\AcOSQRNeGIHpEf4XA,ZQSHCFGIU_CjOSQRXYHYE[WU_Q&Ef4H [^] XORVLf6Hf4HpEaXIOSK6U4f4HYEf4H³f4NC<XYQRHYCLGIHpE\LCLNPHpE2C«rADNPOY
hKhHpE>O[WA[WNWaNPORHf4H#[eA\a\AcOSQRNeGIH¡~/4W#FbagVL[OSNPM6[WNWXHgMFADQ
1/cf4VERG&K6XYa\ANPaM6[WNWGNOSHJf4HJQRXpE,U0[PV6ORNWU_Cf4H[^] XYdjVLAcOSNPU0C¥f6H[eAG&KLAD[WHVLQ[WU_Q&ESd0VLH¡[WHMLA_Ef4HOSHaMLEORHYCLf¥_HYQSE[^] NWC4LC6N^U0VLE_HYQRQSU_CFEJZ[^] -4HQ&GINeGIH ¯ 0/4 µJdjV] NW[C6H#EY] A__NPOMFA_Efl] V6C6H#GU)L^CLGINef4HCFGIH_ •
bycq!K:+3´| w!||k$3 m6s's] vA\aNeE,H#HYC UHV6QSH#f4H#[eA\aXIOSK6U4f4HJf4HpEXY[PXYaHYCjOSEFC6NWEP1M<U_VLQ[PHTM6QSU_L[PYaHTf4HT9HV6a\ADCLCERV6NWcADCjOHYE,OORQSYEERNPaNW[WA_NPQSH
−u′′ + au = ff6ADCFE
]0, 1[u′(0) = α, u′(1) = β.
¯ WYµ0X9ADM6M<H[WU_CLEdjV6H ¯ Ppµ02A_f4aHIOV6CLHBERU_[WV4ORNWU_CV6C6NedjV6HTf6ADCLE
H1(Ω)E,Nf ∈ L2(Ω)
α, β ∈ R
HOa ∈ L∞(Ω)
ORH['d0VLHa(x) ≥ a0 > 0
M MLfLADCLEΩ0U_NWQ[WHJ®2KLA_M6NOSQRH#0L 'Ag«U0QRaJV6[eAcORNWU_CcA_QRNeAcOSNPU0C6C6H[W[WH ¯ 0/_f4Hb[^] A_M6M6QSU¤NWa\AcOSNPU0C¡NWC0OSHQSC6HTf4HNWHCjONeGIN$H4OSQRU0V6_HYQ
uh ∈ VhORH['d0VLH
∫ 1
0
(u′h(x)v′h(x) + a(x)uh(x)vh(x)) dx =
∫ 1
0
f(x)vh(x) dx − αvh(0) + βvh(1),
M<U_VLQ;OSU_V4Ovh ∈ Vh
l?Cf4XpGIU_aM<U0ESADCjOuh
E,VLQ;[eA¡FA_ERHJf4HpE(φj)0≤j≤n+1
<[eA¡«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WHBf6A_CLEVh
QSHNPHYCjOZQSXYERU_VLf6QRHbf6ADCFERn+2 [WHbER`4E>OSaHb[PNWC6XpADNWQRH
KhUh = bh,A¤0HYGUh = (uh(xj))0≤j≤n+1
4HIOV6CLHTC6U_V60H[W[PHba\AcOSQRNeGIHTf4HbQSNP0NWf6NOSXKh =
(∫ 1
0
(
φ′j(x)φ′i(x) + a(x)φj(x)φi(x)
)
dx
)
0≤i,j≤n+1
,
HO(bh)i =
∫ 1
0f(x)φi(x) dx
ERN1 ≤ i ≤ n,
(bh)0 =∫ 1
0 f(x)φ0(x) dx − α,(bh)n+1 =
∫ 1
0 f(x)φn+1(x) dx + β. vU_Q&ERdjV6Ha(x)
C] HpE>OMLA_EV6C6H³«U_CLGIORNWU_C GU_CLE,OSA_CjORH_NW[;HpE>OADVLESE,NBC6XpGIHYESESADNWQRHHYC M6Q&AcOSNWdjV6Hf] V4OSNP[WNeE,HYQJf4HpET«U_QSagVL[PHpEgf4H¡djVLA0f4Q&AcORVLQRH\M<U_V6QJXcAD[WV6HYQ#[PHpEJGUHlWGINWHCjOSEgf4H¡[eAa\AcOSQRNeGIH KhGIU0aaHbC6U0VLE[^] A¤0U_CLE?«rADNPO9f6ADCFE[] HI4HaM6[WHTM6QSXYGXYf4HYC0OM<U_V6Q[WH#ERHYGIU0CLfaHYag6QSHbh
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[ # $ !66# "! # +N = 1
¯ 9u|jyc¢_o¢_| ]%]B^ <<7 *)%$7 2 & ;# 9:9) ;)7 ; & ) 3 P1 2 <>%7 &
−u′′ = f 92 ]0, 1[u(0) = α, u(1) = β,
?;>% @)% * 7 ! 9 " 2 7 & :9) % 7 ! ! & &> # 2<<'2)% 2 C6>492 7 6#"'9 & & )%9 & 7 ;2 %* > 9;#"7 /9u|jyc¢_o¢_| ]%] % ><>% >'9:7 <)%7 & ?9) ) & 2,+ /-1 )#<<>)K23*" 7 25"'( "a(x) = 0 92 Ω / ="(% >% *" 7 2 & 2(% # 9I 6 & 7 ;2 % 6! 9 7 2 & ;# 9 9#;>7 ; & > 3 P1 %7 )%/ ="(%)%@* ,<>>$ ;2 & $% ;#6 9% 7 5 6 & ,7 ;2 % 7 9) ;# ;>% >47 2 ! 9 ":9! & < 2 7 ;
∫ 1
0
f(x) dx = α− β,
!*" =#! #"'9 #6 <>%;6)% ;#= 7 7 69# @% & 7 #,9=*'2'9"% 2(% / & <'2>% >%! %;!7 622 =7 ;#@% & / 0 / /9u|jyc¢_o¢_| ]%] \ <<7 *)%$7 2 & ;# 9:9)9 ;>% >'!# 3 #:+ %57 '2< (% 01 2 <>% !7 & =9 % 7 !+ / 1!/ ?;)% >%?* 2 C ; & 2 !)(% ; 9 @% 9% 9)) " > & =7 ;2 % # % ;#6 9% 2 =7 2 & >& & 2(% ! Kh + # !@ > & 7 <7 2 ?<>% #K1& 2 2 C#"'9 & & >% bh 9 ;>% >#/ )& 4* =<>%7 =<>%7 & 9)> & 2+ /-1!/9u|jyc¢_o¢_| ]%] ! 9)% (n + 2)
& 2 6C# <>"'(( >787 #=+.27 " ;#K1 ( ;## 2 <> xj = j/(n + 1) <>"% 0 ≤ j ≤ n + 1 ! % >7 ;# >(% <'2)%$9# % #C6@% 9 &>& % 2 9>% k > 0 / 2<<7 *" # '2'* & 2 6C <>"'(( >787 @% ! 7 " ('9 '27 fj / 27 <>) 69 <>! 59;><7 2'! & >+7 % ( 9 2)1;'% % :7 ;))% >627 94 & * 7 2 & &: 6>% +.",9 C>)% 2 7 2 9# (% ; & ; 7 >%" ;#K1#/ )%.<>% ;!)% 7 2:% )% 9) 7 2 <>) 9 ;* 7 )%:9 & )>% & 9 ;>7 ; & > 3 /
*KJ*KJ 5?/ #Uv3 Uv/;:>U U1 US016=RM P!14=,5?/ 7 U363NUv83°§U0V6Q\f4XaU_CjOSQRHYQJ[eAGU_C_HYQR0HCLGH¡f4H[WAaXIOSK6U4f4Hf4HYEX[WXaHYC0O&EJLC6NeE
P1HYC V6CLHf6NaHYCLERNPU0Cfl] HpE,MFA_GIHCLU_VLEERV6NW_U0CLE-[eATf4Xa\ADQ&G&K6HHpERdjV6NeERERXHfLADCLE-[eAE(U0VLE, ERHYGIORNWU_C ¯ W_0/4D9U_VLEf4XILCLNWESE,U0CLEORU0V4O9fl] AD<U_Q&f¡VLCqF~?hjycm4nY|k'y¡w Porsªnp|jyp~2qFm4nYoqFs
rhrGU_aaH#f6ADCLE[WH# HYaaH¯ W_ µ0
Yh[Z§s'ornYorq<s %] Q] xh q@|'|jfj b | kvq e jg k m c h e jk~| b q eKcBb h P1 R q@|'| c -q eKcBb h c h q c kvj rh m jH1(0, 1)
m qdhp Vhm' ' h c j o | b g k eb g e v ∈ H1(0, 1) o |Nq4k
(rhv)(x) =
n+1∑
j=0
v(xj)φj(x).
Ecol
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¯ / "+ n " I # $ !6
®2HO,ORH f6XILC6NPORNWU_C A6NPHYCxVLC¥ERHCLEGYADQp!HYCx0HQRORV¥f4V¥ vHaaH FL µ6 µ6![PHpEg«U0CLGOSNPU0CLE\f4HH1(0, 1)
ERU_CjO2GU_CjORNWCV6HYEHIO?[WHV6Q&E-cAD[WHVLQSE?MFU0CLGOSV6H[W[PHpEERU_CjO2f4U0CLG6NWHCf4XLC6NWHYEY0 ?] NWCjORHQSM<UD[WXHrhv
fl] V6C6HT«U0CLGOSNPU0CvHYE,O;ERNPaM6[WHaHCjO9[eA«U_CFGORNWU_C AdW\C6HTMLADQ9aU0QSGHYA_V4djV6N'GU)L^CLGINef4HA¤0HYG
vERV6Q#[WHYEJE,U0aaHOSE#f6VaA_NP[W[eAD_H
xj_U0NPQT[eA !NW_V6QSH ¯ 0/_IXHa\A_QSdjV6U0CLETdjV] HYCV6C6Hf6NPaHCFE,NWU_Cfl] HYERMLA0GIH[^] NWCjORHQSM<U_[WXHHpE>O2f6XILC6NWHM<U_V6Q?ORU0V4ORH«U0CLGOSNPU0Cf4H
H1(0, 1)_HIOC6U_CMLA0EERHVL[PHYaHYCjObM<U_VLQT[PHpEB«U_CLGIORNWU_CLEbQSX0V6[PNWQSHYEbf4H
H1(0, 1)rGHdjV6NE,HYQSA[WH\GA_EbHYCf6NPaHCFE,NWU_CERV6M<XQSNPHYV6QSHp
v
r vh
x0 xn+1=1=0 x2
¯ 0/ DC0OSHQSMFU0[WADORNWU_CP1
fl] V6C6Hb«U_CFGORNWU_Cf4HH1(0, 1)
'A\GIU_C0HQS_HCFGIH;f6HB[eAaXIORKLUf6H#f4HYEXY[PXYaHCjOSELCLNWE
P1QSHM<U0ERHTE,VLQ[PHT[WHaaH#E,VLNPcADCjOp
;2|+3 3´| %] Q] ,w WorsªnY|jy¤~?q<rm4nporq<s" `>bdcferh R b | kvq e jg k m c h e jk~| b q e~cBb h P1 w b g k e b g e
v ∈ H1(0, 1) o c i k c ' jlimh→0‖v − rhv‖H1(0,1) = 0.
j2|> g+p o p c v ∈ H2(0, 1) o q4 b k-p c $j- c p e jEg hjE b hp e qdh e j C c h m' |Njh m qdh e j m j h e jlfj gNj‖v − rhv‖H1(0,1) ≤ Ch‖v′′‖L2(0,1).9U_VLE¡QRHYMFU0VLERERU_CFEaU0aHCjOSA_C6XaHCjO[eA´f4XYaU0CLE,ORQ&AcORNWU_C f4H³GH [WHaaH MFU0V6QXYC6U_CLGHQOSU_V4Obf4HE,V6NPORHJ[PHJQRXpE,VL[O&AcO;MLQRNWCLGINWMLA_['f4HGIHO,ORHE,U0VLE, ERHYGIORNWU_C djV6N!XIOSA_6[WNOB[WAGIU_C0HQS_HCFGIH#f4H[eAaXIORKLUf6H#f4HYEXY[PXYaHYCjOSELCLNWE
P1M<U_VLQ[PHTM6QSU_L[PYaHTf4H#@;NWQRNeG&K6[WHIOY
£'hqFy8:+3´| ]%] `>bdcfeu ∈ H1
0 (0, 1) j e uh ∈ V0h jup p b g e~cBb hNp m j z w! j e z w - ok@jup~|Nj@ e~c i8jt.jh e w %" b k-p o qt le b m j m jlp t9jlh e p ' h c p P1 b h_i8jk j o N jlp e $ m4c kvj gNj
limh→0‖u− uh‖H1(0,1) = 0.
¯ PFj j2|> g+p o p c u ∈ H2(0, 1) z!-j g c jup e i kvq c p c f ∈ L2(0, 1) o q4 b k-p c j - c p e j7g hj7 b hp e qdh e jC
c h m |jh m q4h e j m j h e jfj gNj‖u− uh‖H1(0,1) ≤ Ch‖u′′‖L2(0,1) = Ch‖f‖L2(0,1).
¯ Wp_
Ecol
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[ # $ !66# "! # +N = 1
¯ µ.¡|+3´m4ycilk!| %] Q] vAMLQRHYaNWQSHTGIU0CLGI[WVLERNPU0C ¯ W#Fjf6V10KLXU_QSaH ¯ 0/4 ¯ HYE,OQSA_NvADVLESERNE,N[PHE,HpGIU_CFfaHagLQRH
fADM6MLA_Q,OSNPHYCjOJE,HYV6[WHaHCjOZ
H−1(0, 1)! ?] HpE>OSNPa\AcOSNPU0C ¯ P¤_#NPCLf6NWdjV6H[WANPORHpERERH¡f4H¡GU_C_HYQR0HCLGH\f4H[eAaXIORKLUf6Hf4HpE#X[WXaHYC0O&ETLCLNWE
P1-®2U_aaH¡GIHO,ORHa\A'A>UDQSADORNWU_CHYE,O;MLQRU0MFU0Q,OSNPU0C6C6H[W[WHgZ
hFU0C³f4NPObdjV6H#[eAaXORK6U4f4Hf4HYE;X[WXaHCjO&ELCLNWE
P1GU_C_HYQR0H[PNWC6XYA_NPQSHaHYC0Op
XHa\ADQ&djV6U_CLE9d0VLHT[PH 0KLXU_QSaH ¯ /6 ¯ f4HgGU_C_HYQR0HCLGH;HpE>O;¤A_[WA_6[WHT[PU0QSESdjV6Hb[WA\a\ADORQSNWGHf4H\QRNW_Nef4NPORX KhHO#[PHERHYGU_CLfaHaJ6QRH
bhE,U0CjObXYcAD[WV6XYETH6A_GOSHaHCjOp!®2HM<HCLfLADCjOY[eAaXIORK6U4f4Hf4HpE§X[WXaHCjOSE!FC6NWE
P1GU_C_HYQR0H?ADVFERERN[PU0QSESdjV] U0CgV4OSNP[WNeE,Hf4HpEv«U0QRaJV6[WHYE§f4HdjVLA_f4Q&AcOSV6QSHA_f4Xpd0VFAcORHpEMFU0V6Q9GA_[WGV6[WHQ KhHIObh
r_U_NWQ F µ0 H •
.¡|+3´m4ycilk!| %] Q] U0VLEf4XYaU_CjORQSU_CLE[PH 0K6XYU_QSaH ¯ 0/4 ¯ f6ADCFE-[PH;GA_E-f] V6C¡aA_NP[W[eAD_HV6CLN«U_QSaH02G_] HpE>OR ZD f4NWQSHf6H³M<U_NWC0O&Exj
XYdjV6Nef4NWE,OSA_CjOSEf6A_CLE[WH³ERH0aHYCjO[0, 1]
rADV6ORQSHaHCjOf4NPOxj+1 − xj = h
I HQSXYERV6[O&AcObMFHYV4O#C6XpADC6aU_NWCLETERHJ0XC6XYQSA_[PNeERHQbZf6HYEBa\ADNW[P[eAD0HYEBC6U_CV6C6NP«U_QSaHBa\ADNeEQRXY_V6[WNPHYQSE;rA_VERHCLEf4Hb[eAg@;XILC6NPORNWU_C ¯ µ6W_¤jHIOf6A_CLE2GHBGYA_EhHYE,O[eAf4NWE,OSA_CLGIHaAD4NPaJV6a HCjOSQRHTf4HYV4M<U_NWCjOSE H
h = max0≤j≤n(xj+1 − xj) •
.¡|+3´m4ycilk!| %] Q] ;C M<HV6O«rA_NPQSHV6C6HADCLA_[PU0_NWHHCjOSQRH[WAGIU0C_HQS_HYCLGIHfl] VLC6HaXORK6U4f4Hfl] XY[PXYaHCjOSE?LC6NeEHO2[eAgGIU0C_HQS_HYCLGIHfl] V6C6H;aXIORKLUf6H;f4HBf4NDCªXQSHCFGIHYELC6NWHYEY[X9ADM6M<H[WU_CLE2djV6H_fl] A_M6QSYE[PH 0K6XYU_QSaHbf4HT vAc /40/40/ E j[eAGU_C_HYQR0HCLGH9fl] V6CERG&KLXa\AJA_V4¡f4NC<XYQRHYCLGIHpELC6NWHYEf4XYGU_V6[WHTf4H#ERAE,OSA_6NP[WNPORXbHIO9f4H#ESAGIU_CFE,NeE>O&ADCLGH_ DCLf6NWdjV6U0CLEdjV6H[eEE,U0CjO[WHYEXYdjV6NW¤A_[PHYCjOSET¬«U0Q,aHY[WE&f4HTGIHpENPCL_QSXYf4NWHCjOSEfLADCLE[WHTGIU_CjOSHIORHTf4HpEX[WXaHCjO&ELC6NeEL vHbQCB0[PHbf4HT[eAGU_CLERNeE>O&ADCLGHMFU0V6Q§[PHpE'XY[PXYaHCjOSE'LC6NeE§HYE,O A>U_V6XMLADQ§[eA9MLQRU0M6QRNWXIOSX2fl] NWC0OSHQSMFU0[WADORNWU_Cf4VJ vHaaH ¯ /6 6O&ADCLf6NWEd0VLH;[WHbQCB0[PHBf4Hb[WAE>O&AD6NW[WNOSX;HpE>OORHYCV¡MLA_Q[eAJM6QRU0M6QSNPXORX;f6HbGUHQ&GINWjNPORXBf4Hb[WAg«U_QSaH;LNP[WNPCLXYADNWQSHd0VLNvA_ESE,VLQRHB[WAQSXYERU_[WV4ORNWU_CrE,OSA_6[PH¤f4HbORU0V4ORH#A_M6M6QSU¤4NPa\AcOSNPU0CNWC0OSHQSC6H_ •Yh+3´q<s'tYnycm4nporq<s#]§ H# vHaaH ¯ 0/4MFHYQRaHO9fl] A_M6M6[WNWdjV6HYQ[WH#QRXpE,VL[O&AcO9f4HgGU_C_HYQR0HCLGHBf6V HaaH ¯ W_ µ#djV6NªHCjORQ&A8V¬C6H9NWaaXYf4NeAcOSHaHCjOT ¯ W#Fj°§U_V6QU_4OSHC6NWQT ¯ P¤_jU_C¡aA>A>U_QSH[^] HpE>ORNWaADORNWU_Cf4V HYaaH ¯ P0 /Jf4HJ®2XYA
‖u− uh‖H1(0,1) ≤ C infvh∈Vh
‖u− vh‖H1(0,1) ≤ C‖u− rhu‖H1(0,1),
GIHTdjV6NM<HQSaHO9f4H#GIU0CLGI[WV6QSH;0Q@,0GIHbADV HYaaH ¯ /6 6
9U_VLE-f6U_C6C6U0CLE!a\ADNWC0OSHCLA_CjO![eA;f6XaU_CLE,ORQ&AcOSNPU0CJf4V HYaaH ¯ 0/4;E,U0VLE![WA9«U_QSaHf4Hf4HYV4ADV4OSQRHpE[PHYaaHpEORHYG&KLC6NWdjV6HpE;?|+3 3´| %] Q]K^ j - c p e jg hj b hp e q4h e j C c h m |jh m q4h e j m j h e jfj gNj o | b g k e b g ev ∈ H2(0, 1) o
‖v − rhv‖L2(0,1) ≤ Ch2‖v′′‖L2(0,1), ¯ W ¯
j e‖v′ − (rhv)
′‖L2(0,1) ≤ Ch‖v′′‖L2(0,1). ¯ W8G_
Ecol
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¯ F "+ n " I # $ !6
Yh+3´q<s'tYnycm4nporq<s#]N(U0NOv ∈ C∞([0, 1])
c°-A_Q§f6XILC6NPORNWU_CvD[] NWC0OSHQSMFU0[PXYHrhv
HYE,OV6C6H«U_CFGORNWU_CAdW\C6HbHIOYLMFU0V6QORU0V4Ox ∈]xj , xj+1[
LU_CAv(x)− rhv(x) = v(x)−
(
v(xj) +v(xj+1)− v(xj)
xj+1 − xj(x− xj)
)
=
∫ x
xj
v′(t) dt− x− xj
xj+1 − xj
∫ xj+1
xj
v′(t) dt
= (x− xj)v′(xj + θx)− (x− xj)v
′(xj + θj)
= (x− xj)
∫ xj+θx
xj+θj
v′′(t) dt,
¯ W
MFADQ#ADMLM6[PNeGADORNWU_Cf4H[WA«U0QRaJV6[WHf6HYETA_GYGIQSU_NeERERHaHYC0O&EFC6NWEgA¤_HYG0 ≤ θx ≤ x − xj
HIO0 ≤
θj ≤ h ;CHYCf4XYf4VLNOHYCV6ORNW[PNeERA_CjO[] NPCLX0A_[PNPORXTf4Hg®A_VLG&K`0!(4G&K ADQKJ|v(x) − rhv(x)|2 ≤ h2
(
∫ xj+1
xj
|v′′(t)| dt)2
≤ h3
∫ xj+1
xj
|v′′(t)|2dt. ¯ P 0?CNWCjORXY_Q&ADCjOb ¯ W j?MFADQQ&ADM6M<U_QROZ
xERV6Q[^] NWCjORHYQRcAD[W[WH
[xj , xj+1]6U_CU_6ORNWHCjO
∫ xj+1
xj
|v(x)− rhv(x)|2 dx ≤ h4
∫ xj+1
xj
|v′′(t)|2dt,
GH?djV6N^MLA_Q'ERU_aa\AcORNWU_CbHYCj¤f4U0C6C6HH4A0GOSHaHCjO2 ¯ W ¯ II°-ADQ!f4HCLERNPORX?GHQRXpE,VL[O&AcOvHpE>OvHYCLGIU0QRHQ&ADN4M<U_V6Q§OSU_V4O
v ∈ H2(0, 1)D vAbf4XYaU0CLE,ORQ&AcORNWU_Cf4H# ¯ W8GD'HpE>O§ORU0V4O?ZB«rADNPO?ERNPaNW[WA_NPQSH4H_MFU0V6Q
v ∈ C∞([0, 1])HIOx ∈]xj , xj+1[
U_CXYGIQSNPOv′(x)− (rhv)
′(x) = v′(x)− v(xj+1)− v(xj)
h=
1
h
∫ xj+1
xj
(
v′(x)− v′(t))
dt
=1
h
∫ xj+1
xj
∫ x
t
v′′(y) dy.
[PHY¤A_CjOADVGYADQSQRXGHIORORH9NWC6XY0AD[WNPORX_jADM6ML[PNedjVLADCjO®A_VLG&K`0!(4G&K ADQKJf4HYV4«U_NeEHOERU_aa\ADCjO?HCjU_CU_4OSNPHYCjOT ¯ W8G_6djV6NHpE>OADVFERERNlcAD[WNWf4HTM<U_VLQORU_V6O
v ∈ H2(0, 1)MFADQf4HCFE,NPORX0
;2|+3 3´| %] Q]K^_^ j - c p e jg h>j b hp e q4h e j C c h m |jh m q4h e j m j h e jfj gNj o | b g k e b g ev ∈ H1(0, 1) o
‖rhv‖H1(0,1) ≤ C‖v‖H1(0,1), ¯ 0/ E
j e‖v − rhv‖L2(0,1) ≤ Ch‖v′‖L2(0,1).
¯ 0/4p j2|> g+p o | b g k e b g e v ∈ H1(0, 1) o b hq
limh→0‖v′ − (rhv)
′‖L2(0,1) = 0. ¯ /'/_
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[ # $ !66# "! # +N = 1
¯ Yh+3´q<s'tYnycm4nporq<s#] vHYE2M6QSHV60HYEf4H ¯ 0/ E -HIOT ¯ /6p2E,U0CjOf6A_CLE[PHBa¦aH;HpE,MLQRNPOdjV6HbGIHY[P[WHYEf4V[PHYaaHTM6QSXYGXYf4HYCjOYN(U0NO
v ∈ H1(0, 1)&0'U_V6O9fl] A_FU0QSfU_CA
‖rhv‖L2(0,1) ≤ maxx∈[0,1]
|rhv(x)| ≤ maxx∈[0,1]
|v(x)| ≤ C‖v‖H1(0,1),
HC_HQRORVf4V HYaaH:FF µL µL@J] ADV4OSQRHbMLA_Q,OpGU_aaHrhv
HYE,OA4W\C6H_4HO_Q-,_GH;Z[WAM6QSU_MLQRNWXIOSXFL f4V vHaaH:FL µ6 µJdjV6NvAdW\QRaHTdjV6HvHYE,O6NWHC[WAM6QSNWaNPORNW_HTf4H
v′6U_CA
∫ xj+1
xj
|(rhv)′(x)|2 dx =(v(xj+1)− v(xj))
2
h
=1
h
(
∫ xj+1
xj
v′(x) dx
)2
≤∫ xj+1
xj
|v′(x)|2 dx,
MLADQ®ADVLG&K`j!(4G&K A_QCJ0IGIHd0VLN¤MLA_Q!ERU_aa\AcOSNPU0C#HCjcGU_CLf4VLNO-Zg ¯ 0/ E IY°§U0V6Q'U04ORHYC6NWQ ¯ 0/4¤U_CQRHYM6QRHYCLf[WAf4HYV44NPYaHTXY0A_[PNPORXTf4H\ ¯ P fl] U&[^] U0Cf4Xpf4V6NPO
|v(x) − rhv(x)| ≤ 2
∫ xj+1
xj
|v′(t)| dt.
CX[WHcADCjOADVGYADQSQRX0jHYCV4ORNW[PNeESADCjOB®ADVLG&K`j!(4G&K A_QCJ0_HCNWCjORX0QSA_CjOMFADQQ&ADM6M<U_QROZxLM6VLNWEHCE,U0aaA_CjOHYC
j6U0CU04ORNWHCjO6NWHC ¯ /6pI°-A0ERERU_CLEZ [eAf4XaU_CFE>OSQSADORNWU_C f4H ¯ /'/_I(U0NO
ε > 0-®2U0aaH
C∞([0, 1])HpE>O\f4HCFE,Hf6ADCLE
H1(0, 1)4MFU0V6QOSU_V4O
v ∈ H1(0, 1)NW[HI4NeE>OSH
φ ∈ C∞([0, 1])OSH[vdjV6H
‖v′ − φ′‖L2(0,1) ≤ ε.;Q
rhHpE>OV6C6H#A_M6M6[WNWGYAcOSNPU0C[WNPCLXYADNWQSHbdjV6Nl0XQSNFH ¯ 0/ E 6f4U0CLGBU_CHCf4XYf6V6NO
‖(rhv)′ − (rhφ)′‖L2(0,1) ≤ C‖v′ − φ′‖L2(0,1) ≤ Cε. HG&K6U_NPf6HφHObf4H
εXIO&ADCjO9LX0ªU_C³f4Xpf4V6NPO#f4H ¯ W8G_ADM6ML[PNedjV6XZ
φdjV6H_ªMFU0V6Q
hERV WERA_aaHYC0OM<HIOSNOp
‖φ′ − (rhφ)′‖L2(0,1) ≤ ε.°§A_QGIU0CLE,XpdjV6HCjOY4HYCERU_aa\ADCjOGIHpEORQSU_NeEf4HQSC6NWQSHYENWC6XjAD[WNOSXYEU_CU_6ORNWHCjO‖v′ − (rhv)
′‖L2(0,1) ≤ ‖v′ − φ′‖L2 + ‖φ′ − (rhφ)′‖L2 + ‖(rhv)′ − (rhφ)′‖L2 ≤ Cε,GIHTdjV6NNWaML[PNedjV6H ¯ /'/_
9u|jyc¢_o¢_| ]%] ; & (% >%7 ;* 27 >9 ;#@% & / 0 / 9 !" >% )>'! <>"%7 <>% !7 & :9)> & 2 + /-1!/
Ecol
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re.
¯_¯ "+ n " I # $ !6
*KJ*) O M Uv/21S B/;=RSP2 'AaXORK6U4f4HTf4HpEX[WXaHYC0O&ELC6NeE
P2QSHM<U0ERHBERV6Q[^] HpE,MFA_GIHTf4NeESGIQSHIO
Vh =
v ∈ C([0, 1])ORHY[vdjV6H
v∣
∣
[xj ,xj+1] ∈ P2M<U_VLQORU_V6O
0 ≤ j ≤ n
, ¯ 0/Dµ0
HO9E,VLQE,U0CERU_VLE, HYERMLA_GHV0h = v ∈ Vh
ORHY[vdjV6Hv(0) = v(1) = 0 . ¯ 0/'F
'A¡aXORK6U4f4H\f4HpE;XY[PXYaHCjOSE;LC6NeEP2
HYE,OB[eAaXORK6U4f4H\fl] ADMLM6QRU¤4NWa\AcORNWU_C³¤A_QRNeAcOSNPU0C6C6HY[P[WHgNWC4OSHQSC6H?f4H2[WAn(U_VFE>E,HpGOSNPU0C ¯ P0 /ADM6M6[WNed0VLXH?ZGHYE'HYERMLA_GHYEVh
U_VV0h
D®2HV4GINjE,U0C0O!GIU_aM<U0ERXYEf6H;«U0CLGOSNPU0CLEGIU0C0OSNPCV6HpEMLA_QSA_FU0[PNedjV6HYE2MLA_QaU0QSGHYADV6\djV] U_CMFHYV4OQRHYM6QSXYERHCjORHYQZJ[] ADNef4HTf4H«U0CLGOSNPU0CLEf4HTLA0E,HBOSQRpEE,NWaM6[PHpE
ψj
ψj+1/2
0
1
x x xj x xj-1 j-1/2 j+1/2 j+1
¯ µ¡ vHYE«U_CFGORNWU_CFEf4HTLA_ERHTf4HYEXY[PXYaHCjOSEFC6NWEP2
DCjORQSU4f4V6NeE,U0CLEgORU_V6Ofl] AD<U_Q&f´[WHYEJM<U_NWC0O&EaNW[WNPHYV4 f4HYEERH0aHYCjOSE[xj , xj+1]
f4XLC6NeEMLADQxj+1/2 = xj + h/2
MFU0V6Q0 ≤ j ≤ n ;Cf4XILCLNO9A_VLERERNf4HYV4«U_CFGORNWU_CFE¨>aQSHYE>©
φ(x) =
(1 + x)(1 + 2x)E,N − 1 ≤ x ≤ 0,
(1− x)(1− 2x)E,N
0 ≤ x ≤ 1,0
ERN |x| > 1,HOψ(x) =
1− 4x2 ERN |x| ≤ 1/2,0
ERN |x| > 1/2.
(4Nv[WH#a\ADNW[P[eAD0HBHpE>O;V6C6NP«U_QSaH0LM<U_V6Q0 ≤ j ≤ n+ 1
U_Cf4XLC6NPO;[WHYE«U0CLGOSNPU0CLE9f4HgLA_ERH0U_NWQ[eA !NP0V6QRH ¯ µjψj(x) = φ
(
x− xj
h
)
, 0 ≤ j ≤ n+ 1,HIOψj+1/2(x) = ψ
(
x− xj+1/2
h
)
, 0 ≤ j ≤ n.
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[ # $ !66# "! # +N = 1
¯ G;?|+3 3´| %] Q]K^ jlp~|q'@j Vh
o m' ' h c |Nq4k z w-, l o jlp e g hp b g+p jup~|Nq-j m j H1(0, 1)m jm4c t9jlhNp cBb h 2n + 3 o j e"e b g e j"r b h e~cBb h vh ∈ Vh jup e7m ' h c j m j6t.qdh c + kvj6g h c gNjs|Nq4k)pljlp
i8qdjlg k-pEqdg -9p b t t9j e p (xj)0≤j≤n+1 j e qdg -t c c jg - (xj+1/2)0≤j≤n
vh(x) =
n+1∑
j=0
vh(xj)ψj(x) +
n∑
j=0
vh(xj+1/2)ψj+1/2(x) ∀x ∈ [0, 1].
j"t lt9j o V0ho m' ' h c |Nq4k z dw-, *4 o jlp e g h)p b g+p jlp~|q'-j m j H1
0 (0, 1)m j m4c t9jhp cBb h 2n+1 o
j ee b g e jr b h> e~cBb h vh ∈ V0h jlp e2m ' h c j m j t9q4h c + k@jsg h c gNj|Nq4k"pljlp"i4q4 jlg k-p7q4g - p b tt9j e p(xj)1≤j≤n j e q4g -)t c c jg - (xj+1/2)0≤j≤n
vh(x) =
n∑
j=1
vh(xj)φj(x) +
n∑
j=0
vh(xj+1/2)ψj+1/2(x) ∀x ∈ [0, 1].
.¡|+3´m4ycilk!| %] Q]K^+\ D>GIN§HCLGU_QSH_Vh
HYE,ObV6C³HpE,MLA0GIHfl] XY[PXYaHCjOSE9LC6NeEbf4H 'AD0QSA_C6_HG«>ª[WAXHa\ADQ&djV6H ¯ /6 /0®2U_aaH¡[WHYEg«U_CLGIORNWU_CLEE,U0CjOJ[WUGYAD[WHaHCjO
P2!U_C f6NO\djV6H[^] HYERMLA_GH
Vhf4XILCLNMLADQ# ¯ 0/Dµj4HYE,O[] HYERMLA0GIHTf4HYEXY[PXYaHCjOSEFC6NWEf6H# vAD0QSA_C6_HBfl] U_Q&f4QRH /4 •
Yh+3´q<s'tYnycm4nporq<s#]9°§A_Q\ADM6M6[WNeGAcOSNPU0C f4V HYaaHIFL µ6W Vh
HIOV0h
ERU_CjO6NPHYCxf4HpEE,U0VLE>HYERMLA_GHYEf4HH1(0, 1)
L HYV6Q9f4NWaHCLERNWU_CHIO9[WHYELA_ERHYEM6QSU_M<U0ERXHpEE,HbOSQRU0V6_HYCjO«rA0GINW[PHYaHCjO9HCQRHYaA_QSdjVLA_CjOBdjV6Hψj(xi) = δij
ψj+1/2(xi+1/2) = δij
ψj(xi+1/2) = 0
ψj+1/2(xi) = 0_U0NPQ[eA!NP0V6QRH ¯ µj
@;XYGQRNW_U0CLE[eATychtYq<rk'nYoqFs~!ycm4nYoilk'|f4V¡M6QRU06[WaH9f4HB@9NWQRNeG&K6[WHIOb ¯ GD-MLADQ2[WAgaXORK6U4f4Hf4HYE;X[WXaHCjO&E9FC6NWEP2< 'A\«U0QRaJV6[WADORNWU_C cA_QRNeAcOSNPU0C6C6H[W[WH¡ ¯ /09f4H[^] A_M6M6QSU¤NWa\AcOSNPU0CNWCjORHYQRC6HQRHYjNWHCjOZQRXpE,U0VLf4QSHBfLADCLE
R2n+1 [WH#E,`4E,ORYaHb[WNPCLXYADNWQSHKhUh = bh.
¯ 0/_0°!U0V6QHI4M6[WNWGNOSHQGIHER`4E>OSaH³[WNWC6XYA_NPQSH NW[BHpE>OGIU0aaU4f4Hf4HG&KFADC60HQ¡fl] NWCLf4NeGIHHC C6U_OSA_C0Of4XYERU_QSa\ADNeE[WHYE2MFU0NPCjOSE
(x1/2, x1, x3/2, x2, ..., xn+1/2)ERU_VLE?[WA#«U_QSaH
(xk/2)1≤k≤2n+1HIO[WALA_ERH
(ψ1/2, ψ1, ψ3/2, ψ2, ..., ψn+1/2)f4H
V0hERU_VLE9[WA«U_QSaH
(ψk/2)1≤k≤2n+1ª@BADCLEBGIHO,ORHLA_ERH
Uh ∈ R2n+1 HYE,O§[WH_HpGOSHV6Q-f4HpE-GIUU_Q&f4U_CLC6XHpE'f4H[eABE,U0[PV4OSNPU0CA_M6M6QSU4G&K6XH uhdjV6N40XQSNFH
uh(x) =
2n+1∑
k=1
(Uh)k/2ψk/2(x)A¤0HYG
(Uh)k/2 = uh(xk/2), ¯ 0/ ¯
HIOU_CAKh =
(∫ 1
0
ψ′k/2(x)ψ
′l/2(x) dx
)
1≤k,l≤2n+1
, bh =
(∫ 1
0
f(x)ψk/2(x) dx
)
1≤k≤2n+1
.
HYE#«U_CLGIORNWU_CLEJf4HLA0E,Hψk/2
U_CjOgV6C¨>M<HIOSNO,©ERV6M6M<U_QROY'HIO[WAML[PV6MFADQROJf4HYEJGIUHlWGNPHYC0O&Egf4HKh
E,U0C0Of4U0CLGCV6[eE#9C GA_[WGV6[9ERNPaM6[WHaU_CjORQSH djV6H[WAa\ADORQSNWGH f4HQRNW_Nef4NPORX KhHpE>ONeGIN
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¯ "+ n " I # $ !6
M<HCjO&A_f4NeAD0U_CLA_[PH
Kh = h−1
16/3 −8/3 0−8/3 14/3 −8/3 1/3 0
0 −8/3 16/3 −8/3 01/3 −8/3 14/3 −8/3 1/3
0 −8/3 16/3 −8/3 00 1/3 −8/3 14/3 −8/3
0 −8/3 16/3
.
XHa\ADQ&djV6U_CLEdjV6HbGIHIORORHBa\AcORQSNeGIHBHYE,OM6[WVLE-¨,M6[PHYNPCLH©BdjV6HbGIHY[P[WH;U04ORHYCV6H;MFADQ[WAgaXIORKLUf6Hbf6HYEXY[PXYaHCjOSE9LC6NeEP1ªHIObf4U_CLGJdjV6Hg[eAQSXYERU_[WV4ORNWU_C f6VER`E,ORYaHg[WNPC6XpADNWQRHJGIU_6ORHQ&A\M6[WVLEBG&K6HQBHCOSHaMLE;f4HgGA_[WGV6[^L°§U_V6Q9XcAD[WV6HQ9[PHgERHYGU_CLfaHaJ6QSH
bhU0C AQRHpGIU_VLQSEA_V4a¦YaH#«U_QSagV6[WHYEf6HdjVLA0f4Q&AcORVLQRHU0V «U0QRaJV6[PHpE\fl] NPCjORXY_Q&AcOSNPU0C CV6aXQSNWdjV6H¤d0VLHGH[W[PHpEM6QSXYERHCjORXYHYEf6ADCFE[WAaXORK6U4f4H
P1
£'hqFy8:+3´| ]%]B^ `Qb4cfeu ∈ H1
0 (0, 1) j e uh ∈ V0h jlp9p b g e~cBb hp m jz dwR j e z w-, z w-, @ o kvjup~|Nj- e~c i8jt9jlh e w %" b k-p o q9t e bIm j m jup t.jh e p ' h c p P2 b hNi4jk j o _ jup e $ mdc kvj gNj
limh→0‖u− uh‖H1(0,1) = 0.
j#|Dg+p o p c u ∈ H3(0, 1) z!-j g c jlp e i kvq c p c f ∈ H1(0, 1) o q4 b k-p c %j- c p e jng hj b hp e qdh e jC
c h m |jh m q4h e j m j h e jfj gNj‖u− uh‖H1(0,1) ≤ Ch2‖u′′′‖L2(0,1).
;u|jyc¢_or¢0| ]%] ;> ;>% 27 C2 7 # 2>% " & > <>%;!;9))%=9; & (% >% 7 ;#@% & / 0 / / vH 0K6XYU_QSaH ¯ 0/4W#FaU_CjOSQRH[^] A¤¤A_CjOSAD0H-M6QSNPCLGNPMFAD[0f6HYEXY[PXYaHCjOSEvLC6NeE
P2HpERN_[eAERU_[WV4ORNWU_CHpE>OQSX_VL[PNWQSH_AD[WU_Q&E[eAxGU_C_HYQR0HCLGH f4H[eA´aXORK6U4f4HHYE,Oilk!m6w'ycm4nYoilk'| [eA´NOSHYESE,Hf4HGU_C_HYQR0HCLGHHYE,OMLQRU0MFU0Q,OSNPU0C6C6H[W[WH;Z
h2 A_[PU0QSE2d0VLH;[eAgGU_C_HYQR0HCLGHMFU0V6Q[PHpE2XY[PXYaHCjOSE2LC6NeEP1
HYE,O2E,HYV6[WHaHCjO[WNPC6XpADNWQRH#rM6QSU_M<U_QRORNWU_C6C6HY[P[WHZhI'*NWHC\E,LQ?GIHO2A¤cADCjOSA__HAbV6C\M6QSNIHDNP[F`JAf6HV4«U_NeEML[PVLEfl] NPCFGIU_CLCjVLHYE9rHI6A_GIORHYaHYCjO
2n+1A_V\[PNWHV¡f6H
nM<U_V6Q2[PHpE?XY[PXYaHYCjOSE?LC6NeE
P1f6U_CLG9[WAJa\AcORQSNeGIH9HpE>Of4HYV4\«U_NeE?ML[PVLE0QSA_CLf4H_jHIOHCM6[WVLE?[eAJaADORQSNWGH;AJGINWCLd\f4NeAD_U0CLAD[WHYE?C6U_CCV6[W[WHYEA_VJ[WNPHYVf4HORQSU_NeE-f6A_CLE§[PHGYA_E
P1'XHYa\ADQ&d0VLU_CLE§djV6HERN6[WAbERU_[WV4ORNWU_CC] HYE,OMLA0E§QRXY_V6[WNPYQRH
u ∈ H3(0, 1)NW['Cv] `A¡A_VLGIVLC³A¤cADCjOSA__HbORKLXU_QSNed0VLH¡a\ADNeEBADVLESE,N!M6Q&AcOSNWdjV6H¤ZV4OSNP[WNeE,HYQBf4HpEXY[PXYaHCjOSELCLNWEP2
M6[PV6OCBDO9djV6HP1
*KJ* J³3655 93L= l1 SyT8PO,=>1+P!16= US9U_VLEESA¤_U0CLE¡djV6H[WA¥ERU_[WV4ORNWU_C fl] VLC M6QSU_L[PYaHf4H@;NPQSNWG&KL[PHO0XQSNLH[WHM6QSNWCLGINWMFHf4Va\ADNWagVLa r_U0NPQ[PH 0K6XU0QRYaH40/40//_I?D[BHYE,ONWaM<U_QROSA_CjOf4HERA¤0U_NWQ¡E,NbGIHO,OSH³M6QSU_M6QSNWXIORXHpE>O9GU_CLERHQS_XYH9MLA_Q[] ADMLM6QRU¤4NWa\AcORNWU_CcADQSNWADORNWU_C6C6HY[P[WHBNPCjORHYQRCLH_
Ecol
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[ # $ !66# "! # +N = 1
¯ by¤qF~?q<tYornYoqFs %] Q]K^ ~!ycors!¢_or~?|w'k3 m4uoB3k$3 w!ortp¢Dyc|jn " xh pug || b pj gNj f ≥ 0|kvjlp gNj9|qdk eb g e m q4hp ]0, 1[ w %" b k-p o q p b Dg eKcBb h uh
m j R q-||>k b - c t9q e~cBb h i8qdk c q e~cBb h_hjfjz w us|qdk7q9t e bIm j m jup t.jh e p ' h c p P1 i k c ' j uh ≥ 0
m qdhp [0, 1] wYh+3´q<s'tYnycm4nporq<s#](U_NPO
uh[eAE,U0[PV6ORNWU_Cf4H ¯ P0p4?C0HQRORVf4V HYaaH ¯ /6P0jU0CA
uh(x) =n∑
j=1
uh(xj)φj(x),
U&[PHpE«U0CLGOSNPU0CLEφj
E,U0C0O[WHYE«U_CLGIORNWU_CLE;f4HTLA_ERH#f4HpEX[WXaHYC0O&ELC6NeEP1
f6ADCFEV0h
HOUh =
(uh(xj))1≤j≤nHpE>OERU_[WV4ORNWU_Cf4VE,`4E,ORYaHb[WNPCLXYADNWQSH
KhUh = bh. ¯ 0/'G_
HYE«U_CLGIORNWU_CLEφj
ERU_CjO2f6HYE-«U0CLGOSNPU0CLE'¨RG&KLA_MFHpADV4j©br_U0NPQ[eA !NW_V6QSH ¯ Wp§djV6NªERU_CjOMFUjE,NPORNW_HpEHNP[lERV WOf4U0CLG9f6H;aU_CjOSQRHYQ2djV6H;ORU_V6ORHYE[WHYEGIU0aM<U0ESADCjOSHYE?f6V¡_HpGOSHV6QUh = (U j
h)1≤j≤nE,U0C0OMFUjE,NPORNW_HpETMFU0V6QgMLQRU0V6_HYQgdjV6H[eA«U_CLGIORNWU_C
uhHpE>OgM<U0ERNPORNW_H¡ERV6Q
[0, 1]XADMLMFHY[PU0CLEgdjV6H0vHCMFUjERA_C0O
U0h = Un+1
h = 04[WH#E,`4E,ORYaHb[WNPCLXYADNWQSH ¯ 0/'G_?HpE>OXpd0VLNPcAD[WHCjO9Z
−U j−1h + 2U j
h − Uj+1h = hbjh
M<U_V6QOSU_V4O1 ≤ j ≤ n. ¯ /
(U_NPOU j0
h = minj Ujh
[WAM6[WVLEBMFHORNPORH\GIU0aMFUjERA_C0OSHJf4HUh
HlEY] NW[-` AM6[WVLERNPHYV6Q&E;ML[PVLEBM<HIOSNOSHYEGIU_aM<U0ESADCjORHpEªU0CG&KLU_NeE,NPO#GIH[W[WHf4H\M6[WVLETM<HIORNPOgNWCLf4NeGIHj0(N
j0 = 0vA_[PU0QSE
U jh ≥ U0
h = 0MFU0V6QORU_V6Oj6GHBdjV6NlHpE>O[PHbQSXYERV6[POSAcOQSHYG&K6HYQSG&K6X0+(4N
j0 ≥ 14AD[WU_Q&E
U j0h < U0
h = 0HIOGIU0aaH
Un+1h = 0
U0CHCf4XYf4VLNO;djV6Hj0 ≤ n
F®2U_aaHbjh =
∫ 1
0fψj dx ≥ 0
MLADQK`MFU_ORK6pE,HTERV6QfU_CMFHYV4O;AD[WU_Q&Ef4XYf6V6NPQSHTf4HT[eAJQSH[eAcOSNPU0C ¯ 0/ 2M<U_V6Q
j0djV6H
(
U j0h − U
j0−1h
)
+(
U j0h − U
j0+1h
)
≥ 0,
GIH;djV6NFHYE,O2V6CLH;GIU0CjORQ&A_f4NeGOSNPU0CA¤0HYG[WH9GYADQ&A_GOSQSHaNPC6NWa\AD[§rE,ORQSNWGIO&?f4HU j0
h
j°-ADQ2GU_CLERXYdjV6HYCjO[WAaXIOSK6U4f4H#f4HYEX[WXaHCjOSEFC6NWEP1
_XQSNPLHb[PHbM6QSNPCFGINWMFH#f4Va\Ac4NWagV6a f4NWESGIQSHIOp9U_VLE!A¤_U0CLEvf6XaU_CjORQSX?f6A_CLE'[PHpE'ERU_VLE,E,HpGORNWU_CFElM6QSXYGIXpf4HCjOSHYEvf6HYE'QRXpE,V6[POSADOSEvORK6XYU_QSNWdjV6HpEvf4HGIU_C0HQS_HCFGIH_I9U_VLEvMFU0V6_U0CLEz<hjy¤oBZ!|jy9slk3´hjycorilk!|+3´|sªn2r|tzornY|tYtp|tw'|#¢0qFsªz<|jy8A<|s'¢0|M6QRXpf4NPORHYEHYCQSXYERU_[WcADCjO2[WH;MLQRU06[PYaHBf4HT@;NPQSNeG&K6[PHOb ¯ GD?MLADQ[eAgaXORK6U4f4HTf4HpE2XY[PXYaHCjOSELCLNWEA¤_HYGBf4HpEa\ADNW[P[eAD0HYEf4HBOSA_NP[W[PHpEf4NeE>OSNPCFGORHpEF®2U0CLERNWf4XYQRU0CLE[] HI4HYaML[PH#ERV6NW¤A_CjO
− ((1 + x)u′)′ + (1 + cos(πx)) u = fM<U_V6Q
0 < x < 1u(0) = u(1) = 0
¯ 0/*jA¤_HYG
f(x) = −π cos(πx) + sin(πx)(1 + cos(πx) + π2(1 + x))vf4U0CjO;[eAERU_[WV4ORNWU_CHI6A_GIORHHYE,O
u(x) = sin(πx)2 ] NWf4XpAD[ERHQ&ADNPOf4HGA_[WGV6[WHQ[] HQSQRHYV6QHI6A_GIORH ‖u − uh‖H1(0,1)
!a\A_NWEGIH[eACLXYGIHpERERNPORHf4Hg«rA_NPQSHJf6HYEBGAD[eGIVL[WEBM6QSXYGNWEBfl] NPCjORXY_Q&AD[WHYEYFGHdjV6N§C] HYE,OBMFA_EBGIU_aaU4f4HE,N![WAE,U0[PV4OSNPU0CuHYE,O?GIU0aML[PNedjV6XH0_C\M6Q&AcOSNWdjV6H#rHIO2G_] HpE>O?GHdjV6HC6U0VLE§«rADNeERU_CLE§NeGIN¬§U_C\ERHGIU0C0OSHCjORHf4HGA_[WGV6[PHYQ-[] HQSQSHV6Q§M6QSU>A>HORXHf6A_CLE
Vh0GD] HYE,O,Zcf4NPQSHd0Vv] U0CGYAD[eGIV6[WH ‖rh(u−uh)‖H1(0,1)
U0C
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odui
re.
4G E "+ n " I # $ !6
f6NO9A_VLERERNdjV] NP[vEY] AD_NPOf4Hb[WAs!qLy43´|w!ortp¢Dy4:jnY|¡f6ADCLEVhL ?] NWCjORXYQR¦Of6H#GIHIORORH#A_M6M6QSUG&KLH;HpE>OdjV6HT[^] U_CM<HV4O;GYAD[eGIV6[WHQ9HI6A_GIORHaHYC0O[WHYENPCjORXY_Q&AD[WHYEM6VLNWESdjV6H
rh(u− uh) = rhu− uh ∈ VhGIHY[WAQSHNWHCjO;ZC6H#MFA_EOSHC6NWQ;GU_aM4ORHJf4HYEHYQRQSHVLQSEfl] NWC0OSHQSMFU0[WADORNWU_CHCjOSQRHH1(0, 1)
HOVhI
;COSQSA0GIHgGHIORORHgHYQRQSHVLQ9f4NeERGQRORH ‖rh(u− uh)‖H1(0,1)
HC«U0CLGOSNPU0C³f4V MLA0E9f4V³aA_NP[W[eAD_Hh vU_Q&ERdjV6Hg[WAERU_[WV4OSNPU0C HYE,O;QSX0V6[PNWQSH_F[PH 0K6XU0QRYaH ¯ 0/4 ¯ M6QSX0U_NPO9V6C6HGIU_C0HQS_HCFGIHT[WNPC6XpADNWQRHrHC
h-f4H;[] HQSQSHV6Q-MLA_Q?[eATaXIOSK6U4f4H;f4HpEXY[PXYaHYCjOSE-FC6NWE
P10OSA_CLf4NeE?djV6H[WH 0K6XYU_QSaH ¯ /6PFMLQRXY_U_NPO2VLC6HTGIU_C0HQS_HCFGIH9djVLA0f4QSADORNedjV6HrHC
h2 f4H;[^] HQSQRHYV6QMLADQ[WAaXIOSK6U4f4HTf4HYEXY[PXYaHYCjOSEFC6NWEP24@;A_CLE[WHBGYA_Ef4HB[] HI4HaM6[WH\ ¯ / 0?U_COSQSA0GIHbGIHO,ORHBHQSQSHV6QMFU0V6Qf4NC<XYQRHYCjORHYEcAD[WHVLQSEf6H
hERV6Q[eA 'NW_VLQRH ¯ FHYCXYG&K6HY[P[WH[PU00A_QRNPORK6aNedjV6HpI_ HpEGIQSU_NPgU_V[WHYE-QSU_CFf6EGIU0QRQSHYERM<U_CLf4HYCjOZf6HYETQSXYERV6[POSAcO&E#f4HGYAD[eGIV6[^[WHYET[WNP0C6HYEgE,U0C0Ogf6HYE#f4QSU_NPORHpETf4HQSXI«XYQRHYCLGIH\GIU0QRQSHYERM<U_CLf6A_CjObA_V4«U0CLGOSNPU0CLE
h2 HOh3 QRHpE,M<HYGIORNW_HYaHYCjOYXHYa\ADQ&d0VLU_CLE#djV6H¡[] V4ORNW[WNWESAcOSNPU0C´fl] VLC6H¡XpG&K6H[W[PH¡[PU00AcQSNPORK6aNed0VLHMFHYQRaHOgf4H6NWHCNeE,VLA_[PNeERHQT[WHYETNPORHYESERHYE#f4HGIU_C0HQS_HCFGIHGIU_aaH\[eAMFHYCjORH¡f4V[WU_jADQSNOSK6aH2f4H[^] HYQRQSHVLQ'HYC#«U_CFGORNWU_Cf4V[PU00ADQSNPORK6aH2f6H
h ;CgU0LE,HYQR0H2f4U0CLG2V6CJMLK6XC6U0aC6Hf6H2tpkv~?|jy =S¢0qFsªz<|jy8A<|s'¢0|<pG_] HpE>OR ZD f4NWQSH?djV6H?[WHYE'X[WXaHCjO&ElLC6NeE§GIU0Cj0HQS_HYC0OlM6[WVLE'QSA_M6Nef4HaHCjOdjV6HbGIHTdjV6NªHYE,OM6QSXV¡MLA_Q2[eAgORK6XYU_QSNPH H4[] HQSQSHV6QHYE,OHYC
h2 MFU0V6Q[WAaXIOSK6U4f4H P1HOh3 MFU0V6QGH[W[PH
P2'®2HjADNWC³HpE>OTf4Z¡[^] VLC6N«U0QRaNPORX\f4Va\ADNW[P[eAD0HJHObA_VG&K6U0N³f6HJ[eAC6U_QSaHJf4NeERGQRORHfLADCLE
Vh
−310
−210
−110
010
−910
−810
−710
−610
−510
−410
−310
−210
−110
010
+
+
+
+
+
++
++
Ο
Ο
Ο
Ο
Ο
Ο
Ο
ΟΟ
¯ FJ®A_E§fl] V6C6HE,U0[PV4OSNPU0CgQSX0V6[PNWQSH,HcH4HaM6[WHB ¯ 0/*0IcU_QSaH2f4NeESGIQSIORHH1 f4H[] HQSQSHV6QHYC «U0CLGOSNPU0Cf4VMLA_E
hf4VaA_NP[W[eAD_H[WHYEbGIQSU_NP GIU0QRQSHYERMFU0CLf4HYC0OBADV4 XY[PXYaHYCjOSE;LC6NeE
P1ª[WHYEQSU_CFf6EADV4X[WXaHCjOSEFC6NWE
P24[WHYEf6QRU0NOSHYEERU_CjO[PHpEORQ&A_GIXpEf4H
h→ h2 HIO h→ h3 (Nª[eAgERU_[WV4ORNWU_C\Cv] HpE>OMLA_E2QRXY_V6[WNWQSH__NW[<`A#ORU0V'A>U_V6Q&E?GU_C_HYQR0HCLGHa\ADNeE?A¤_HpGV6C6H;NOSHYESE,HML[PVLE?«rADNW6[WH;djV6HBGIHbd0VLNFHpE>OM6QRXpf4NPOf6A_CLE?[WH;GYA_E?QRXY_V6[WNPHYQ?MLA_Q2[WHYE 0K6XU0QRYaHYE ¯ /6 ¯ HIO ¯ 0/4W#FL°§U0V6QU_4OSHC6NWQV6C6HTERU_[WV4ORNWU_CC6U_CQRXY_V6[WNWQSH_U_CM6QRHYCLfV6CERHYGU_CLf¡aHagLQRHTf6A_CLE
H−1(0, 1)djV6N<C] ADM6MFADQRORNWHCjO2MLA0E?ZL2(0, 1)
jC¡VLC6H;f4NWaHCLERNWU_C¡fl] HYERMLA0GIH9U0C\MFHYV4OADNWCLE,N<M6QSHCLf6QRH;V6C6H
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[ # $ !66# "! # +N = 1
8GaA0ERERHTf4HT@9NWQ&A_GDF®2U0CLERNWf4XYQRU0CLEf4U_CFGb[] HI4HaM6[WH
−u′′ = 6x− 2 + δ1/2M<U_V6Q
0 < x < 1u(0) = u(1) = 0
¯ µ E A¤_HYG
δ1/2
[eA³a\A_ESERHf4H@9NWQ&A_G¡A_V´M<U_NWC0Ox = 1/2
'f6U_CjO[eA³ERU_[WV4OSNPU0C´HI6A_GIORHHYE,Ou(x) =
1/2− |x− 1/2|+ x2(1− x) ;CORQ&A_GIH9[] HQSQRHYV6Q ‖rh(u−uh)‖H1(0,1)
HYC«U0CLGOSNPU0Cf4HhE,VLQ2[eA !NW_V6QSH ¯ \rHCXpG&K6H[W[PH[PU00ADQSNPORK6aNed0VLHpY HYE!GIQSU_NP;U0VT[PHpEQSU_CLfLEvGIU0QRQSHYERM<U_CLf4HYCjOvZf4HYEvQRXpE,VL[O&AcOSE'f4H2GYAD[eGIV6[<
P1U_V
P2QRHpE,M<HYGIORNW_HaHYC0OF[PHpE[WNP0C6HYE;E,U0C0OBf4HYE;f4QRU0NOSHYEf6HgQSXI«XQSHCFGIH#GIU0QRQSHYERM<U_CLf6A_CjOA_V4«U_CLGIORNWU_CLE√hHIOhQSHYERMFHpGOSNP0HaHCjOY ;C\_U0NO2djV6H;[PHpEXY[PXYaHYCjOSE-FC6NWE
P1HIO
P2GIU0Cj0HQS_HYC0OZT[eA#a¦YaHjNPORHpERERHBM6QRU0MFU0Q,OSNPU0C6C6HY[P[WHBZ √
h6GHTd0VLNlHYE,OLNPHYCNWC4«XQSNWHV6QZ[WAJNOSHYESE,HTf4H
hrU_V¡a¦YaH
h2 M6QRXpf4NPORHf6ADCFE§[PHGA0E!QRXY_V6[WNPHYQY¤CMLA_Q,OSNWGV6[WNPHYQY¤NW[6C] `gAbADVLGV6CJNWCjORXYQR¦O§ZBV4OSNP[WNWERHQ-[WHYE§XY[PXYaHCjOSELC6NeEP24ML[PV4OKBDO9djV6H
P1Lf6A_CLEV6CORHY[GA_EY
−310
−210
−110
010
−310
−210
−110
010
++
+
+
+
++
++
ΟΟ
Ο
Ο
Ο
ΟΟ
ΟΟ
¯ ³®A0E;f] V6CLHERU_[WV4ORNWU_C³C6U_C QSX0V6[WNPYQRH HFHHYaM6[PH ¯ µ E I<9U_QSaHf4NeERGQRORHH1 f4H[] HQSQRHYV6QHC «U_CLGIORNWU_Cf4V MLA0E
hf6VaA_NP[W[eAD_Hr[PHpE9GQRU0NGU_QSQRHpE,M<U_CFf4HCjO9ADV6X[WXaHYC0O&ELCLNWE
P16[WHYEQSU_CFf6EADV4X[WXaHCjOSEFC6NWE
P26[WHYEf6QRU0NOSHYEERU_CjO[PHpEORQ&A_GIXpEf4H
h→√hHOh→ h
I°§U0V6Q#GA_[WGV6[PHYQT[] HQSQSHV6QTf6A_CLEb[PHpE !NP0V6QRHpE ¯ FHIO ¯ ¡C6U_VLETA¤0U_CLEBV4OSNP[WNeE,X\V6CLHERU_[WV4OSNPU0CHI6A_GIORH_®2HM<HCFf6ADCjOYªERN!GH[W[PH GN'C] HYE,ObMLA_EBGIU0C6CV6H_<U_C M<HV6OB[eAQSHaM6[eA_GHQ9MFADQ9[eA¡ERU_[WV4OSNPU0CADM6M6QSU4G&K6XHU04ORHYCjVLHA¤0HYG[WHaA_NP[W[eAD_H[WHM6[WVLE-LCE,V6MLMFUjE,XYH¦IORQSH[eA;ML[PVLE2GIU0Cj0HQS_XYHp_®2HO,ORHM6QRU4GXYf4V6QSHf4H¢0qFsªz<|0y4A<|s'¢0|slk3´hjycorilk!|TMFHYV4OA_VLESE,N6¦ORQSHaNWERHHCUHVLjQSHM<U_V6Qfl] A_V4ORQSHYEaXORK6U4f4HpECV6aXQSNWdjV6HpE`GIU_aM6QSNeETHIO9ERV6QRORU0V4O&2[WU_Q&ERdjV] U_CC6H#f4NeE,M<U0ERHTfl] A_VLGIVLCOSK6XU0QRYaHf4HBGU_C_HYQR0HCLGH_j®;] HYE,OE,U0V6_HYC0O[PHbERHV6[ªaUc`_HYCJ¨>KLHV6QSNWE,ORNedjV6H©;f6H;0XQSNLHYQ2ERNªV6CA_[P0U_QSNOSK6aHGIU_C0HQS_H9HO9Zd0VLH[W[PHTNPORHYESERHBMFADQQSA_M6M<U_QROA_VQ&A4W\C6HYaHCjOf4VaA_NP[W[eAD_H0
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
4Gd/ "+ n " I # $ !6
*KJ* O M Uv/21S B/;=RS´7 <Uv3NM =1 U±9M6QRpE;A¤0U_NWQf4XIFC6N!f6HYE;X[WXaHCjOSELC6NeE
P1HIO
P2<[PHJ[PHpGOSHV6Q;NPa\AD0NPCLHT«rA_GNP[WHaHCjObGIU_aaHYC0O0XC6XYQSA_[PNeERHQHIO;f6XILC6NWQ9f4HpEX[WXaHCjO&ELC6NeE
PkA¤_HpG
k ∈ N∗ ª®2HpEX[WXaHCjO&ELC6NeE<f4NPOSEf6H¡ vA__Q&ADC60H_lV4OSNP[WNWERHCjOf4HYE#«U_CLGIORNWU_CLE#f6H¡LA0E,HdjV6NE,U0CjOgE,HYV6[PHYaHCjOJGIU0CjORNWCjVLHYE#a\A_NWETMFA_EGU_CjORNWC6aHCjOf4XQSNW¤A_6[WHYEY 0'U_V6ORHI«U0NWEYcNW[4HYE,O-G[WA_NPQ-djV6Hf4HpE!MFU0[P`C"B0aHpE-f4HP3
MFHYV6_HYCjO-ERHQSA0GGU_Q&f4HQf4HBaA_C6NWQSH;GIU0CjORNWCjLaHYCjOf4XYQRNWcAD6[WHYEYj@BADCFEGH;GYA_E2[PHpE?cAD[WHVLQSEf4HYEf4XYQRNW_XYHYE2E,HYQRU0C0OA_VLESE,N!V4ORNW[WNWERXHpE;M<U_VLQbGYADQ&A_GIORXQSNeE,HYQ[PHpE;«U_CLGIORNWU_CLE0U_NWQ;[eA6XHa\ADQ&djV6H ¯ /6 /0 ;C³NWC0OSQRU4f4VLNOf6U_CLGV6CLHaXORK6U4f4Hf6HYEhrh+3´|sªnptZ§s'otBw!| |jy43´onp|P3
djV6N6QSHM<U0ERHE,V6Q-[^] HYERMLA_GHf6NWESGIQSHIOVh =
v ∈ C1([0, 1])ORHY[vdjV6H
v∣
∣
[xj ,xj+1] ∈ P3M<U_V6QOSU_V4O
0 ≤ j ≤ n
. ¯ µ6p
D[«rADV4O9LNPHYC«rADNWQRHTAcORORHYC0OSNPU0C djV6H#f6ADCFE[WA\f6XILC6NPORNWU_C ¯ µ6pf6HVh
U_Cf4Ha\A_CLf4H#ADV6¡«U_CFGOSNPU0CLEBfl] A_M6MLA_Q,OSHC6NWQ;ZC1([0, 1])
<HIO;CLU_C M6[WVLEBE,HYV6[WHaHCjOBZC([0, 1])
l®;] HYE,OBGHJdjV6N'«rADNPO;[eAf6NDCªXQSHCLGHbHCjORQSHb[PHpEX[WXaHCjO&ELC6NeEf4H HYQRaNPORHTHIO9f6H# vAD0QSA_C6_H00QSHYERM<HYGOSNP0HaHCjOp
0
1
j
ψj
φ
xjx xj-1 j+1
¯ ¯ ¡ vHYE«U_CFGORNWU_CFEf4HTLA_ERHTf4HYEXY[PXYaHCjOSEFC6NWEf] 9HQSaNPORHP3
;CMFHYV4OQRHYM6QSXYERHCjORHYQ2[WHYE2«U_CFGORNWU_CFEf4HVh
ZJ[] ADNef4Hbf4HB«U_CLGIORNWU_CLEf4HbLA0E,H;ORQSYEERNPaM6[WHYEY;Cf4XLC6NPO9f4HV4«U_CLGIORNWU_CLE¨,aQSHYE>©
φ(x) =
(1 + x)2(1− 2x)E,N − 1 ≤ x ≤ 0,
(1− x)2(1 + 2x)E,N
0 ≤ x ≤ 1,0
ERN |x| > 1,HOψ(x) =
x(1 + x)2E,N − 1 ≤ x ≤ 0,
x(1− x)2 E,N0 ≤ x ≤ 1,
0E,N |x| > 1.(4Nv[WH#a\ADNW[P[eAD0HBHpE>O;V6C6NP«U_QSaH0LM<U_V6Q
0 ≤ j ≤ n+ 1U_Cf4XLC6NPO;[WHYE«U0CLGOSNPU0CLE9f4HgLA_ERH0U_NWQ[eA !NP0V6QRH ¯ ¯
φj(x) = φ
(
x− xj
h
) M<U_VLQ0 ≤ j ≤ n+ 1, ψj(x) = ψ
(
x− xj
h
) M<U_V6Q0 ≤ j ≤ n+ 1.
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[ # $ !66# "! # +N = 1
8Gcµ;?|+3 3´| %] Q]K^ jlp~|q'@j Vh
o m' ' h c |Nq4k z w u o jlp e g hp b g+p jup~|Nq-j m j H1(0, 1)m jm4c t9jlhNp cBb h 2(n + 2) w b g e jr b h eKcBb h vh
m j Vh jup e"m' ' h c j m j9t9q4h c + kvjg h c gNjn|Nq4k tp|jtz6m6r|k'y¤t|jn¢_|r|t¡w!|tpm´w!hjy¤orzFh| qdg -6p b tt9j e p (xj)0≤j≤n+1o j eb h q | b g k eb g e x ∈
[0, 1]
vh(x) =
n+1∑
j=0
vh(xj)φj(x) +
n+1∑
j=0
(vh)′(xj)ψj(x). ¯ µ'/_
Yh+3´q<s'tYnycm4nporq<s#] vHYEg«U_CLGIORNWU_CLEf4HVh
XOSADCjOf4HG[WA0ERERHC1 'NW[E] AD0NOJ6NPHYCxfl] V6C¥E,U0VLE>HYERMLA_GHgf4H
H1(0, 1) ;C0XQSNFHT«rA_GNP[WHaHCjObd0VLHg[WHYE
(φj , ψj)«U_QSaHCjO9V6C6HJLA_ERHgf4H
VhHYCQRHYaA_QSdjVLA_CjOTdjV6H
φj(xi) = δijψj(xi) = 0
φ′j(xi) = 0
ψ′
j(xi) = δijr_U_NWQ#[eA !NP0V6QSH¯ ¯
;C³MFHYV4OBLNPHYC³E,LQ;V4OSNP[WNeE,HYQ;[^] HpE,MFA_GIHVh
U0V³f4V aU0NPCLEBERU_C ERU_VLE,^HpE,MFA_GIH#f6HYE9«U_CLGIORNWU_CLEd0VLN!EY] A_C6CV6[WHCjO;HYC E HIOpMFU0V6Q9QSXYERU_VFf4QRHg[PHJM6QRU06[WaHgf4H@9NWQRNeG&K6[WHIO ¯ GDIFa\A_NWEBGIHJC] HpE>OMLA_E[^] V6ORNW[PNeERADORNWU_C [eAML[PVLE¡GIU_VLQSA_C0OSHf6HVhC M6Q&AcOSNWdjV6HU_C V4OSNP[WNeE,H
VhM<U_V6Q\ychtYq<k!wvyc| Philk!m4nYoqFs w'|t~§rm6ilk!|t´0U_NWQ³60G E JHIO[WH®2KLA_M6NPORQSH¤2U_V¥M6[PV6OCBDOf4HYE\M<U_V6ORQSHYE\HYCf4NPaHYCLE,NWU_C
N = 1
u′′′′ = ff6ADCFE
]0, 1[u(0) = u(1) = u′(0) = u′(1) = 0,
¯ µ_µjrdjV6N-A_f4aHOBV6CLHJV6CLNWdjV6HE,U0[PV6ORNWU_C
u ∈ H20 (0, 1)
E,Nf ∈ L2(0, 1)
IFCHlCªHIOpVh
Cv] HpE>OBMLA_EE,HYV6[PHYaHCjOBVLCERU_VLE,^HpE,MFA_GIH#f6HH1(0, 1)
<a\A_NWE;HYE,ObA_VLESE,N!V6CERU_VFE> HYERMLA_GHgf4HH2(0, 1)
GIHd0VLN-C] HpE>OTMLA0EB[PHGA0E;M<U_VLQB[WHYEbXY[PXYaHCjOSE;LC6NeETf4H\ vA__Q&ADC60HpIL°§U_V6QbQSXYERU_VLf6QRH ¯ µ_µjCLU_VLEADV6QSU_CLE<HYERU_NWCf6VERU_VFE> HYERMLA_GHV0h = v ∈ Vh
ORH['d0VLHv(0) = v(1) = v′(0) = v′(1) = 0 . ¯ µ)Fj
;?|+3 3´| %] Q]K^ jlp~|q'@j Vho j e p b hp b g+p jlpK|Nq-j V0h
m ' h c |Nq4kEz w *4 o p b h e#m jlpp b g+p jup~|Nq-jup m j H2(0, 1) o j e9m j H2
0 (0, 1) kvjlp~|j- eKc i4jt.jh e o m j m4c t9jhp cBb hNp 2(n + 2) o j e 2nkvjup~|Nj- e~c i8jt9jlh e w b g e j%r b h> e~cBb h vh
m j V0h jlp em ' h c j m jt9q4h c + kvjg h c gNj$|Nq4kpjupi4q4jg k-pj e @jfjlp m j7pq m k c i qdg -)p b tt9j e p (xj)1≤j≤n
o j eb hqs| b g k e b g e x ∈ [0, 1]
vh(x) =n∑
j=1
vh(xj)φj(x) +n∑
j=1
(vh)′(xj)ψj(x).
Yh+3´q<s'tYnycm4nporq<s#](U0NOvh ∈ Vh
HFH[W[PHJHYE,OBf4HJGI[eA_ESE,HC1 E,VLQ [0, 1]
HIOC2 MFADQ9aU0QSGHYADV6ª@9U0CLG\ERAf4XYQRNW_XYH
v′hXOSA_C0O#GU_CjORNWCV6HHIO
C1 MLA_QTaU_Q&GIHYA_V4llADM6MLA_Q,OSNPHYCjObLNPHYCZH1(0, 1)HC_HYQ,OSVf4V HYaaH FF µLP 0j°-ADQGIU0CLERXYdjV6HCjOp
vhHpE>OV6CX[WXaHCjOf4H
H2(0, 1)4 vHBQSHYE,ORHf4V[PHYaaHTHpE>OERHagL[WA_6[PHTA_V vHaaH ¯ 0/4W ¯
@;XYGQRNW_U0CLE§6QSNPY_HYaHYCjO![eABQRXpE,U0[PV4OSNPU0CgM6Q&AcOSNWdjV6Hf4H[^] XpdjVLAcOSNPU0Cf4HpE§M6[eA_djV6HYE ¯ µ_µjvMFADQ§[WAaXORK6U4f4H#f6HYEXY[PXYaHYCjOSELCLNWEf] 9HQSaNPORHP3L 'A«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WHbf6HT[] ADM6MLQRU¤4NPaADORNWU_CNPCjOSHQSC6HbHYE,O
OSQRU0V6_HYQuh ∈ V0h
OSH[vdjV6H ∫ 1
0
u′′h(x)v′′h(x) dx =
∫ 1
0
f(x)vh(x) dx ∀ vh ∈ V0h. ¯ µ00
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
4G F "+ n " I # $ !6
;CJf4XYGU_aMFUjE,Huh
ERV6Q![WA9LA_ERHf6HYE(φj , ψj)1≤j≤n
HO!U0C#C6U_ORHUh = (uh(xj), u
′h(xj))1≤j≤n[WH¡0HYGIORHVLQJf4HE,HpEgGUjU0QSf6U_C6C6XYHYE#fLADCLEGIHO,ORHLA0E,H0' vA «U_QSagV6[eAcOSNPU0CcADQSNWADORNWU_C6C6HY[P[WH³ ¯ µ0_QSHNWHCjOZQRXpE,U0VLf4QSHBf6A_CLE
R2n V6CER`4E>OSaHb[PNWC6XYA_NPQSHKhUh = bh.
;u|jyc¢_or¢0| ]%] 27 >7 )% @<7 & )47 2 & 2(% !:9 % 9 ; Kh <>"%?+ / - 1!/
¡, " g%!( ($I( $ ('$#N ≥ 2
9U_VLETCLU_VLETML[WAGIU_CFETa\ADNWC0OSHCLA_CjObHYC´f4NWaHYCLERNPU0Cf] HpE,MLA0GIHN ≥ 2
HYCM6QSADORNedjV6HN =
2, 3I_°§U_VLQERNPaM6[WNPLHQ[] HI4M<U0ERX_jGIHQROSA_NPCFEQSXYERV6[O&AcO&E?C6HBE,HYQRU0CjO2f4XaU0C0OSQRXpE?djV] HC¡f6NPaHCFE,NWU_C
N = 2La\ADNeE9NW[eE;EY] XORHYCLf4HCjObZ\[WAf4NWaHYCLERNPU0C
N = 3rA_V M6QSNl<MLADQR«U_NeE<f4HJGU_aM6[WNWGYAcOSNPU0CLEOSHYG&K6CLNWdjV6HpEHIOM6Q&AcOSNWdjV6HpENPaM<U_QROSADCjOSHYE&9U_VLEGU_CLERNef4XQSU_CLE[WHbM6QRU06[WaHbaU4f4Y[PH#f6Hb@;NWQRNeG&K6[WHIO
−∆u = ff6ADCLE
Ωu = 0
E,V6Q∂Ω,
¯ µ ¯ f6U_CjO9C6U0VLEESA¤_U_CFEd0Vv] NW['A0f4aHIO9V6CLHgE,U0[PV6ORNWU_CV6C6NedjV6Hgf6A_CLE
H10 (Ω)
<E,Nf ∈ L2(Ω)
r_U0NPQ[WH®2KFADM6NPORQSH#_@BADCLE?ORU0V4OGIH;djV6NªERV6NOC6U0VLE?ERV6M6M<U0ERHQSU_CLE2djV6H9[WH;f4U0a\ADNWC6HΩHpE>O~?qFrFhw'y¤oilk'|rMFU0[P`j0U_CLA_[_ERN
N = 2IYG_] HpE>OR ZD f4NWQSHd0VLH
ΩHYE,OvV6C6H2QRXYV6C6NWU_CBLC6NWH2f4H?MFU0[P`0Yf4QSHYEf6H
RN IXA_M6MFHY[PU0CLEdjV] V6CM<U_[W`_Yf6QRHBHYE,OV6C6HTNWC0OSHQ&E,HpGOSNPU0C¡LCLNPHTf4H#f4HYaN HYERMLA_GHYEf4HRN HO9d0VLHB[WHYEMFADQRORNWHYEf4HERU_C FU0QSfxd0VLNA_M6MLADQRORNWHCLC6HCjO\Z V6C¥E,HYV6[Kj`M<HQSM6[eADC ERU_CjOA_M6M<H[WXHYEERHYEJ«rA_GHYEY! 'A Q&ADNeE,U0Cf6H#GIHIORORH#K`M<UDORKLYERHTHYE,O9d0Vv] NW[vC] HYE,OM<U0ESE,NW6[WH#f4H#a\ADNW[P[WHQH6A_GOSHaHCjO9djV6H#f4H#ORH[eEU0V6_HYQ,O&E9U_VLE;f4NWQRU0CLEM6[WVLE;[PU0NPC GHgdjV6N'ERH#MLA0ERERH#M<U_V6QBf4HYE;f4U_a\ADNWC6HpE_XYC6XQ&ADV4Z\<U_Q&f6E?¨RGIU_VLQR<HYE>©r_U0NPQ[eA9XHa\ADQ&djV6H ¯ µ6W I
*)*,+ O M Uv/21S B/;=RS143L=RP§/ ?8ORP-=>3US0'U_V4OGIU0aaHYCLGIH9MLADQ[WAgf4XILCLNOSNPU0Cfl] V6C¡a\A_NP[W[WA__Hf4Vf4U_a\A_NPC6H
ΩMLADQf4HYE?ORQSNeADC60[PHpEHCf6NPaHCFE,NWU_C
N = 2HO'f4HYElORXIOSQSA_Yf4QSHYEªHYC#f4NWaHYCLERNPU0C
N = 3 ;CTQSH0QRU0V6MFH§[WHYEªOSQRNeADCL_[WHYEªHIO'[PHpEOSXIOSQSA_Yf4QSHYE§f6A_CLE![WA;«rADaNW[P[WHM6[WVLE§_XYC6XQ&AD[WHf6HYE
NE,NWaM6[WHI4HYEY ;CA_M6MFHY[P[WH
NE,NWaM6[PH4H
Kf4H
RN [^] HYC_H[WU_MLMFHGIU_C0HI4H2f4H(N+1)
M<U_NWCjOSE(aj)1≤j≤N+1
f4HRN ¤A_M6M<H[WXYE§E,U0aaHOSE!f4H K
*NWHC\ER6QV6C2 ERNPaM6[WHI4HHYE,O?ERNPaM6[WHaHCjO?VLCJOSQRNeADCL_[WHHIOV6C
3ERNPaM6[WHI4HV6CORXORQ&ADpf4QRHTr_U_NWQ[eA !NP0V6QRH ¯ j ;Cf4NPO?d0VLH[PH
NERNPaM6[WHI4H
KHYE,OC6U_Cf4X0XC6XYQRXERN4[PHpE-M<U_NWC0O&E
(aj)1≤j≤N+1Cv] A_M6MLADQRORNWHCLC6HCjOMLA_EZV6Ca¦aHTK`jM<HQSM6[eADCf4HRN [WHBORQSNWA_C6_[WHbU_V¡[WHbORXORQ&ADpf4QRHBHYE,OC6U_C¨,M6[eAcO,©>I#(NU_C¥C6UDOSH
(ai,j)1≤i≤N[WHYE\GUjU0QSf6U_C6C6XYHYEf4V¥_HYGIORHYV6Q
aj-[WAGIU0CLf4NPORNWU_C f4HC6U_C
Ecol
e Po
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chni
que.
Ne
pas
repr
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re.
[0 # $ !66# "! # +N ≥ 2
8GDf4X0XC6XYQRHpERGHCLGHBf4H
KHpE>OdjV6HT[eAJa\ADORQSNWGH
A =
a1,1 a1,2 ... a1,N+1
a2,1 a2,2 ... a2,N+1 aN,1 aN,2 ... aN,N+1
1 1 ... 1
¯ µ[G_
E,U0NONWC_HQ&ERNP6[WHrGIH#djV6HT[^] U_CE,VLM6MFUjE,HYQSAJOSU_V'A>U0V6QSEMFADQ[eA\E,V6NPORH¤N9CNE,NWaM6[PH4HbAA_V4OSA_C0Of4Hb«rA_GHYEdjV6HTf4H#ERU_aaHIOSEY6djV6NvERU_CjOH[W[PHpE> a¦aHYEf4HpE
(N − 1)ERNPaM6[WHI4HYEY
¯ G ¡-4HYaML[PH#f4Hba\ADNW[W[WA__HBORQSNWA_C6_VL[WA_NPQSH;HYCf6NPaHCFE,NWU_CN = 2
Yh[Z§s'ornYorq<s %] \%]K^a`Qb4cfe
Ω g h b g i4jlk e b hNh>j -+j| b -m k c gNj m j RN wh 3´m6orrmNAF| e k c q4h g q c kvj b g9g h>j e k c qdh g q e~cBb h m j Ω jlp e g h jlhNpjlt9l j Thm j N p c t|>j -+jlp7zh b h m#[ h k p
(Ki)1≤i≤n g c i k c ' jh e
'w Ki ⊂ Ω j e Ω = ∪ni=1Ki
o,+w7R c h e jk-pj- e~cBb h Ki ∩ Kj
m j m jlg - N p c t|j-+jup mdc p eKc h> e p6jup e g h m p c t|>j -+j o qdi8j-0 ≤ m ≤ N − 1 o m'b h e"eb g+pjlp p b tt9j e p p b h e q4g+p-p cm jup p b t t.j e p m j Ki j e Kj wz 2h mdc t.jhp cBb h N = 2 o R c h e jk-plj- eKcBb h m j m jg - e k c q4h jlp7jup e p b4cfe i cBm j o p bdcfe k -m g cfe j$ g h p b t t9j e b tt g h o p b4cfe g hj7qdk e jE b tt g hj |sªnpoB:jyc| jlh m4c t9jlhNp cBb h N = 3 oR c h e jk-pj- e~cBb h m j m jlg - ee k@q + m kvjlp"jlp e p bdcfe i cBm j o p b4cfe g h6p b tt9j e b tt g h o p bdcfe g hjqdk e jE b tt g hj jlh eKc + kvj o p b4cfe g hjrlq'-j b tt g hjjlh eKc + kvj w
jlp tpq_3 3 |0nptb g s'q|k!w't m gt.q c fq j Th p b h e jlp7p b t t.j e p m jlp N p c t|>j-[jlp Ki g c
j b t| b pjh e w q4k7 b h_i8jh e~cBb h o j|Nq4k@q4t + e kvj h m p c h>j7 j7t9q - c t g t m jlp m4c q4t + e kvjlp m jlpN p c t|>j -+jlp Ki wD['HYE,O;GI[eADNWQ9djV6H#[eA\@;XILC6NPORNWU_C ¯ µLPTCLH#MFHYV4O;EY] A_M6M6[WNWdjV6HYQ9djV] ZV6CU_V60HQROMFU0[P`0XYf4QSNed0VLHHIO\MLA_E\ZV6C¥U_VL_HQROdjV6H[eGIU0CLdjV6H_- vA@;XILCLNOSNPU0C ¯ µLPGIU0CjORNWHCjOVLC GHQROSA_NPC¥C6U_aJ6QSHf4H
Ecol
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chni
que.
Ne
pas
repr
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re.
4G ¯ "+ n " I # $ !6
QSHYE,ORQSNeGORNWU_CFE#E,VLQ#[PH¡aA_NP[W[eAD_H H!f6ADCFEgGIHGYA_EgU_CMLADQS[WH¡E,U0V6_HYCjOTf6H63´m6ormAF|¢0qFs1 qLy43´|<9CHI4HaM6[WH#f4Hba\ADNW[P[eAD0HbGU_C4«U0QRaHBHYE,O9f4U_CLC6XTZ[WA !NP0V6QSH ¯ 0G4OSA_CLf4NeEd0VLHT[WA !NP0V6QSH ¯ MLQRXpE,HYC0OSHTf4HYEERNOSVLAcOSNPU0CLENWC0OSHQ&f4NPORHYEMFADQ[eA@;XIFC6NOSNPU0C ¯ µ6W_
K i
K j
K i
K j
¯ -4HYaML[PHpEf4H#ERNOSVLAcOSNPU0CLENPCjOSHQ&f4NOSHYEM<U_V6QVLCa\A_NP[W[WA__HBORQSNeADC60V6[WA_NPQSH_.|+3´m4ycilk'| ] \] U0VLE!C6Hf6NWERU_CLE!QSNPHYC#NeGIN6f4HpE'A_[P0U_QSNOSK6aHYE!d0VLNjM<HQSaHIORORHCjOf4HGIU0CLE>OSQRVLNQSHV6C³a\ADNW[W[WA__HJORQSNWA_C6_V6[eADNWQSH_ª®2U0CjORHCjOSU_CLE;C6U_VFEBf6Hf6NPQSHdjV6H0lE] NP[§HpE>OTQSH[eAcORNW_HYaHCjO9«rA0GINW[PHf6HJa\ADNW[W[PHYQbf6HYETf4U0aA_NPCLHYE;M6[WA_CLENW[-HI4NeE>OSHf6HJC6U0ag6QSHV6 [PU0_NeGINWH[eE9[WNPLQRHpEBdjV6N§M<HQSaHIO,OSHCjOf6H[WH«rADNWQRH¤lNW[HYE,OTHCLGU_QSHA0ERERH#JGIU0aM6[PNedjV6X\f4Ha\ADNW[P[WHQ#f4HpETf4U_a\A_NPC6HpE;ORQSNWf4NWaHCLERNWU_C6C6HY[WEY9U_VLEQSHC0Uc`_U_CFE2Z[^] U0V6QSA__HEF0/Dµ4Hv[PHT[WHYGIORHYV6QNPCjOSXQSHYESE,XbMLA_QGH#E,V>A>HIOY •
;u|jyc¢_or¢0| ] \]B^ @ Th & 2 787 2 ,9 Ω <>"% Ω " )% ? & <7 & >5! ':<>7 %"'279 R2 / > nt 7 & >% $9)$(% 2 "7 # 9 Th nc 7 & )%?9 2'! "#>;# 9#3(% 2 "7 #+6 !); ! && # 9)) (% 2 "7 #6 # & <; * C>7 1 ns 7 & >% 9)C && !9" & 2 787 2 &! n0s 7 , & )%9)C && # ;>% >%$9 & 2 787 2 )4+.* 65< 2 #% ∂Ω 1#/ ; & "(%)% 7 # % >7 2 %'9 ?9 7 >%" nt +ns = nc +1 ! 3nt +ns = 2nc +n0s /@BADCLExV6C
N ERNPaM6[WHI4H
KNP[HYE,O¥GU_aaU4f4H fl] V4OSNP[WNWERHQ¥f4HYE¥GIUU0QSf4U0C6C6XYHYELA_QR`4GHC4OSQRNedjV6HYEA_V[WNWHVf4HYEGUU_Q&f4U_C6CLXHYEGA_Q,OSXYERNPHYC6C6HpE2VFE,V6HY[P[WHYEYNXA_M6M<H[WU_CLEdjV6H06E,N
KHYE,OV6C
NERNWaML[PHHBC6U0Cf4XY_XCLXQSX;f4HTERU_aaHIOSE
(aj)1≤j≤N+1[PHpE¢0qqFy¤w!q<s's!h|jt !m6yp¢0|sªnycorilk!|t
(λj)1≤j≤N+1f6H
x ∈ RN ERU_CjOf4XLC6NWHYEMLA_QN+1∑
j=1
λj = 1,
N+1∑
j=1
ai,jλj = xiM<U_VLQ
1 ≤ i ≤ N, ¯ µ djV6N<A_f4aHO?6NWHC\V6C6H9V6C6Ned0VLH9E,U0[PV6ORNWU_C\GA_Q[WAba\ADORQSNWGH
A0f6XILC6NWHMLA_Q9 ¯ µ+GDIDHYE,O?NWCj0HQ&E,NW6[WH_
XHa\ADQ&djV6U_CLEdjV6HT[WHYEλj
ERU_CjOf4HpE«U_CLGIORNWU_CLEAdW\C6HYEf4Hx ;C_XQSNPLHBA_[PU0QSEdjV6H
K =
x ∈ RN ORHY[vdjV6Hλj(x) ≥ 0
M<U_VLQ1 ≤ j ≤ N + 1
,HOdjV6H;[PHpE(N +1)
«rA0GIHYEf4HKE,U0CjO?[PHpE?NWCjORHYQSERHYGIORNWU_CLEf6H
KHOf4HpEK`MFHYQRM6[eADCFE
λj(x) = 0
1 ≤ j ≤ N + 1 ;CMFHYV4OBA_[PU0QSE9f4XILCLNPQ;V6CHCLERHaJ6[WHgf4H#M<U_NWCjOSE9f6H
KdjV6N!_U_CjO$A>U0V6HQ;V6C
Ecol
e Po
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chni
que.
Ne
pas
repr
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[0 # $ !66# "! # +N ≥ 2
8GGΣ1 Σ2 Σ3
¯ 0vQSHNW[P[WNeETfl] U0QSf6QRH_/6lHIOgµM<U_V6QTVLCOSQRNeADCL_[WHrHCKLA_V4O&BHO#V6CORXIOSQSA_Yf4QSHHYCLA_E&6 vHYEQSU_CLf6EQSHMLQRXpE,HYC0OSHCjO[PHpEMFU0NPCjO&Ef4VORQSHNW[P[WNWEYQCB0[PHJMLADQRORNeGIV6[WNWHQBMFU0V6Qb[WAE,V6NPORHIHlM<U_V6Q;ORU0V4OBHYCjORNWHQ
k ≥ 1U_CA_M6M<H[W[PHnyc|orrortw WqLycw'yc|
k[] HCLERHaJ6[WHΣk =
x ∈ K OSH[vdjV6Hλj(x) ∈ 0,
1
k, ...,
k − 1
k, 1 M<U_V6Q 1 ≤ j ≤ N
. ¯ µ j
°!U0V6Qk = 1
NW['EY] A__NPO;f4Hg[^] HCLERHaJ6[PHJf4HYE;E,U0aaHOSE9f6HKFHIO9M<U_V6Q
k = 2f6HYEBE,U0aaHOSEHOf4HYEBM<U_NWC0O&E9aNW[PNWHV6³f4HYEbADQS¦IOSHYE;QRHY[PNeADCjObf4HV4³ERU_aaHIO&EJr_U0NPQB[WA!NP0V6QSH ¯ 0I<@BADCFE;[WHGA_E_XCLXQ&AD[^
ΣkHpE>OV6CHCFE,HYag6[WHBLC6Nvf4HTM<U_NWCjOSE
(σj)1≤j≤nk
9U_VLEgf4XILCLNWESE,U0CLEba\ADNWCjORHCFADCjOT[] HCLERHaJ6[WHPk
f6HYEbM<U_[W`jCB_aHYETZGUjHWGINWHCjOSEbQSXHY[WETf4HRN f6ADCFE
Rf6Hgf4H0QRXbNWC4«XQSNWHV6QU0VXY0A_[vZ
kLGD] HYE,O,Zcf4NWQRHbdjV6HbORU0V4O
p ∈ PkEY] XYGIQSNPO9E,U0VLE[WA«U_QSaH
p(x) =∑
i1,...,iN≥0
i1+...+iN≤k
αi1,...,iNxi11 · · ·xiN
N
A¤_HYGx = (x1, ..., xN ).
] NPCjORXYQR¦O§f6H[eA;C6UDOSNPU0CJf4H2ORQSHNW[P[WNeEΣk
fl] V6CN ERNPaM6[WHI4H
KHpE>O-djV] NP[4M<HQSaHIO-f4HGADQ&A_GIORXYQRNeE,HYQORU_VFE[PHpEMFU0[P`C"B0aHYEf4H
PkU0Cf6NO9djV6H
ΣkHYE,O9k's!ortpq<z6m6sªnbM<U_V6Q
PkI
;?|+3 3´| %] \%] \ `Qb4cfeK g h N p c t|j-+jIw b g k2g hjh e~c jlk k ≥ 1 o p b4cfe Σk j e kvj c f c p m b k m kvj
k o m ' h c |Nq4k z dw o mb h e jlp"| bdc h e pEp b h e h bde p (σj)1≤j≤nkw %" b k-p o e b g e | b hdt.j m j
Pk jup eme jkut c h "m j"t.qdh c + kvjg h c gNj|qdkpjupi4q4jg k-pq4g -| bdc h e p (σj)1≤j≤nk w&%"g e kvjt9jlh em4cfe o c $j- c p e j g h>j-q8pj (ψj)1≤j≤nk
m j Pke jfj gNj
ψj(σi) = δij 1 ≤ i, j ≤ nk.
Ecol
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4G "+ n " I # $ !6
Yh+3´q<s'tYnycm4nporq<s#]v vH#GA_QSf4NWCLA_[ªf6HΣk
HO[WAf4NWaHCLERNPU0Cf4HPk
GIU)L^CLGNWf4HYCjOcard(Σk) = dim(Pk) =
(N + k)!
N ! k!r[PH_XYQRNPLHYQ;HYC0V6NWERHJf] HHYQSGNWGH_ªADVaU0NPCFE;M<U_VLQk = 1, 2
®2U_aaH[] ADM6ML[PNeGADORNWU_CdjV6N^lZOSU_V4OM<U_[W`jCB_aH¡f4HPk'«rADNPOGU_QSQRHpE,M<U_CFf4QRHERHYEJcAD[WHV6Q&EgERV6Q[PH¡ORQSHNW[P[WNeE
ΣkHYE,OJ[WNPC6XpADNWQRH0'NW[ERV WO9f4HbaU_CjORQSHQdjV] H[W[WHBHpE>ONWC'A>HpGORNW_HTM<U_VLQaU0CjORQSHQdjV] HY[P[WHbHYE,O6N A>HYGOSNP0H__(4U_NPOf6U_CLGbV6CM<U_[W`C"B_aH
p ∈ PkdjV6NvEY] ADC6CV6[WHTE,V6Q
Σk U0C0OSQRU0CLEY4MLADQQSXYGV6QRQSHCFGIHTE,VLQ[WA\f4NWaHCLERNWU_C
NdjV6H
pHYE,ONWf4HYCjORNed0VLHaHCjO?CV6[LERV6Q
RN _°§U0V6Q N = 1_NP[FHYE,O?GI[eADNWQ?djV] VLCM<U_[W`C"B_aHf4H;f4H0QRX
kdjV6NEY] A_C6CV6[PHbHYC
(k + 1)MFU0NPCjOSE9f4NWE,ORNWCLGIOSEHYE,OCV6[(V6M6M<U0ERU_CLE[WHTQRXpE,VL[O&AcOjQ&ADNlZ[^] U0QSf4QSH
N−1Y®2U0aaH
xf4XM<HCLf#[PNWC6XYA_NPQSHaHYC0Ovf6HYEGUjU0QSf6U_C6C6XYHYEªLA_QR`4GHCjORQSNWdjV6HpE
(λj(x))1≤j≤N+1U0CMFHYV4Of6XILC6NWQ¡VLC MFU0[P`C"B0aH
q(λ) = p(x)f4H³f4HY_QSX ADV ML[PVLE
kHC [eAcADQSNWA_6[WH
λ ∈RN+1 _(Nl[^] U0C64HBVLC6H#GIUU_Q&f4U0C6C6XH λj
f6A_CLE[^] HYCLE,HYag6[WH 0, 1/k, ..., (k − 1)/k, 1 HIO9djV6H[^] U_CMFUjE,Hλ = (λ′, λj)
vU0CU04ORNWHCjO#V6CM<U_[W`jCB_aHqj(λ
′) = q(λ)djV6Nf4XYMFHYCLff4H
N − 1cA_QRNeAD6[WHYENWCLf4XYMFHYCLf6A_C0OSHYE GA_QU_CA[WAQSH[eAcOSNPU0C ∑N+1j=1 λj = 1
HOdjV6NHpE>OCV6[BE,V6Q\[eAERHYGIORNWU_C f4V OSQRHYNP[W[WNWEΣk
GIU_QSQSHYERMFU0CLf6ADCjOZ[eAcAD[WHV6Q\LXYH³f4Hλj®2U_aaH³GIHO,ORHE,HpGORNWU_CHpE>OA_VLERERNl[WHbORQSHNW[P[WNWEf] U0QSf4QSH
kf] V6C
(N − 1)ERNPaM6[WHI4HTf6ADCFE[] K`jM<HQSM6[eADC
λj64X06U_CMFHYV4OA_M6M6[WNWdjV6HYQv[^] K`M<UDORKLYERH2f4HQRXpGIV6QSQSHCLGH-HIO§HCgf4Xpf4V6NWQRHdjV6H
qj = 0¤±V4OSQRHYaHYCjO!f4NPOY¤[WH«rA_GIORHYV6Q
λj(λj − 1/k) · · · (λj − (k− 1)/k)(λj − 1)f4NWNWERH
q6GHBdjV6NlHpE>OV6CLHBGU_CjORQ&A_f4NeGOSNPU0CA¤0HYG[WH«rA_NOBd0VLHT[PH#f4HY_QSX#f4H
q(λ)HpE>ONWC4«XQSNWHV6QU0VX0A_[vZ
kLESADV6ǤERN
q = 0LGIH#djV6N'HYE,O[PHTQSXYERV6[POSADOf6XYERNPQSX_
;2|+3 3´| %] \%] `>bdcfeK j e K ′ m jg - N p c t|>j-[jlp q q4h e g hjrlq-j b tt g hj Γ = ∂K ∩
∂K ′ w `Qb4cfe g h6jlh eKc jk k ≥ 1 w&%" b k-p o jg k-p e kvj c f c p m b k m k@j k o Σk j e Σ′k b h cBm jh e pug k-j eKe j
rlq'-j Γ w j|> g+p o e qdh em'b hNh pK j e pK′m jlg -| b h4t9jlp m j Pk
o qsr b h e~cBb h v m ' h c j|Nq4k
v(x) =
pK(x) p c x ∈ KpK′(x) p c x ∈ K ′
jlp e b h e~c hNgNj7p-g k K ∪K ′ o p c j e pljg jt.jh e p c pK j e pK′b h em jlp7i8qdjlg k-p g c b h> cBm jlh e
qdg -s| b4c h e p m g e kvj c f c pspug k qrlq'-j b tt g hj Γ wYh+3´q<s'tYnycm4nporq<s#]3D[ªHYE,OG[WA_NPQdjV6H;[eA#QSHYE,ORQSNeGORNWU_CZ#V6C6H;«rA_GH;f4H
Kf6HBERU_C\OSQRHYNP[W[WNWEfl] U_Q&f4QRH
ΣkHpE>OADVLESERN6V6C\OSQRHYNP[W[PNeE?f] U0QSf4QSH
kf6A_CLE[^] Kj`M<HQSM6[eADCGIU0CjORHCFADCjO?GIHO,ORH9«rA_GIH00djV6N<C6H9f4XYMFHYCLfdjV6Hf6HYEgERU_aaHIO&E#f4H¡GHIORORH«rA_GH_'°§A_QgGIU0CLERXYdjV6HCjOpl[PHpETORQSHNW[W[PNeE
ΣkHIO
Σ′k
GU)L^CLGINef4HYC0O#ERV6Q[WHVLQ;«rA0GIH\GIU0aagV6C6HΓ(4N
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[0 # $ !66# "! # +N ≥ 2
8G9u|jyc¢_o¢_| ] \] @ K N ! & <7 9 6 && ! (aj)1≤j≤N+1 / ="(%)%4*" I"<>"7 & p ∈ P1 C & ! 6" 7 2:@% &
p(x) =
N+1∑
j=1
p(aj)λj(x),
7 # (λj(x))1≤j≤N+1 67 !@@% 9) ;# '2>% #>(% * # 9) x ∈ RN /9u|jyc¢_o¢_| ] \] \ @ K N ! & <7 ' 9 6 & & ! (aj)1≤j≤N+1 / 9); ,7 <> & 7 > (ajj′ )1≤j<j′≤N+1 9)2>% 9 K < 2)%?7 >% #@% 9";# '2)% '!)(% *" #
λj(ajj′ ) = λj′(ajj′ ) =1
2, λl(ajj′ ) = 0 <>% l 6= j, j′.
?;>% @)% *" Σ2 #6 <>%; C; & )?!6 (;49)56 && #?!59#,<> & 7 ):9)2)% #5#*" ",<>"7 & p ∈ P2 6 & # C" ,7 2 @% &
p(x) =
N+1∑
j=1
p(aj)λj(x) (2λj(x) − 1) +∑
1≤j<j′≤N+1
4p(ajj′ )λj(x)λj′ (x),
7 # (λj(x))1≤j≤N+1 67 !@@% 9) ;# '2>% #>(% * # 9) x ∈ RN /U_VFEA¤_U_CFE2a\A_NPCjORHYCLADCjOORU0VLE[WHYEU0V4ORNW[WEM<U_VLQf4XILCLNPQ[eAaXIORKLUf6H#f4HYEXY[PXYaHYCjOSELCLNWEPk
Yh[Z§s'ornYorq<s %] \%]2e q4h eEmb hNh g h t9q c fq j Thm g h b g i8jk e b h_hj-[j7| b -m k c gNj Ω o
q t le b m j m jup t9jh e p ' h c p Pko b g hh[3 |jsªnptZ!s!ortnYy¤om6sAFk!rm6oryc|jtw!| ;2mAFy¤mLs$A<|w WqLycw'y¤|
k o q8p-p b cB j $6-j7t9q c fq j o jup e2m' ' h c j|qdk7R jlpK|Nq-j m4c pkvj eVh =
v ∈ C(Ω)e j gNj v |Ki ∈ Pk | b g k e b g e Ki ∈ Th
. ¯ F E
x2h<q-||Njlf j s!q|k!w'tw!|t´w'|+AFychjt w'| ro 2|0y¤nph R jhpjlt9j m jup | b4c h e p (ai)1≤i≤ndl
m jlpe kvj c f c p m b k m kvj k m jE qg h m jup N p c t| j -+jlp Ki ∈ Th wxhhj b t| e j g g h>j"pjlg jr bdc pjup"| b4c h e p g c b h cBm jh e j e ndl jup e jh b t.kvj m j m j k p m j9 c -jk e6m j)q t le b m j m jlp t9jlh e p ' h c p Pk wnx2h q-||Njlf j w'|+AFy¤htw!| ro ?|jy¤nYh m g hjr b h e~cBb h v ∈ Vh R jhpjt.jm jlpsi4q4 jlg k-p m j v jlh@jlp h b jlg m p (ai)1≤i≤ndl
w2xh m ' h cfe qdg+p@p c j7p b g+p jlpK|Nq-j V0h |qdkV0h = v ∈ Vh
e j gNj v = 0 pug k ∂Ω . ¯ FFp vU_Q&ERdjV6H
k = 1[WHYEgC6UHVLf6EJf4HYEJf4H0QRXpE#f4H[PNWFHYQ,OSX¡GIULCFGINef4HCjOJA¤0HYG[WHYEgERU_aaHIO&Egf4VaA_NP[W[eAD_H0I U0QSESdjV6H
k = 2GIHpElC6UHVLfLEE,U0CjOvGIU_CFE>OSNOSV6XYEf] V6CLHMLADQROvf4HpEE,U0aaHOSEf6VTaA_NP[W[eAD_HHIO9fl] ADV4OSQRHbMLA_Q,Of4HpEMFU0NPCjO&EaNP[WNPHYV4f6HYEADQS¦IOSHYEQSH[WNWA_CjOf4HYV4ERU_aaHIO&E
.¡|+3´m4ycilk!| %] \%] ] ADMLMFHY[P[eAcOSNPU0C¨,X[WXaHYC0O&ELC6NeE³f6H vA__Q&ADC60H©GIU0QRQSHYERM<U_CLf ADV4 X[WXIaHYCjOSEgLC6NeEf6U_CjOJ[WHYEf4HY_QSXYEgf6H¡[WNP<HQRORXE,U0CjOf4HpEgcAD[WHVLQSEJMFU0CLGOSV6H[W[WHYEf4HYEJ«U_CFGORNWU_CFEJf4H[] HYERMLA_GHVh ;CM<HV4Of4XLC6NWQfl] ADV4OSQRHpE?O>`M<HYEfl] X[WXaHCjO&ELC6NeE4MLA_QHI4HaM6[WHb[PHpE2XY[PXYaHCjOSELC6NeE?f4H HQSaNOSHg0U_NWQ?[eAE(U0VLE>ERHYGOSNPU0C ¯ 0/4_!M<U_V6Q2[PHpERdjV6HY[WE?[PHpE?f4HY_QSXYEf4H;[PNW<HQRORX9ERU_CjO?[WHYE¤A_[PHYV6Q&EMFU0CLGOSV6H[W[PHpEf4HT[eAg«U0CLGOSNPU0CHIOf4H#ERHYEf4XYQRNW_XYHYEY •
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)E "+ n " I # $ !6
bycqF~?q<tYornYorq<s %] \%] jup~|Nq-j Vho m' ' h c |Nq4kz dw * u o jlp e g h)p b g+p jup~|Nq-j m j H1(Ω)
mb h e q mdc t.jhp cBb h jlp e ' h c j o # qdj q4g9h b t9lk@j m j m j k p m j" c -jlk e w j|> g+p o c >j- c p e jg hjn-q8pjm j Vh (φi)1≤i≤ndl
m ' h c j|Nq4kφi(aj) = δij 1 ≤ i, j ≤ ndl,e jfj gNj
v(x) =
ndl∑
i=1
v(ai)φi(x).
Yh+3´q<s'tYnycm4nporq<s#]4 HpEvXY[PXYaHCjOSE§f4HVh¤XIO&ADCjO!QSX0V6[WNPHYQSE'E,V6Q-G&KLA0djV6H?a\ADNW[W[PH
KiHOGIU_CjOSNPCVLEERV6Q
ΩADM6MFADQRORNWHC6C6HYCjO\Z
H1(Ω)_U0NPQ\[WH³ vHaaH FL µ6W 0IBQ@,0GIH A_V vHaaH ¯ µ6 F[PHpEXY[PXYaHCjOSEf4H
VhERU_CjOHI6A_GIORHYaHYCjOU_6ORHCVLEHYC A0ERERHaJ6[WA_CjOE,VLQG&KFA_djV6H
Ki ∈ Thf6HYEM<U_[W`C"B_aHpE9f4H
Pkd0VLN'GIULCFGINef4HCjO;ERV6Q9[WHYE;f4H0QRXpEf4HJ[PNWFHYQ,OSXgf4HpE«rA_GHYEJrGIHdjV6NvM6QSU_VL_H#ADVMFA_ESERA__Hd0VLH
VhCv] HpE>OMLA0E§QRXpf4V6NPO?ADV4E,HYV6[WHYE!«U_CFGORNWU_CFE-GIU_CFE>O&ADCjORHpESIDC4LCvDHC\A_ESERHagL[WA_C0O[WHYEgLA_ERHYE
(ψj)1≤j≤nk
f4HPk
ERV6QJG&KLA0djV6Ha\ADNW[P[WHKi
¬«U0V6QRCLNPHpE#MLA_Qg[WH¡ HYaaH ¯ µ6 µ0BU_CU04ORNWHCjO[eALA_ERHTADC6C6U0CLGIXYH(φi)1≤i≤ndl
f6HVh
.|+3´m4ycilk'| ] \] ;CU_4OSNPHYC0O#VLCQSXYERV6[POSADOgE,HYag6[eAD6[WHMFU0V6Q#[WHE,U0VLE,^HpE,MLA0GIHV0h
'f4XLC6NMFADQ ¯ FFpIªdjV6N?HYE,ObVLCERU_VFE> HYERMLA_GHJf4HH1
0 (Ω)f4Hf6NPaHCFE,NWU_CLC6NWH\XjAD[WHA_VC6U0ag6QSH\f4Hf6H_QSXYEf4HT[WNWFHYQ,OSXTNPCjORXYQRNWHVLQSEbU0CC6HTGIU_aM4OSHTMLA_E[WHYECLUjHYVLf6EERV6Q[WHbFU0QSf
∂Ω •
;u|jyc¢_or¢0| ] \] @ Th & 2 787 2 ,9 Ω <>"% Ω " )% ? & <7 & >5! ':<>7 %"'279 R2 / ) nt 7 & >%I9) (% 2 "7 =9 Th nc 7 & >% 9 2'##=" !>;=9#(% 2 "7 # +6 !);,! && # 9)) (% 2 "7 #6$# & <;* C>7 K1 ns 7 : & >%9 6 && !:9 & 2 787 2 ) ! n0s 7 & >% 9 6 & & ! ;>% >%:9 & 2 787 2 )/ ="(% >%* =7 9 & ) " 9# !<'2'! Vh # V0h C"
dimVh =k(k − 1)
2nt + kns − k + 1, dimV0h =
k(k + 1)
2nt − kns + k + 1.@;XYGQRNW_U0CLEv[eA;y¤htpqFkvnporq<s~'ycm4nporilk!|#f4VJM6QRU06[WaH2f4H@;NPQSNeG&K6[PHO ¯ µ ¯ lMLA_Q'[eA9aXORK6U4f4Hf6HYE;X[WXaHCjOSE9LC6NeE
Pkl vA«U0QRaJV6[eAcORNWU_CcA_QRNeAcOSNPU0C6C6H[W[WH ¯ /0f4HJ[] ADMLM6QRU¤4NWa\AcORNWU_CNWCjORHQSC6Hf6HNPHYCjONeGIN5H
ORQSU_VL_HQuh ∈ V0h
ORHY[vdjV6H ∫Ω
∇uh · ∇vh dx =
∫
Ω
fvh dx ∀ vh ∈ V0h. ¯ F[/_
;Cf4XpGIU_aM<U0ERHuh
ERV6Q[eALA_ERHBf6HYE(φj)1≤j≤ndl
HIOU0CMLQRHYCLfvh = φi
GH#d0VLNvf4U_CLC6Hndl∑
j=1
uh(aj)
∫
Ω
∇φj · ∇φi dx =
∫
Ω
fφi dx.
?CC6UDO&ADCjOUh = (uh(aj))1≤j≤ndl
bh =
(∫
Ω fφi dx)
1≤i≤ndl
6HIOHYCNPCjORQSU4f4V6NeERA_CjO[eA93´m+=nYycor¢_|³w'|ycoBA<orw!onphKh =
(∫
Ω
∇φj · ∇φi dx
)
1≤i,j≤ndl
,
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[0 # $ !66# "! # +N ≥ 2
[WAB«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6C6HY[P[WHf6ADCFE
V0hQSHNWHCjO?ZBQRXpE,U0VLf4QSHfLADCLE
Rndl[WHE,`4E,ORaH[WNPCLXYADNWQSH
KhUh = bh.®2U_aaHJ[WHYE;«U_CFGORNWU_CFEBf6HJLA0E,Hφj
U_CjObV6C¨,M<HIORNPO>©\E,VLM6MFU0Q,ObADV4OSU_V6QTf4VC6UHVLfai
r_U0NPQB[WA'NW_VLQRH ¯ W E I_[^] NWCjORHYQSERHYGIORNWU_C¡f4HpE2ERV6M6M<U_QROSEf4H
φjHIOφiHpE>OE,U0V6_HYCjO?NWf4HBHIO[eA#M6[WV6MLA_Q,Of4HYEGIUHlWGINWHCjO&Ef4H Kh
E,U0C0OCV6[eE ;Cf4NPO;djV6Hb[WAa\AcOSQRNeGIH KhHYE,O9¢Dyc|k!tY|<
¯ W E 6U_CFGORNWU_Cf4HTLA0E,HP1
HYCf6NPaHCFE,NWU_CN = 2
°§U0V6QGA_[WGV6[PHYQ§[PHpE-GIUHlWGINWHCjO&E-f4H Kh
cU_CM<HV4OV4OSNP[WNeE,HYQ-[WA;«U_QSagVL[PHfl] NPCjOSX_Q&AcOSNPU0CgH4A0GOSHE,V6NWcADCjORH0 ;CC6UDOSH(λi(x))1≤i≤N+1
[PHpE2GUjU0QSf6U_C6C6XYHYE?LADQS`4GIHCjOSQRNedjV6HYE2f4V¡M<U_NWCjOGU_V6Q&ADCjOxfl] VLC
N ERNPaM6[WHI4H
K4°§U0V6QORU0V4O
α1, ..., αN+1 ∈ N4U_CA
∫
K
λ1(x)α1 · · ·λN+1(x)
αN+1 dx = Volume(K)α1! · · ·αN+1!N !
(α1 + ...+ αN+1 +N)!.
¯ F0µj°!U0V6QGA_[WGV6[WHQ-[WHERHYGU_CLfJaHaJ6QRH
bhHIOa¦YaHXY_HYC0OSV6H[W[WHaHCjO-[WAba\ADORQSNWGH Kh
DU0CV6ORNW[PNeE,Hf4HYE.1 qLy43k'|tw!|ilk'mLwvycm4npkvyc| rU_Vx«U_QSagV6[WHYEfl] NPCjOSX_Q&AcOSNPU0C´CV6aXQSNWdjV6H¤gdjV6Nf4U0C6C6HYC0OV6C6HA_M6M6QSU¤4NPa\AcOSNPU0Cf6HYEJNPCjORXY_Q&AD[WHYETERV6QJG&KFA_djV6HNERNPaM6[WHI4H
Ki ∈ Th!°-ADQJHI4HaM6[WH_!E,N
KHYE,O2V6C
NE,NWaM6[PH4H9f4H;ERU_aaHIO&E
(ai)1≤i≤N+1j[PHpE«U0QRaJV6[PHpE?ERV6NW¤A_CjORHYE?_XYC6XQ&AD[WNWERHCjO[WHYE«U_QSagV6[WHYEHYCf4NWaHCLERNWU_C_6f4NPORHpEf4V¨,MFU0NPCjOaNP[WNPHYV©THIOf4HYE2¨OSQSA_MFJHYE>©=H
∫
K
ψ(x) dx ≈ Volume(K)ψ(a0), ¯ F)F
A¤_HYGa0 = (N + 1)−1
∑N+1i=1 ai
6[PHTFADQS`GHCjORQSH;f6HK4HIO
∫
K
ψ(x) dx ≈ Volume(K)
N + 1
N+1∑
i=1
ψ(ai). ¯ Fj0
®2U_aaH [PH³aU0CjORQSH[PHpE-4HQ&GINeGIHpE ¯ µ6 ¯ HIO ¯ µ6 ?GHYE\«U0QRaJV6[WHYEERU_CjOHI6A0GORHpE\M<U_V6Qf4HYE«U_CLGIORNWU_CLE§A4W\C6HpE§HIOERU_CjO§f6U_CLGADMLM6QRU4G&K6XYHYE!Z9[^] U_Q&f4QRH/9HChMFU0V6Q-f4HYE!«U0CLGOSNPU0CLE§QRXY_V6[WNWQSHYEY
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/ "+ n " I # $ !6
'A;GU_CLE,ORQSVLGOSNPU0Cgf4H[WA;a\AcOSQRNeGIH KhHpE>O-ADM6M<H[WXHm6tptY|+3 !mA<|#w'|rm93 m4nycor¢0|<p 'A9aNeE,HHYC¡UHVLjQSH9NWC4«U0QRa\AcOSNWdjV6HBf4HbGIHIORORHBXIO&ADM<HBf4VGA_[WGV6[FM<HV6O¦ORQSH;A0ERERH#JBGIU_aM6[WNed0VLXH_ja\ADNeE2ERU_CGU_4OTHYCOSHQSaHf4HORHaMLEgf4HGA_[WGV6[-HYE,OT«rADNW6[WH_!®2HC] HYE,O#MLA_Eb[WHGA0ETf4H\[eAQSXYERU_[WV4OSNPU0Cf4VER`4E>OSaH[WNPCLXYADNWQSH KhUh = bh
d0VLNLHYE,O?[^] XOSA_MFH[eAT~§rk!tT¢_qvnp|k'tp|Jf6H[WA#aXORK6U4f4H9HC\OSHaMLEf6H2GA_[WGV6[ªHIO!HC#ML[WA0GIH?aXYaU0NPQSHpIY?CgMLA_Q,OSNWGV6[PNWHQp[WHYE!GA_[WGV6[eElORQSNef4NPaHYCLE,NWU_CLC6H[eE'ERU_CjOvHYCLGIU0QRHOSQRpEG&K6HYQSEf6HC6U0E A>U_V6Q&E?f4pEdjV6H[^] U0C\V4ORNW[WNWERH9f4HpEa\ADNW[P[eAD0HYE!LCFEj ] -4HQ&GINeGIH ¯ µLP0M<HQSaHIOf6HJEY] HYC QRHYCLf4QSHgGIU0aM4ORH0 HYV6QSHVLERHaHCjOpF[eA\a\AcOSQRNeGIHf4HgQSNP0NWf6NOSX KhHYE,OB¢_yc|jk!tp|GD] HYE,O,Zcf6NPQSHTd0VLHB[eAM6[WV6MLADQROf4HTE,HpEX[WXaHCjO&EE,U0C0OCV6[eESI6GIHTdjV6NM<HQSaHIOf4HbaNPC6NWaNWERHQ9[PHpEGAD[eGIVL[WErMFU0V6Q9ML[PVLEBf4Hf4XIO&ADNW[WE9_U_NWQ[WHYEBAD[W_U0QRNPORK6aHpEf4H#QSXYERU_[WV4ORNWU_C f6HJER`E,ORYaHYE[WNWC6XYA_NPQSHYEfLADCLE9[WA
(4HYGOSNPU0Cpµ6W;f4Hb[^] A_C6C6HH¤XA_M6M<H[WU_CLEdjV6Hb[eAga\AcOSQRNeGIH KhHYE,OC6XYGHYESERA_NPQSHaHCjONPC_HYQSERNW6[PHMFADQADMLM6[PNeGADORNWU_Cf4V HaaH ¯ W_WBHIO9djV] H[W[PHTHpE>OER`jaXORQSNWdjV6H0
;u|jyc¢_or¢0| ] \] ; & (% >%$7 2 @% & 7 + / -15>=9 & ) " N = 2 /;u|jyc¢_or¢0| ] \] (% >% * 7 @% & 7 # + / &1! + / 16>2'(# <>% ψ ∈ P1 /;u|jyc¢_or¢0| ] \] @ K (% 2 "7 9 R2 9 6 && ! (ai)1≤i≤3 # 9 '2)% '!)(% a0 / (aij)1≤i<j≤3 7 # <> & 7 > 9# 6 & ) 9 (% ; &: ;# ai, aj / (% >%,* 7 2@% & 7 :9):* 2'9% 2(%
∫
K
ψ(x) dx ≈ Aire(K)
3
∑
1≤i<j≤3
ψ(aij)
>2'( <>"% ψ ∈ P2 62 9 *" 7 2:@% & 7 ∫
K
ψ(x) dx ≈ Aire(K)
60
3
3∑
i=1
ψ(ai) + 8∑
1≤i<j≤3
ψ(aij) + 27ψ(a0)
>2'( <>"% ψ ∈ P3 /;u|jyc¢_or¢0| ] \] @ (bi)1≤i≤I 9) <> 9 N ! & <7 ' K ! (ωi)1≤i≤I 9# <> 9)% ;#)7 / @ 4 4@% & 7 :9):* 2'9% 2(%
∫
K
ψ(x) dx ≈ Volume(K)
I∑
i=1
ωiψ(bi)
* 6 '2'!=<>% ψ ∈ Pk / (% >% * &3<>%$ ! " % ;"7 >% ψ 21
Volume(K)
∫
K
ψ(x) dx =
I∑
i=1
ωiψ(bi) +O(hk+1),
h #647 :9 2 & !(% 9) K /
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[0 # $ !66# "! # +N ≥ 2
µ1 5 2
8 69
4 7 3
¯ W_\HYaM6[PH#f6HTaA_NP[W[eAD_HBHIO9f6HTCjVLaXYQRU_OSADORNWU_Cf6HYEC6UHYVLf6EY9u|jyc¢_o¢_| ] \] #" 9 >% 7 2>% %; Ω =] − 1,+1[2
& 2 787 ; ! 2 7 2 "% // 27 >7 )% 7 2 & 2(% # 9 % 9 ; Kh 9) ;)7 ; & ) 3 P1 2<<7 *" ;# 2 >2<7 2' ) 2 !"'9 "2 7 & #9)) & 2 + 7 C>% 2 7 & ;!(% # 9" & 2 787 2 ) 1!/9u|jyc¢_o¢_| ] \]B^ <<7 * >% 7 2 & ;! 9) 9),;>7 ; & > 3 P1 2 <>% 7 & 9) % 7 #+ / -1,92 7 #2)%.% ; Ω =]0, 1[2 2 7 & 2 787 2 ) (% 2 "7 2 % @% & 9) 7 2 % /0/ (% >% *=7 2 & 2(% # 9 % 9 ; Kh #47 2 &)& & 2(% #:*" :!>787 * 7 ": >'9"% 2 < 2)%?2<<7 #2 9 7 2 & ;! 9) 9) 9 ;>% >'!# 3 + # 2'(>% & 7 <7 2 h2 <>% # 1 & 2 *" 7 :6#"'9 & & )% bh 9 ;>% >#/
¯ P8/ A_NP[W[WA__H;ORQSNWA_C6_VL[WA_NPQSH;VLC6N«U0QRaHTfl] V6CGA_QRQSX_9u|jyc¢_o¢_| ] \]B^^ % ><>% >'9 7 )62 " ?9)7 '>% #/ -/ / > n 7 & >% ,9)<> 9 & 2 787 2 !%I!>; 9 2>% %; +.!<<>6; (% 7 &)& I<>"% 2'* #>;1#/ & ;>% > K7 " <'2>%$7 :7 # @) 99 & 2 787 2 )=+ 7 #9) "% ;#9) 7 >>%;1#/ (% >% *7 2 & 2(% !9 % 9 ; Kh 9);>7 ; & > 3 P1 #6 9462 787 :9 7 @% 9"% 9 n2 # 9 7 2>% ))%9) '2 9:9) 7 @% 9% 9 2n +<>"% n "% 2 9@1!/
="(% >%* 7 2 &>& & ;! 9):# 7 &>& <> 9 & 2 787 2 ) <>%7 ' > Ω =]0, 1[3
!"'9 C> # & 2(% #=9 62 787 9) 7 @% 9% 9) n3 #49 7 2)% >%9 2'9) 9) 7 @% 9% 9)2n2 + n ,7 & >%:9 @>'947 7 % 9 :2)% :9"=> > Ω 1!/.¡|+3´m4ycilk!| %] \%] ®2U_aaH³[WHaU0CjORQSH [] HI4HYaML[PHf4H³[^] -4HQ&GINeGIH ¯ µLP E [WA a\ADC6NWQSHf4HCjVLaXYQRU_ORHYQ¡[WHYECLUjHYVLf6Ef4HYEf4HY_QSXYE¡f6H³[WNP<HQRORX¥U_V f4H «rAGIU_CXpdjV6NPcA_[PHYC0OSH³[WHYE«U_CLGIORNWU_CLEf4HLA_ERHpAV6C6HNWCLV6HYCLGIH ERV6Q\[eAE>OSQRVLGIORV6QSHGIQSHVLERHf4H[WAa\AcOSQRNeGIH Kh
2GD] HYE,O,Zcf4NPQSHERV6Q[] HaM6[eA_GHaHCjOf6HYEE,HpE?XY[PXYaHCjOSE2C6U_CCV6[WEY6®2U_aaH9HI4M6[WNed0VLXBf6A_CLE2[WA (HYGIORNWU_C³pµ6Wg0U_NWQY
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F "+ n " I # $ !6
MFADQHI4HaM6[WH_-[WH HYaaHYµLP0 F-GIHO,OSHE,ORQSVLGIORV6QSHGIQSHVLERHf6H[eAa\AcORQSNeGIHAVLC6H_Q&ADCLf4HNWCLVLHCLGHERV6Q\[eA³M<HQR«U_QSa\ADCLGHf6H[eAQRXpE,U0[PV6ORNWU_Cxf6V¥ER`4E>OSaH[PNWC6XYA_NPQSH KhUh = bh°-ADQH4HaM6[WH_ªE,N![] U_C³QRXpE,U0V4OBGHJER`E,ORYaH#[PNWC6XpADNWQRHJMLADQ;V6C6HaXORK6U4f4Hf4V O>`M<HB¨,X[WNPaNWCLAcOSNPU0C³f4H
bA_VLESE©6NW[vHYE,O;A¤cADCjO&AD_HYV4f4HJG&K6U_NeE,NWQV6C6HJCV6aXQSUDOSADORNWU_CdjV6NvQSH0QRU0V6MFHT[WHYEX[WXaHCjOSE9C6U_CCV6[eEM6QSYEf4HT[eAf4NeAD_U0CLAD[WH_ •.|+3´m4ycilk'| ] \]B^ °§U0V6QJERNPaM6[WNPLHQ[] ADCFAD[W`ERH³ADVLESE,N26NWHC djV6H[WA³aNeERH¡HC UjHYV6QRH¤bU_CM<HV6OV4ORNW[PNeERHQV6C6HBORQ&ADCFE>«U0QRa\AcOSNPU0C¡A4W\C6HbM<U_V6QQ&ADaHYC6HQ2ORU0V4O
N ERNWaML[PHH
Kf4VaA_NP[W[eAD_H
ThZTV6C
N ERNPaM6[WHI4Hf4H¨>QSXI«XYQRHYCLGIHI©
K0_°-ADQ2GIH9ERNPaM6[WH;G&KFADC60HaHCjO?f4HcA_QRNeAD6[WH_DORU_VFE[WHYEGYAD[eGIV6[eEERHBQ&ADaYC6HCjOZf4HYEGYAD[eGIV6[eEE,VLQK0
6?CMLQSADORNed0VLH_4U_CG&K6U0NWERNOERU_V60HCjOK0 =
x ∈ RN ORHY[vdjV6H N∑
i=1
xi ≤ 1, xi ≥ 0 1 ≤ i ≤ N
, ¯ F ¯
HOgU_C0XQSNLH\ESADCLEbM<HNWC6H¡djV6HORU_V6ON ERNWaML[PHH
KHpE>O#[^] NPa\AD0H\MLADQTVLC6H\ORQ&ADCFE>«U0QRa\AcOSNPU0CAdW\C6Hf4H
K0?CHlCªHIOp[PHpE\GIUU_Q&f4U0C6C6XHpELADQS`4GIHCjOSQRNedjV6HYEY-f4XILCLNPHpE\MLADQ ¯ µ JERU_CjO\[WHYEa¦YaHpE-M<U_V6QKHIOK0
HOY0HC\C6U_OSADCjOλ = (λj)1≤j≤N+1
x = (x, 1)
[WHMFU0NPCjO2GU_V6Q&ADCjOf6ADCFEKx0 = (x0, 1)
[PHbM<U_NWCjO9GIU_VLQSA_C0Of6A_CLEK0
4U_CAAλ = x
6HOA0λ = x0
4U'&[PHpEaADORQSNWGHYEAHIOA0
ERU_CjO;f6XILC6NWHYE;MLA_QJ ¯ µ+GDHO;NWC_HQ&ERNP6[WHYEY ;CHYC³f4XYf6V6NObf4U_CFGgdjV6Hx = AA−1
0 x0G_] HpE>OR ZD f6NPQSHd0Vv] NW[LHNeE,ORHV6CLHa\AcORQSNeGIH
B0NPC0HQ&E,NW6[WHf] U0QSf4QSH
N0HIO?V6C\0HYGIORHVLQ
b ∈ RN ORH[eEdjV6Hx = Bx0 + b
•
¯ Ppµ A_NP[W[eAD_H;ORQSNeADC60V6[WA_NPQSHBM6[PVFELCd0VLH#GIH[WV6Nf4HT[eA !NP0V6QRH ¯ 0G4 ?] HHYQSGNWGHTE,V6NWcADCjOaU_CjORQSHTdjV6HT[WAaXORK6U4f4H#f4HpEX[WXaHYC0O&ELC6NeE
P10XQSNFHB[WHTM6QSNPCFGINWMFHf6Va\ADNWagVLa
;u|jyc¢_or¢0| ] \]B^& 9 * & 2(% # 2>% % ;# %;>787 B = (bij)1≤i,j≤n # !& 2(% # 3<>"% " i
bii > 0,
n∑
k=1
bik > 0, bij ≤ 0 ∀ j 6= i.
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[0 # $ !66# "! # +N ≥ 2
(% >% * " ! & 2(% !#6 )% 7 #* " 7 #?!@ >9 6 >% 6,6"<>) 7 /9u|jyc¢_o¢_| ] \]B^ \ I6 <7 2'! > 9 & > " N = 2 / @ uh 7 2 C"78 " 2<<>% @ ; 9"<>%7 & 9 % 7 !:+ / - 1) <'2>% 7 2 & ;# 9 9),;)7 ; & ) 3 P1 / #<<>)C*" " 7 2 "7 9# (% 2%7 # Ki ∈ Th 6 ;)% )% " ; 2# π/2 / ="(% >%5*uh(x) ≥ 0 92 Ω f(x) ≥ 0 92 Ω / 9 #2 & "(%)% 2 * & <>"%" ε > 0 Kh + ε Id , ! & 2(% !& Kh # 7 2 & 2(% #9) % 9 ;/
¯ PF 0'HQSaH#ERU_VLQSGHff6ADCLE[^] XYdjVLAcOSNPU0C ¯ µ ¯
D[!C] `AXNWf6HaaHCjO;ADVFGIV6C6HJf4NDWGV6[OSXJZ\XORHYCLf4QSH#[WAaXORK6U4f4Hf4HYE9X[WXaHCjO&ELC6NeEPkZf] A_V4ORQSHYEM6QSU_L[PYaHpEdjV6H\ ¯ µ ¯
9u|jyc¢_o¢_| ] \]B^ <<7 * >%57 2 & ;# 99);>7 ; & > 3 Pk 2= & 9 7 ;)7 2 ;+ / 1!/ ="(%)%3) < 2)% '7 >% *" =7 2 & 2(% ! 9) % 9 ; Kh #6 92!:#2 9 @% 9"% Nndl
N # 7 2 9 & ) "9 ##< 2'# ! ndl #647 & )%:9= @>'9) 9:9) "% ;# 9) 7 >)% ;/9u|jyc¢_o¢_| ] \]B^ @<7 >%?7 2 & 2(% !:9 % 9 ; Kh > =<'2)% 2<<7 #2 9) 7 2 & ; ! 9 9#;)7 ; & ) 3 Pk 2 <>% 7 & 9> & 2
−∆u+ au = f 92 Ω∂u∂n = g !% ∂Ω,
¯ F+G_2 f ∈ L2(Ω) g ∈ L2(∂Ω) "! a ∈ L∞(Ω) )7@* a(x) ≥ a0 > 0 < / < / 92 Ω /
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_¯ "+ n " I # $ !6
;u|jyc¢_or¢0| ] \]B^ ="(%)%*" I7 2 & 2(% ! 9 % 9 ; Kh >I<'2>%52<<7 #2 " 9I7 2& ;! @9,9#$;)7 ; & ) 3 Pk 2=<>%7 & ,9,! ! "%! 9 "9)7 '>% #/ 0/ 0 #6 >% 7 & 2 4<'2 & ;!(% */;u|jyc¢_or¢0| ] \]B^ 6 <>% "<>6 9 %;6 9% & ;>% * & > 7 ;*'2 " 9# <7 2'*" #+ / 1 < 2)% & ;# 949 ;)7 ; & ) ,+ 9)4 <> )% &: 1$) 9 & ) " N = 2 / "% & 2 787 2 )4(% 2 "7 2 % Th (% 9" ,7 ##< 2'#:9 K'% !
Vh =
v ∈ C1(Ω) )7* v |Ki ∈ P5 <>% " Ki ∈ Th
.
="(% >%* " <>"7 & p ∈ P5 #6,#2)% 2'!;>% 6; 9 & 2 >% * !% (% 2 "7 K<'2)%$7 #0 27 >% % ;#)787 #! 2#
v(aj),∇v(aj),∇∇v(aj),∂p(bj)
∂nj = 1, 2, 3,
¯ F (a1, a2, a3) 6"7 # C && ! 9 K (b1, b2, b3) 7 & 7 ) 9) !);# 9 K !! ∂p(bj)/∂n9;# " 7 2 9);)% ;# @% & 27 2 !); 9) bj / ="(% >%* Vh 6 ! !<'2'!9 H2(Ω)9"57 ;)7 ; & ) v C"#2)% 2'(;>% 6;$9) & 2 >% *" 4< 2)% 7 # 27 >% + / 1?<>"%'2'*C && ! ! & 7 > 9 2>% 9 & 2 787 2 / 9);9" % & ;! 9) 9 ;>7 ; & > + 9 9 % % K14<>%?% ;#C"'9"% + / 1!/9U_VLE!OSHQSaNWC6U0CLE-GIHO,ORHERU_VFE>E,HpGOSNPU0C#HC[^] NW[P[WVLE,ORQ&ADCjO-MLADQ§V6CQSXYERV6[POSAcO§CV6aXQSNWdjV6HU04ORHYCjVMFADQT[eAaXORK6U4f4Hf4HpETX[WXaHYC0O&EBLC6NeE
P1A_M6M6[WNWdjV6XYHA_VM6QRU06[WaH\f4H\@9NWQRNeG&K6[WHIO¡ ¯ µ ¯ I HERHYGU_CLfaHaJ6QSHHYE,O2f4U0C6C6XMLA_Q[eA !NP0V6QRH ¯ PFL ;C\M<HV6O2NWC0OSHQSM6QSXIORHYQ?GIH9M6QRU06[WaHGIU0aaH[eAaU4f4XY[PNeERADORNWU_C³f6Hg[eA¡f4NC<VFE,NWU_Cf6ADCLE;[^] ADORaU0ERM6K6YQRHgf] V6CMFU0[P[WVLA_C0OBXaNeE9MLA_Q;VLC6HJERU_V6Q&GIH[WU4GA_[PNeE,XYH__ vH;f4U0a\ADNWC6H9f4H9GYAD[eGIV6[FQRHYM6QSXYERHCjORHV6CLH9QSX_NWU_C\A_V4ORU0V6Q?f4H;[WA#ERU_VLQSGH#[eA#f4NWQRHpGORNWU_C0HQRORNeGA_[PHHYE,OJ¨>aUc`0HC6C6XYH©HO¡ADLERHCjOSHf6V GYAD[eGIV6[¬HIOU0C¥ERV6M6M<U0ERHdjV6H[WAGU_CLGHCjORQ&AcOSNPU0CHpE>OJCjVL[P[WH¡ERV6QgERU_C<U_Q&flv@;HV4a\ADNW[P[eAD0HYETU0C0OJXIORX¡V4ORNW[PNeERXYE H'[PH¡aA_NP[W[eAD_H#¨,_QSU0ESE,NWHQ,©f4H¡[WA
!NW_V6QSH ¯ G§HIO\[WHa\ADNW[W[WA__H\¨LC4©f4H[WA !NP0V6QSH ¯ Ppµ6§ HYEQSXYERV6[POSAcO&EGIU_QSQSHYERMFU0CLf6ADCjOJERU_CjOaU0C0OSQRXpEERV6Q[eA !NP0V6QRH ¯ P¤4QSHYERM<HYGOSNP0HaHCjOp ;CQRHYa\ADQ&d0VLHdjV6H³[WAcAD[WHVLQ¡a\Ac4NWaA_[PHf6HJ[eAERU_[WV4OSNPU0C CV6aXQSNWdjV6Huh
HpE>ObM6[WVLEBX[WH_XJMFU0V6Qb[PHa\ADNW[P[eAD0H#LCdjV6HJM<U_VLQB[WHaA_NP[W[eAD_H0QRUjERERNWHQ;r[PHpEXYG&K6HY[P[WHYE2C6HbE,U0C0OMLA0E?[WHYEa¦aHYE&F®;] HYE,O[WAJa\ADC6NP«HYE,OSADORNWU_C¡f4V«rADNPOd0VLH;[WH;MFA_Ehf6Va\ADNW[W[WA__HB_QSU0ESE,NWHQC] HYE,OMFA_EHCFGIU_QSHbA_ESE,HJ;M<HIOSNOM<U_V6QdjV6Hb[eAERU_[WV4OSNPU0CA_M6M6QSUG&KLXH
uhA_NOGU_C_HYQR0X9_HYQSE[eAE,U0[PV4OSNPU0CH6A_GOSH_(N[^] U_CQSA>A>U_V4OSH;HYCM6[WVLEf6HB[eAf4NC<VLERNWU_CV6CHlCªHIO9f4HGU_C_HpGOSNPU0C¥raU4f4XY[PNeESADCjO#V6C_HYCjOTGU_CLE,OSA_CjOgf6ADCLE#[WAf4NWQSHYGOSNPU0CK6U_QSN JYU_CjOSA_[PH0ª_U_NWQ6Ppµ0RU0CM<HV6O9_U0NPQ[^] HC<HO9f4HTMLADCFA_G&K6HTADNWCLERNlM6QSUf6V6NOERV6Q[eA\GIU_CFGIHCjOSQSADORNWU_CfLADCLE[eA 'NW_VLQRH ¯ W ¯rU_4OSHCV6H\A¤_HpGg[WH\a\ADNW[P[eAD0HgFCFv 'A¤A_[PHYV6QTa\Ac4NWaA_[PH\f4H[eAE,U0[PV4OSNPU0CHpE>O#ML[PVLEbM<HIOSNOSHHCMLQRXpE,HYCLGIHTfl] V6CORHYQRaH#f4HTGIU0C_HYGIORNWU_C6GHTd0VLNvGIU0QRQSHYERMFU0CLfLNPHYCZ[^] NPCjORVLNOSNPU0CMLKj`4ERNWdjV6HTdjV6H[WHT_HYCjO¨Rf4NW[PVLH©B[WHYEGU_CLGHCjORQ&AcOSNPU0CLEX[WH0XHYEf4HTM<U_[W[WVLADCjOY*)*KJ 5?/ #U3 Uv/9:>U Ul1 US014=vM P'16=,5?/ 7 U363U'83
9U_VLE'f4XaU_CjOSQRU0CLEª[WAGU_C_HYQR0HCLGH!f6HYElaXIOSK6U4f4HYE'fl] X[WXaHCjOSElLC6NeEPk
M<U_V6Q[WHM6QSU_L[PYaHf6Hg@9NWQSNWG&K6[WHIOg ¯ µ ¯ "DCLERNeE>OSU_CLEERV6Q[WHb«rADNPO;djV] NP[!E] AD0NO9ERHVL[PHYaHYCjO9fl] VLCM6QRU06[WaHTaUf6[WH_
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[0 # $ !66# "! # +N ≥ 2
G
¯ P¤ (U0[PV6ORNWU_C³A_M6M6QSUG&KLXHuh
f6Hg[^] Xpd0VFAcORNWU_Cf4HJf4NCªVLE,NWU_Cx ¯ µ ¯ M<U_V6QB[PHJa\ADNW[P[eAD0H_QSU0ESE,NWHQf4HT[eA !NW_V6QSH ¯ GKLA_V4O&HOMFU0V6Q[PHba\ADNW[W[WA__HBLCf4Hb[WA!NW_V6QSH ¯ WYµLA0ESI
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"+ n " I # $ !6
¯ W ¯ 6(U_[WV4ORNWU_CA_M6M6QSUG&KLXHuh
f4HT[^] XYdjVLAcOSNPU0Cf6HbGU_C_HpGOSNPU0C4 f6NDCªVLERNPU0C6Ppµ0RHObdjV6HGIHpE;aXORK6U4f4HpE;GU_C_HYQR0HCjO9M<U_V6Qbfl] ADV4OSQRHpE9M6QSU_L[PYaHpEªGIU0aaHGIHY[PV6N-f4HHYV6a\ADC6C ¯ F+G__9U_VLE2AD[W[PU0CLEA¤_U0NPQ§<HYERU_NWCfl] Kj`M<UDOSK6YERHYE-0XU0aXORQSNWdjV6HpE-E,VLQ?[eATd0VFAD[WNOSX9f4V\a\ADNW[P[eAD0H_°§U0V6Q9OSU_V4O
NE,NWaM6[WHI4H
KU_CNPCjORQSU4f4V6NPObf6HV4 MLA_QSA_aIORQSHYE9_XYU_aXIOSQRNedjV6HYE H<[WHw'om3 :jnyc|
diam(K)HIO[WAycq<s'w!|kvy
ρ(K)f4XIFC6NPHGU_aaH[WHf4NWA_aIORQSHf4H[WAM6[WVLE_Q&ADCLf6H¡<U_V6[WHGU_CjORHYCV6HTf6ADCLE
K
diam(K) = maxx,y∈K
‖x− y‖, ρ(K) = maxBr⊂K
(2r).
*NWHC ER6QpLU_C A\ORU_V>A>U_V6Q&Ediam(K)/ρ(K) > 1
ª®2H#Q&ADM6M<U_QRO9HpE>O;f] A_V4OSA_C0O;M6[PVFE90QSA_CLfd0VLHK
HpE>O2¨RADM6[eAcOSN © H6NP[vaHpE,VLQRHbHCdjV6HY[WdjV6H#ERU_QRORHB[WAOSHCLfLADCLGHbZ[eAf4X0XC6XYQRHpERGHCLGHBf6HK4CMLQSADORNed0VLH_6GIU0aaHbHYCOSK6XU0QRNWH_4NW[l«rA_V4OXNPORHQf] V4OSNP[WNeE,HYQf4HYE
N ERNPaM6[WHI4HpE
KOSQRU0MA_M6[WADORNeE
Yh[Z!s!onporq<s %] \%]K^_^<`>b4cfe(Th)h>0 g hj2pug cfe j m jt.q c fq jlp m j Ω wxh mdcfe g c p q cfem g hj
pug cfe j m j 3 m6orrmA<|tTych[A<k!ror|jyct p cw7qp-g cfe j h = maxKi∈Th
diam(Ki)e jh m i8jk-p o
,[w c j- c p e j g h>jE b hNp e qdh e j C e jlf j gNj o | b g k e b g e h > 0 j e2eb g e K ∈ Tho
diam(K)
ρ(K)≤ C. ¯ F 0
.|+3´m4ycilk'| ] \]B^&?C f4NWaHCLERNPU0CN = 2
[eA GIU0CLf4NPORNWU_C ¯ FjHYE,OXpd0VLNPcAD[WHCjOSHZ´[eAGU_CLf6NOSNPU0C³E,VLNPcADCjOSHJERV6Q;[WHYEbA_C6_[WHYE;f4V OSQRNeADC60[PHK
HªNW['HNeE,ORHV6C³ADCL_[WHgaNPCLNPaJV6aθ0 > 0djV6N'aNPC6U0QRHrV6C6NP«U_QSaXaHCjO9HYC
hOSU_VLE[PHpE9ADCL_[WHYEf4HTOSU_V4O
K ∈ Th"DCLERNWE,ORU0CLE9ERV6Q[WHT«rA_NO
Ecol
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[0 # $ !66# "! # +N ≥ 2
diam(K)
ρ(Κ)
¯ P4G ¡@;NWA_aORQSHdiam(K)
HOQRU0CLf4HVLQρ(K)
fl] VLCOSQRNeADCL_[WHK
d0VLH[eAGIU0CLf4NPORNWU_C ¯ F 09HYE,OTORU_V6OgADVLESE,N§NWaMFU0Q,O&ADCjORHHCM6Q&AcOSNWdjV6H\djV6HM<U_V6Q#[^] ADCLA_[P`4ERHf4HGIU_C0HQS_HCFGIH;djV6Nv¤AERV6NWjQSH_ •9U_VLEbM<U_VL_U_CFE;a\A_NPCjORHYCLADCjOTXCLU_CLGHQb[PHQSXYERV6[POSAcObM6QSNPCFGINWMLAD[f4H\GHIO,OSH\E,U0VLE>ERHYGOSNPU0CdjV6NA4W\QRaH2[WABGIU0Cj0HQS_HYCLGIHf4H[eAaXIOSK6U4f4Hf4HYE'X[WXaHCjO&EvFC6NWEPk
HIOdjV6N4f4U_C6CLH?V6C6HHYE,ORNWa\AcOSNPU0Cf4HT[WANPORHYESERHbf6H#GIU_C0HQS_HCFGIH;ERNl[eAERU_[WV4OSNPU0CHpE>OQSX0V6[PNWQSH_ £'hqFy4:[3 | ] \]B^ \ `Qb4cfe
(Th)h>0 g hjpug cfe j m j t.q c fq jlp k # g c jk-p m j Ω w `Qb4cfe u ∈H1
0 (Ω) o qp b g e~cBb h m g |k b l + t9j m j c k c j e z w - o j e uh ∈ V0ho -jlfj m j p b h<q@|
|k b - c t9q e~cBb h c h e jkuhjz dw *,l)|qdk)qt le b m j m jup t9jlh e p ' h c p Pk w %" b k-p)qt e b m jm jlp t.jh e p ' h c p Pk b hNi4jlk j o N jlp e $ m4c kvj gNjlimh→0‖u− uh‖H1(Ω) = 0.
¯ E j| g+p o p c u ∈ Hk+1(Ω) j e p c k + 1 > N/2 o qd b k-p b h q9R jlp eKc t9q e~cBb h m jkukvjg k
‖u− uh‖H1(Ω) ≤ Chk‖u‖Hk+1(Ω), ¯ 4¤
b)C jlp e g hj b hp e q4h e j c h m |jh m q4h e j m j h j em j u w
.¡|+3´m4ycilk!| %] \%]K^ H 0K6XYU_QSaH ¯ µ6WYµEY] ADM6M6[WNed0VLHHCb«rA_NO§ZORU0V4ORHaXORK6U4f4Hfl] XY[PXYaHCjOSELC6NeEf4H³O>`MFH vAD0QSA_C6_H´rMLADQHI4HaM6[WH_[WHYEX[WXaHCjOSE¡LC6NeEQRHpGOSA_C6_VL[WA_NPQSHYEf6H³[eA (U0VLE>E,HpGORNWU_C ¯ µ6 µ0!C¥HlCªHIOp![WHE,HYV6[ADQS_V6aHCjOV4ORNW[PNeERXHpE>O[WAGIU0CLE>OSQRVFGORNWU_Cxfl] VLC U_M<XQ&AcOSHV6Qfl] NWCjORHYQRM<U_[eAcOSNPU0C³LA0E,XE,VLQB[eAGADQ&A_GIORXYQRNeERADORNWU_C f4HpEB«U_CLGIORNWU_CLEbf4HVh
MLADQb[WHV6Q&EBcAD[WHV6Q&E;A_V4C6UHVLf6Ebf4HpEbf4H0QRXpE;f4H[WNP<HQRORX0lGIHd0VLN-HYE,O;OSU_V'A>U0V6QSE;M<U0ESE,NW6[WHJM<U_V6QTf4HpEBX[WXaHCjOSE;LCLNWETf4HO>`jM<H; 'AD0QSA_C6_H#r_U_NWQ?[eAEXHa\ADQ&djV6H ¯ µ6 ¯ I'XHYa\ADQ&d0VLU_CLEdjV6H00M<U_VLQ2[WHYEGA0EM6K`4E,NedjV6HaHYC0OMFHYQ,OSNPC6HYCjOSEN = 2
U_VN = 3
6[eA\GIU0CLf4NPORNWU_Ck + 1 > N/2
HpE>OOSU_V'A>U0V6QSE9ERADORNeE>«rA_NOSH#f4YEdjV6Hk ≥ 1
•.¡|+3´m4ycilk!| %] \%]K^ ] HYE,ORNWaADORNWU_C fl] HQSQSHV6Q# ¯ 4pf4V 0K6XU0QRYaH ¯ µ6WYµC] HYE,O9QSA_NPH#djV6HE,N2[WAERU_[WV4ORNWU_CH4A0GOSH
uHYE,O#QSX_VL[PNWQSH_vGHdjV6N?C] HYE,OgMLA0EbORU0V'A>U_VLQSET[WH¡GYA_EY$(N
uC] HYE,O#MLA_E
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E "+ n " I # $ !6
QSX0V6[WNPYQRH0U_C´GU_CLE,OSADORHHYC´M6Q&AcORNedjV6H¡djV6H[eA GU_C_HYQR0HCLGHHpE>OM6[WVLE#[WHCjORH0U_NWQ#[WA !NP0V6QRH¯ HCV6C6HJf6NPaHCFE,NWU_Cfl] HpE,MFA_GIH¤ª@g] ADV6ORQSHJMLA_Q,OpF[eA¡GU_C_HYQR0HCLGH ¯ E ªdjV6N-A[WNPHYVf6A_CLE[^] HYERMLA_GH§¨Rfl] XC6HYQR0NPHI©&¤C] NPaM6[WNed0VLHMLA_E§[eAbGIU_C0HQS_HCFGIHM<U_CLGIORV6HY[P[WHf4Huh
U_Vf4HE,HpE-f4XQSNP0XHpE 'A!NW_V6QSH ¯ W NW[W[PVLE,ORQSHGIH«rADNPOTMFU0V6Qb[PHM6QSU_6[WaHf4H\@9NWQRNeG&K6[WHIO ¯ µ ¯ 9A¤_HYGf ≡ 1
f6A_CLE[eA_XU0aXIORQSNWHf4V¨RGIU_NWC³QSHCjOSQSA_C0O,©gU&[WAERU_[WV4ORNWU_CHYE,ObERNPCL_V6[WNPYQRH0U_NWQb[PH\ HYaaH\40/4 µ_µj9V6aXQSNWdjV6HYaHCjOYª[WHaU4f4V6[WHf4V0QSA0f4NWHCjOBf4Huh
GIQSU_NPOB0HQ&E9[^] NWC4LCLNf6ADCLEB[WHGU_NWC³[WU_Q&ERdjV6HhORHCFf_HQ&E JYXQSUr[PHgaAD4NPaJV6a f4H |∇uh|
cA_V4O0.92
M<U_V6Q[WH#a\ADNW[W[WA__HTf4HT0A_VLG&K6H#Z0 GERU_aaHIO&E1.18
MFU0V6Q[WHaA_NP[W[eAD_Hf6VaNW[PNWHVZ F ¯)E0¯ ERU_aaHIO&E_HIO1.50
MFU0V6Q?GH[WV6NFf4Hf6QRU0NOSHZ 'G/#ERU_aaHIO&ES •
¯ W U4f4V6[WH¡f4V´0QSA0f4NPHYCjOgf4Huh
M<U_V6QgORQSU_NeE#a\ADNW[P[eAD0HYE\f4H¡M6[WVLEgHC´M6[WVLE#LC f4HjADVLG&KLHBZf4QSU_NPORH¤ 'AJf4XYaU_CLE,ORQ&AcOSNPU0Cf4V0KLXU_QSaH ¯ µLPpµTQSHM<U0ERH;ERV6Q2[eAf4XILCLNOSNPU0CERV6NPcA_C0OSHBf] V6CqL~2h=ycm4np|kvyw WorsªnY|jy¤~?q<rm4nporq<s
rhHO?E,VLQ[PHQSXYERV6[POSADO?fl] NWCjORHYQRM<U_[eAcOSNPU0Cf4H[eAb°QSU_M<U0ERNOSNPU0C ¯ µ6W ¯
X9ADM6M<H[WU_CLE?d0VLHC6U_VFE?A¤_U0CLE§C6U_ORX(ai)1≤i≤ndl
[eAb«rADaNW[P[WH9f4HpEC6UHYVLf6E?f6HYE?f6H_QSXYE?f4H[WNP<HQRORXHO(φi)1≤i≤ndl
[eAgLA0E,HBf4HV0h
f4H;[eAJaXORK6U4f4Hbf4HpE2XY[PXYaHCjOSE2LC6NeEPk
r_U_NWQ[WAJ°QSU_M<U0ERNOSNPU0C¯ µL0G_4°§U_VLQORU_V6ORHB«U_CLGIORNWU_CGIU0CjORNWCjVLHv6U0Cf4XLC6NPO9E,U0CNWCjORHYQRM<U_[WXH
rhv(x) =
ndl∑
i=1
v(ai)φi(x). ¯ '/_
'A¡f4NCªXQSHCLGHgM6QSNPCFGINWMLAD[WHJA¤0HYGT[^] XORVLf6Hg«rA_NOSHgHCf4NWaHYCLERNPU0CN = 1
HYE,OTd0VLH_<[PHpE«U0CLGOSNPU0CLEf6HH1(Ω)
Cv] XOSADCjO#MLA0E#GIU0CjORNWCjVLHYEb[WU_Q&ERdjV6HN ≥ 2
l[^] U0MFXYQSADORHVLQ#fl] NWCjORHYQRM<U_[eAcOSNPU0Crh
Cv] HpE>OMFA_E9f6XILC6N§ERV6QH1(Ω)
r[PHpEcAD[WHVLQSE9MFU0CLGOSV6H[W[PHpE9fl] V6C6Hg«U_CLGIORNWU_C f4HH1(Ω)
C] U_CjO;AM6QRNWU_QSNMFA_Ef4HJE,HYCLESILXpADC6aU0NPCLEY6HO;GD] HYE,O9[eAQSA_NWERU_Cf4H#[^] K`M<UDORKLYERHk + 1 > N/2
rh
|tn §or|sw!h[Z!s!otpkvyHk+1(Ω)
GYADQ[WHYE«U_CLGIORNWU_CLE;f4HHk+1(Ω)
E,U0CjO9GIU_CjOSNPCV6HpE#Hk+1(Ω) ⊂ C(Ω)f] A_M6QRpE[PH 0K6XYU_QSaH FL µ60/_0
bycqF~?q<tYornYorq<s %] \%]K^ `>b4cfe(Th)h>0 g h>jnpug cfe j m jst9q c fq jlp k # g c jlk-p m j Ω wxhpug || b pj gNj k+ 1 > N/2 w %" b k-p o | b g k e b g e v ∈ Hk+1(Ω) R c h e jlk~| b j rhv jup e c jh m ' h c j o j ec
j- c p e j7g hj b hp e q4h e j C o c h m |jh m q4h e j m j h j em j v o e jlf j gNj‖v − rhv‖H1(Ω) ≤ Chk‖v‖Hk+1(Ω).
¯ Dµ0
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 # $ !66# "! # +N ≥ 2
6±f6aHO,O&ADCjO9M<U_V6Q[^] NPCLE,OSA_CjO[WA°QSU_M<U0ERNPORNWU_C ¯ µ6W ¯ C6U_VFEMFU0V6_U0CLEGIU0CLGI[WV6QSH#djVLADCjOZ[WAGIU_C0HQS_HCFGIH;f6HT[WAaXIOSK6U4f4H#f4HYEXY[PXYaHCjOSEFC6NWE
Pk
Yh+3´q<s'tYnycm4nporq<s w!k £'hqFy4:[3 | ] \]B^ \] ;CA_M6M6[WNWdjV6H[PHGA0f4QSHA_LE>OSQSA_NObf4H[WA (U0VLE>E,HpGORNWU_C ¯ W_0/4L°§U0V6QBf6XaU_CjORQSHQ ¯ E U0C V4ORNW[WNWERHJ[WHJ vHaaH ¯ P0 µ¡A¤_HYG V = C∞c (Ω)
djV6NHYE,O6NPHYC¥f6HCLERH f6ADCFEH1
0 (Ω)®2U0aaH
C∞c (Ω) ⊂ Hk+1(Ω)
[^] HpE>OSNPa\AcOSNPU0C ¯ Dµjf4H[WA°QRU0MFUjE,NPORNWU_C ¯ µLP ¯ M<HQSaHIOf6H _XYQRNPLHQ¡[] K`jM<UDOSK6YERH´ ¯ 0\f4V vHaaH ¯ W_ µ M<U_V6Qf4HYE«U_CLGIORNWU_CLE-QSX0V6[WNPYQRHpE§U_C\C] AbMLA_EFHpE,U0NPCf4H9[WATGU_CLf4NPORNWU_Ck+1 > N/2
f6ADCLE[WAT°?QRU0MFUjE,NPORNWU_C¯ µ6W ¯ I°§U0V6Q-U_4OSHC6NWQ-[^] HpE>OSNPa\AcOSNPU0Cf] HYQRQSHV6Q ¯ 4pvU_CV4OSNP[WNWERH[WH vHaaHf4H9®2XYA ¯ W_0/9djV6N4CLU_VLEf4NO9djV6H‖u− uh‖H1(Ω) ≤ C inf
vh∈V0h
‖u− vh‖H1(Ω) ≤ C‖u− rhu‖H1(Ω),
E,Nrhu
ADM6MFADQRORNWHCjO'6NWHCZH1(Ω)
p°-A_Q§A_M6M6[WNWGYAcOSNPU0Cgf4H[WA;°QSU_M<U0ERNOSNPU0C ¯ µ6W ¯ ZuU_CgU04ORNWHCjO ¯ 4¤
G:J" > 0 P0(E 2 .(;" 99Quh
2 (Y+(9% ? "54 ""9R% -9% %P99 C 0 DTF " V0h 0 (U , )++%9 ,-, ) )R"-,)9 9"9 %9 - S%(9)9. $-PS ""S( /+9%(,) Kh
)S(+), ;9bh 0 " "/9% Q"N
).-("3$-R-,)9 ,/'9 ,)"+ K "3%9 /9%BP9% Q" 0 -%" +%"B9%(%+ 7689 " 3 "'Q$ P-99 ?%+$-; $9(X)9J"P "< " 7 97#$"%Pk
,)P.M)9' ", 9C .%& " % F0" B$/+9%, %)M+%"('6O9X@ ":QP P-9"939%(%+ @",, "): %P9) I+""
N = +%"(R" K 2@)R",9*@()3 C#)@ "
P2k−29@54 29% -9% 4 )+9 " C 0 E/F)2:9 293.-(; C uh
23(S+"9% "%+,) > 9*,, " .M , %(9 --9% P)9%(Q F0 -R") " )P9S#$"%(P1
" "9%%(+ ()B6O9X Q$ /9"9 C 0 DDF C 0 D F 9"@)+9! "*9 , %+% "'.%&9 ++S *, P.M9 ", 0 •
ΩΩ h
¯ P ¡±MLM6QRU¤4NWa\AcORNWU_CMLADQVLCf6U_a\ADNWC6HbMFU0[P`0XYf4QSNed0VLHΩh
f] V6CU0V6_HYQ,OQSX_VL[PNWHQΩ
"%(Q"X;9% > .M X, Q"%7+ ++ (-99Q " -%Ω
G4 2 $ C 9%(Q C %+ 9- '9 %) F0 , , S$/X9,J")
Ω$/!Y " -%
C 9%(QΩh
QP"*54 -%(*-@ %( Th C .M%&@(ED1%"9 0EMF0 @,J"%+%& Ωh
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
'/ "+ n " I # $ !6
)'+ % - C /. , hX "%( > +9! /R"%X -%( F "!9 X -% > 9SQG4 %1)R%29S", "29
C C Q%! $ Q ( , 9;9 " Ω F .M)9%&#$dist(∂Ω, ∂Ωh) ≤ Ch2.
- uh
( +(9% "U 4 -"9R% -9% .-9%(-9%" " 54 )+$,)Vh
-++, % % - Th
) K 54 S#$%Pk 0 " ")5 +%7 ,J"%+%&
Ωh ⊂ ΩVh
< ='? = < =' = = # "!54 )+$,) ; .
V " Q" N ,J")9,J(+"9% R-,)9
u C $/)R +%<() , $ %9% R %%&9 + " - F ,)QP"%@, "(%(Q +)9% + )P 4 -$ C+ 0% -%" B "% +% N = 2
" P+9 C .M%& " % F Q-())() 9#$%P1"+%
u ∈ H2(Ω)$-9< X9 "9
‖u − uh‖H1(Ωh) ≤ Ch‖u‖H2(Ω), C 0 DMF$ % Q""@ < 9:#%
P2"+%
u ∈ Hk+1(Ω)-9 + "()
‖u− uh‖H1(Ωh) ≤ Ch3/2‖u‖Hk+1(Ω). C 0 F -, "+Q -, )+939JP "7 2 9-9%265-%99'" "))() 91#$%
P1-"%(9Q":(*, P.M) = ", C 0 DF 2 "X 9:QP" C 0 E/F % "), .97) "X"9% -(7" " )P9#$%
Pk.M ,
k ≥ 2 0 *9 %) K , )+9 +%&9-9%Y Y%+9P ""%9! ") )() 9 #$"%%(+$/)+9%(Q $ F % 4 %&S "S %(( -+9%B Ω
> * ; $/ ")* % K ; " ,)"9;)$ ;"9 P"7$/@ "6O9 /9% "N = +%"(R"):2 - C , )+9' ")689 /9%( )23"' =)-(%9/9%( "B( +-2689 -9% "B%+9P "%&9 K V /Q 0 0E F0 -S)R " "% +%
N = 2 9%(%++".M 3"'+-26O-9 -9% C#% - " 9 T Q% "6O9 +9% -(* "'9)68)9 ", * ?+9% -( "3 ")3, 9 3)2@ -9, '$/;( 0 ( =)* !%()"R -9,J")*(B689P9% > 9 ∂Ω
-∂Ωh 0 )"'9! +9)<"X)29%( /9%
W4 )+9 "9% P9!QP" C 0 E/FC .M%& " E % " % F0 •
++ -%(9 K "2+-9% Y( 9+%&9% 0 0E Q54 " "++9 "9 % > 9 ,9"9 0" X++!$/'(, "2+9,)9% W4 )-9)" 4 %(9)9(-9%(, 3 "!,J$QP" % " -%(( 0 P%& K
"N = +%)R B+9 %(%B 4 9 k
Σk 0 ")#%&@54 )-9)"' W4 %P99(-9%
rK""@9"9'68,9%(
v, 9%P*+
K
rKv = p ∈ Pk9 GQ
p(x) = v(x) ∀x ∈ Σk. C 0 F 4 " > ( G 0 P0 X9-%&QP"I9" 1C : Pk
2 ")99%("7 7 -% > 97"%(QI$-+ :. "9:"R %P97 "Σk F -3, +Q" C 0 F ")#%&3;"%() rK
, " "%(, -9%C (%"%&9S "*+9,)9 ( F0 & - /@ ? " $ ( 24) &., 49 k + 1 > N/2 : &5 .+ 45 # - (+ 5 & 01)+-&.(
rK2+ 0 - (2)- 52 "&)(9+- (4 # Hk+1(K) # .(9 Hk+1(K) +%-0 .- 2+ 4( 2&.(2+ .(9+
C(K)+ ,0 0 49 &)45 + &)4+ v ∈ Hk+1(K)
&)(
‖v − rKv‖Hk+1(K) ≤ C(K)|v|Hk+1(K), C 0 $ F& |v|Hk+1(K)
"+ 01 ,76- ( &.5"7 # (9- .5
|v|2Hk+1(K) =∑
|α|=k+1
∫
K
|∂αv|2dx = ‖v‖2Hk+1(K) − ‖v‖2Hk(K).
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 # $ !66# "! # +N ≥ 2
_µ; <>='? & ?A@B< 1"
k+1 > N/2 :J" > D"0 0 T % "%(Q:Q" Hk+1(K) ⊂ C(K)
, .- "9<",)9 " @6O,)9%*
Hk+1(K)+';"% ")#% <, 6O9) %-%&9
,)9%" 0 /7,)+ QP" rKv 27"B C#3 "I , B, % 9: ") $ (%"%&9
!, 9% ' "v ∈ Hk+1(K)
W)!, , % "W4 %(-+9BQ))+$,)Hm(K)
.M ,m ∈ N 0
""%:Q rK 27%("%&9*,)9% "
Hk+1(K) 0 +9 %9 :54 % %&9‖v‖Hk+1(K) ≤ C(K)
(
|v|Hk+1(K) + ‖rKv‖Hk+1(K)
)
, C 0 MF) 9, -3$/3, + "%,)9% C , @:54 .M"@ " K 6O%&@@ W4 +9 :%)M%&9 ! .M%&-R")1(Y " 2+-9% C5D0E F " 54 % %&9 " 1%, -9 F0 7)R%(29 ", " +%&9vn ∈ Hk+1(K)
9 !QP"1 = ‖vn‖Hk+1(K) > n
(
|vn|Hk+1(K) + ‖rKvn‖Hk+1(K)
)
. C 0 MFW'9)9' "'",J" " C 0 MF %%(Q'QP"'(X+"%&9 vn
2@;9" "Hk+1(K) 0 /<"% =, -9% " :J - > * V %(,J D0 0 TPE $%)R%29*"'+" = +%&9 vn′
Q%$, P.M)9* "Hk(K) 0W*9)9 ! 9%&9! C 0 F %$ %(QP"Q( +"%9S < ")9%.M ) ∂αvn′
"<9"< "&9% = %$ %, |α| = k + 1
N, P.M)9.M)9 9 "L2(K) 0 -X, "+Q vn′
, .)9 "Hk+1(K).)93"*%(%&9
vQ%.M)9%&# C +9-P K ( %%&9! - C 0 F+F
|v|Hk+1(K) = 0, ‖rKv‖Hk+1(K) = 0. C 0 FGS9 % > 9')M%&9! " C 0 F +9 QP" v ∈ Pk
,/K
2@, R" C $/@"%,-9% 9 %&9)9 ( 9+%&9% D0 T0 F0 / ( )#$"%&9%( C 0 F " rK
rKv = v"
v ∈ Pk 0 N "R% > *)M%&9X " C 0 F +9 ",!Q rKv = v = 0$, SQ%W 2<", + %,)9%.M ,
(%%9< B9)9@ "@M-,J " C 0 MF0 17;"9 "%& C 0 $ F "%(QP" C 0 F7K (v−rKv)
9) -Q$-P:QrK(v − rKv) = 0
<QP" |v − rKv|Hk+1(K) = |v|Hk+1(K)
"%9QP" < )9%.M 4 9
k + 1 W4 C#' "
Pk+P<P 0
4 %, . % 7 " W 0 P0(E <HIX; = '%;)+ 23Q"3 , 2-P9< "54 %(" M-%9C 0 $ F Y
K "% > 9 "U)R%, %&9 0 U)Y9 , %+B( $ "", ""(X
+"%.-P 0 $ ( "4!&)' 49 k + 1 > N/2
+ 49 diam(K) ≤ 1 : 0 .- 2+ 4(*E 2&.(92+ .(+ C@< < <?
K+ ,0 0 4 &.45 + &.4+ v ∈ Hk+1(K)
&)(
‖v − rKv‖H1(K) ≤ C(diam(K))k+1
ρ(K)|v|Hk+1(K). C 0 E/F
; <>='? & ?A@B< "9%%(+ ( V /QP" 0 P0E Q%@ 9 Q 9"N = +%)R K
254 % Y- " +"26O-9 -9% BU "
N = +%)RU " 9)68)9 ", K0
3 ")#%'- C 0 D F0 "+9 )P: "%&"%(W)R%29*"* -+9%, %P.M)9+%;"(B)<" .M),)9 "
b C " -P3 " K F 9)NQ"39"x ∈ K
%NR"%29x0 ∈ K0
.M)9%&#-Px = Bx0 + b. C 0 TF
1"';"9)%& C 0 E/F W-+ X54 % %&9 C 0 $ F )-;%B " K0)'U"%(Q! ,J$" =)P3 "*.-9%(;" C 0 TF0 (B.S)9)++9 '+9".M)@(B $ "", *$/@-+ K K
"*(,)2-9 -@, )+9*%" M-(%&9 0 @ )%"<@, ,-, G "P:"@%( "%(Q3+% =)P1()9%, %$- I C B, ":;)+%3(),)9 ""+', +"9" % F0 WM, ;% ,J$- I "@./9%(;@)-P
det(B)M)I : )9%.M I -
K 4 ;"9 K -+9%&: : )9%.M
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
>F "+ n " I # $ !6
"K0
$-7, +%&9%( /. ,B−1 M%(R"%29<" , 2-P9 C
P%$ $ "9< "KM9 *Q
"39"9*68,9%( 9 % > 9 v(x).M ,
v0(x0) = v(Bx0 + b)
|v0|Hl(K0) ≤ C‖B‖l | det(B)|−1/2 |v|Hl(K)
|v|Hl(K) ≤ C‖B−1‖l | det(B)|1/2 |v0|Hl(K0).
" "%&< ,* " C 0 $ F|v − rKv|H1(K) ≤ C‖B‖k+1‖B−1‖|v|Hk+1(K)
‖v − rKv‖L2(K) ≤ C‖B‖k+1|v|Hk+1(K).
- % "9 .M9%#'65-, % QP"
‖B‖ ≤ diam(K)
ρ(K0), ‖B−1‖ ≤ diam(K0)
ρ(K).
" , X;%$-P@,) 39 +&-9@ ;9%()P C 0 E/F0
; <>='? ?A@B< - =A@B?A@B< N /',)2+9",)9%W$+%v ∈ Hk+1(Ω)
$+ %P9 = rhv
9 2+9 %9!N = +%(")R K
2<+% rKv 0 - , +Q)P
‖v − rhv‖2H1(Ω) =∑
Ki∈Th
‖v − rKiv‖2H1(Ki).
-"(%(QU( 2--9% C 0 E/F K ,JQU %Ki C .M),Y( , "29
C"
9"9) F ):,)@ -%((' 2I9 "%(354 %" M-(%&9 C 0 D MF 9)I "< 9)3"%&6O-9 (*--+diam(Ki)/ρ(Ki) 0 " %&
‖v − rhv‖2H1(Ω) ≤ Ch2k∑
Ki∈Th
|v|2Hk+1(Ki)≤ Ch2k‖v‖2Hk+1(Ω)
, !Q% 2<*9 +&- +%&9 0
;u|jyc¢_or¢0| ] \]B^ ="(%)%@*<>"%?# 9) & 2 787 2 )%; "7 >% !#<>"%@9# ;)7 ; & )3 P1 7 "<>;)% 2)%9 )%.<>7 2 " rh ;)% :> 9 & ) " N = 2 " 3
‖v − rhv‖L2(Ω) ≤ Ch2‖v‖H2(Ω).
*)*) O M Uv/21S B/;=RS3U':<1+P-/ ?8ORP§=,3US(N[WH¡f4U_a\A_NPC6H
ΩHpE>Of4H\O>`MFHQRHpGO&ADC60V6[WA_NPQSHrG_] HpE>OR ZD f4NWQSHdjV6H
ΩHpE>OJV6CU_V60HQRO#MFU_[W`_Xpf4QSNWdjV6H\f4U0C0O#[WHYET«rA_GIHpETE,U0CjO#MFHYQRM<HCLf6NWGV6[WA_NPQSHYETA_V4AD4HYE&lU_CM<HV4OJ[PH\a\ADNW[W[PHYQ#MLADQ#f6HYEQSHYGIOSA_C6_[WHYEr_U_NWQ[eA !NP0V6QRH ¯ / E !HO?V4ORNW[WNWERHQVLC6HaXIOSK6U4f4H9fl] X[WXaHCjO&E-LC6NeE?A0f6ADM6ORXH0_U0VLEA_[P[WU_CFEBf6XILC6NWQTf4HYEBXY[PXYaHYCjOSEBLC6NeETf4HJO>`MFH\ vA__Q&ADC60H¡rG_] HpE>OR ZD f6NPQSHJf4U0CjOb[WHYEbf4HY_QSXYEBf4H[PNP<HQRORXTERU_CjOf4HYEcAD[WHVLQSE2MFU0CLGOSV6H[W[WHYEf4HB«U_CFGORNWU_CFESI6f4NO&EX[WXaHCjO&ELC6NeEQk
F®2U_aaHCGU_CLEMFADQf4XLC6NWQV6CN^QSHYGIOSA_C6_[WH
Kf4H
RN GIU_aaHb[PHTMFA¤_XC6U0C4 f6X_XYC6XQSXp ∏Ni=1[li, Li]
A¤_HYG−∞ < li < Li < +∞ ;CC6UDOSH
(aj)1≤j≤2N
[PHpEE,U0aaHOSEf4HK
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 # $ !66# "! # +N ≥ 2
0
¯ / E -4HYaML[PH#f4HTa\A_NP[W[WA__HBQRHpGO&ADC60V6[WA_NPQSHBHCf4NPaHYCLE,NWU_CN = 2
Yh[Z§s'ornYorq<s %] \%] Q^<`>bdcfe
Ω g h b g i8jk e b hNhj-+j2| b -m k c gNj m j RN w$h 3´mLorrmA<|yc|¢4=nYm6sAFk!rmLoyc|m j Ω jlp e g hjhpjt.j Thm j N kvj- e qdh jlpzh b h m#[ h k p (Ki)1≤i≤n
g ci k c ' jh e'w Ki ⊂ Ω j e Ω = ∪n
i=1Kio
,+w7R c h e jk-pj- e~cBb h Ki ∩ Kjm j m jlg - N kvj- e qdh jup m4c p e~c h e pjup e g h m kvj- e qdh j o qdi8j-
0 ≤ m ≤ N − 1 o m'b h e"eb g+pjlp p b tt9j e p p b h e q4g+p-p cm jup p b t t.j e p m j Ki j e Kj wz 2h m4c t9jlhNp cBb h N = 2 o R c h e jk-pj- e~cBb h m j m jlg -kvj- e qdh jlpjup e p b4cfe i cBm j o p b4cfe g hp b t t9j e b tt g h o p bdcfe g h>jrlq-j b t tg h>jjh e~c + k@jIw jlp tYq3 3´|jnptb g s!q|k'w!tm g t9q c fq j Th p b h e jup p b t t9j e p m jlp N kvj- e qdh jup Ki
g cj b t| b pjh e w q4k7 b h_i8jh e~cBb h o j|Nq4k@q4t + e kvj h m p c h>j7 j7t9q - c t g t m jlp m4c q4t + e kvjlp m jlpN kvj- e q4h jup Ki w9U_VLEbf4XLC6NeERERU_CFE;[^] HCLERHaJ6[PH
Qkf4HpEBMFU0[P`C"B0aHpEbZGUjHWGINWHCjOSE;QRXYH[eEbf4H
RN f6A_CLERf4H9f4HY_QSXNPC6«XQSNPHYV6Q?U0V\X0A_[LZ
k~§m4yJy¤m6~'~?qFy¤n ¢¤£'mLilk'|z6m4ycorm§r|<GD] HYE,O,Zcf4NPQSHdjV6HOSU_V4O
p ∈ QkE] XYGQRNPOE,U0VLE[eAg«U0QRaHp(x) =
∑
0≤i1≤k,...,0≤iN≤k
αi1,...,iNxi11 · · ·xiN
N
A¤0HYGx = (x1, ..., xN ).
XHa\ADQ&djV6U_CFEdjV6H[WH9f4H0QRXOSUDO&AD[Ff6HpM<HV4O¦IOSQRH9ERV6M<XQSNPHYV6Q?Z
kjGIH9djV6Nªf4NC<XYQRHYCLGINWH[^] HpE,MFA_GIH
Qkf4H
Pk°§U0V6QORU0V4OHCjOSNPHYQ
k ≥ 1U_Cf4XLC6NPO[PHTnYy¤|orrotJw PqFycwvyc|
kf4V
N QRHpGO&ADC60[PH
KGIU_aaH[] HCLERHaJ6[WH
Σk =
x ∈ K ORH[djV6H xj − ljLj − lj
∈ 0, 1
k, ...,
k − 1
k, 1 M<U_V6Q 1 ≤ j ≤ N
. ¯ ¯ µj
°!U0V6Qk = 1
NW[?EY] A__NPO#f4H[] HCLERHaJ6[WHf4HYETERU_aaHIO&ETf4HKlHO#MFU0V6Q
k = 2HION = 2
f4HYEE,U0aaHOSEYLf4HYEMFU0NPCjOSE9aNP[WNPHYV4f4HYEA_QR¦ORHpEQRHY[PNeADCjO9f4HYV4ERU_aaHIO&EFHIO9f4VLA_QR`4GHCjORQSH\_U0NPQ[WA !NP0V6QSH ¯ /6pI vH-ORQSHNW[W[PNeEΣk
fl] V6CN QSHYGO&ADC60[PH
KHpE>O'k!s!ortpqFrz6m6sªn-MFU0V6Q
QkpGD] HYE,O,Zcf4NWQRH-djV] NP[0MFHYQRaHIOf4H#GA_QSA0GORXYQRNeERHQOSU_VLE[PHpEMFU0[P`C"B0aHpEf4H
Qk
Ecol
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lyte
chni
que.
Ne
pas
repr
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re.
¯ "+ n " I # $ !6
Σ1 Σ2 Σ3
¯ /6 0'QRHYNP[W[PNeEgf] U0QSf4QSH_/4'HIOµ M<U_V6QJV6C´QSHYGO&ADC60[PH[WHYEJQRU0CLf6E#QSHMLQRXpE,HYC0OSHCjO#[WHYEM<U_NWCjOSEf4VOSQRHYNP[W[PNeE&;2|+3 3´| %] \%] `>b4cfe
K g h N kvj- e qdh jIw `>bdcfe g h jh e~c jlk k ≥ 1 w %" b k-p o e b g e | b hdt.jm j Qk jlp em'le jlkut c h 9m jt.qdh c + kvj g h c gNj|qdk pjupEi4q4 jlg k-pqdg -E| bdc h e p m g e kvj c f c p m b k m k@jk o Σk
o m ' h c |qdk9z w ldwYh+3´q<s'tYnycm4nporq<s#] ;C0XQSNLHbdjV6HT[WHbGYADQ&f4NWCLAD[f4H
ΣkHIO[eAf4NWaHYCLERNPU0Cf4H
QkGIULCFGINef4HCjO
card(Σk) = dim(Qk) = (k + 1)N .®2U0aaHB[^] A_M6M6[WNWGYAcORNWU_CdjV6N4ZgORU0V4OM<U_[W`C"B_aHbf4HQk
j«rADNPOGU_QSQRHpE,M<U_CFf4QRH;E,HpE2cA_[PHYV6QSEERV6Q[PHOSQRHYNP[W[WNWEΣk
HYE,O[PNWC6XYA_NPQSH_0NP[lERV WOf] HKLNP<HQV6C6HBLA_ERH;f6HQk
f4U0CjO2[WHYE2X[WXaHCjOSEcAD[WHCjOB9HCVLC³M<U_NWC0O#f4V³ORQSHNW[P[WNeE;HO E ADNW[W[PHYV6QSEBM<U_V6Qbf4XYaU0CjORQSHQB[PHQSXYERV6[O&AcOp>(U0NOTV6CMFU0NPCjOxµ f4H
Σkf6XILC6NvMLADQxµ
j − ljLj − lj
=µj
k
A¤0HYG0 ≤ µj ≤ k, ∀ j ∈ 1, ..., N.
;Cf4XLC6NPO[PHTM<U_[W`C"B_aHp ∈ Qk
MLA_Q
p(x) =
N∏
j=1
k∏
i=0i6=µj
k(xj − lj)− i(Lj − lj)(µj − i)(Lj − lj)
A¤_HpGx = (x1, ..., xN ).
;C_XYQRNPLHT«rA0GINW[PHYaHYCjObdjV6Hp(xµ) = 1
OSA_CLf4NeEbd0VLHpEY] ADC6CV6[WHERV6Q;ORU0VLE9[WHYEbA_V4ORQSHYE;MFU0NPCjO&Ef6H
ΣkLGIH#djV6NlHpE>O[WHbQRXpE,V6[POSADOf4XYERNPQSX_
®2U0aaHf6ADCLE;[WHJGA0EOSQRNeADCL_V6[eADNWQRHTCLU_VLEBA¤_U0CLE[eA¡GIU0CLf4NPORNWU_C³ERV6NWcADCjORHf4HJGU_CjORNWCV6NOSXJZOSQSA¤0HQ&EV6CLH?«rA0GIHbC6U0VLE![WA_NWESE,U0CLE[eABf4XaU_CFE>OSQSADORNWU_CYORU0V4O§Z9«rADNPOE,NWaNP[eADNWQSH2Z9GH[W[PHf4V HYaaH¯ µL FF4ADV[PHpGOSHV6Q;2|+3 3´| %] \%] N\ `>b4cfe
K j e K ′ m jg - N kvj- e qdh jlpnq qdh e g h>jrlq-jE b t t g hj Γ = ∂K ∩∂K ′ w `>bdcfe g h jh e~c jlk k ≥ 1 w/%" b k-p o jg k e kvj c f c p m b k m kvj k Σk j e Σ′
k b h cBm jh e pug ks-j eKe j
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 # $ !66# "! # +N ≥ 2
[G
rlq-j Γ w j|Dg+p o le q4h emb h_h pK j e pK′m jg -E| b hdt.jlp m j Qk
o qnr b h eKcBb h v m' ' h c j|qdk
v(x) =
pK(x) p c x ∈ KpK′(x) p c x ∈ K ′
jup e b h e~c hNgNj7pug k K ∪K ′ o p c j e pjlg jt9jlh e p c pK j e pK′b h em jlp7i8qdjg k-p g c b h cBm jh e
q4g - | bdc h e p m g e k@j c f c pspug k qrlq-j. b tt g hj Γ w?CM6Q&AcOSNWdjV6H0ªU_C³V6ORNW[PNeE,HERV6Q,OSU_V4Ob[WHYEBHYERMLA_GHYEQ1
HOQ2
l vA'NW_VLQRH ¯ 0//aU0C0OSQRHV6C6H«U_CLGIORNWU_C³f4HgLA_ERHQ1
HYC³f4NWaHCLERNPU0CN = 2
U_C³MFHYV4O;`_XYQRNPLHYQ9djV6H#[WHYE9«U_CLGIORNWU_CLE;f4HQ1C6H#E,U0CjOMFA_EA4W\C6HpEMLADQaU0QSGHYADV6¡GIU_aaHTGIHY[P[WHYEf4H
P1
¯ /'/ LU_CLGIORNWU_Cf4HTLA_ERHQ1
HCf4NWaHYCLERNPU0CN = 2
9u|jyc¢_o¢_| ] \]B^ @ K = [0, 1]2 7 ' > ; > 9 & > " N = 2 9) 6 && !a1 = (0, 0) a2 = (1, 0) a3 = (1, 1) a4 = (0, 1) / 9;3 x3 = 1− x1 x4 = 1 − x2 ! i ! && 7 2 27 )%9) i & @97 / 5%'!#=6 )62 " @'2'*" :6 && # ai #69;3 < 2)% xi = xi+1 = 0 / ?;)% >%*" 7 # ! " 9 2 C9) Q1 6
pi(x) = xi+2xi+3 <>"% 1 ≤ i ≤ 4,
! * !>787 9) Q2 6Pi(x) = xi+2(2xi+2 − 1)xi+3(2xi+3 − 1) <>"% 1 ≤ i ≤ 4Pi(x) = −4xi+2(xi+2 − 1)xi+3(2xi+3 − 1) <>"% 5 ≤ i ≤ 8P9(x) = 16x1x2x3x4.
.¡|+3´m4ycilk!| %] \%] C³M6Q&AcOSNWdjV6H0LU0C QRHYaM6[WA0GIH#MFADQR«U_NeE[^] XY[PXYaHYCjO9LC6NQ2
MFADQ9VLCADV6ORQSHX[WXaHCjO!LC6NM6[WVLE'ERNWaML[PH0pHO'OSU_V4O-ADVFERERN0HWGA_GH_pC6UDOSXQ∗
2
¤CJf4NWaHCLERNWU_CN = 2
¤[] X[WXaHCjOLC6NQ∗
2
HYE,O?f4XILCLNLMLA_Q-[PHpE «U_CLGIORNWU_CLE?f4HLA_ERH(pi)1≤i≤8
f4H[^] HYQSGNWGH ¯ µ6W U0CA;HYC6[WH_X[WAf4HQSC6NWQSHp9I ;C _XYQRNPLHdjV6H[WHYEf4HY_QSXYEf4H[PNW<HQRORXf4H
Q∗2
E,U0C0O[WHYEE,U0aaHOSEHIO\[WHYE
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
"+ n " I # $ !6
aNW[PNWHV6 f4HpEADQS¦IOSHYE\f4VQSHYGIOSA_C6_[WHa\ADNeEMLA0EERU_C LADQS`4GIHYC0OSQRH¤-C f6NPaHCFE,NWU_CN = 3
[^] X[WXaHCjO9FC6NQ∗
2
HYE,OBf4XLC6N!MLADQBE,HpE9f4HY_QSXYEf6Hg[WNP<HQRORXd0VLN!ERU_CjO9[WHYE E,U0aaHOSE9HIO;[WHYEJ /aNW[PNWHV6f4HpEADQS¦IORHpEf4VGIV6<H\NW[vC] `¡AMLA0Ef4H#f4H0QRXpEf4HT[PNW<HQRORX#Z[] NWC0OSXQSNPHYV6Q •
Yh[Z!s!onporq<s %] \%] 2e qdh emb hNh g ht9q c fq j.k@j@ e q4h g q c kvj Thm g h b g i4jlk e Ω o q)t
e b m j m jlp t9jh e p ' h c p Qk jup em ' h c j|Nq4k7R jlp~|q'-j mdc plkvj e
Vh =
v ∈ C(Ω)e jl gNj v |Ki ∈ Qk | b g k e b g e Ki ∈ Th
. ¯ ¯ Fj
xh)q-||Njlfj"h b jg m p m jlp w!|+ALycht9w!|Jo ?|jy¤nYh R jlhNpjlt9l j m jup$| bdc h e p (ai)1≤i≤ndl
m jup e kvj c f c pm b k m kvj k m j qg h m jup N k@j@ e q4h jlp Ki ∈ Th w®2U0aaH?fLADCLEv[PH?GYA_EªOSQRNeADCL_V6[eADNWQRH0[WA9@9XLC6NPORNWU_C ¯ µL /02AVLCgE,HYCLEl0Q@,0GIHZ[eAM6QSU_M<U0ERNOSNPU0CERV6NWcADCjORH\rf6U_CjOC6U_VFE[WA_NWESE,U0CLE[WAf4XYaU_CLE,ORQ&AcOSNPU0CA_V[WHYGIORHVLQHC_V6NeERHbf] HHYQSGNWGHpIbycqF~?q<tYornYorq<s %] \%] jlpK|Nq-j Vh
o m ' h c |Nq4k z w*4 o jlp e g h<p b g+p jup~|Nq-j m j H1(Ω)m'b h e q mdc t.jhp cBb h jup e jh b t9kvj m j m j k p m j. c @jk e ndl w j"|> g+p o c j- c p e j.g h>j9-q8pjm j Vh (φi)1≤i≤ndl
m ' h c j|Nq4kφi(aj) = δij 1 ≤ i, j ≤ ndl,
e jfj gNjv(x) =
ndl∑
i=1
v(ai)φi(x).
®2U0aaH[WHYE#X[WXaHCjOSETLC6NeEQk
E,U0CjOgf4HYETXY[PXYaHCjOSEbFC6NWEJf4HO>`MFH vAD0QSA_C6_H0ªU_CMFHYV4Of6XaU_CjORQSHQ?[PHpEa¦aHYE?QRXpE,VL[O&AcOSE?f4H9GIU0C_HQS_HYCLGIHdjV6HM<U_VLQ?[eAbaXORK6U4f4HBf4HYE?X[WXaHCjO&E-LC6NeEPk-U_VFEg[eADNeERERU_CFEgADVx[PHpGOSHV6Q[PHERU_NWCxf4H_XYQRNPLHQdjV6H[eA³f6XaU_CLE,ORQ&AcOSNPU0C f4V 0K6XU0QRYaH¯ µLPpµE] ADMLM6[PNedjV6H¨,agV4O&AcOSNWE¡agV4O&ADCLf4NeE>© A_V ORK6XYU_QSaH ERV6NPcA_C0O[eA´@;XILC6NPORNWU_C ¯ µ6W_ f4Ha\A_NP[W[WA__HpE2QSX0V6[WNPHYQSEEY] XORHCFfA_NWERXaHYC0OA_V4¡a\ADNW[P[eAD0HYEQSHYGO&ADC60V6[eADNWQRHpESI
£'hqFy8:+3´| ] \] `Qb4cfe(Th)h>0 g h>jp-g cfe j m jnt9q c fq jlp"k@j@ e q4h g q c kvjup"k # g c jk-p m j Ω w`Qb4cfe
u ∈ H10 (Ω) o q p b g eKcBb hj-[q' e j m g |>k b + t.j m j c k c j e z dw - o j e uh ∈ V0h
o qyp b Dg eKcBb h q-||k b j)|Nq4k q t le b m j m jlp t9jh e p ' h c p Qk w %" b k-p q t e bIm j m jlp t.jh e p ' h c p Qk b h_i8jk j o N jlp e $ m4c kvj gNj
limh→0‖u− uh‖H1(Ω) = 0.
j2|> g+p o p c u ∈ Hk+1(Ω) j e p c k + 1 > N/2 o q4 b k-p b h q9R jlp eKc t9q e~cBb h m jkukvjg k‖u− uh‖H1(Ω) ≤ Chk‖u‖Hk+1(Ω),
b*)C jup e g hjE b hp e qdh e j c h m |jh m q4h e j m j h j e2m j u w
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 # $ !66# "! # +N ≥ 2
"I ")%+)<) (!-9%( -%((-'9 ,(%&9 )@ W4 =)P9*#%
Qk) "9%%9X "9% " +-26O-9 -9% 0 -
N = $/ > 954 % $/" -"(%,/9% F
,); %&9[0, 1]N C 2 = - > 9 2" $ =-()(- F0 .M)9%&# QG4 "
N = $/ > 9 2N
65,) $/ > T KYT )BQP"++9)%(%X)2!;% 54 % "Y+9)%(% Y,);B"%&9 0 !-9X -%()SY " %Ω-
N = $-- > (-9 ') ")#%&<9JP "! 4 93#$%@;$-+ !+" 54 % - F (Qk)
-,J$Q
N = $/ > 9 " 4 +, Qk"X( , "; "%&9 C ))-
F (Qk) 6= Qk F0 )"<"++%G9%(%+)' "):+2689 /9%(<3,)%(Q ) C " F8F ,4 )2 ( 9JP "* " )() 9S#% %+--)+9%(QP" C .M%&B( V /QP" 0 0E K , + 2) F0 /B)R "1 % =)+%N = 2
$54 "9%%(9/9%Y "+"26O9 /9%Q1
)9) "! -%() -%("S/. ,S " Q -()7Q)(,)$Q" C K 65-, 7"B--( > F0 1""7 @ ")-%( "9).CM K " % 0• <"++%N %)<"$-+9%!
Ω)
N = +%)R ")<"+9 N = 9),)" 7I,)2+9"%&9<"3)9J"P "3 4 9 #$% ("-I 7 " R! C Pk
)Qk 01"<"@ "! "% $:9 P.MAC3 ", 9 K " % 0 •
" ")#%& " 9 #% %9)9 %(%&9 )P+9Pk
)Qk
% +%
N = 3<- L )P9 #$"% "9%+ /9% Q" 4 - 9
k 0 P+" Q
Ω =ω×]0, L[
.M),ω.M)+7 "
R2 0 % ω$-: " +9%(
Ti)
]0, L[$-: " I+ )P9
[zj , zj+1] 0 "#$%& -9* ")<9%+ ' R3 , '(!9P "%&
Ti × [zj , zj+1].M , 9Q)(
-%(Ω 0 , 2+9%&@-9< " I6O",)9%"@ "';+ %(9)9 "%(%&9 3 +9<, < " Pk
)Qk+<, @"9%+ 0 1<< "* ")-%( ":9 P.MAC" K " % 0 •
*)* O M Uv/21S B/;=RS T5283 2145U$S 'A_XYC6XQ&AD[WNeERADORNWU_C f4H [eAaXIORKLUf6Hf4HpEX[WXaHYC0O&E\LCLNWEZ´f4HYEE,`4E,ORYaHpE¡fl] XYdjVLADORNWU_CLEADV4 f4XYQRNW_XYHYEJMLADQRORNWH[W[WHYErGIU0aaH[WHER`4E>OSaHf6H[^] X[eA_E,ORNeGINPORX[PNWC6XYA_QRNeERXHp#C6HMFUjE,HMLA_Ef4HM6QRU06[WaHYEbMLA_Q,OSNWGV6[WNPHYQSEY'®2H\C] HYE,O#MLA_Eb[WHGA0ETMFU0V6QT[WHE,`4E,ORaHf4HpETXYdjVLADORNWU_CLETf6H)(jORU _HYE4 G¤ZBGA_VLE,Hf4H[WABGIU0CLf4NPORNWU_Cfl] NWCLGU_aM6QSHYESE,NW6NW[PNPORXf4V FV6NWf6HbrU_VGIU_CFf4NOSNPU0CJf4Hf4NW_HYQR0HCLGHCjVL[P[WHMFU0V6Q-[eAbjNPORHpERERHpID ] NPaM<U_QROSA_CLGIHM6Q&AcOSNWdjV6HGU_CLERNWf6XQ&AD6[WHf6HYEERNPaJV6[eAcORNWU_CFE§CjVLaXYQRNedjV6HYEHCaXYGYADC6NedjV6Hgf4HpE LV6Nef4HpE9NWCLGIU0aM6QRHpERERNW6[PHpEA>VLE,ORNPLHJdjV6HJC6U_VLEMLADQS[PNWU_CFE6QSNPY_HaHYC0OBf4HgGIHGA_EMLADQRORNeGIV6[WNWHQ?d0VLN0M<HQSaHIO'A_VLERERN_f4H2aU_CjORQSHQvdjV6H-[^] ADCLA_[P`4ERH-CjVLaXYQRNedjV6HC] HYE,OvMLA0ElOSU_V'A>U0V6QSEGIHb[PU0C6 LHVL_HBORQ&ADCLdjV6NW[W[PHTdjV6HT[^] U0CNWa\AD_NWC6HTZ[WA[WHYGIORV6QSHTf4H#GIHbM<U_[W`4GIU_MLNPX¤X9ADM6M<H[WU_CLETdjV6H0f6ADCFEbV6Cf6U_a\ADNWC6HFU0QRCLXGU_C6C6H4H
Ω ⊂ RN lHCM6QSXYERHCFGIHf4H«U_Q&GIHYEHIORXYQRNWHV6QSHYEf(x)
4HIOMFU0V6Qf4HYEGIU0CLf4NPORNWU_CLEADV6[WNWaNPORHpEfl] A0f4K6XYQRHYCLGIHBf4V LVLNWf4HbADV6MFADQSU_NeE[PHpEXYdjVLAcOSNPU0CLEf4HE(jOSU 0HYEE] XYGQRNW_HYCjO
∇p− µ∆u = ff6A_CLE
Ωdivu = 0
f6A_CLEΩ
u = 0ERV6Q
∂Ω
¯ ¯ 0U&
µ > 0HYE,O[eA NeERGU0ERNOSX¡f4V LV6Nef4H_-@BADCLEJ[WA(U_VFE>E,HpGOSNPU0C 4 µ60/C6U0VLEJA¤0U_CLE#M6QRU0MFUjE,XGIU_aaHB«U_QSagV6[eAcOSNPU0CcA_QRNeAcOSNPU0C6C6H[W[WHBf4H ¯ ¯ 00'QRU0V6_HYQ
u ∈ V ORHY[vdjV6H ∫Ω
µ∇u · ∇v dx =
∫
Ω
f · v dx ∀ v ∈ V, ¯ ¯0¯
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ E)E "+ n " I # $ !6
U'&VHpE>O[^] HpE,MFA_GIHTf4H 9NP[W<HQRO9f4XILCLNMLADQ
V =
v ∈ H10 (Ω)N ORH['d0VLH
divv = 0M MvLf6ADCFE
Ω
. ¯ ¯ GD
®2U0aaHVGIU0C0OSNPHYCjO[WAGU_CjORQ&ADNWCjORHfl] NWCLGIU0aMLQRHpERERNPLNP[WNOSX
divv = 0YNP[0HYE,OORQSYE'f4NW\GNP[WHHYCTM6QSADOSNWdjV6HBf4H;GU_CLE,ORQSV6NWQRH;f4HYEADM6MLQRU¤4NWaADORNWU_CLE¤A_QRNeAcOSNPU0C6C6HY[P[WHYE?NPCjORHYQRCLHYE?f6HJ ¯ ¯0¯ ?GIU_aaHC6U0VLE[^] A¤_U_CFE§«rADNPO A>VLESdjV] NeGIN^j°[PVFE?M6QSXYGNWERXaHYC0Op_[eA#f4NWGIV6[PORX;HYE,O2f4H;f6XILC6NWQ2ERNWaML[PHYaHYCjOBH4M6[PNeGINPOSHaHCjO?VLCERU_VLE, HYERMLA_GH
Vhf6HVf4HTf4NWaHCLERNPU0CLCLNPHTf4U0C0O[WHYEX[WXaHCjO&EEY] XYGIQSNW_HCjO_Q-,_GHA_V4«U_CFGORNWU_CFEBf6HFA_ERHJf4HpEbX[WXaHCjOSE;LCLNWE
PkU_V
Qkl°§A_QbHI4HaM6[WH_lERN Th = (Ki)1≤i≤nHpE>OVLCa\ADNW[W[WA__HBORQSNWA_C6_VL[WA_NPQSHBf6HB[^] U_V60HQROGU_C6CLHI4HbMFU0[P`0XYf4QSNWdjV6H
Ω6U0CM<HV4O9f6XILC6NWQ
Vh =
v ∈ C(Ω)N OSH[vdjV6Hdivv = 0
f6A_CLEΩ, v |Ki ∈ PN
k
M<U_V6QOSU_V4OKi ∈ Th
,a\A_NWETNW[?Cv] HpE>O#MLA0E#GI[eADNWQgdjV6HVh
CLHE,U0NO#MLA0E9¨>ORQSU_MMFHORNPO>©HIOf4H¡djV6H[W[WH\aA_C6NWQSH\U_CMFHYV4OGYADQ&A_GIORXQSNeE,HYQBERHYETXY[PXYaHCjOSETHYCORHYQRaHYETf6H¡f4H0QRX\f4H\[WNWFHYQ,OSX_v?CMLADQRORNeGIV6[WNWHQpl[WAGU_CLf4NPORNWU_Cdivv = 0
f6A_CLE'[WABf4XIFC6NOSNPU0CJf4HVh
aX[eADC60H?OSU_V4OSHYE'[PHpE!GU_aM<U0ESADCjORHpE'f4Hv¤GHdjV6NQSHCLfTORQSYEf6NDWGINW[WHgHIObGIU0aML[PNedjV6Xg[eAGA_QSA0GOSXQSNWESAcOSNPU0Cfl] V6C6HJLA_ERH#HML[PNeGINPORHf4H
Vh ;C C] V4OSNP[WNWERHJf6U_CLGMFA_E[eAg«U0QRaJV6[eAcORNWU_CcADQSNWADORNWU_C6CLH[W[PH ¯ ¯ GD2MFU0V6Qf4XLC6NWQV6C6HbaXIOSK6U4f4H#fl] X[WXaHCjOSELC6NeEC M6QSADORNedjV6H_U0CNPCjOSQRU4f4V6NPOgV6CLH¡ADV4OSQRH«U0QRaJV6[eAcORNWU_CcADQSNeAcORNWU_CLC6H[W[PHf6HYE#XpdjVLAcOSNPU0CLE#f4H
(ORU 0HYE!djV6N6GIU0CLE,NeE,ORH2Z;C6HMLA0E«U0QSGHQ§[] NWCLGIU0aMLQRHpERERNPLNP[WNOSX2f6A_CLE![WABf4XLC6NPORNWU_Cf4H2[^] HYERMLA_GH2HO§ZjADQ&f4HQ-[eA#MLQRHpERERNPU0C\GIU_aaHNWCLGIU0C6CV6H9f6A_CLE[eAb«U0QRaJV6[WADORNWU_CcADQSNWADORNWU_C6CLH[W[PH0_CaJV6[PORNWM6[PNeADCjO[eAM6QSHaNWQSHXpdjVLAcOSNPU0C¥f4H´ ¯ ¯ 0gMFADQV6CLH«U_CLGIORNWU_C¥ORHpE>Ov ∈ H1
0 (Ω)N HIO[eAf4HV6NWaHXpdjVLAcOSNPU0C¥MFADQV6CLH³ADV4OSQRH«U0CLGOSNPU0C¥ORHYE,Oq ∈ L2(Ω)
2U_C U_6ORNWHCjOADMLQRpENPCjORXY_Q&AcOSNPU0C¥MFADQMFADQRORNWH H6ORQSU_V60HQ(u, p) ∈ H1
0 (Ω)N × L2(Ω)/RORH['d0VLH
∫
Ω
µ∇u · ∇v dx−∫
Ω
pdivv dx =
∫
Ω
f · v dx
∫
Ω
qdivu dx = 0,
¯ ¯
M<U_VLQ;OSU_V4O(v, q) ∈ H1
0 (Ω)N × L2(Ω)/RQCNPCjOSXQS¦IO#E,VLM6M6[WXaHCjOSA_NPQSHJf6H ¯ ¯ HpE>OTdjV6H[eA\M6QSHYESERNPU0CC] `HpE>O9MFA_EXY[PNWaNPCLXHJGU_aaHJf6A_CLEg ¯ ¯_¯ D[§ERHQ&A\f4U_CLGgMFUjERERNW6[PHJf4HJGYAD[eGIV6[WHQGHIORORHcADQSNeAD6[WH#M6K`4ERNWdjV6HYaHYCjO?NPaM<U_QROSA_C0OSHp-A¤0HYGT ¯ ¯ I_U0VLE[eADNeESE,U0CLEADV[WHYGIORHVLQ?[WH9E,U0NPCf6HT_XQSNPLHQHC_VLNWERHTfl] H4HQ&GINeGIHb[PHTQSXYERV6[POSADOERV6NWcADCjOY
;2|+3 3´| %] \%] \%^<`>b4cfe(u, p) ∈ H1
0 (Ω)N × L2(Ω)/R w j b g |j (u, p) jlp e p b g eKcBb h m jz w u)p c j e pjg jlt9jh e p c jlp e p b g e~cBb h z0rlq c j o qdg pjlhNp m g -b k + t9j w [w u m jlp gNq e~cBb hNp m j `_e b jup9z dw w
D[ªHYE,OA_[PU0QSE«rA_GNP[WH;f4HBGIU0CLE>OSQRVLNPQSHV6C6HBADM6MLQRU¤4NWaADORNWU_CcADQSNWADORNWU_C6CLH[W[PHNWCjORHYQRC6HBf4H ¯ ¯ ;CNWC0OSQRU4f4VLNO[WHYEHpE,MLA0GIHpEf4NWESGIQSHIO&E
V0h =
v ∈ C(Ω)N OSH[vdjV6Hv |Ki ∈ PN
k
M<U_V6QOSU_V4OKi ∈ Th
HOv = 0
ERV6Q∂Ω
,Qh =
q ∈ C(Ω)/RORHY[vdjV6H
q |Ki ∈ Pk′M<U_V6QOSU_V4O
Ki ∈ Th
,
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 # $ !66# "! # +N ≥ 2
/ E d0VLN4_XQSNPLHCjO§6NPHYCdjV6H
V0h×QhHYE,O§VLCERU_VLE, HYERMLA_GH?f4H
H10 (Ω)N×L2(Ω)/R
f6Hf6NPaHCFE,NWU_CLC6NWH_L ?] A_M6M6QSU¤NWa\AcOSNPU0CcADQSNWADORNWU_C6C6HY[P[WHbNPCjOSHQSC6HTf4H ¯ ¯ ?HYE,O9E,NWaM6[PHYaHCjO
∫
Ω
µ∇uh · ∇vh dx−∫
Ω
phdivvh dx =
∫
Ω
f · vh dx
∫
Ω
qhdivuh dx = 0,
¯ ¯ j
MFU0V6QORU_V6O(vh, qh) ∈ V0h ×Qh
4-4M6[PNedjV6U_CFEGU_aaHCjOU_CQRXpE,U0V4OB ¯ ¯ 0?HCM6Q&AcORNedjV6H_CC6UDO&ADCjOnV
[eAgf4NWaHCLERNPU0Cf4HV0h
HIOnQ
GIH[W[WH;f4HQh
jU0CNWC0OSQRU4f4VLNO[WAJLA0E,H(φj)1≤j≤nV
f4HV0h
HIO#[eA¡FA_ERH(ψj)1≤j≤nQ
f4HQh
GIU0CLE>OSQRVLNOSHA¤0HYGg[PHpEB«U_CLGIORNWU_CLEbf4HLA0E,Hf4HYEbXY[PXYaHCjOSELC6NeE#0U_NWQ[eA°QSU_M<U0ERNOSNPU0C ¯ µL0G_ ;Cf4XYGU_aMFUjE,Huh
HIOph
E,V6QGHYEFA_ERHYEuh(x) =
nV∑
j=1
uh(aj)φj(x), ph(x) =
nQ∑
j=1
uh(a′j)ψj(x),
C C6U_OSA_C0OUh = (uh(aj))1≤j≤nV
HOPh =
(
ph(a′j))
1≤j≤nQ
U_C U_4OSNPHYC0O[PHE,`4E,ORYaH[PNWC6XYA_NPQSHTE,VLNPcADCjO(
Ah B∗h
Bh 0
)(
Uh
Ph
)
=
(
bh0
)
, ¯ G E
U&B∗
h
HpE>O[eAa\AcORQSNeGIHTA_f A>U0NPCjOSHrU_VORQ&ADCLERM<U0ERXHpf4HBh
bh =
(∫
Ω f · φi dx)
1≤i≤nV
6HOAh =
(
µ
∫
Ω
∇φi · ∇φj dx
)
1≤i,j≤nV
, Bh =
(
−∫
Ω
ψidivφj dx
)
1≤i≤nQ, 1≤j≤nV
.
HYE?G&K6U0ERHYE-ERHGIU0aML[PNedjV6HCjO[WU_Q&ERdjV] NP[FE] AD0NOf4H9ESA¤_U0NPQ-ERN6U_CM<HV4O?ORU_V>A>U_V6Q&E-QSXYERU_VLf4QSH[PH9ER`4E>ORaH[WNPC6XpADNWQRHT ¯ G E 'f4Ha\ADC6NWQSHV6C6NedjV6H_dXHYa\ADQ&d0VLU_CLE§djV6H[WA;a\AcOSQRNeGIHAh
HpE>O?ER`aXIORQSNed0VLHf4XILCLNPHMFUjE,NPORNW_H¡f] U0QSf4QSHnV
§djV6H[eAa\AcOSQRNeGIHBh
HYE,OgQRHpGO&ADC60V6[WA_NPQSH\f4HO&ADNW[P[WHnQ × nV
HIO;djV6H0LE,N'[WAa\ADORQSNWGHT_[WU_LA_[PHTf4H¡ ¯ G E HYE,O9E,`aXIOSQRNedjV6H#fl] U_Q&f4QRHnV + nQ
LH[W[WH#C] HYE,OMLA_Ef4XILCLNPHTM<U0ERNOSNP0H_69XYADCLaU0NPCFEU_CAJ[WHTQSXYERV6[O&AcOERV6NPcA_C0Op;?|+3 3´| %] \%] \ jp p e + t9j c h q c kvjEz dw u q m t9j e$eb g b g k-p2g hjp b g e~cBb h (Uh, Ph)
m q4hNpRnV ×RnQ w jsi8j- e jg k Uh jup e g h c gNj o e q4h mdc p gNj Ph jlp e g h c gNj $ R q m'm4cfe~cBb h.|>k + p m g h t9jlh em j KerB∗
h wYh+3´q<s'tYnycm4nporq<s#]#®2U_aaH( KerBh)⊥ = ImB∗
h
NW[HYE,O\«rA_GNP[WH f4H_U0NPQdjV6H´ ¯ G E gHpE>OXYdjV6NW¤A_[PHYCjOZORQSU_V60HQ
Uh ∈ KerBhOSH[djV6H
AhUh ·Wh = bh ·WhM<U_V6QOSU_V4O
Wh ∈ KerBh.
D['ERV WO9AD[WU_Q&Efl] A_M6M6[WNWdjV6HYQ[WH 0K6XU0QRYaHbf6Hg vADjNW[P0QSA_a µ6 µ6WbMFU0V6QU_6ORHCLNPQ[^] HI4NWE,ORHYCLGIHHIO;[^] V6C6NeGINPORXJf6HUh
f6ADCLEKerBh
F°-ADQ;GIU_CFE,Xpd0VLHCjOY! ¯ G E A_f4aHOBADVaU0NPCLE9V6C6HgERU_[WV4OSNPU0C(Uh, Ph)
f6ADCLERnV × RnQ
!®2U0aaHUh
f4U_NPOgADM6MFADQRORHCLNPQTZKerBh
NP[?HYE,ObVLC6NWdjV6HfLADCLE
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ E / "+ n " I # $ !6
RnV_°§A_QADNW[P[WHVLQSEYcU_C_XYQRNPLHADNeE,XYaHCjOd0VLH
PhHpE>OV6CLNWdjV6HZB[^] A0f6f4NPORNWU_CM6QSYE-fl] V6CX[WXaHCjOf6H
KerB∗h
'ZJU'&[PHpEG&K6U0ERHYEERH#GIU_aM6[WNed0VLHCjOY6G_] HpE>OdjV6HT[WHBCLUc`0ADVKerB∗
h
C] HYE,O$A,ADa\A_NWEQSXYf4VLNOADV0HYGIORHVLQ?CV6[LHOdjV] NP[FMFHYV4OMLADQR«U_NeE-¦IOSQRHORQSYE'¨,_QSU0E>©*0vU0V4Of4XYMFHYCLff4VG&K6U_NPdjV6N<HYE,O«rADNPOf6HYEU0QSf6QRHpEkHIOk′f4HYEXY[PXYaHCjOSEFC6NWEM<U_VLQ[WANPORHYESERHBHOMFU0V6Q[eAJMLQRHpERERNPU0C
;2|+3 3´| %] \%] \_\ jnh b q4g KerB∗h b h eKc jh e qdg)t b4c hNpjsi4j- e jlg k 1I
m j RnQm'b h eeb g e jlp
jup b t| b pq4h e jlp7p b h e q4 jup $ 1 w %"g e k@jlt9jlh emdcfe o q7|kvjlp-p cBb h m4c pk + e j ph jlp e q4g t b4c hNpm' ' h c j $9g h>jE b hNp e q4h e j|k + pwYh+3´q<s'tYnycm4nporq<s#]$(4U_NPO
rh ∈ QhHOwh ∈ V0h
4°-A_Qf4XILCLNOSNPU0CWh · B∗
hRh = BhWh · Rh =
∫
Ω
rhdivwh dx.
;Qrh = 1
A_M6MLADQRORNWHCjOORU0V'A>U_VLQSEZQh
6HIO9GU_aaH∫
Ω
divwh dx =
∫
∂Ω
wh · n ds = 0
M<U_VLQORU_V6Owh ∈ V0h
6U0CHYCf4XYf4VLNO9djV6HRh = 1I = (1, ..., 1)
A_M6MLADQRORNWHCjOZKerB∗
h
0 -1 +1 0
+1
-1
0+1-1
-1 +1 0
-1 0+1
0
¯ /_µ U4f4HJNPCFE>O&AD6[WHJf4HJ[WA¡M6QSHYESE,NWU_C ERV6Q9VLC³a\ADNW[P[eAD0HTORQSNeADC60V6[WA_NPQSHTV6C6NP«U_QSaHrX[WXIaHYC0O&ELC6NeEP1
M<U_V6Q[eANOSHYESE,HbHIO[eAM6QSHYESE,NWU_CFI;2|+3 3´| %] \%] \ b k-p gNj k = 2 j e k′ = 1 z t9jlh e p ' h c p P2 | b g k q i cfe jup-pj j e P1| b g k)q.|>kvjlp@p cBb hd o j6h b q4g KerB∗
h jlp esm j mdc t.jhp cBb h g h o jlh jh m k |qdkj)i8j- e jg k 1Iz!qdg e kvjt9jlh em4cfe o qs|kvjlp-p cBb h mdc plk + e j ph jlp e g h c gNj $g hj b hp e q4h e j|k + pdw$b k-p gNj k = k′ = 1 z t9jlh e p ' h c p P1 | b g k q i cfe jup-pj j e q|>kvjup-p cBb h o j6h b q4gKerB∗
h jlp e jh ' h kvqd m j mdc t.jhp cBb h p e k c e jt.jh e |> g+p kvq4h m j gNjng h z!q4g e kvjt.jh e#mdcfe o q|>kvjlp@p cBb h m4c pk + e j ph hQ jup e |Nq8p7g h c gNj o tt9j $g hjE b hp e qdh e j2|>k + pwYh+3´q<s'tYnycm4nporq<s#]$(4U_NPO
rh ∈ QhHOwh ∈ V0h
4°-A_Qf4XILCLNOSNPU0CWh · B∗
hRh = BhWh ·Rh =
∫
Ω
rhdivwh dx = −∫
Ω
∇rh · wh dx.
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
[0 # $ !66# "! # +N ≥ 2
/ E µ U_Q&ESd0VLH
k = 2HOk′ = 1
6[WHB0QSA0f4NWHCjO ∇rh HpE>OGIU0CLE,OSADCjOf6A_CLEG&KLA_djV6Hba\ADNW[W[PHKi
6f4U_CFG∫
Ω
∇rh · wh dx =n∑
i=1
∇rh(Ki) ·∫
Ki
wh dx.
;Q[WAJ«U_QSagVL[PHTf4H#djVLA0f4QSADORV6QSHbf4Hb[] -4HQ&GINeGIH ¯ µ6 GgC6U_VFEf4NO9djV6H04MFU0V6Qwh ∈ P2
∫
K
wh dx =|K|
N(N + 1)/2
∑
j
wh(aij)
U&[WHYE(aij)
ERU_CjO?[WHYEN(N +1)/2
M<U_NWCjOSE2aNP[WNPHYV4f4HYEADQS¦IOSHYE-QSH[WNeADCjO?[WHYE2ERU_aaHIO&EaiHOajf4H
KFC QSH0QRU0V6MLADCjO[WA¡E,U0aaHgERV6NWcADCjO;GIHpEM<U_NWC0O&EaNW[PNWHV4´d0VLN'ERU_CjO9GIU0aaJV6C Zf4HYV4aA_NP[W[WHYE&4U_CU04ORNWHCjO
∫
Ω
∇rh · wh dx =∑
aij
wh(aij) ·( |Ki|N(N + 1)/2
∇rh(Ki) +|Kj |
N(N + 1)/2∇rh(Kj)
)
.
C M6QSHCLA_CjOwh
djV6N9cA_V4O³ERV6Q[WH³M<U_NWCjO¡aNP[WNWHVaij
HO E A_NP[W[WHV6Q&E?U_CHC f6XYf4V6NPOdjV6HWh ·B∗
hRh = 0NWaML[PNedjV6H#djV6H
|Ki|∇rh(Ki) + |Kj |∇rh(Kj) = 0. ¯ Gp
vA«U0CLGOSNPU0Crh
Axf4U_CLG³V6C 0QSA0f4NWHCjOf4Hf4NWQRHpGORNWU_C GIU_CFE>O&ADCjORH a\A_NWEf4U_CjO[] U_QSNPHYCjOSAcOSNPU0CG&KLADC60HTf4HgERHCLEf] V6CLH#a\ADNW[P[WH#Z[] ADV6ORQSH_<®2U0aaHrh
ADMLMLADQRORNWHCjO9ZP1
HO9HYE,O;GIU0CjORNWCjVLHbERV6QΩDE,U0CJ0QSA0f4NPHYCjO!O&ADC60HCjORNWH[HpE>O?GU_CjORNWCVZB[^] NWCjORHQR«rA_GHHCjORQSHf4HV4JaA_NP[W[WHYEYc°§A_QGIU0CLE,XpdjV6HCjOYNP[ªHYE,OCV6[HO[eAE,HYV6[PHBM<U0ESE,NW6NW[PNPORXbf6ADCFEB ¯ G¤-HYE,Od0VLHB[WHB_Q&A_f4NWHCjOf6H
rhE,U0NOCV6[ªMLADQRORU0V4OYGD] HYE,O,Zcf4NPQSHTdjV6H
rhE,U0NO9VLC6Hb«U_CLGIORNWU_C GU_CLE,OSA_C0OSH_6?C GIU_CFGI[WVLE,NWU_Cv6U_CAaU_CjORQSX#djV6H
Wh ·B∗
hRh = 0MFU0V6QORU0V4O
WhNWaML[PNedjV6H#djV6H
Rh =®E,ORH
1I6djV6NHYE,O[PHTQSXYERV6[POSADOQSHYG&KLHQ&G&K6X_@;U_CLC6U_CLEBV6CGIU0C0OSQRH^H4HaM6[WHgHCf4NWaHYCLERNPU0Cf4HYV4³fl] HYERMLA_GHg[WU_Q&ERdjV6H
k = k′ = 1 ;CQRHYM6QRHYCLf[PH9aA_NP[W[eAD_HOSQRNeADC60V6[eADNWQRHV6C6NP«U_QSaH9f4VGYADQSQRX
Ω =]0, 1[20U_NWQ[eA 'NW_VLQRH ¯ P8/_ ;Cf4XILCLNOB[WA«U0CLGOSNPU0C
p0 ∈ QhFA¤_HpG
k′ = 1LMFADQ;ERHYE¤A_[PHYV6Q&E −1, 0,+1
ADV4ORQSU_NeEERU_aaHIO&Ef4H#G&KLA_djV6H;ORQSNeADC60[PHKi
r_U_NWQ[WA !NW_V6QSH ¯ 0/Dµ0I&0vU0V'A>U_VLQSEMLA_Qf4XILCLNOSNPU0C6U_CABhWh · Rh =
∫
Ω
rhdivwh dx.
ADNeEGU_aaHwh
HYE,OAdWCLH³MLA_Q¡aU_Q&GIHpADV4E,VLQG&KLA0djV6H a\ADNW[P[WHKiERA f4NP0HQS_HYCLGIH HpE>OGIU_CFE>O&ADCjORHTf6A_CLEG&KLA_djV6H
KiHOU_CA
∫
Ω
rhdivwh dx =
n∑
i=1
divwh(Ki)
∫
Ki
rh dx
d0VLN-¤A_V4OJXYQRUM<U_V6Qrh = p0
GYADQ ∫Kip0 dx = |Ki|
3 (0 + 1 − 1) = 0l°-ADQ#GU_CLERXYdjV6HYC0O
p0HC60HCLf4QSHbV6CC6U_VL_HYA_V¡0HYGOSHV6Qf6HKerB∗
h
6HYCM6[PVFEf4H1I
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
/ E F "+ n " I # $ !6
@BADCLE[eAM6Q&AcORNedjV6H_w$:t¥ilk'| m w!oB3´|s!tYoqFs w!|KerB∗
h
|jtYnxtYnyco¢DnY|+3´|sªnx~!k'tAFycm6s!w'|´ilk!| k!s m 3 h0np£'qvw'|w!|jthh[3 |jsªnptZ§s'ot¢_qFy¤y¤|tY~?q<s'w!m6sªnY|x|tnorslk'nYorof=tpm§r|]-C´HC<HOY§E,N
dim( KerB∗h) > 1
[PHGAD[eGIVL[?CV6aXQSNWdjV6Hf4HYEJE,U0[PV4OSNPU0CLE#f4V´ER`4E>OSaH[WNWC6XYA_NPQSHJ ¯ G E ?cAJGU_CLf6V6NPQSH;Zf4HpE2UjERGNP[W[eAcORNWU_CFE?CV6aXQSNed0VLHYEE,V6Q[eAgMLQRHpERERNPU0C H[] AD[W_U0QRNPORKLaHKLXYERNOSH\HCjORQSHM6[PVFE,NWHV6Q&ETM6QSHYESERNPU0CLEbf4NWESGIQSIOSHYEPh
f4U0CjO#[WAf4NC<XYQRHYCLGIHA_M6MLA_Q,OSNPHYC0O#ZKerB∗
h
;Cf4NPOgdjV6H[WAaXIOSK6U4f4H\HYE,O#ors!tnpm!|<%XHa\ADQ&djV6U_CFE5A>VFE>OSHaHCjOgdjV6H[] X[WXaHYC0O
p0f4H[WAf6XaU_CLE,ORQ&AcOSNPU0C³f4V HaaH ¯ µ6 µ>F¡EY] NPCjORHYQRMLQRORHJGU_aaHJVLC6HJUjERGNP[W[eAcORNWU_Cf4H[WA¡M6QRHpERERNWU_C³Z[^] XYG&K6HY[P[WHBf6V¡a\ADNW[P[eAD0H_ (N
dim(KerB∗h) = 1
U_C¡XY[PNWaNPCLH9«rA0GINW[PHYaHYCjO[^] NWCLf4XORHYQRaNWCLAcOSNPU0CERV6Q[eAM6QSHYESE,NWU_Cf4NeERGQRORHPh
HCNPaM<U0ESADCjO#ERAcAD[WHVLQTHCV6CC6UHVLfU_V6NWHCHCM6QSXYGNWESADCjO#ERAaUc`0HC6C6HERV6Q#[WHf6U_a\ADNWC6HΩv@BADCLEBOSU_VLET[WHYETGYA_EYlNP[?C] `AMLA_ETfl] NWCLf4XIOSHQSaNPCLADORNWU_CE,VLQT[WANPORHpERERH
UhdjV6N-HpE>Ogf4XLC6NWHf4H\a\ADC6NWQSHV6C6NedjV6H_°§U0V6QTM6[WVLE#f4Hf4XOSA_NP[eETE,VLQT[PHpETaXIORKLUf6HYEf] XY[PXYaHYCjOSELC6NeEHYCaXpGADCLNWdjV6HTf4HpE LV6Nef4HYEY6C6U0VLEQRHYC_Uc`_U0CLEZ6F µ>FdH^
9U_VLE9C] A¤0U_CLEQSNPHYCf4NO;MFU0V6Q[^] NWCLE,OSADCjO;f6HT[WAQSXYERU_[WV4ORNWU_CM6Q&AcOSNWdjV6H#f4V ER`4E>OSaHT[WNPC6XpADNWQRH ¯ G E j°!U0V6QGIH[eA#U0CV4OSNP[WNWERHBV6CA_[P0U_QSNOSK6aH_f4NOfl] JpA AL0NeERERV¡f4HB[eATORKLXU_QSNWH;f4HB[] U_M6ORNWaNPESAcOSNPU0C¤d0VLH2C6U0VLE'_HQSQSU_CLE'ADV®2KFADM6NPORQSH9 E D[EY] AD_NPO'[eZBfl] VLC#ORQSYE!<H[HI4HYaML[PHfl] NPCjORHYQSA0GOSNPU0CHYCjORQSH[^] ADCLA_[P`4ERHJCV6aXQSNed0VLHHO#[] U_M6ORNWaNeESAcORNWU_Cv-4M6[WNWdjV6U0CLEb6QSNPY_HaHYC0OT[^] Nef4XYHMLQRNWCLGINWMLA_[PHrC6U_VFET` QSHNPHYCLf4QSU_CLEbHYCf4XIO&ADNW[?ADV´®2KLADMLNOSQRH E ;CESADNPO#d0VLH\[PHpETXYdjVLADORNWU_CLETf4H)(ORU 0HYEERU_CjO!XYdjV6NW¤A_[PHYCjORHYE'ZV6CJM6QSU_6[WaH2f6H?aNPCLNPaNeERADORNWU_Cgfl] V6C6HXCLHQS_NWH90U_NWQv[^] -4HYQSGNWGH4 µ6W E 9U_VLE#_HQSQSU_CLEbdjV6H_'f4H\[WAa¦YaH\a\A_C6NPYQRH0l[WAQSXYERU_[WV4OSNPU0Cf4V´ER`4E>OSaH[PNWC6XpADNWQRH ¯ G E ;HpE>OXpdjV6NPcA_[PHYC0OSHTZJ[eAaNPC6NWaNWESAcOSNPU0CE,V6NWcADCjORHJ(Uh) = min
Vh∈KerBh
J(Vh)A¤_HYG
J(Vh) =1
2AhVh · Vh − bh · Vh.
?] A_[P0U_QSNOSK6aH#fl] JYA A\M<HQSaHIO;MLQRXpGINeE,XYaHYCjOBf4HJQRXpE,U0VLf4QSH#GIHgMLQRU06[PYaHgf4HgaNWC6NWaNeESAcORNWU_CERU_VFEGIU_CjOSQSA_NPCjORH0°§U_VLQ9f4HYE9QSA_NWERU_CFEf6HgGIU04O9f4HgGYAD[eGIV6[^6U_CC] V4OSNP[WNWERHgdjV6HTOSQRpEQ&ADQSHaHCjO[WAaXORK6U4f4Hgf6HYEXY[PXYaHCjOSEFC6NWE
P2M<U_V6Q[eANOSHYESE,HHO
P1M<U_VLQ[WAMLQRHpERERNPU0C ;C [PVLN9M6QSXI«YQRHV6CLH³ADV4OSQRHaXORK6U4f4H0Lf4NPORH
P1 6V6[W[PHbM<U_V6Q[eANOSHYESE,HbHIO P1M<U_VLQ[WAM6QSHYESE,NWU_CvD['E] AD0NOf4HT[eAJaXORK6U4f4Hf6HYE'X[WXaHCjOSEvLC6NeE
P1M<U_VLQ'[eANPORHYESERH?HIO![WA9M6QSHYESE,NWU_C#f6A_CLE'[WA0d0VLH[W[PH2U_C#HYC6QRNeG&K6NPO'[^] HpE,MFA_GIH
V0hf6HYENOSHYESE,HpE2HYCA>A>U_V4O&ADCjOf6ADCLEESAgFA_ERH;M<U_V6QG&KFA_djV6H9a\A_NP[W[PHbHOM<U_VLQG&KFA_djV6HbGIU_aM<U0ESADCjORHfLADCLERN VLC6Hg«U0CLGOSNPU0C §k'r|f4XILCLNPHGIU0aaHJ[PHJM6QRU4f4VLNO
λ1(x)...λN+1(x)ªU&³[PHpE
λj(x)ERU_CjOg[WHYEJGIUU_Q&f4U0C6C6XHpEbLADQS`4GIHYC0OSQRNedjV6HYEgf4HxfLADCLEg[WAa\A_NP[W[PH
Ki-®2U_aaH¡GIHO,OSH«U_CFGORNWU_CLV6[P[WHHYE,O-CjVL[P[WHE,V6Q-[WHFU0QSfJf4H
KiHIO-M<U0ERNOSNP0HZB[^] NWCjORXYQRNWHV6Qp¤U_C[PVLN6A_ESE,U4GINWHGIU_aaHf4H0QRXf4H[WNWFHYQ,OSX[WHLADQS`4GIHYC0OSQRHf4H[WABa\ADNW[P[WH__®2HO,ORHaXORK6U4f4HHYE,O-E>O&AD6[WHGIU_aaH[PHaU_CjORQSH[] HI4HQ&GINeGIHERV6NWcADCjOY
;u|jyc¢_or¢0| ] \] ="(%)%$* <>"% 7 2 & ;! 9) 9),;>7 ; & > 3 P1 787 <>"% 7 2 ) !6C # P1 <>%57 2 <>% #C ":=2 dim(KerB∗
h) = 1 / vHYE2NPCLE,OSA_6NW[PNPORXpE?f4H;M6QRHpERERNWU_C\C6HBE,U0C0O2MLA_E?[] ADMLA_CLAD0Hf4HYE?aXIORKLUf6HYEfl] XY[PXYaHCjOSE?LC6NeED[H4NWE,ORHBADVLESE,N<f4HYE?aXIORKLUf6HYEf4H;f4NCªXQSHCLGHYE?LC6NWHYEd0VLNFM6QSXYERHCjOSHCjO?[WH9a¦aHO>`M<H;fl] NWCLGIU0Cj0XICLNPHYC0OpLGIU0aaHb[WHTaU_CjORQSHb[] HI4HQ&GINeGIHbERV6NPcA_C0Op
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[0 # $ !66# "! # +N ≥ 2
/ E 9u|jyc¢_o¢_| ] \]%^ ! 9)% 7 ;* 2 9(# + / 1 )=9 & > N = 1 + !& 9)7 = 2=2'' ;>% < C* K2 C"78 ":@<7 :#6 u = 0 ! p <>% & 9 f & 2 73<>>% & !,9 > ! & <>%) 9% 7 #:<)%7 & #9 9 C'% ;! K2 ">1!/ ""% Ω = (0, 1) "!" 9 >% =7 & 2 787 2 49 <> xj = jh 2 h = 1/(n+ 1) ! 0 ≤ j ≤ n+ 1 / 9;3 7 2 & ;# 9 99 ;)%) ## 3 !>(% ;## + 9 @% 9% 01! 2
µ−uj+1+2uj−uj−1
h2 +pj+1−pj−1
2h = f(xj) <>"% 1 ≤ j ≤ nuj+1−uj−1
2h = 0 <>"% 1 ≤ j ≤ nu0 = un+1 = 0.
(% >%* !I & I9 ;* 2 27 ; >% * #6 & 27?<>6; #=> <'2>% '7 >% * 7 2<>%6 " (pj) #659); 3 # 7 2'99 ":9 4#" 62<>% #":9 & 7 <7 49 <>% #C "9); < 2)% C# ! & <>)K2 (1, 0, 1, 0, ..., 1, 0) /.¡|+3´m4ycilk!| %] \%] \ ] NWf4XYHf6H[eA«U0QRaJV6[WADORNWU_C´cA_QRNeAcOSNPU0C6C6H[W[WH³ ¯ ¯ #EY] XIORHYCLf ESADCFEgM6QSUD6[PYaH#ADV vADML[WA0GINWHCU_VZgOSU_V4OU0MFXYQSADORHYV6QH[W[PNWM4ORNedjV6H_L°!U0V6QQSXYERU_VLf4QSH
−div (A∇u) = ff6A_CLE
Ωu = 0
ERV6Q∂Ω,HCMFUjERA_CjO
σ = A∇u 4U0CNWCjORQSU4f4V6NPO[eA«U_QSagVL[WADORNWU_C¡cA_QRNeAcOSNPU0C6C6H[W[WH
−∫
Ω
divσv dx =
∫
Ω
fv dx
∫
Ω
A−1σ · τ dx+
∫
Ω
udivτ dx = 0,
MFU0V6QTORU0V4O(v, τ) ∈ L2(Ω) × H(div)
! vAaXIOSK6U4f4Hfl] X[WXaHCjOSE#LC6NeE#djV6N?HYCf4XYGU_V6[WH\HYE,Of4NDCªXQSHCjOSHf4H¡GIHY[P[WHYEgd0VLN?U_CjO#XORX\HI4M<U0ERXHpE#f6ADCFE#GIHG&KLA_M6NPORQSH_[W[WHHYE,OgADM6M<H[WXH¡aXORK6U4f4Hf4HYEXY[PXYaHYCjOSELCLNWEaNPjOSHYEY •
*)* =RS8POR=RS[P!16=>5?/ 7US 3 S8O>1+P!1S´/8M 3F=RT8US@BADCFETGIHIORORH\ERU_VLE, ERHYGIORNWU_C CLU_VLEbf4NeE,U0CLEBQSA_M6Nef4HaHCjObd0VLH[ed0VLHYEBaUDOSETERV6Qb[eA¡zotYk!m6rotYm =nYorq<sf4HpETQRXpE,VL[O&AcOSEBU04ORHYCjVFETMLADQb[eAaXIORKLUf6Hf4HYEbXY[PXYaHYCjOSEbLCLNWEYv HpEBL_VLQRHpEBGNf4HpERERU_VLEU_CjOXIORXbOSQSA0GIXYHYEZ[] ADNef4H#f4V[WU__NeGINWH[-r[PNW6QSHp_Q&ADMLK6NWdjV6H;6f6µ_f 'ANWERVLA_[PNeERADORNWU_CTf4HpEvQSXYERV6[POSAcO&ElM<U_V6Q'V6C#M6QSU_6[WaHESGA_[WA_NPQSHU'&T[] NPCFGIU_CLCjVLHHYE,O!ZcA_[PHYV6QSEf6ADCLE
RHYE,O9A_ESERH#J#ERNPaM6[WH_FC³f4NWaHCLERNPU0C
N = 2U_CM<HV4O;ORQ&A_GIHYQ9f4HpE[PNW_CLHYE9f] NeE,UccA_[PHYV6QSEHIO U_Vf4HpE#GA_Q,OSHYE#f4H\0QRNeERXf] NWCjORHCFE,NPORX0'GIU0aaH¡U_CMFHYV4Og[WH_U0NPQ#ERV6Q#[WHYE 'NW_VLQRHpE ¯ WpHIO¯ W ¯ D[«rADV4OC6XpADC6aU_NWCLEJ«rADNWQSHAcORORHYC0OSNPU0CxA_V4´XYG&K6HY[P[WHYEf4HcAD[WHV6Q&EGIU_aaH¡[WHM6QRU0V6_H[WA
'NW_VLQRH ¯ WpgU&[WHYEf4HYV4ORQ&A_GXYEbrGU_QSQRHpE,M<U_CFf6ADCjOADVa¦YaHTM6QRU06[WaHBERV6Qf4HYV4¡a\ADNW[W[WA__HYEf4NWE,ORNWCLGIOSE&gERU_CjOf] A_[P[WV6QSHGIU0aMLADQ&AD6[WHa\ADNeEfl] XpG&K6H[W[WHYEf4H¡QSXI«XYQRHYCLGIH¡ORQSYEf4NDCªXQSHCjOSHYEY'Cf4NPaHYCLE,NWU_CN = 3
U0C³OSQSA0GIH\f4HpEbNWERU0ERV6QR«rA_GIHpErERV6Q,«rA0GIHpE;U'&[] NPCFGIU_CLCjVLHHpE>O#GU_CLE,OSA_C0OSHp;U_V6NPHYCf4HYEGU_V6M<HYEM6[eADCLHYEY
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/ E_¯ "+ n " I # $ !6
¯ 0/'F ®ADQRORHbfl] NWERUccAD[WHV6Q&Ef6ADCFEV6C6H#GIU0V6M<HTM6[WA_C6HTfl] V6CM6QRU06[WaHTf4H#f4NC<VLERNWU_Cµc@
¯ /0 D>E,UjE,VLQ,«rA0GIHYEMFU0V6Q¡V6C M6QRU06[WaH³f4Hf4NDCªVLERNPU0C µc@ rcAD[WHV6Q&E¡f4NWaNPCVLA_CjOf4HjADVLG&KLHBZf4QSU_NPORH¤
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[0 # $ !66# "! # +N ≥ 2
/ E G
?C0V6NeE,Hfl] HI4HaM6[WH_cC6U0VLE§GIU0CLE,Nef4XYQRU0CLEvVLCJM6QSU_L[PYaHf4Hf4NC<VFE,NWU_Cf6ADCLE§[^] HYERMLA_GHfl] V6C6HGIU_CFGIHCjOSQSADORNWU_C rV6CM<U_[W[WVLADCjObMLADQBHI4HaM6[WHp;XYaNeEBMLADQbV6CLHERU_VLQSGHg[WUGYAD[WNWERXHE,VLQB[WHE,U0[r[WALA_ERH?f4VGIV6<HpI;C#QSXYERU_V6OvMLADQ![eAaXIORKLUf6Hf4HpEXY[PXYaHYCjOSEvLC6NeEQ1
[PH2M6QRU06[WaH2f4H@9NWQSNWG&K6[WHIO ¯ µ ¯ A¤_HpGTV6CE,HpGIU_CFfaHYag6QSHgCV6[!HIObV6C6HGIU0CLf4NPORNWU_C³A_V4[PNWaNOSHYEBf4HJ@;NPQSNeG&K6[PHOBMFADQRORU_V6OERA_V4«E,V6Q2[WA#LA_ERH9f4VGV6FH0j±9V¡GIHYCjORQSH9f4H[eATLA0E,HU0C\NPaM<U0ERHV6C6HBGIU0CLf4NPORNWU_CADV6[WNPaNPORHpE2f4H@9NWQRNeG&K6[WHIO¨>C6U0C4 K6U_aU_0C6HI©u = 1
LHIObE,VLQ9[WHgQSHYE,ORHgf4HJ[WA¡LA_ERH#V6CLHJGIU0CLf4NPORNWU_C A_V4[PNWaNOSHYEf4HTHVLaA_C6C6 'A !NW_V6QSH ¯ />FgQSHM6QSXYERHCjOSH;[WHYEcAD[WHV6Q&Ef4Huf6ADCLEVLC6HTGIU_VLMFH0jHO[WA !NP0V6QSH¯ 0/_Jf4HYENeERU0ERV6Q,«rA0GIHpE2f4H
u 'A NeE,VFAD[WNWESAcOSNPU0C f4HYEQSXYERV6[O&AcO&E¡M<U_V6QV6C M6QRU06[WaH³0HYGOSU_QSNPHY[JrU& [^] NPCLGU_C6CV6HHYE,OZ¤A_[PHYV6Q&E!fLADCLE
RN 'HYE,O-f6NDCªXQSHCjORH0c°QSHC6U0CLEcMFADQ§HI4HaM6[WH_c[WHGA0E-f4VER`4E>OSaHf4H[] X[eA_E,ORNeGINPORX_U0NPQ?[WAE(4U_VLE, ERHYGIORNWU_C4 µ6WpI ;C\M<HV4O2ORQ&A_GIHYQ?f4HpE LpG&K6HYE-QSHMLQRXpE,HYC0O&ADCjO[WH_HpGORHYV6Q?GYAD[eGIV6[WX_aA_NWE!GIHO>`jM<Hfl] NWa\AD0HHYE,O!f4NWGINW[PHZ[WNPQSH?HO!Z9NPCjORHYQRMLQRXORHQp#D[jHYE,O'LNPHYC#M6[WVLEvMFADQS[WA_C0O!f4HOSQSA0GIHQ[WA ¨Rf4XI«U0QRaXYH© f4V f4U_a\A_NPC6H³HCV4OSNP[WNWESADCjO[^] NWCjORHYQRM6QSXIO&AcOSNPU0C¥MLKj`4ERNWdjV6Hf4H [eAE,U0[PV4OSNPU0C HYCf4NPaHYCLE,NWU_CN = 2
U_V3'X9ADM6M<H[WU_CLE2djV6H9[WH_HpGOSHV6Q2NPCLGU_C6CV
u(x)HYE,O2[WH9f4XYM6[WA0GIHYaHYCjOf4VMFU0NPCjO
xE,U0VLE§[^] A0GORNWU_Cf4HpE!«U_Q&GIHYE§H4HQ&GIXHpE HcMLADQGU_CLERXYdjV6HYC0Opc[PHM<U_NWCjO?GIU_VLQSA_C0O-f4Vf4U_a\A_NPC6Hf4XI«U0QRaXHYE,O
x+u(x)D9U_VLE!NW[P[WVLE,ORQSU_CLE§GIHpE§f4HV4JaA_C6NWQSHYE§f4HQRHYM6QRXpE,HYCjORHQ![WHYE§QSXYERV6[O&AcO&E'ERV6Q[] HI4HaM6[WH#fl] V6C6HTM<U_V4OSQRHTHYCLGA0E>OSQRXYHTE,V6QERU_C<U_Q&f0HQRORNeGA_[l0ADVFG&K6HGIU0CLf4NPORNWU_CADV4[PNWaNOSHYEf4H@9NWQSNWG&K6[WHIO!HIO[WNPLQRHERV6Q-[PHpEADV6ORQSHYE§<U_Q&f6ErGU_CLf4NPORNWU_CADV6J[WNPaNPORHYE?f4HHVLaA_C6CF-ERU_VLaNeERHZ¡ERU_CM<U_Nef6E9MLQRU0M6QRH[eA\«U_Q&GIH
fHYE,O;V6C_HYGIORHYV6Q;GU_CLE,OSA_C0O;_HQRORNeGA_[e _U0NPQ;[WA!NP0V6QRH ¯ 0/ ¯ ] A¤¤A_CjOSAD0Hgf4HORQ&A_GHQb[eAGU_C4L0V6Q&AcORNWU_Cf4XI«U0QRaXHHYE,OTd0Vv] U0CM<HV4OT`³ERV6M<HQSMFUjE,HYQ;[WHORQ&A_GXfl] VLC6H#ADV4OSQRHb0QSA_CLf4HVLQESGA_[WA_NPQSHbGIU_aaHb[WAC6U0QRaHTf4VORHYCLERHV6Qf4HpEGIU0C0OSQSA_NPCjOSHYEY
¯ / ¯ @9H¡0ADVFG&K6HZ f4QSU_NPORH0vf4XML[WA0GIHaHYC0Op'GU_C4L0V6Q&AcORNWU_Cf4X«U_QSaXYH_'HIOgC6U0QRaH¡f6VORHCFE,HYV6Q9f4HpEGIU0C0OSQSA_NPCjOSHYEb[WHYEcA_[PHYV6QSE[PHpEX[WH_XYHYEERU_CjO[WHYEM6[WVLE«U0CLGIXYHYE&f6ADCFEV6C6HTM<U_V4OSQRHHCLGYA_E,ORQSXHbE,U0V6aNWERHTZERU_CM<U_Nef6EM6QSU_M6QSH_
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/ E "+ n " I # $ !6
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!"!#$&%('*),+-/.102.10 354/687:9,;=<>?6:@A9B4C/DFEHGJILK:MONQPRDBDTS1NUEV5WJSQILEPQXBY=Z[IFNQG:XV5PRM[D/SRK?DTENQPQILZ\D!]8DTSXT^`_JIaNRM[VLWJSIa_8b]8XTPRM[cLXTDTSdKJIaPRNRM[DZ[Z\DeSfEag DTS1N1hiYjhi]8M\PQD&YZkg XNR_J]:Dl]8DeSmcjIaZ[D_JPQSBK:PRV5K:PQDTS2DN!]8DeSmnoV5WJENQM\V5WJSmK:PQVLKJPRDeSm]8DEDTS/XT^`_JIaNRM[VLWJSTpLqdIr VLNRM[cjIjNRM[VLWs]8DtEDNRNRD$XNR_J]8DuDTS1Nv]8VL_Jw:Z\D5pdxvg _:WJDyKJIaPRNTfEDZzI*cjI*W:VL_SKDTP r DN1NRPQDu]|g XNQ_J]8M[DP]8DTSvS1V5Z\_8NQM\V5WJSKJILP1NQM[E_:Z[M\TPRDeSfd]:MONQDTSV`SREM\Z[ZzIaW`NRDeSDW~NQD r KJSoVL_scM[w:PHIaW`NRDeSQfd]8DTSK:PQVLw:Z[ r DTS]|g XTcLV5Z\_8NQM\V5WyILSQS1V8EM\XeSmYEDTSmXe^`_JIjNQM\V5WJSp`xg Ia_:NRPQD=KJILP1NefLW:V5_JSBDTWu]8Xe]8_:M[PRV5WJSm_:WJD r XNQG:V8]8D]8DPRXeS1V5Z\_8NQM\V5W$LXTW:XPHIaZ[D=]8DlEDTS rr DTS/K:PQVLw:Z[ r DeS/]|g Xc5VLZ[_8NRM[VLWy^`_:DlW:V5_JS r DN1NRPQVLWSmDWtVD_:cPQD]:IaWJSFZ[DvC/GJILK:MONQPRDJpxVLWJW:VLWJSNQVL_8N]8DS1_:M\NRD_:WDb8D r K:Z[Dv]8DUJd::tUUvKV5_:PZ\DqdIaKJZ[I5EM[DWuIc5DTE&EVLWJ]8M\NRM[VLWtIa_8byZ[M r M\NRDeS!]:DlxM\PQMzEHG:Z\DNTp8M Ω DTS1NF_:WuV5_:cLDTP1NBw?VLPQW:X]8D RN
V5WEHG:DPHEHG:DlZ\DeS&EV5_:K:Z[DTS(λ, u) ∈ R×H1
0 (Ω)fJIc5DTE
u 6= 0f:S1V5Z\_:NRM[VLWJS,]8D
−∆u = λu]JIaWJS
Ωu = 0
SR_:P∂Ω.
kp[eq DPRXTDZ
λDeSAN$IaKJKDTZ\Xt::sjj?fDNZ[InoVLWJENRM[VLW
u(x)¡ ¢¡jj£?¥¤¦§U¨j©ªT« ?§¬UU?pJqBg DTWJSRD r wJZ\D]8DeS,cjIaZ[D_:PHSFK:PQVLK:PQDTS!DTS1N=IaKJKDTZ\XZ[DS1K?DTENRPQD]8Dyk8p\pW*K?D_8NnIaM[PRDlZ®g IaWIaZ[VLLM[D=DTW`NRPQD¦p[eBDN,Z[DlKJPRV5w:Z\ r DK:Z[_JS,SRM r K:Z[D]8D]8XNQDP r M\WJIaNRM[VLW]8DTSFcjIaZ[D_JPQS!DNcLDTENRDT_:PQS!K:PQVLKJPRDeS!] g _:WJD r IaNRPQM[ED A ]|g V5PQ]:PRD n f
Au = λuIcLDeE
(λ, u) ∈ R× Rn,kp°¯L
DWIa±$P r IaW`N^5_JDvZkg V5KXTPQIaNRDT_:P −∆DeSAN_:W:D²1LXWJXPHIaZ[M[SQIjNQM\V5W³DTW]8M r DWJSRM[VLWM[W8´JWJM\D$]|g _:W:Dr IaNRPQM[ED A DWµ]8M r DWJSRM\V5W£´JWJM\D5p&qdI¶PRXeS1V5Z\_8NQM\V5W·]8D¬k8p\uSRDPHI¡_:NRM[Z\D¸KV5_:P*PRXeS1V5_J]8PQDZ[DTS
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¯:T¹ ! "$#%&' ()*
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-!.102.@A BDCE`9F1<&68@19B4 ;GEH=7:9IJF,KLMGEs@R4&E56%NU68@19B4=4&N2@A7OGE2cjIaW`N]8DW:VL_JSZzIaWEDP]:ILWJSZ[DTS]8XTcLDZ[VLKJKD r DTW`NQSIawJS1NRPHIaM\NQS]8DZzIuKJPRV8EHGJILM\W:DSRDTENQM\V5W fr V5W5NQPRV5WJSDWy^`_:V5M:ZzIPRXeS1V5Z\_8NQM\V5W$]|g _JW$K:PRV5w:Z[ r D,Ia_8bcILZ\DT_:PHS2K:PQVLK:PQDTSK?DP r DN/]8D=PRXeS1V5_J]8PQDIL_JSQS1M:_JWyK:PQVLw:Z[ r D&]|g Xc5VLZ[_8NRM[VLW pQPVL_:P!EDZzIW:V5_JSBILZ\Z[VLWJS2nIaM[PRD&_JW:D=IaWIaZ[VLLM[D&IcLDeE!Z[IPQXTSRVLZ[_8hNQM\V5W]:DS8+8SANQ r DTSF]8MSR XPQDW`NRM[DZzSFDW*]8M r DTWJSRM\V5Wy´JW:M[DLpxILWJS/NRVL_:N,ED^5_JM S1_:M\N A ]8XTSRM\5W:D_:W:Dr IaNRPQM[EDST+ r XNQPRMz^`_:DPRXTDZ[Z\D5fJ]8X´W:M\DK?V5SRM\NRM[cLDLf]|g V5PQ]:PRD n p W*W:VLNRD λk
S1DeS,cjIaZ[D_:PHSFK:PQVLK:PQDTSDNrkSRDTSFcLDeENQD_:PHS!K:PRV5K:PQDTSTf
1 ≤ k ≤ n f8NRDTZ[S,^`_:D Ark = λkrkpW EV rr DTWJEDKJILP,_:W ST+8S1NR r D]8MSR XPQDW`NRM[DZ ]8_*K:PQD r M[DP,V5PQ]8PQD
∂u
∂t+Au = 0
K?VL_:Pt ≥ 0
u(t = 0) = u0,
¦p 95VQU
u(t)DeSANl_:W:DvnoVLWJENRM[VLW]:DEZ[I5SRSRD
C1]8D
R+]:IaWS
Rnf DN
u0 ∈ RnpWViZDTS1Nlw:M[DWEVLWJW`_^`_:D¦p 95$I5] r DNu_JW:D S1V5Z\_:NRM[VLW _:W:Mz^`_:D VLw:NRDW_:DDW ]8MzIaLV5WJIaZ[MzSRILW5N$ZzI r IjNRPQMzED A pPmZ[_JSKJPRXeEMzS1X r DTW`NTf|ZzI]:VLW:W:XTDM\W:M\NRMzIaZ[D$S1D$]8XeEV r K?V5SRD$S1V5_JSlZ[ItnoV5P r D u0 =∑n
k=1 u0krkfdED$^`_:M]:VLW:W:D
u(t) =
n∑
k=1
u0ke
−λktrk .
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5 )<#-< "&1* ¯85 W ]8D_:bM[ r DDb8D r KJZ\DDeSAN,Z[DST+8S1NR r D]8MSR?XTPRDTW`NRM[DZd]8_*]8DT_8b8M\ r DV5PQ]:PRD
∂2u
∂t2+Au = 0
K?VL_:Pt ≥ 0
u(t = 0) = u0,
∂u
∂t(t = 0) = u1,
kp =`
V Uu(t)DeSAN2_:WJDFnoVLWJENRM[VLW]8D&EZzILSQS1D
C2 ]8D R+ ]:ILWJS Rn faDN u0, u1 ∈ RN p :mW$]8XTEV r KV`SRILW`NZ\DeS]8V5W:W:XTDTSM\W:M\NRMzIaZ[DTSlSRVL_JS=Z[IynoVLP r D u0 =∑n
k=1 u0krkDNu1 =
∑nk=1 u
1krkfU¦p =`=IL] r DNEV rr DS1V5Z\_:NRM[VLW_:W:Mz^5_JD
u(t) =
n∑
k=1
(
u0k cos
(
√
λkt)
+u1
k√λk
sin(
√
λkt)
)
rk .
ViZ,DTS1N$EZ[ILM\PySR_:P$EDeS$]8D_:b¡Db8D r K:Z[DTS$^`_:D*ZzI~EVLWJWJIaMzSRSQIaWEDt]:_ SRKDeENQPRD ]8D*ZzI r IaNRPQM[ED AKDTP r DN]8DvPRXeS1V5_J]8PQDZ\DeS=KJPRV5w:Z\ r DTS=] g XTcLVLZ[_8NQM\V5W¡k8p 9`,DNk8p =p 2&_JSQS1MXTc`Mz]8DTW5NS1V5M\DTW`N1h¦M\ZzSfEDTSDbD r K:Z\DeS$S1V5W`NvNQVL_8NyYnIaM\NPQDKJPRXeS1DTW5NHIjNQMOnS]8D*ZzI]8X r IaPHEHG:D^`_:D*W:V5_JSIaZ[Z\V5WJSS1_JM\cPQD]:IaWJSZ[IS1_:M\NRD5p(0=VL_JSvIaZ[Z\V5WJSPQD r K:ZzILEDTPZzI r IjNRPQMzED A KIaPZkg VLK?XPHIjNRDT_:P −∆fdZkg DeS1KILED
RnKJIaP2Zkg DeS1KILED,]8D7=M\Z[w?DPRNL2(Ω)faDNBW:VL_S2IaZ[Z\V5WJS ²1]:M[ILLVLWIaZ[M[SRDP1³mZ\D&qIaK:ZzILEM\DTWvK?VL_:PBPRXeS1V5_J]8PQDZ®g XT^`_JIaNRM[VLW*]8DZ[IEHGJILZ\DT_:PFVL_*Zkg Xe^5_IjNRM[VLW*]8DeSFVLWJ]8DeSp
2&´JW]:DvSRDEV5W`cjILM\WJEPRD^`_:Dk8p\FDTS1Nw:M[DWZ[I²1w?VLW:W:D³noVLP r _:ZzIjNQM\V5W ]8_ K:PQVLw:Z[ r DIL_8bcILZ\DT_:PHSK:PQVLKJPRDeSK?VL_:PZ[D qdIaKJZ[I5EM[DW fUVLW¶KDT_8N$KJILSQS1DTPvKIaP_:W¶ILPR5_ r DW`N$]8D²RSRXKJILPQIaNRM[VLW]8DTSlcjIaPQM[ILw:Z[DTSA³v]:IaWJSlZkg Xe^`_JIjNQM\V5W]8DZzItEHGJILZ\DT_:PlVL_¸Z®g XT^`_JIaNRM[VLW]:DTSlVLWJ]:DTS^`_:DW:VL_JS]8XeEPQMOhcLVLWS2noVLP r DZ[Z\D r DW`NTpQ:mWuZkg ILwJSRDWJED=]8D&NQDP r DSRVL_:PHED5faDNFDWv²1VL_Jw:Z\MzIaW`N1³K:PRVjcMzS1V5M\PQD r DTW5NUZ[IEVLW]8MONQM\V5W M\WJMONQM[ILZ\DvDN=Z[DTSEVLW]8MONQM\V5WJS=IL_8b*Z\M r MONQDTSTfW:V5_JS=EHGJDPHEHG:VLWJS&_:W:DSRVLZ[_8NRM[VLWu
]8DEDTSXT^`_JIjNQM\V5WJS,^`_:M STg XeEPQM\c5DlSRVL_JSFZzIvnoVLP r Du(x, t) = φ(t)u(x),Eag DTS1N1hiYjhi]8M\PQDy^`_:DuZkg VLW¡SRXKJILPRDuZ\DeScjIaPQM[ILw:Z[DTS]:DyNRD r KJSDN] g DeS1KJI5ED5pUM
u
DeSANS1V5Z\_:NRM[VLW¡]8DZ®g XT^`_JIaNRM[VLW*]8DZ[IEHGJILZ\DT_:P∂u
∂t−∆u = 0,
¦p/.LVLWtNRPQVL_:c5D®Ia_ r VLM[WJS!noVLP r DZ[Z\D r DTW`NHF^`_:D
φ′(t)φ(t)
=∆u(x)
u(x)= −λ
V Uλ ∈ R
DTS1N,_:W:DEVLWJS1NQILW5NQDM[WJ]8XTKDTWJ]:ILW5NQD]8DtDN=]8D
xp WtDTW]8Xe]8_:M\N^`_:D
φ(t) = e−λtDN=^`_:Du]8VLM\N NQPRDSRVLZ[_8NRM[VLW*]8_*K:PQVLw:Z[ r DIa_:bucjIaZ[D_:PHSFK:PQVLK:PQDTS
−∆u = λu¦p 5
r _:W:M]8DEVLW]8MONQM\V5WJS,Ia_:btZ[M r M\NRDTS,I5]8XT^`_JIaNRDeSp
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¯:e¯ ! "$#%&' ()*
xDZ[I rTr D r IaWJM\TPRD5f:S1Mu
DeSAN=SRVLZ[_8NQM\V5W]8DZkg XT^`_JIjNQM\V5W]:DTSFVLW]8DTS∂2
u
∂t2−∆u = 0,
k8p°LV5WuNQPRV5_:cLD^`_:D
φ′′(t)
φ(t)=
∆u(x)
u(x)= −λVQU
λ ∈ RDTS1N_:WJD*EVLWJS1NQILW`NRDLpmC/DNRNRDtnoVLMzSAhiEM!VLW¶DW ]8XT]8_JMONy^`_:DLf2SRM
λ > 0®ED*^5_JMFSRDPHIDR?DeENQM\c5D r DW`NZ\D$ETILSHf IaZ[VLPHS φ(t) = a cos(
√λt) + b sin(
√λt)DN^5_JD
u]8V5MONDWEVLPQD NRPQDSRVLZ[_8NQM\V5Wu]8D$k8p `p,D r ILPQ^`_:V5WJSB^`_:D5fSRM Z\DEV r KV5P1NQD r DW`NFDWuDeS1KILEDl]8DlZ[IS1V5Z\_:NRM[VLW
u
DeSANZ[D rTr DlK?VL_JP!Zkg XT^`_JIjNQM\V5Wt]8DlZzIvEHGJILZ\DT_:PFDNFKV5_:PFZ®g XT^`_JIaNRM[VLW]8DTS!VLW]8DTSTfM\Z|W g DWDTS1NFKJILSF]8DrTr DK?VL_:PSRVLW~EV r KV5P1NQD r DW`NDW¸NRD r KS DZ[Z\DV`SREM\Z[Z\DDW¸NQD r KJSK?VL_:PZ[DTSV5WJ]8DTSILZ\V5PQS^`_ g DZ[Z[D]:XTEPQVzN,Db8K?VLW:DTW5NQM\DTZ\Z[D r DTW5N&DTWuNQD r KJS®EILP λ > 0!KV5_:P,Z[IEHGJILZ\DT_:Pep
8 ? D0&V5_JS!cLDWJVLWJS!]8DlcLVLM[P/^`_:DlZkg Xe^5_IjNRM[VLW]8DTS!V5WJ]8DTS!IL] r DNF]:DTSFS1V5Z\_:NRM[VLWJSV`SREM\Z[ZzIaW`NRDeS/K?XPQM[V]:M[^`_:D r DTW`N&DWtNRD r KJS=]8_N,+`K?Du(x, t) = e−iωtu(x),VQU
ω =√λDNZzInoPRXe^5_JDWJED]8DeSV5SQEM[Z\ZzIjNQM\V5WJSvDN
u(x)DTS1NZ[D_JPyI r K:Z[MONQ_J]8DLp ViZ,STg ILLM\NZzY] g _:WJDETIaPHILENQXPQM[S1NRMz^`_:DLXTW:XPHIaZ[D]8DeS=Xe^`_JIjNQM\V5WJSIL_8b ]8XPQM\c5XDeS&KJILP1NQM\DTZ\Z[DTSZ\M[W:XeIaM[PRDeS=]8DvN,+KDG-+K?DPQwV5Z\Mz^`_:DLpLC/DTS2SRVLZ[_8NQM\V5WJSUV5SQEM[Z\ZzIaW`NRDeSUVLW`NU_JWvM[W`NRXP N2K:G-+8SRM[^`_:D!XcMz]8DW`N2DNmM[WJ]8XTKDTWJ]:IaW`N]:DyZ[I PRXeS1V5Z\_8NQM\V5WsLXTW:XPHIaZ[D$]|g _JW:DuXe^5_IjNRM[VLW] g XTcLVLZ[_8NQM\V5W¸G-+K?DPQwV5Z\Mz^`_:DLp(:mZ\Z[DTS r V8]8XZ[M[SRDW`NN,+K:Mz^`_:D r DTW5NB]:DTS2cM[w:PQIaNRM[VLWJS,KJIaP2Db8D r K:Z[DLfaXTZ[I5SANQM[^`_:DeSQUV5_]8DeS2VLW]8DTS&KJIaP2Db8D r K:Z[DLfaXTZ\DeEhNQPRV r IaLWJXNRMz^`_:DTSHf5DNFDZ[Z\DeS/SRDlPRDTWJEV5W5NQPRDTW`NBLXTW:XPHIaZ[D r DW`NBDTWyZkg ILwJS1DTWJEDl]8DNRDP r DSRVL_:PHEDDNILK:PQTS!_:WNRD r KS&]|g XNQILw:Z[M[SQS1D r DW`N&^5_JM KDTP r DN/²1]|g VL_Jw:Z\M[DP1³ZzIEVLW]8MONQM\V5W*M\W:M\NRMzIaZ[DLp •
`j¨ « ¨5 Ω = RN !#"
u(x) = exp(ik ·x) %$&'"()*$&#+,"-. #0/(2143 576$ |k|2 = λ 98 ):;<+,+ :$&#+,"-. #=%$;'>7?(?< + @%:/(A?7+ >7) PmPQDWJVLWJS,_JW*IL_8NRPQDDb8D r KJZ\D5f:Y$SRIc5VLM[PFT> 6-? U ªT« §¬¢UB ¨ 4 C ¢ « §ED?`MzSRSR_:D]8DZzI r XhETIaW:Mz^`_:D^`_JIaW`NRMz^`_:D$oc5VLM[P,Z\DC/GJILK:MONQPRDp
`j¨ « ¨5 GFH "7I?<<; -. +J@-K"7+ <V (x) L G<M!")(NO$P
u(x, t) = e−iωtu(x)-$;Q$&+,". 2/(i∂u
∂t+ ∆u− V u = 0
/(>7)$RN × R+
∗ ,¦p 5
>7+ G$u(x)
%$&L$R+,". 2/
−∆u+ V u = ωu/7>7)$
RN .¦p º5
WyPQDNQPRV5_:cLD=Z\D rTr D&N,+KD]8DK:PRV5w:Z[ r D=SRKDeENRPHIaZ ^`_:Dv¦p 5f`YZkg IL]:]8M\NRM[VLWt]|g _JWyNRDTP r D] g V5PQ]8PQDTSXPQV K:PReSp PV5_:P Z®g XT^`_JIaNRM[VLW ]8D¡8EHG:P.U8]8M\WJLDPZzI¶cjILZ\DT_:P K:PQVLK:PQDωSg M[W5NQDPQK:PQNRDEV rr D/_JW:DFXW:DTPR5M\D5pqI=KJZ\_JSUK?DNQMONQD,cILZ\DT_:PKV`SRSRM[w:Z\DF]8DFEDN1NRDFXTW:DPQLM[D!EVLPQPRDeS1K?VLW]YZkg XW:DTP1h5M\DF]8DFZkg XNQIjNnoV5WJ]:I r DTW5NHIaZ8]8_ST+8S1NR r D,]8XeEPQMON2KJILP,kp 5pjq DTS2IL_8NRPQDTScjIaZ[D_:PHSTfeKJZ\_JS5PQILWJ]8DeSf
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5 O -J & ¯859]8VLW:WJDW`NmZ\DeS2XWJDPQLM[DTS2]8DeS2XNHIjNQSmDb:EM\NRXeSp58VL_JSm]:DTSmEVLWJ]8M\NRM[VLWSd²APHIaMzSRVLW:WJILw:Z[DTSA³/S1_:PBZ\D,K?VaNQDW8hNRM[DZ
Vf|EDTSW:M[cLDeIa_8b ]|g XW:DTPR5M\DSRVLW`N]:M[SQEPQDNHSDTWW:V r w:PQDM\W8´W:Mm]8XWJV r wJPQILw:Z\D*ED^`_:M2DeSANEVLGJXPQDW`NFIc5DTElZzIvcMzS1M[VLWK:G-+8S1Mz^`_:D]8DTS Tp
`j¨ « ¨L V (x) = Ax · x >
A > $ @ !#" G@%<+,+ /(@ < ?< $ . #-G<J!")
u(x) = exp(−A1/2x · x/2)-$;:"7A$&+,". /( 1 3 (6 $ ω = tr(A1/2) O8 ;<+,+ :$&#+,"-. #:-$;'>7?(?< + @%:@ &> /(> <&>7+
! #" +d'%$B&"U)xlIaWSEDNRNRDySRDTENRM[VLWWJVL_JSM[W`NRPQV]:_:M[SRVLWS_JW:DNRG:XTVLPQM\DS1K?DTENRPHIaZ[DILwJS1NRPHIaM\NRD]:IaWJSZ[DTSDeSAhKJILEDTSU]8D7=M\Z[wDTP1N,oc5VLM[PdKJILPDb8D r K:Z[D'°¯ )(opjq D/w:_8N_:Z\NRM r DF]8DeS]8XTcLDTZ\V5K:KD r DW`NQSU^`_:M`SR_:M[cLDTW5NDTS1N=]8DLXTW:XPHIaZ[M[SRDP,YZzI$]8M r DTWJS1M[VLW*M[W8´JW:M[DZ[DPRXeS1_:Z\NQIaN&w:M[DW EV5W:W_DTW ]8M r DTWJS1M[VLW´JWJM\D^`_:MIj±$P r D$^`_:DNRV5_8NRD r IaNRPQM[EDS8+ r XNRPQMz^5_JD$PRXTDZ[Z\DDTS1N]8MzIa5VLWJILZ\MzSRILw:Z[Dv]:ILWJS_:W:D$wILSRDV5P1NQG:VahW:VLP r XDLp%:BWKJPRD r M[PQDZ\DeENQ_:PRDV5WtK?VL_:PQPHIvI5] r DN1NQPRDNQVL_JSFZ[DTSFPQXTSR_:Z\NQIaNQS,]8DEDN1NQDS1DeENQM\V5W p
-/[email protected] * Cd4C 7 NFR@i6%CExlIaWSyNQVL_8NtED^`_:MS1_:M\N
V]8XTSRM\5W:D _:W£DeS1KILED ]:D 7&M[Z[wDTP1NPRXTDZ r _:W:M]|g _JW£K:PQV]:_:MONSRETIaZzIaM[PRD 〈x, y〉 p
+ 6 ;§ «ªT« ?§ 9 F94-,/.01A243567698:0<;=>?0<.>@8:019A=01B3C;=.>D0135E73
VEF>HG
V IJ K=6F638183L >8M32BN64BO.64BO35E73 A 2PBOA=38λ ∈ R
38QSR 018&3T70UG=V3W2XA8MAYZ3S4.P28x ∈ V 20 L AB[0 \3
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VEFG
V i E01V3ZFEj.01V3 i V3818M3C3
〈Ax, y〉 = 〈x,A∗y〉 ∀x, y ∈ V.+ 6 ?§ ª j ªe« ?§ (PVL_:P
y ∈ V ´JbX5fJSRVLM\N L ∈ V ′ ZzIvnoVLP r DZ[M[W:XTILM\PQDEVLW`NQM\W_:D]8X´JW:M[DKJIaPL(x) = 〈Ax, y〉 p(P2ILPvIaKJK:Z\MzEIaNRM[VLWs]8_lkFG:XTVLPQ r D e¯:p\5p\e*]8D ,M[DTSRSLfM\Z/Db8M[S1NRD$_JW_:W:Mz^`_:Dz ∈ V
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L(x) = 〈Ax, y〉 = 〈x,A∗y〉 p
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+ 6 ;§ «ªT« ?§ 9 F9rc,/.01A243567698:0<;=>?0<.>@8:019A=01B3C;=.>D0135E73
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K i . 6323TFDBO>01BO3243WG. G G[201V3 un′;=. L 3B 3 3_EFG V I
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9 F9 CHGW2X3G?6F;=3CEF3#018=3B[VE735E01Y 3HG[0<.n01 \S0<3 i 8=.28M3201VA3B
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p C/DNRNRDSR_:M\NRDILK:KJILP1hNQM\DTW`N&w:M\DTW YvZzIwV5_:Z[D_:W:M\NRXnoDP r XD5p%PILP&IaM[Z\Z[D_JPQSTf8KV5_:P n 6= pVLW*I
‖en − ep‖2 = ‖en‖2 + ‖ep‖2 − 2〈en, ep〉 = 2,ED^5_JM K:PQVL_:c5Dl^`_ g Ia_JE_:W:DSRVL_JS1hiS1_:M\NRD]8DenWdg DeSAN&_:WJDS1_:M\NRD]:DC!IL_JEHG-+Lp
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i ≥ 1 )( / @ < =+ >7?(?7+ %>. # + )@-> GA
?><Ax = (aixi)i≥1
* @ !#" A %$& %. (") #!")A
%$& ?>;'$P J$&<"7+ <$limi→+∞ ai = 0
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b 4(6 j`¡ GF,4.>01V
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(uk)k≥1EF3
V.B[Y A3 E73 L 3=; 32B=G
69BO.=69BO3G_EF3A i L 3=;
Auk = λkuk6.2B
k ≥ 1.
8 ? UI9 F9 C/V rr DEVLWJSRXT^`_:DTWJED]8_^kFGJXVLPQ r Dp°¯8p :fJDNIc5DTElZ\DeS rr DeS&W:VLNQIjhNRM[VLWJSTf8VLW*VLw:NRM[DW`N,Z[I¢ 6 ¨5B «oªe« ?§¡TB¨ ª Jy]8DNQVL_8N,XTZ\X r DTW`N v ∈ Vv =
+∞∑
k=1
〈v, uk〉ukIc5DTE ‖v‖2 =
+∞∑
k=1
|〈v, uk〉|2.
• `j¨ « ¨LGFMr ( G<? )/ + %$M)<&>. #$ %Q+ -$(?< $&%$ /#"@- 3 %# # #-G<!")(N ?<#"(
v ∈ V N + @!">. Au = v
>/ %0"7 "7< !#") $R+,". u ∈ V $P Q$&<"7+ <L$P
v@
+∞∑
k=1
|〈v, uk〉|2λ2
k
< +∞.
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`j¨ « ¨LGF H V = L2(0, 1)
%A
+ >7?7?(+ ->. + @> /(V
/7>7$V
/ @ ?)> (Af)(x) = (x2 +1) f(x) * @ !#" A %$& %. (")(N / @ 2?<($P . )(N<>7"; >/ " ; > $?)>)$% ?>; #-G<'!")
A >I?>)$2/( )>7+ "(G$ ?<G? -$ ( ?<"74 >!@ '>7")$&$!#"
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V [ \L]_^a` b<cd]fe&]/g YTh-i ]_j Z j'k)] hml \ X \j)k'e3[ YZ b3[A^*]_[ hon ]'c Z ]_[j\2[ h p ] Z A [\P]bd`<` h Y c)b ZTYaX \ h Y \Pk'b Y j']qc X \ ZTY \2[2]b3[ ZrXs b<eut X3Y \ Z ]vc XGw ` b<c Z ]fe&] V e<b3\^ V x1ym\Ne&kzJ\ YZ
m = infu∈V \0
〈Au, u〉〈u, u〉 , ] Z M = sup
u∈V \0
〈Au, u〉〈u, u〉 .
h-X j_^)| ‖A‖ = max(|m|, |M |) |] Z ^ X3YZ m |~^ XGYZ M |m]_^ Zmn b h ]_[j`Fj X `2j)]e&] A x
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¯: ! "$#%&' ()*
K ~O $ ;<@ %T:*@ +-("!#(*$%&2( |〈Au, u〉| ≤ ‖A‖‖u‖2 3$ max(|m|, |M |) ≤ ‖A‖ 6 :3%')(E2:G'% "3!#!#(A
("9 %:G%) :3! 3@-$% $ =%)@-("$&%,E?3,%) %u, v ∈ V
4〈Au, v〉 = 〈A(u + v), (u + v)〉 − 〈A(u− v), (u− v)〉≤M‖u + v‖2 −m‖u− v‖2≤ max(|m|, |M |)
(
‖u + v‖2 + ‖u− v‖2)
≤ 2 max(|m|, |M |)(
‖u‖2 + ‖v‖2)
.
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:GD:3@-+-+-(")9 "!#!#(m ≤ M 2$ ("9 ("C*:G9 9'2@-;:G$&%)9 :+-@-(",+ 9'@ % ‖A‖ = M ≥ 0 9'@ % ‖A‖ = −m
:;<(m ≤ 0
6.-m$29'@ *)$9#+ (:39 ‖A‖ = M ≥ 0 / +' :3 %')(0:39 ‖A‖ = −m("9 %1"!#E+-4*%)("!#("$&%932 !#*%')@4&2(>("$D)(*!#E2+ :5:G$&%A
EF: −A 6 67@ %>2$( 9'2@ %)( (un)n≥1(B;<(*%)(")9
2$2@ %d:G@-)(*9 (V&@ (*9 %! : C @ !#@-9):G$%)( 2:G$29,+ :f82$2@ %)@-$ (
M 9 ("9 %: <;= u @ )(>(
limn→+∞
〈Aun, un〉 = M(*% ‖un‖ = 1.
-m!#!#(A
(*9 %?*!#EF:*%)( @-+m(*C @-9 %)( 2$2(9'293 r9'@-%)(#%)("+-+-(@&2(Aun′
"$;<(*"(3( 2:G$29V
;<(_)9q$2(+ @-!#@ %)(
v6 :3 %')(EF:G'% $ :
〈Aun, un〉 ≤ ‖Aun‖ ≤ ‖A‖ = M,
% 9& +' $ 2@ %&2(limn→+∞ ‖Aun‖ = M 9 ("9 %: <; @ )(>&2( ‖v‖ = M
6BA@-$F:G+-("!#("$&% "!#!#(
‖Aun −Mun‖2 = ‖Aun‖2 + M2 − 2M〈Aun, un〉,$ = %)@-("$&%C&2(
limn→+∞ ‖Aun −Mun‖ = 06~ + : 9'3293 r9'2@ %)(
n′ "("+ :A@-!#E2+-@4&2(D&2(un′"$;<(*"(3( ;<(*)9
v/M / L!#@-$9 9'@ M 6= 0 E + ($:G9 M = 0(*9 %v%')@-;@ :3+mE2@-9"&F @-+m@-!#E2+-@4&2(D&(
A = 0 6 6 : "3$&%)@ $@-%)> ( A $#(*$D @ %1 $&2(Aun′
"3$;<(*"((:39'9'@F;<(*)9Av/M / ("$#E2+-9
(f;<(_)9v 6 6GH 2$2@I"@ %) (+ : +-@-!#@-%)(!#$&%')(C&2(
Av/M = v 9 ("9 %: <;= u @ )(D&2( v("9 %/$;(*%)("
E)E )( / $23$ $2+: ‖v‖ = M 6= 0 6 :G9'9'J"@-;v+R:v;:G+ (*>E)3E)( M6
H IKLKINM?ORQ OTSFS V?XGYZV [\ ]_^r` bc_]Be<]g Yh-i ]_j Z ] Z A [\?] b_`<` h Y cdb ZTYaX \ h Y \Pk'b Y j']c X \ ZTY \2[2] c X3w ` b<c Z ]
e&] V e<b3\^ V xFK X [j ZrX [ Z j)k)] h δ > 0 | YTh \L ]:M Y ^ Z ]fb3[#` h [^?N*[OL [\\ X3w#i j']1zJ\ Y e&] n b h ]_[ jd^,`Fj X `Fj)]_^]_\Ne&]"P X jd^/e&] h L Y \ Z ]_j n b hh ] ]− δ, +δ[ | ] Z h ]^ X [^ s ]_^a` b<cd]fe&]_^ n ]'c Z ]_[ jd^`Fj X `Fj)]_^qbG^d^ X c Y k_^RQcSP b<c*[\P]e&]vcd]_^ n b h ]_[jd^`Fj X `Fj)]_^v]_^ Z e&]qe Yw ]_\2^ YaX \qz~\ Y ]xK ~OT 3$&%')$29&F $ $(E?("%EF:G9 :;<3@ $2(@-$82$2@ %)/% "+-"!#("$&%)9>(
V +-@-$2:G@ )( !#("$&% @-$F "E?("$22:3$&%)9 &2@o9'@-("$&%;<(_%)(")9BE )E )("9( AE?("9B;3:3+-(" )9BE)3E)("9
λ%)("+-+-("9>&2(
|λ| ≥ δ > 06U $ E)J*4(EF: D"$&%'d:3@I*%)@-$ 6U7 2EE?9'3$29f $V&F @-+1(*C @-9 %)( $2(9'2@ %)(@ $W8F$@ (
(uk)k≥1% "+-"!#("$&%)9 (
V +-@ $:3@ )("!#(*$% @-$F "E?("$F:3$&%)9 (*%m$2(9'@-%)((;3:3+-(" )9 E )E )("9 (λk)k≥1%)("+-9X&(Auk = λkuk
(_% |λk| ≥ δE?3%)3%
k ≥ 1. $$3%)(Ek
+-( 9'3293 r("9'EF:"( ;(*%)3)@-("+P("$(("$2 )>EF: + :>:3!#@-+-+ ((u1, u2, ..., uk)
6Y-m!#!#(Ek−1
("9 %@ $"+-29,9 %')@I*%)("!#("$&%,:3$9
Ek @-+ (*C @ 9 %)( 2$ ;(*%)(" ,2$2@ %d:G@-)(
vk ∈ Ek&2@P("9 %>3'%)09(3$F:G+#;
Ek−16
-m!#!#( |λk| ≥ δ + :N9'2@ %)(vk/λk
("9 %f=?3)$2*( 2:G$29V (*% E2@-9"&2( A
("9 %D"3!#EF:_% $ E?("%(*C%'d:3@ )(2$(B9'93 9'@ %)(B%)("+-+ (>&(
Avk′/λk′"$;<(_"((/2:G$29
V6Z-m(*E?("$F:3$&% E?3 j < k
3$ E?(" %*)@ )(
Avk
λk− Avj
λj= vk + (A− λk Id)
vk
λk− Avj
λj. / 56I[\ 6
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5 O -J & ¯8j $#;<_)@I82( T:*@ +-("!#(*$%&2(
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(A−λk Id)Ek ⊂ Ek−1 $,+-("9 (" C# (*)$2@-(*)9
%)(_)!#("9O;q)@ %)(>( / 56I[\ 6 :GE2E2:G'%)@-("$2$("$&%U; Ek−16Y-m3!#!#(
vk("9 % 3'%)09(3$F:G+;
Ek−1 3$ 2@ %
( / 56I[\ 6∥
∥
∥
∥
Avk
λk− Avj
λj
∥
∥
∥
∥
≥ ‖vk‖ = 1,
*(>@P("9 %>2$2("$&%'d: @I*%)@-$A:;<(+ :*$;<(*"(3("$"(/ (B+ :9'93 r9'2@ %)(Avk′/λk′
6
@K= YB2KNIDMPORQ O OG (1G ("!#!#(5&6 76I[\/!#$&%')( &2(m+ ("$9'("!=+-(,("9J;:G+-(")9E )E )("9/(
A$% (*9 %/EF:39;@ ( %d:3$F @-9>(+-(DG ("!#!#(f5&6 76I[9[#!#3$&%')(C&2(?"(*%("$9'("!=+-(("9 %/9'3@-%
82$2@ 9'@ %q@-$8F$@~"$!v=d:3=+-(f:;<(D\@"!#!#(v9'("2+~E?3@ $&%q :"!2+ : %)@ 3$ 6 :Gq:G@-+ +-(" )9 "3!#!#(A
(*9 % 8F$2@-(/E?9'@ %)@-;<( %)%)(*9 +-("9 ;3:3+-(" )9E )E )("9 9'3$%9 %')@I*%)("!#("$&%E?9'@ %)@-;<("9"6 3%)3$29(λk)
+-("9;3:3+-(" )9vE )E )("9f (
A(_%
Vk = Ker(A − λk Id)+ (*99'93 r("9'EF:"( E )E)(*9:G9'9'J"@-"9 / +-(VG ("!#!#(
5&6 76I[9[$2329 @ %1:G29'9'@&2( d0F:&2(Vk
("9 %m(>@-!#("$9'@ 3$C8F$@ ( 6 6Y $)("! : S&2(1&2(,+-("9m9'3293 r("9'E2:"(E )E )("9
Vk9'$&%>3'%)02(3$F:3 C ("CV;f ("C +2(*$ (P(*% 9'@ vk ∈ Vk
(*%vj ∈ Vj
:;<(k 6= j :G+-3)9 *!#!#(
A("9 %:3 %) :=! @-$&% 3$ :
〈Avk, vj〉 = λk〈vk, vj〉 = 〈vk, Avj〉 = λj〈vk, vj〉,
9&+ 3$ @-%&2( 〈vk, vj〉 = 0E2@-9"(
λk 6= λj67 @ %
W+' : 02*)("$"(:3$29
V(+ $2@-$D("9
Vk
W =
u ∈ V, ∃K ≥ 1%)("+%&2(
u =
K∑
i=1
uk, uk ∈ Vk
.
$$"3$29 %')@-%UT:"@-+ (*!#("$&%2$(>=2:39'(02@-+-=?(*'%)@-("$2$(B(W
EF: 1)"2$@-$("9 =F:G9'("91G'%)02$3)! :G+ (*9(d0F:&2(
Vk / _02:"$ (1 @ !#(*$29'@-$>8F$@ (m(*%~3'%)09(3$F:GCq("$&%')(m(" C 6 6T $&%')$9U&% (*$ :3@ %W = V
/ *(&2@oE);<(*d: :39'9'@H&2(/+ :9'@-%)((λk)
("9 %@-$82$2@-(qE22@-9"&2(V
("9 %(v@-!#("$29'@-3$A@ $W8F$@ ( 6 6% $@-$&%')@ %,+ G'%)029(3$2:3+ (W
8F$2@PE2:G
W⊥ = u ∈ V%)("+%( 〈u, v〉 = 0 ∀ v ∈W .
-m3!#!#(W
("9 %9 %d:G=2+-(AEF: A / AW ⊂ W 6 $ ;*)@ 8F( &2(
W⊥ ("9 %D:39'9'@ 9 %d:G=2+-(AEF: A:G
〈Au, v〉 = 〈u, Av〉 = 09'@
u ∈ W⊥ (*%v ∈ W
6O $ E?(" %# $D 8F$2@ v+ :)("9 %')@I*%)@-3$ (A;
W⊥ &2@ ("9 % :39'9'@o2$(:3EE2+-@I:G%)@-3$N+-@-$2":3@ )(>"$&%)@-$2(/:3 %) :=! 3@ $&%)(>"3!#EF:_%)(6F : B:GE2E+ @I: %)@-$ G ("!#!#(N5&6 76I[\ 9'@ W⊥ 6= 0 "(*%'%)()("9 %')@I*%)@-$ :3!#(*% :39'9'@2$2(;3:3+-(" E)3E)(N(*% $;(*%)(" /E)E )(
u ∈ W⊥ &@J9'3$&%:G29'9'@ ;:G+ (*q(*%q;<(*%)(*/E)3E)("9q (A6#-X (*9 %=@ (*$ 9q$2(
*$&%'d: @4_%)@ 3$ :;<( +-(X:3@ %&2( E2:G, 8F$@-%)@-3$ W "$&%)@-("$&% !:;/%)29m+ (*91;<(*%)(*)91E )E )("9(A(_%>&2(
W ∩W⊥ = 0 6 : R"3$29'&2(*$% $:D$2"("9'9):3@ )("!#(*$%W⊥ = 0 (*%R*!#!#(
W(*9 %
(*)!#B3$ ("$ @ %(W = 0⊥ = V
6
#IK IMPORQPOTS Q Go:"!#3$29 %'d: %)@ 3$NN.102"G)4"!#(56 76 8 ("9 % ("$*3)(;3:3+ :G=2+-(9'@A
$F ("9 % EF:39 8F$@-(mE?9'@ %)@-;<(m:3C )("9 %')@I*%)@-$29~9'@ ;3:3$&%)("9 E)4"9+ +-("9 ;3:3+-(" )9E)3E)("9o$2(J9'$&% EF:G9 $2*("9'9):3@ )("!#("$&%E?39'@ %)@ ;("9 +-("9m;3:3+-(" )9 E )E )("9 $$#$2+-+-("9JE?("2;("$&%_%')(>(*$#$2!v=)(.82$2@ (_% KerA / +-(,9'293 r("9'E2:"(E )E )(:39'9'J"@-?;+ :;3:3+-(">E )E)( $ +-+ ( 6 E?(*%*%')(/ (@-!#("$9'@ 3$ @-$82$2@-(6 •
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¯:T ! "$#%&' ()*
)!"U+ '!"' "U,+t ' ")&% ,))'t t-!.F.10 7:9IF,K(L G<N7J@TNU68@19B4=4&GF
0=VL_JSBPRDTcLDWJVLWJS2IL_ETIL]8PQD,cjIaPQM[IaNRM[VLW:WJDZ8M[W5NQPRV8]8_JMON/IL_uC/GJIaK:M\NRPQD9:p5qmg M\W`NRXTP N/]8D&ED=EI5]8PRDI5SRSRD%S=LXWJXPHIaZ:DeSANF^5_dg M[Z?STg IaK:K:Z[Mz^5_JDPHIY]8D=WJV r wJPRDT_8b r V8]8TZ\DeSB]8MSR?XTPRDTW`NQSTp`xlIaWJS/_:WyDeS1KILED]:D 7=M\Z[wDTP1NVW:VL_JSEVLWJSRM[]:XPQVLWJS=_:W:DvnoVLP r Dvw:M[Z\M[W:XTILM\PQD a(·, ·) f 6`ª «@? ?fEVLW`NRM[W_:DDNEVDPHEM[cLDLf8ELg DeSANRh¦Yah¦]8M[PQDl^`_:D
a(w, v) = a(v, w)fDN&M[Z Db8MzSANQD
M > 0DNν > 0NRDZzS,^`_:D
|a(w, v)| ≤M‖w‖V ‖v‖VK?VL_:P&NRVL_:N
w, v ∈ VDNa(v, v) ≥ ν‖v‖2V
KV5_:PFNRV5_8Nv ∈ V.
PV5_:PK?VL_:c5VLM[PIaK:K:Z[Mz^5_JDPZ[DTSPRXeS1_:Z\NQIaNQS]:DyZ[I SRDTENRM[VLW~K:PQXTEXT]8DTW5NQDLfW:VL_JSM\W`NRPQV8]8_:MzS1V5WJS_:WWJVL_:c5DZmM[W:LPQXT]:M\DTW5NefY SQIcLV5M\P_JWIa_8NQPRDyDeS1KILEDy]:D 7&M[Z\w?DPRNHp(0&V5_JSnILM[SRVLWSZ®g G-+KVLNRG:eS1DnoV5WJ]:I r DTW`NQIaZ[DS1_JM\cjIaW`NQD
V ⊂ H IcLDTEM[WADTENRM[VLW EV r KILENQDVDTS1N,]:DWJSRD]:IaWS
H.
¦p[LeqBg Db8K:PRDeSRSRM[VLW²AM[WADeENRM[VLWtEV r KJI5ENQD³&cLDT_8N!]8M[PQD=K:PQXTEM[SRX r DW`N/^`_:DZ®g VLK?XPHIjNQD_:P!]|g M\WJEZ\_S1M[VLW I^`_:M!Y
v ∈ VILSQS1V8EM\D Iv = v ∈ H
DeSANvEVLW`NRM[W_~DNvEV r KJILENuoc5VLM[PZ[I xX´W:MONQM\V5W p°¯8p ap2=_8NRPQD r DTW5N]8M\NTf|Zkg G +K?VaNQG:TSRD¦p[L,M r KJZ\Mz^`_:D$^5_JD]8DvNRV5_8NRD$SR_:M\NRDwV5PRWJXDv]:D V VLWKDT_8NDbNRPHIaM[PRD_:WJDS1V5_JS1h¦SR_:M\NRDEVLWcLDTPR5DW`NRD]:IaWS
Hp q DeS&DTSRKJI5EDTS
HDNVW:DKJILP1NHIaLDTW`N&KJILS=Z\DrTr DBK:PRV8]8_JMONSQEILZ[ILM\PQDLfDNdW:V5_JS Z[DTSdW:VaNQDPQVLWJS 〈·, ·〉H DN 〈·, ·〉V K?VL_:PXcMONQDP|NQVL_8NQDBEV5W8no_JSRM\V5W p0=VL_JS2EVLWJSRM[]:XPQVLWJSdZ\D!K:PQVLw:Z[ r D!cILPRMzIjNQM\V5W:W:DTZ`]8DFcjIaZ[D_JPQSK:PQVLK:PQDTSUS1_JM\cjIaW`N,oV5_vK:PQVLwJZ\ r DSRK?DTENQPQILZz08NQPRV5_:cLDTP
λ ∈ RDNu ∈ V \ 0 NQDZzS,^5_JDa(u, v) = λ〈u, v〉H ∀ v ∈ V. k8p\¯L
W ]8M[PHI^`_:DλDTS1N&_:W:DcjIaZ[D_:P,KJPRV5K:PRD]8_*K:PQVLwJZ\ r DcjIaPQMzIjNRM[VLWJW:DZm¦p[e¯LlVL_ ]8DZzInoV5P r DwJM\Z[M\WJXTIaM[PQD
a!DN,^`_:D
uDeSAN&Z[DcLDTENRDT_:P,K:PQVLK:PQDlI5SRSRV8EM[XLp
8 ? O VL_SZkg G +K?VaNQG:TSRDkp[L|Z[DTSDeS1KJI5EDeSHDNVW:D/K?D_JcLDW`N 1I r ILM[SIcLV5M\PZ[D rr DK:PQV]:_:MON=SQEILZ[ILM\PQDLp8M[W:VLW*M[Z[S,SRDPHIaM[DW`NFX`Ia_8buKJ_:M[SQ^`_:D V DTS1N=]8DWJSRD]:IaWS H p IaMzSEDZzI DeSANM r K?V5SQS1M[w:Z[DETIaPvILZ\V5PQSZ®g M[WADTENRM[VLW¶]:D
V]:IaWJS
HSRDPHIaM\NZkg Mz]8DTW5NQMONQX^`_:MFW g DTS1NvKILSEV r KJI5ENRD$cLVLM[PFZ®g :2b8DPHEMzEDv8p ¯:p\p •xVLW:WJVLWJSNRV5_8N]8DvSR_:M\NRD_:WDbD r K:Z\DEV5WJEPQDNlDNlN,+K:M[^`_:D]|g _:W:DvNRDZ[Z[DSRMONQ_JIjNQM\V5W pWPUV5_:P_JWV5_:cLDTP1N2w?VLPQW:X
ΩfaV5WK?V5SRD
V = H10 (Ω)fH = L2(Ω)
fLDNmZ[IlnoV5P r DFw:M[Z\M[W:XTILM\PQD&S8+ r XNRPQM[^`_:DDeSAN=]:X´JW:M[DKJIaPa(u, v) =
∫
Ω
∇u · ∇v dx.C/V rr D C∞
c (Ω)DTS1N=]8DWJSRDYZ[InoV5M[S,]JIaWJS
H10 (Ω)DN=]:IaWS
L2(Ω)f:DN&LPLEDlIL_PkFGJXVLPQ r D
=p 9Jp ¯:]8D,DZ[Z[M[EHG f Zkg G-+K?VaNRGJTSRD¦p[Le&DTS1NcLXPQM\´JXD5fDNlV5WIyc_~Ia_¸C/GIaK:M\NRPQD!.t^`_:DEDNRNRD
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5% #%&' ( * )1D(1 < ( ¯8eºnoVLP r Dlw:M[Z\M[W:XeIaM[PRD a DTS1N!w:M[DWEV5W5NQM\W_:DlDN&EV`DTPQEM\c5D=SR_:P V p-P2IaPF_:W:DS1M r K:Z[DlM[W`NRXTLPHIjNRM[VLWuKJIaPKJIaPRNRM[DLf8V5Wtc5VLM\N!nI5EM[Z\D r DW`N=^`_:D$kp[e¯5/DTS1N&XT^`_:M[cILZ\DTW`N,Y
−∆u = λu]JIaWJS
Ωu = 0
SR_:P∂Ω,Eag DTS1N1hiYjhi]8M\PQD^5_JD
λDNuSRVLW`N,cILZ\DT_:P,K:PQVLK:PQDDN,noV5WJENQM\V5WtK:PQVLKJPRD]8_ qdILK:Z[I5EM[DW pqdDTS&SRVLZ[_8NQM\V5WJSF]8Dyk8p\¯L!S1V5W`N,]:VLW:W:XTDTSFKJILP,Z\DPQXTSR_:Z\NQIjN&SR_:M[cILW`NTp
b 4(6 j`¡ OGFc,4.>01V3H
EF3FT 3G?6F;=3GeEF3 018=3B[ BOA=38 GeE73eE>01YZ3HG[0<.> 01 \S0<3 ISJ G=676H.G33V ⊂ H L 3=;%01j3=;?0<.>!;=.Y6H7;V33 3 V 3G= EF3HG3gEF>HG
H I ,4.>01 a(·, ·) 243.B[Y 3 018:019A=>01BO3 GY ADB[0<3 ;=.?013 3d;=. 3BO;0 L 3aG[2B V I 8M.B=GZ8M3G L 8M32B=Gd69BO.=69BO3G EF3 I .B[Y 3o2935G[201 3C;B.0UG=GV3(λk)k≥1
EF3_BOA=38 Go6.)G[01?0 GZ20SV39E L 3B=G8 R 01 \p0 i 3018 3OT70UG[V3W293G3v 0183B[D0<393CEF3H (uk)k≥1
EF3 L 3;V32B=Gp69BO.=69BO3G5)G=G. ;0<A[G i ;R 3G= ` E>01BO33uk ∈ V,
3a(uk, v) = λk〈uk, v〉H ∀ v ∈ V.
_3o698: G i (uk/√λk)k≥1
3[G[293G32018=3B[D0<393CEF3V6.2B#8M3%69B. E>201pG;=8M>01BO3
a(·, ·) I+ 6 ?§ ª j ªe« ?§ 3PVL_:P
f ∈ H fJW:VL_JSFPQXTSRVLZ[cLV5WJS/Z\DK:PQVLwJZ\ r DcjIaPQMzIjNRM[VLWJW:DZNQPRV5_:cLDTPu ∈ V NQDZ ^`_:D a(u, v) = 〈f, v〉H
K?VL_JPFNRVL_:NRDnoVLWENRM[VLWv ∈ V. ¦p[ 95
ViZ|DTS1N!nI5EM[Z\D]8Dc5XPQMO´JDTP/Z[DTSFG-+K?VaNRGJTSRDTSF]8_PkFG:XV5PR r D9:p 9:p[]8DlqIjbh M\Z[LPHI r K?VL_:P¦p[ 9`^5_JMIL] r DN]8V5WJE_:W:D_JW:M[^`_:DS1V5Z\_:NRM[VLW u ∈ V p W ]8X´JW:M\N=_:W:DvIaK:K:Z[MzEIjNQM\V5W Z\M[W:XeIaM[PRD A ]8DH]:ILWJS
V^`_:M|Y
fILSQS1V8EM[D=ZzIS1V5Z\_8NQM\V5W
u = Af p%2&_8NQPRD r DTW`N,]:MONef8Z®g IaK:KJZ\MzEIaNRM[VLWyZ[M\WJXTIaM[PQD ADTS1N=]8X´JWJM\DKJILPAf ∈ V NQDZd^`_:D a(Af, v) = 〈f, v〉H
KV5_:PFNRV5_8Nv ∈ V. ¦p[=`
:mW*K:PRDTWJIaW`Nv = Af ]:ILWJSk8p\ =`f8VLWVLw8NQM\DTW5N
ν‖Af‖2V ≤ a(Af,Af) = 〈f,Af〉H ≤ ‖f‖H‖Af‖H ≤ C‖f‖H‖Af‖VEIaP=Z®g VLK?XPHIjNQD_:P=] g M[WADTENRM[VLW I ]8D V ]JIaWJS H DTS1NEV5W5NQM\W_ p3P2IaPEVLWJSRXT^`_:DTW`NTfJZkg IaK:K:Z[MzEIjNQM\V5WZ\M[W:XTILM\PQD A DTS1NEVLW`NRM[W_:D]8D H ]:IaWS V p|W ]8X´JWJMON r ILM\W`NQDWJILW5N&_JW:DIaKJK:Z\MzEIaNRM[VLW*Z[M\W:XeIaM[PRDA = IA ]8D H ]:IaWJS H f8^`_:M?DTS1N!w:M[DWEV5W5NQM\W_:D5p8C/V rr D I DeSAN,EV r KJI5ENef5Z[D=K:PQV8]8_:M\N A DTS1NIa_JSQS1M EV r KJI5ENocLV5M\P,Zkg :2b8DPHEMzEDp°¯8p 95p%PV5_:P r VLW`NRPQDP,^`_:D A DTS1N&Ia_8NQVahiIL]AV5M\W`NTfV5WtK:PQDW]v = Ag ]:ILWJS¦p[=`BDN&V5WtV5w8NRM[DW`Nef:KV5_:PFNRV5_8N f, g ∈ H f
〈f,Ag〉H = 〈f,Ag〉H = a(Af,Ag) = a(Ag,Af) = 〈g,Af〉H = 〈g,Af〉H ,YyEIL_JS1Dv]8DZzIyS8+ r XNRPQM\D]:D a f EDv^5_JMK:PQVL_JcLD^`_:D A DTS1NIa_:NRVahiIL]AVLM[W5N=]:X´JW:MUK?V5SRMONQMOnB]:ILWJSHp8W$K?D_:N!]8V5WJE=ILK:K:Z[M[^`_:DTPBZ[DkFG:XV5PR r Dlp°¯8p YZkg V5KXTPQIaNRDT_:P A ^`_:M?DWucLXPQM\´JD,NRV5_8NRDeSmZ[DTSG +K?VaNQG:TSRDTSTp ViZUDb8MzSANQD_:W:DS1_JMONQDv]8XeEPQVLMzSRSQIaW`NRD
(µk)k≥1]8DPQXDTZ[S,K?V5SRM\NRM\nSl^`_:MdNRDTWJ] c5DPHS&¹:fDNFM\Z?Db8MzSANQD=_:W:DlwJI5S1DG:M\Z[w?DPRNRM[DW:W:D
(uk)k≥1]8DHnoVLP r XD=]8DlcLDeENQD_:PHSmK:PQVLKJPRDeSB]8D A fIcLDeE
Auk = µkukKV5_:P
k ≥ 1.
Ecol
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¯5¯a¹ ! "$#%&' ()*
&D r IaPH^`_:VLWJS=^5_JDLf?KJIaPlEDNRNRDvX5ILZ\M\NRX5fJZ[DTScLDeENRDT_:PHS,K:PQVLK:PQDTS ukILK:KJIaPRNRM[DWJW:DW`N=W:V5W S1DT_:Z[Dhr DTW5N$Y H r IaMzSIa_JSQS1M,Y V p ,DTcLDWJVLWJS r IaM[W5NQDWJILW`NIL_¡K:PQVLw:Z[ r DIa_8bcjIaZ[D_JPQSvK:PQVLK:PQDTS¦p[e¯5/^`_:M|K?D_:NSg XTEPRM[PQD
a(u, v) = λ〈u, v〉H = λa(Au, v) ∀ v ∈ V,YETIa_JSRD]8DZ[I]8X´JW:M\NRM[VLWsk8p\ =`f:Eag DTS1N1hiYjhi]8M\PQDa(u− λAu, v) = 0
f:]:VLWJEu = λAu = λAu.
P2ILPUEVLWJSRXT^`_:DTW5NefTZ[DTScjIaZ[D_:PHSK:PRV5K:PQDTS(λk)k≥1
]8_K:PQVLwJZ\ r D/cjIaPQM[IaNRM[VLW:W:DTZ?¦p[e¯5 SRVLW`NDb8I5EhNQD r DW`NZ\DeSlM\WcLDTPQSRDTSl]8DeScjILZ\DT_:PQSK:PQVLK:PQDTS (µk)k≥1]:D
Af|DNZ\DT_:PQSlc5DTENQD_:PHSK:PRV5K:PQDTS=SRVLW`NZ[DTS rTr DeSp WtK?V5SRD
λk =1
µk
DNvk =
uk√λk
.
P2ILPEVLWJS1NRPQ_JENQM\V5W fZ\DeSc5DTENRD_JPQSK:PQVLK:PQDTSuknoVLP r DW`Nv_:WJDwILSRDuG:M[Z\w?DPRNRM[DWJW:D ]8D H pmWc5XPQMO´D^`_:D
a(vk , vj) =a(uk, uj)√
λkλj
= λk〈uk, uj〉H√
λkλj
= δkj ,
DNFEV rr DZkg VLPRNRG:V5LVLWIaZ]:DTS (vk)k≥1]JIaWJS
VDTS1N,EVLW`NRDTW_u]:IaWS/Zkg VLPRNRG:V5LVLWIaZ]:DTS
(uk)k≥1]JIaWJSH®^`_:MDeSAN/PRXe]8_:M\N/Ia_cLDeENQD_:P/W`_JZzfLVLW$DTWy]8XT]:_:MON!^5_JD&Z\DeS
(vk)k≥1noV5P r DTW5Nm_JW:D=wJI5S1DGJM\Z[wDTP1NQM\DTW:W:D]8D
VK?VL_:P,Z[DK:PQV8]8_:M\N=SRETIaZzIaM[PRD
a(u, v)p
8 ? O ViWJSRM[S1NRV5WJSFS1_:P,Z[DnIaM\N=^5_JDZkg V5KXTPQIaNRDT_:P A f:]8X´JWJM|KJIaPk8p\ =`f`DeSAN,Zkg V5KXhPHIjNQD_:P&]:DPRXeS1V5Z\_:NRM[VLW*]8DZzInoVLP r _:ZzIjNQM\V5WcjILPRMzIjNQM\V5W:W:DZ[Z[DLf:Eag DTS1N1hiYjhi]8M[PRD^`_ g M\ZDTS1N&DW ^`_:DTZ[^`_:DSRVLPRNRDF,> « § ?5je]8DFZzI=noV5P r DFw:M[Z\M[W:XeIaM[PRD a p5Cg DeSANmK?VL_:PmEDNRNRD,PHIaMzS1V5W^`_:D,Z\DeScjIaZ[D_:PHSUK:PRV5K:PQDTSλk]8D ZzI¸noVLP r _JZ[IaNRM[VLW¬cjIaPQM[IaNRM[VLW:W:DTZ\Z[D SRVLW`NyZ\DeS$M[W`c5DPHS1DeS$]8DTSucjIaZ[D_:PHS$K:PQVLK:PQDTS µk
]:D A pP2ILPDb8D r KJZ\D5fdDW¶]8M r DTWJS1M[VLW~´W:M\DtZ[I noVLP r Duw:M\Z[M[W:XTILM\PQDuSTg XeEPQMON a(u, v) = Ku · v DNvV5WIA = K−1
p`xD rTr D5fKV5_:PFZ\DlqdIaKJZ[I5EM[DWyV5WuI A = (−∆)−1®S1DT_:Z Z®g M[W`c5DPHS1D]8_tqIaK:ZzILEM\DTWDeSANEV r KJILENTf KJILSlZ[D$qdIaKJZ[I5EM[DW Z[_:M\h rr D c5VLM[PZkg :mbDTPQEM[EDup°¯8p 95pW:mWnIaM\NTf ELg DeSANZ\D`IaM[W]:DtPQX5_:Z[ILPRM\NRXt]8DtZzISRVLZ[_8NQM\V5W¡]8_¶qIaK:ZzILEM\DTWKJILPvPHIaK:K?VLPRNvIL_¶SRDTEV5WJ] r D r w:PQDt^`_:MFDTS1NYZkg VLPQM\5M\W:D]8DZzIEV r KJI5EM\NRX]8DZkg V5KXTPQIaNRDT_:P (−∆)−1 p •
`j¨ « ¨5OJ@ #-G<"() >< >7-;L/"BB@- 3 % + G ?(+ >%M+-7?<< $&/(X%7 U) < ;@/( + >n < + )@-> G
a(·, ·) ?><A+-(?< $& ?7+,"$Z> 7+ I!#"!, +J( $;;I/(<"%)$;&>7;%$0?< $ . -$
η > 0 ν > 0
;<+,+ -$'!#")
a(v, v) + η‖v‖2H ≥ ν‖v‖2V?<#"(9;#"-
v ∈ V.1 >7$e C%>)$=+ -$g>7+ <"7$ ?<#?<G%$
(λk)k≥1) $R ?>)$g 1 @ <:?< $ . -$-N > $%)@< -$R<"7+ - λk + η > 0 6
0=VL_JSm]:VLW:W:V5WJSIa_KJI5SRSQIa5DB_:W:D&ETIaPHILENRXPQMzSRIaNRM[VLWNQPReS_:NRM[Z\D&]8DeS2cjIaZ[D_:PHSUK:PQVLK:PQDTS]8_K:PQVahwJZ\ r DlcjIaPQM[IaNRM[VLW:WJDZ2k8p\¯LfIaKJKDTZ\XTDlU « §U¨ « Bt¢U¬ « §d©Q¡8?¬¢vj:§ ª © « 4 `p
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5% #%&' ( * )1D(1 < ( ¯L¯:PUV5_:PEDZzI&V5WM[W5NQPRV8]8_JMONZ\DF^`_:VaNQM\DTW`NU]8D'&I+Z\DTM\5G]8X´W:M®fK?VL_:P2EHGJI5^`_:D2noVLWENRM[VLW
v ∈ V \0 fKJIaPR(v) =
a(v, v)
‖v‖2H.
B? «ªT« §9Urvj:§ ª © « 4 `l,4.01V3H
EF3FTa3GD6H7;3GeE73 #018=3B[ BOA=38 GWEF3E01Y 3G=0<.>P01 \S0<3 ISJ G[6F6.)G3#3 V ⊂ H L 3=;e01 j3;D0<.^;=.Y6H7;V33 3 V 3G[SEF3HG3EFG
H I ,4.01 a(·, ·) 293.>B[YZ3 018:019A=01B3GY ADB[0<3_;=.?0133;=. 3BO;0 L 3G[2B V I .>2Bk ≥ 0
. 9.> 3 Ek 8 R 3G3Y 83^EF3GaG. G 3G?6F;=3G L 3;V.B[0<38 GPE73 E01Y 3G=0<.> k E73V I5J 9.V3
(λk)k≥18MG=201V3/¨aj « Te:§ ª ^EF3G L >8M32B=G 64BO.64BO3GdEW64BO. 8YZ3 L B[0<>?0<.>938 I >I 8M.B=G i 6.2B#V.2 k ≥ 1 i 8M k Y 3 L 8M32B69B.69B3C3G[E7.4A=3o6B
λk = minW∈Ek
(
maxv∈W\0
R(v)
)
= maxW∈Ek−1
(
minv∈W⊥\0
R(v)
)
.¦p[5.5
6B[D0<;28h0<3B i 8e69B3YC0B3 L 832B64BO.64BO3 L AB[0 \N3
λ1 = minv∈V \0
R(v),¦p[ 5
3N .>2/6.01%EF3_YC0101YC2Y E7HG I 3G[N2 L 3=;V32Bp64BO.64BO3Z)G=G. ;0<AZ` λ1 I+ 6 ?§ ª j ªe« ?§ VLM\N
(uk)k≥1Z[IwILSRD,G:M\Z[w?DPRNRM[DW:W:D]8D
HnoVLP r XD&]8DeSmc5DTENQD_:PHSK:PQVLK:PQDTS]8D k8p\¯Lp|xvg ILK:PQTSZ[DakFG:XV5PR r Dyp°¯8p :f (uk/
√λk)k≥1
DTS1N_:W:DwJILSRDG:M[Z\w?DPRNRM[DW:WJDy]8DVpWK?D_:N/]8VLWE&EILPQI5ENRXTPRMzSRDPUZ\DeSmDTSRKJI5EDTS
HDNVYKJILP1NQM\Pm]8D=Z\DT_:PB]8XeEV r K?V5SRMONQM\V5W$S1K?DTENRPHIaZ[DocLV5M\PFZzI ,D r ILPQ^`_:Dp°¯8p º5
H =
v =
+∞∑
k=1
αkuk, ‖v‖2H =
+∞∑
k=1
α2k < +∞
,
V =
v =
+∞∑
k=1
αkuk, ‖v‖2V =
+∞∑
k=1
λkα2k < +∞
.
WPQD r ILPQ^`_:D,Ia_KJI5SRSQIa5D!^`_:D5fLEV rr D,Z[DTSBcjIaZ[D_:PHSK:PQVLK:PQDTS λkS1V5W5N r M[W:VLPQXDeSKJIaP λ1 > 0
fEDNRNRD*EILPQI5ENRXTPRMzSQIjNRM[VLW~nILMONw:M[DW¬IaK:KIaPHI)zNRPQDVEV rr Dt_JW SRVL_JS1h¦DTSRKJILEDu]8D H pBW¡K?D_8NIaZ[VLPHS/PQXXeEPQM\PQDlZ\D^`_:VaNQM\DTW`N=]8D=&I+Z\DTM\5G
R(v) =
∑+∞k=1 λkα
2k
∑+∞k=1 α
2k
,
ED^`_:M ]8X r V5W5NQPRDlM rr Xe]8M[IaNRD r DTW`NFZ[DPRXeS1_JZONHIjNFKV5_:P,Z[IK:PQD r M[PQDcjILZ\DT_:PFK:PRV5K:PQDLp ViW5NQPRV8]8_JMOhS1V5WJSFZ\DSRVL_JS1h¦DTSRKJILEDWk ∈ Ek
DW:5DWJ]:PRXKJILP(u1, u2, ..., uk)
p?W IR(v) =
∑kj=1 λjα
2j
∑kj=1 α
2j
∀ v ∈WkDN
R(v) =
∑+∞j=k λjα
2j
∑+∞j=k α
2j
∀ v ∈W⊥k−1,
Ecol
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¯5¯L¯ ! "$#%&' ()*
] g VQUtZkg V5W ]8XT]8_JMONλk = max
v∈Wk\0R(v) = min
v∈W⊥k−1
\0R(v).
8VLM\NW_:W¸S1V5_JSAh¦DTSRKJI5ED^`_:DZzEV5WJ^`_:D]:ILWJS Ek pdC/V rr D W DTS1Nl]8D]8M r DTWJS1M[VLW k DN Wk−1]:Dl]8M r DWJSRM[VLW k− 1
fZ®g M[W5NQDPHS1DeENQM\V5WW ∩W⊥
k−1
Wdg DeSAN,KJI5SBPQXT]8_JMONQDY 0 pQP2ILPFEVLWJSRXT^`_:DTW`NTfmax
v∈W\0R(v) ≥ max
v∈W∩W⊥k−1
\0R(v) ≥ min
v∈W∩W⊥k−1
\0R(v) ≥ min
v∈W⊥k−1
\0R(v) = λk ,
ED=^`_:M?K:PRV5_:cLD,ZzIKJPRD r M[PQD,X`IaZ[MONQX=]:ILWJS=¦p[5.Lp5xD rr D5f`SRM W DeSAN!_:WuSRVL_JS1hkDeS1KILED,]:ILWJSEk−1f:ILZ\V5PQS
W⊥ ∩WkW g DTS1N&KJI5SFPRXe]8_:M\N=Y 0 fDN
minv∈W⊥\0
R(v) ≤ minv∈W⊥∩Wk\0
R(v) ≤ maxv∈W⊥∩Wk\0
R(v) ≤ maxv∈Wk\0
R(v) = λk,
ED/^`_:MK:PQVL_:c5DBZzI=]8DT_8b8M\ r D/XT5IaZ[M\NRXB]:ILWJSFk8p\.LpjV5MON r ILM\W`NQDWJILW5N u _:WK?VLM[W`N]8D r M\W:M r _ r]JIaWJSFk8p\ 5p5PV5_:Pv ∈ V fjVLWvM\W`NRPQV8]8_:M\N2Z[I=noVLWJENRM[VLW f(t) = R(u+tv)
]|g _:W:DFcjIaPQM[ILw:Z\D!PQXDZ[Z[Dt ∈ R^`_:MI5] r DN&_:W r M[W:M r _ r DW t = 0
pOP2IaPEVLWS1Xe^5_JDW`N&SRI$]8XTPRM[cLXTDSg IaWJW`_JZ\DDTWt = 0p
:BWtNQDWJILW`N=EV r K8NQD]8DED^`_:D f(0) = λ1fJ_:WSRM r KJZ\DETIaZzE_:Z r VLW`NQPRD^`_:D
f ′(0) = 2a(u, v)− λ1〈u, v〉H
‖u‖2H.
C/V rr D v DTS1N^`_:DTZ[EVLWJ^`_:D*]:ILWJS V f2Z[I¸EV5WJ]8M\NRM[VLW f ′(0) = 0W g DTS1N$PRM[DW¶]|g IL_8NRPQD*^5_JDtZzInoV5P r _:ZzIjNRM[VLWucILPRMzIjNQM\V5W:W:DTZ\Z[Dk8p\¯Lf`ELg DeSANRh¦Yah¦]8M[PQD^`_:D u DeSANF_:Wuc5DTENRD_JP/K:PQVLKJPRD=I5SRSRV8EM[XYZ[IcjILZ\DT_:P,K:PQVLK:PQD
λ1p
-!.F.@A N F,Gd<=7 E H7:9H=7 GE¡;< NHF,N2>|@TGd4WKDT_8NlM rr XT]8MzIjNQD r DW`NIaK:K:Z[Mz^5_JDPlZ\DZkFG:XV5PR r Dp°¯8p yYuZ[IunoVLP r _:ZzIjNQM\V5W cjIaPQM[IaNRM[VLW8hWJDZ[Z\D¸]8_·qdILK:ZzILEM[DW£Ic5DTE EVLWJ]8M\NRM[VLWStIL_8b Z[M r M\NRDeSt]:DxM\PQM[EHGJZ\DNTfFED^`_:MlW:V5_JSu]8V5W:W:DZ\DPQXTSR_:Z\NQIaN,SR_:M[cjIaW`NTp
b 4(6 O c,4.>01Ω2 .> L 3B[ =.B[4ACBOA >28:0<3BCE73a;8M)G=G3 C1 E73
RN I f 8 3TF0UG[V3_293G[201V3P;BO.>0UG=G>V3(λk)k≥1
EF3!BOA=38 G#6.)G[01D0 G 20d 39E L 3B=Ga8 R 01\S0 i 3W018o3T70UG=V3!293 =)G32018=3B[D0<393ZE73
L2(Ω) (uk)k≥1 i 3818M3C3C;*F3 uk676H>B[?0<3g`
H10 (Ω)
3 L AB[0 \3
−∆uk = λkuk6 I 6 I EF>HG Ω
uk = 06 I 6 I G[2B ∂Ω.
¦p[a+ 6 ?§ ª j ªe« ?§ 3PUV5_:PmZ[D&qdILK:Z[I5EM[DW$IcLDeEFEVLW]8MONQM\V5WJSmIL_8bvZ[M r M\NRDeSB]8D=xM[PRMzEHG:Z[DNTf5VLW$EHG:V5MOhSRM\N
V = H10 (Ω)fH = L2(Ω)
fJDN&ZzInoV5P r Dw:M[Z\M[W:XeIaM[PRDS8+ r XNRPQM[^`_:DDTS1N=]8X´JW:M[DlKIaPa(u, v) =
∫
Ω
∇u · ∇v dx,
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5% #%&' ( * )1D(1 < ( ¯L¯ 9DN&Z[DlKJPRV8]8_:M\N=SQEIaZzIaM[PQDlSR_:P
L2(Ω)DTS1N,w:M[DW SJP〈u, v〉H =
∫
Ω
uv dx.
Wc5XPQMO´DILM[SRX r DTW5N,Z[DTS,G-+K?VaNRGJTSRDTS,]8_^kFG:XTVLPQ r Dp°¯8p :p lPLEDIL_ kFG:XV5PR r D=p 9Jp ¯:l]8D,DZ[Z\MzEHG fVDTS1NwJM\DTWEV r KILENQD r DW`N=M[WJEZ\_JSl]:ILWJS H p|C/V rr D C∞
c (Ω)DTS1N]8DTWJSRDvY$ZzInoVLMzS]:IaWJS
HDNv]:ILWJS
VfVDTS1Nw:M[DW]:DWJSRDu]:ILWJS
Hp(:mW8´JW fVLWI*c_Ia_¡C/GJILK:M\NRPQD . ^`_:D$Z[InoVLP r Dw:M[Z\M[W:XeIaM[PRD a DTS1NwJM\DTW¸EVLW`NRM[W_:DvDNEVDPHEM[cLDS1_:P V p P2ILPEVLWJSRXT^`_:DTW`NTf M\ZDbMzS1NRD_:W:DS1_:M\NRDEPQVLMzSRSQIaW`NQD
(λk)k≥1]8DtPRXTDZzSK?V5SRMONQMOnS^`_:MmNQDWJ]sc5DPHSZkg M[W8´JW:MkfDNM[Z!Db8M[S1NRDu_:W:DuwILSRDG:M\Z[w?DPRNRM[DW:W:D]:D
L2(Ω) (uk)k≥1f:NRDZzS&^`_:D
uk ∈ H10 (Ω)DN
∫
Ω
∇uk · ∇v dx = λk
∫
Ω
ukv dx ∀ v ∈ H10 (Ω).
PILPy_:W:DS1M r K:Z\DM\W`NRXTLPHIjNQM\V5W¶KJILPuKJILP1NQM\D®]8_ rr D N,+K?D^`_:DEDTZ\Z[D K:PHIjNQM[^`_:XTD ]:IaWS$Z[I]8X r VLWSANQPQIaNRM[VLW]8_5kFGJXVLPQ r D.:p ¯:p ¯5 VLWVLw8NQM\DTW5NFk8p\jap,D r ILPQ^`_:V5WJSd^`_:DmWJVL_JSW g _8NRM[Z\MzSRVLWJSZ[IPQX5_:Z[ILPRM\NRX]8DΩ^`_:DKV5_:P&K?VL_:c5VLM[P&IaK:KJZ\Mz^`_:DP,Z[DkFG:XV5PR r D]8DNRPHILEDJ=p 9Jp\59DN]8V5W:W:DP_:W S1DTWJSm²AKJPRDeSR^`_:DlKJIaPRNRV5_8NA³YZ[IEV5WJ]8M\NRM[VLW Ia_:buZ\M r MONQDTS&]:DxM[PRMzEHG:Z[DNTp
8 ? UI9U £qmg G-+KVLNRG:eS1DySR_:PZ[DyEIaPHILENRTPRDwV5PRWJX]:D$Z®g VL_:c5DPRNΩDeSANILwJS1V5Z\_ r DTW`NnoVLWJ]:I r DW`NQILZ\D]:ILWJS&Z[D5kFG:XTVLPQ r Dvp 9:p/.8p?MdDTZ\Z[DW g DTS1N=KJI5S=SQIjNRMzS1nIaM\NRDLfZ\D_kFG:XV5PR r D)=p 9Jp ¯:]8D=,DZ[Z\MzEHG~SR_:P,Z®g M[WADTENRM[VLW*EV r KJI5ENQD]:D H1(Ω)
]:ILWJSL2(Ω)!DeSAN,DTWt5XW:XTPQILZnIL_8b|f8DN,VLWKDT_8N r V5W`NRPQDP^`_:DZ[DCkFG:XTVLPQ r D8p 9Jp .yW g IuKJI5S=Z[M[D_ p :mWnIaM\NTf M[ZSRDKDT_8N^5_dg M[ZDb8M[S1NRD_:W:DM\W8´W:MONQXtW:VLW]:XW:V r w:PHIaw:Z[De&]8DcjIaZ[D_:PHS&KJPRV5K:PRDeSB²1LXWJXPHIaZ[M[SRXDeSi³Ia_ SRDWJSV U Z[DTS&noVLWJENRM[VLWJSK:PRV5K:PQDTS!W g IaK:KJILP1NQM\DTW:W:DTW5N,KJI5S,Y
L2(Ω)p32µZzIZ\_ r M[PQD]8DZkg :mbDTPQEM[EDp[Lp[V5W r Xe]8MONQDPHIZ\DEILS,]:_*qIaK:ZzILEM\DTWt]:ILWJS
Ω = RN p • `j¨ « ¨LOGF / $ N = 1
NE #$ / Ω =]0, 1[ >7+ "7+ <'?(+ ) ; -;";%$ + -$>7+ <"7$ ?<G?<G%$A + -$5 #) . )$ ?<G?<G%$ /" $<>7?7+ > >) #)/ . )$ >7"T+
;%$ /( &7+ =143 3 6% + > /( /( + >/ @ ?< $ . 2$ ?< >7+ /(5 ? (+ 1 ) L+ > > G!#"=3 % 76 N G<E!#" + >A$&@
+∞∑
k=1
ak sin(kπx)
)<GK /7>7$L2(0, 1)
$ % $R<"7+ - $P ∑+∞k=1 a
2k < +∞ N% /7>7$
H1(0, 1)$P 9$& "(+ <$P ∑+∞
k=1 k2a2
k < +∞ `j¨ « ¨LO ( %)$P / G "(?)> >7+,+ @<+ @<? ? /(
Ω =]0, L1[×]0, L2[× · · · ×]0, LN [NO
+ %$(Li > 0)1≤i≤N
$&#- /(%$p%)$;&>7;%$'?< $ . -$ >7+ "7+ < ?(+ ) ; -J;#"-;-$'+ %$ >7+ <"7$?<#?<G%$ % + -$ . #$?<#?<G%$ /#" $<>7?7+ > >X%/ . #$ >7" + ;-$ /( (+ %143 3 6 qdDCkFGJXVLPQ r D8p 9Jp .yS1DLXTW:XPHIaZ[M[SRDIaMzSRX r DW`NlIL_EI5S] g IL_8NRPQDTSlEV5WJ]8M\NRM[VLWJSlIa_8b Z[M r M\NRDeSp0&VL_SFZ[ILM[SQS1V5WJSFIa_*Z[DTENQD_:P,Z[DSRVLM[W]8D]8X r VLW`NRPQDP,Z[DEVLPQVLZ[ZzIaM[PRDlS1_:M[cjIaW`NTp
Ecol
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¯5¯ = ! "$#%&' ()*
vj® « OGc,4.>01Ω
2m.> L 3B[.>B[9AnBOA F28:0<3B EF3RN EF.> 8M =BO.?0BO3
∂ΩG3
E7A=;.>Yg6.)G3 3 E73FTZ6H>B[?0<3GaE0UG1j.01 3GCBA F28:0BO3G∂ΩN
3∂ΩD
L .01BZ8Mp0F2BO3 I FI f 83T70UG[ 3W243dG[201 3;BO.>0UGG>V3(λk)k≥1
EF3#BA38 Gp6H.G=01D0 G.P28 GW20 V34E L 3B=Gd8 R 01\S0 i 3S0183T70UG[ 3g293 =G3B 0183B[D0<393WEF3L2(Ω) (uk)k≥1 i V38183d3;"H73 uk
6F6B[D0<3 `H1(Ω)3 L AB[0 \N3
−∆uk = λkukE7G
Ωuk = 0
G[2B∂ΩD
∂uk
∂n = 0G[2B
∂ΩN .
8 ? OxILWJSvZ\DEILSv]|g _:W:DtEVLWJ]:MONQM\V5W¡Ia_8b~Z[M r M\NRDTSvK:_:PQD r DTW5N]8D 0=D_ r ILW:W fELg DeSANRh¦Yah¦]:M\PQD^`_:D∂ΩD = ∅ f2ZzInoV5P r D*w:M[Z\M[W:XeIaM[PRD*W g DTS1NyK:Z[_JS$EVDPHEM[cLD*SR_:P H1(Ω)
pPV5_:P]:X r VLW`NRPQDPFZ[DvC/VLPQVLZ[Z[ILM\PQD8p 9Jp°M[Z|nIa_8N=ILZ\V5PQS!_8NQM\Z[MzS1DTP,Z®g :2b8DPHEMzEDv8p 9Jp\5p • `j¨ « ¨5OMr ( ^ )$P / G )")->7"T"( ")<
Ω?)> >7+,+ @<+ @<? ? / !#" % /7>7$
+ <1 23 >7+ <"(+ O?7+ ; < ;";%$:+ -$o)>7+ "(G$:?<G?<G%$'% + -$o . #$ ?<G?<G%$/#"$<>7?7+ > < >) #%/ . #$Q>7" + ;%$M/( " >772$ "7;#"-0+ < /
∂Ω qIuEILPQI5ENQXPQM[SQIjNQM\V5W]8DeS&cjIaZ[D_JPQS=K:PRV5K:PQDTS&KIaP=Z[DK:PQM[WJEM[KD]8DC/V5_:PHIaW`N1hUM[SRG:DP=DTS1NlSRVL_8hc5DW`NFnoVLPRN,_8NRM[Z\D5fEV rr DlZ[D r VLW`NRPQDZ®g Db8DPHEMzEDSR_:M[cILW`NTp `j¨ « ¨5O ( G<?<G</ + -$M)<&>. )$ '+ -$ 7?<< $R%$ /#"B)@% 3 # # G<!#") + > +,+ <"7 1 + > ?(+,")$ ?< . ; 6 #$&&>7-; C /(>7)$ + )@%K7>7+ ;@ / )-><G@ 1 ) A+ >EG?< $ . #76 -$;0?<@ $R@ -0+ > ?<G W>7+ <"7 ?<G? λ1
/ 143 3 6%WK?D_8N/IL_JSQS1M r VLW`NQPRDTPm^5_JD,Z\DeSUnoVLWJENRM[VLWJS2KJPRV5K:PRDeS]8_$qdIaKJZ[I5EM[DW fLIc5DTE!EV5WJ]8M\NRM[VLWJS2IL_8bZ[M r M\NRDeS&]8DxM\PQMzEHG:Z\DN,VL_*]8D'0&DT_ r IaW:W fS1V5W5N,PQX5_:Z\M[PQDTSTp jB? «ªT« ?§9U c,/.01
Ω2 .> L 3B[ =.B[9AeBA F28:0<3BdEF3#;8MGG3 C∞ I 8M.>B=G8M3G .4;D0<.G
64BO.64BO3G#G.>8:2D0<.HGEF3 I =6F6B[D0<393%` C∞(Ω) I+ 6 ?§ ª j ªe« ?§ ,VLM\N
ukZzIkh¦ r DnoVLWENRM[VLW~K:PQVLKJPRD$SRVLZ[_8NRM[VLWs]:ILWJS H1
0 (Ω)]8D¦p[LpWK?D_8NEVLWJSRMz]8XPQDP,^`_:D
ukDTS1N=SRVLZ[_8NRM[VLW*]8_*K:PQVLw:Z[ r DIa_8bZ\M r MONQDTS,SR_:M\cjILW5N
−∆uk = fk]:ILWJS
Ωuk = 0
SR_:P∂Ω,Ic5DTE
fk = λkukpC/V rr D fk
ILK:KJILP1NQM\DTW5N&YH1(Ω)
f:KIaP,IaK:KJZ\MzEIaNRM[VLW*]8_ kFGJXVLPQ r D'.8p°¯8p°¯ ]:DPQX5_:Z[ILPRM\NRXVLWyDTW]:XT]8_:M\N,^5_JDZzIvSRVLZ[_8NQM\V5WukIaK:KIaPRNRM[DW`N!Y
H3(Ω)p:x_EVL_:K f`Z[DSRDTEVLWJ]r D r w:PQD fk
DTS1NmK:Z\_S2PRXTL_:Z[M[DP2ED&^`_:M:K?DP r DNB]|g Ia_J r DW`NRDTP2DWJEVLPQD/ZzIPQXL_JZ[ILPRM\NRXF]8D ukp P2IaP_JW:DvPQXTE_:PQPRDTWJEDnI5EM[Z\DvVLW r V5W5NQPRDvIaM[WJS1M^`_:D uk
ILK:KJIaPRNRM[DW`NYHm(Ω)
KV5_:P=NQVL_8Nm ≥ 1
p:BWc5DPRNR_]8_ kFGJXVLPQ r D=Jp 9:p°¯ .S1_JP!ZzIEV5W5NQM\W_:M\NRX]8DeS!noVLWJENRM[VLWJSF]8D Hm(Ω)
cLVLM[P,Ia_JSQSRM?ZzI&D r IaPH^`_:D=Jp 9:p°¯ 5fVLWDW ]8XT]:_:MON=^`_:D uk
ILK:KJILP1NQM\DTW5N&]8V5WJEC∞(Ω)
p
0=VL_JS,]8X r VLW`NRPQVLWJS r IaM[W`NRDTWJIaW`NF_:W*PRXeS1_:Z\NQIaNF^`_JILZ\M\NQIaNRM\ndNQPReSFM r K?VLPRNQIaW`N&YvK:PRV5KV`SF]8DZ[IKJPRD r M[PQDcjIaZ[D_:P,K:PQVLKJPRD5p
Ecol
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5% #%&' ( * )1D(1 < ( ¯L¯Q.b 4(6 j`¡ OA¢U* « § © ª :§ J aB3 64BO34E#8M3G%9.> >?0<.>HGg3 8M3[G 6H.>T [G3GEHA=.B Y 3 I IFI%J XG=676H.G3C358 R . L 3B[ Ω
3G[o;=.93T 3 I 8M.>B=G#8MW64BO3Y_0BO3 L >8M32B69BO.=69BO3λ1
3G[pG[01Y698M3 0 I 3 I 8M35G. G 3[GD6F;=3%69BO.=69BO3 ;=.B[B3[GD6.9E7o3G[oE73CE>01YZ3HG[0<.> 38M3o64BO3Y_0<3B L 3;V32BN69B.69B3%632?BO3_;*.0UG[0/6.)G[01D0 69B3[G3o6B[V.2oEFG Ω I
8 ? UI9U4 q D kFG:XV5PR r D$8p 9Jp\e¹u]8D PQDM[W8h &_8N r IaWDTS1NS1K?XTEMO´^`_:DIa_~EI5S]8DTSXT^`_JIjNQM\V5WJSm²RSRETIaZzIaM[PRDeSi³®Eag DTS1N1hiYjhi]8M[PRD^`_:DZkg M[WJEV5W:W_:DuDeSANYcjILZ\DT_:PQS,]JIaWJS
Rp C/DPQXTSR_:Z\NQIjNDTS1NnIL_8bDWLXTW:XPHIaZSRM`Zkg M\WJEVLW:W_:D
uDTS1NmY=cjIaZ[D_JPQSUcLDeENQVLPQM\DTZ\Z[DTS!oc5VLM[PK:Z[_JSZ[VLM[WZkg Db8D r K:Z[DF]:_ST+8S1NR r D]8DZ®g XZzILS1NRMzEM\NRXpqIPQILM[SRVLW ]8DEDNRNRDv]:M R XPQDWJEDDW`NRPQDZ[DEI5S&SRETIaZzIaM[PRDDN=cLDeENRV5PRM[DZDTS1Nl^`_:DvEDNRGJXVLPQ r DSTg IaK:K:_JM\DS1_:PZ\DvK:PQM\WJEM\K?Dv]:_ r Ijb8M r _ r oc5VLM[P=Z[D_kFG:XTVLPQ r D .8p°¯8p°¯L¯5^5_JM W g DTS1NFcjILZ[ILw:Z\D^`_:D]:ILWJSFZ[DEILSFSQEILZ[ILM\PQDLp •
+ 6 ?§ ª j ªe« ?§ ,V5MONu ∈ H1
0 (Ω)_:W~cLDTENRDT_:PK:PQVLK:PQDW:VLW~W_:ZBI5SRSRV8EM[XYZzI*K:PQD r M\TPRDcILZ\DT_:PK:PRV5K:PQD
λ1pxg IaK:PQTSvZ[Dtq D rr D.8p°¯8p°¯ = V5W¡SQIaM\N^`_:D u+ = max(u, 0)
ILK:KJIaPRNRM[DW`NYH1
0 (Ω)DN ∇u+ = 1u>0∇u
®]8D rr D KV5_:P u− = min(u, 0)p PILPyEV5WJS1Xe^`_:DW`NTf2ZzInoVLWJENRM[VLW |u| = u+−u− IaK:KJILP1NQM\DTW`N!Y H1
0 (Ω)DNFVLWtI ∇|u| = sign(u)∇u p-:mWuc5DPRNR_u]:DZzIPmPRV5KV`S1M\NRM[VLW p 9:p =]8DvC/VL_:PHIaW`NRh UMzS1G:DTP/V5WI
λ1 = minv∈H1
0 (Ω)\0
R(v) ≡∫
Ω |∇v|2dx∫
Ωv2dx
,
DNNRV5_8NKV5M\W`N]8D r M[W:M r _ r DeSANc5DTENQD_:PK:PQVLKJPRD5p P λ1 = R(u) = R(|u|) f ]8V5WJE |u| DTS1NIa_JSQS1M|c5DTENRD_JP,K:PRV5K:PQDI5SRSRV8EM[XlYλ1p C/V rr D u+
DNu−SRVLW`N&EV r w:M[WJILM[SRVLWJSFZ[M[W:XTILM\PQDTSF]8D uDN |u| f:EDSRVLW`N&Ia_JSQSRM ]:DTSFnoV5WJENQM\V5WJSFK:PQVLK:PQDTSFILSQSRVEM\XTDTSFY λ1p
:BW£nIaM\NTf!VLW·KDT_8N r VLW`NQPRDTPt^`_:D u W:DSTg IaW:W_:Z[D KJI5St]:ILWJS Ω5P 5ED IL_£K:PQM\WJEM\K?D]:_r Iab8M r _ r ²AnoVLPRNA³^`_:MIj±$P r D^`_:DLfJSRM w ∈ C2(Ω)
c5XPQMO´D−∆w ≥ 0
]:ILWJSΩ,DNw = 0
SR_:P∂Ω,IaZ[VLPHS!SRVLM\N
w ≡ 0]:IaWJS
ΩfS1V5MON
w > 0]:IaWS
Ωp W*IaK:K:Z[Mz^5_JDEDPQXTSR_:ZONHIjN=Y
u+DNu−^`_:MS1V5W5N2PQX5_:Z\M[PQDTSYETIa_JSRD,]8D,Z[IPBPRV5KV`S1M\NRM[VLWyp 9:p º5U^`_:M:W:D,K?D_JcLDW`NmKJI5S NQPRD!NRV5_8NRDeS2Z[DTSm]8DT_8bW:VLW W_:Z[Z[DTSTf]8V5WJEZkg _JW:Dv]8DeS]:D_8b DTS1N=W_:Z[Z\D5p 8_:K:K?V5SRVLWJSlIaZ[VLPHS&^`_:DZ[DSRVL_JS1h¦DTSRKJILEDK:PRV5K:PQDILSQS1V8EM[X&Y
λ1SRVLM\N/]8Dl]8M r DTWJSRM\V5WuSANQPRMzENQD r DW`NBKJZ\_JS/LPHIaW]8D&^`_:D5pWyK?D_8N*+NRPQVL_:c5DPB]8DT_8bnoVLWJENRM[VLWJSFK:PQVLKJPRDeS
u1DNu2VLPRNRG:V5LV5WJIaZ[DTSTf`ELg DeSANRh¦Yah¦]:M\PQDl^`_:D
∫
Ω
u1u2 dx = 0,
ED^`_:M DTS1N&M r K?V5SQSRM\w:Z[DK:_:MzSR^`_ g DZ[Z[DTS,SRVLW`N&]8DS1M[LWJDEVLWJS1NQILW`NTf8KJILP1NQVL_8N,W:V5WtW_:Z[Z\DeSp
`j¨ « ¨LO Ω
"7 ")< <4)@ G@%K"7+ <Q #() L #-G<'!") + >?<G )>7+ "(0?<#?<G/#" $<>7?7+ > /(>7)$Ω
>a #)/ . >7"+ ;-$:/( <" >77 %$&A7"(+,+ %!" +,+ :%$;Q$P ?7+
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¯5¯ ! "$#%&' ()*
8 ? OF£qBg DTWJSRD r wJZ\D]:DTS&PQXTSR_:Z\NQIaNQS=]:DvEDN1NQDvSRVL_JS1h¦SRDTENRM[VLW SRDLXTW:XPHIaZ[M[SRDSQIaWJS]:MO±yE_JZONQXBIa_8bVLK?XPHIjNQD_:PHS DZ[Z[M\K8NQM[^`_:DeS 5XW:XTPQIL_8bl]8_SRDTEVLWJ]lV5PQ]8PQDLfTEag DTS1N1hiYjhi]8M\PQD2Ia_K:PQVLwJZ\ r DIL_8bucjIaZ[D_JPQSFK:PQVLKJPRDeS!SR_:M[cjIaW`N
−div(A∇u) = λu]:IaWJS
Ωu = 0
SR_:P∂ΩVQU
A(x)DTS1N&_:W:D r IjNQPRMzEDST+ r XNQPRMz^`_:DEVDPHEM[cLD$cLVLM[PFZ[I$V5_JS1h¦SRDTENRM[VLW.8p°¯8p 95p •
-!.F. <,687OGEL 9,;&KF,GEqBg Db`NQDWJSRM[VLW¶]:DTSPRXeS1_JZONHIjNQS]8DZzISRVL_JS1hiS1DeENRM[VLWsKJPRXeEXT]:DW`NRD*Y]:DTSXT^`_JIaNRM[VLWJSIa_8b¡]:XhPQM[cLXDeSKJIaPRNRM[DZ[Z[DTSDZ[Z[M\K8NQM[^`_:DeSyK:Z[_JSyEV r K:Z[M[^`_:XTDTSy^`_:D Z[D qdILK:ZzILEM[DW¬W:D K?V5SRD KJI5Sy]8D K:PQVahwJZ\ r DeSvEVLWJEDK8NQ_:DZzSvW:VL_:c5DTIL_8b|p(0&V5_JS]8XTEPRM[cLV5WJSw:PQM\TcLD r DTW`NEDN1NQDuLXTW:XPHIaZ[MzSRIaNRM[VLW~KV5_:P]:D_8bDbD r K:Z\DeSvSRM[LW:M\´EIaNRM\nS UZ[DtS8+8SANQ r D*]8DtZkg XZzILS1NRMzEM\NRXZ\M[W:XTILPRMzSRXDyDNZ[DTSXT^`_JIaNRM[VLWJS]8DNRV5DTSTpqdDTSXT^`_JIjNQM\[email protected]/. `|]8D/Z®g XZzILS1NRMzEM\NRX/Z\M[W:XTILPRMzSRXD/]:XTEPQM[cLDW`NdDTWnIaM\NZ\D/PRXTLM r D/SANHIjNRM[VLWJWJIaM[PRD]:DTSFXT^`_JIaNRM[VLWJS,]O+`WI r M[^`_:DeS,S1_JM\cjIaW`NQDTS NQPReS,S1D r w:ZzIawJZ\DeS,YvZkg XT^`_JIjNQM\V5W]:DTS,V5WJ]8DTSH
ρ∂2u∂t2 − div (2µe(u) + λ tr(e(u)) Id) = f
]:ILWJSΩ× R+
∗u = 0
SR_:P∂Ω× R+
∗ ,
¦p[T5VQU
ρ > 0DTS1N/ZzI]8DTWJS1M\NRXcLV5Z\_ r Mz^`_:D=]8_ r IaNRXTPRMzIa_$DN e(u) =
(
∇u+(∇u)t)
/2p7&IaKJKDTZ\V5WJS^`_:D=Z\DeSBEVD±$EM\DTW`NQSB]:D=qdI r X]8_ r IjNRXTPRMzIa_$c5XPQMO´DW`N µ > 0
DN2µ+Nλ > 0
p :mWyZkg IawJSRDWJED]:D=noV5PQEDTS/DbNRXTPRM[D_:PQDTSfDN,DWtW:DlNRDWIaW`N!KILS!EV r K8NQD]|g XTcLDTW5NQ_:DZ[Z[DTS!EVLW]8MONQM\V5WJS!M\WJMONQM[ILZ\DeSQV5WK?D_8NmIL_JSRSRM8EHG:DPHEHG:DTP]8DeSUSRVLZ[_8NQM\V5WJSUV5SQEM[Z\ZzIaW`NRDeSdDTWNQD r KJS2]8Dk8p\e5EV rr D!W:VL_JSUZkg Ic5VLWJS]:XTEPQM\NFK?VL_JP!Zkg XT^`_JIjNQM\V5Wt]8DeSFVLWJ]:DTSF]:IaWS&VL_JS1hiS1DeENRM[VLW*p[Lp°¯8pJC/DZzIvEVLWJ]:_:MON&YEHG:DTPQEHG:DTP!]:DTSSRVLZ[_8NQM\V5WJS
(`, u)]8_*K:PQVLw:Z[ r DIa_8btcILZ\DT_:PHSFK:PRV5K:PQDTSFS1_JM\cjIaW`N
−div (2µe(u) + λ tr(e(u)) Id) = `u]:ILWJS
Ωu = 0
SR_:P∂Ω,
¦p[Tº5VQU` = ω2
DTS1NUZ[DFEIaPQPQX/]8D!Z[I=noPQXT^`_:DWED/]:D!cM[w:PHIjNRM[VLWtoW:V5_JSIcLVLWSdEHGJILW:LX/Z[IW:VaNHIjNQM\V5W]8D!Z[IcjILZ\DT_:PmK:PQVLK:PQDFKV5_:PmXTc`M\NRDTPm_:W:DEVLW:no_JS1M[VLW$Ic5DTE!Z\DEVD±yEM[DW`NB]8DqdI r X λ pQ:mW r XTEILW:Mz^5_JDLfZzIvnoVLWJENRM[VLW*K:PQVLK:PQDuDTS1N=Ia_JSQS1M ILK:K?DZ[XD ¢UuUjJUj ¢U « Uj ªT« §/p
:mWS1_:M[cjIaW`NtZzI r XNQG:V8]8DILK:K:Z[M[^`_:XTDEM\h¦]:DTSQS1_JSuIa_£qdILK:ZzILEM[DW V5W KDT_8N]:X r VLW`NRPQDPuZ\DPQXTSR_:Z\NQIaN,SR_:M[cjIaW`NoW:V5_JS,ZzIaMzSRSRVLWJS!Z[DTS,]8XNQILM\ZzS,Ia_*Z[DTENQD_:P,DTW*L_:MzS1D]|g Db8DTPQEM[EDep jB? «ªT« ?§9U4 c,4.01
Ω2 . L 3B[ =.B[4A BA F28:0<3B!EF3 ;8MGG3 C1
E73RN ISf 83T70UG=V3243%G[201V3W;BO.>0UG=G>V3
(`k)k≥1E73eBOA=38 G 6.)G[01D0 Ge20&V34E L 3B=Gg8OR 01 \S0 i 3 01843T70UG=V3d293 =)G3
2018=3B[D0<393ZE73L2(Ω)N (uk)k≥1 i V38183_3_;"H73 uk
6F6B[D0<3g`H1
0 (Ω)N 3 L AB[0 \N3
−div (2µe(uk) + λ tr(e(uk)) Id) = `kuk6 I 6 I EF>HG Ω
uk = 06 I 6 I G[2B ∂Ω.
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Z\DeS'= K:PQD r M[DPHS r V]:DTSKJPRV5K:PRDeS2]|g _:W:D/²ANRV5_:PA³&]8V5W5N/Z[IwJI5S1D,DeSANB´:b8XD®EVLW]8MONQM\V5W$Ia_8bZ\M r MONQDTS/]8D=xM\PQM[EHGJZ\DNHDN/]8VLW`NZ[DTSdIa_8NQPRDeS|KJIaPQVLMzS|SRVLW`N Z\M[w:PQDTSm®EV5WJ]8M\NRM[VLWIa_8bZ[M r M\NRDTSd]8D0&DT_ r IaW:W?pTqdDTSd¯,K:PRD r M\DTPQS r V8]8DTSTf]:MONHSm]8DFwJIjNRNRD r DTW`NTfLEVLPQPRDeS1K?VLWJ]:DW`NYlZzI rTr D!cjILZ\DT_:P2K:PQVLK:PQDlM\ZzSmSRVLW`N2M\W]8XK?DWJ]JIaW`NQS r IaMzSS8+ r XNRPQMz^5_JDTS KIaPUPRVLNQIjNQM\V5W]8D 90S1_JM\cjIaW`NUZ®g Ijb8D
zmcLVLM[PUZ[I UM[L_:PQDF8p\BDNZ\DkILw:Z\DeIa_8p 9Jp 9`pqIK:PRD r M\TPRDcjIaZ[D_:P,KJPRV5K:PRDlDTS1N&]8VLWE&²1]8V5_:w:Z[D³p
&ILW:$]8_ r V8]8DKJPRV5K:PRD ¯ 9 = IaZ[D_:P,KJPRV5K:PRD T¹5¯:p . = T¹`¯8p/. = T5Q.8p°¯ ¯aº :Lp°¯
p[ ILZ\DT_:PQS!K:PQVLKJPRDeS!EVLPQPRDeS1K?VLW]:IaW`N,Ia_8b r V8]8DeSFK:PRV5K:PQDTSF]8DZzI UM[L_:PQDp[Lp0=VL_JSuKJILSQS1V5WJS r IaM[W`NRDTWJIaW`NtIL_8b¶Xe^5_IjNRM[VLWSu]8DNRV5DTS .8p ^5_JMlSRVLW`Ny_:W:DcLDPHSRM\V5WS1NQIaNRM[VLW:WIaM[PRDB] g _:WvK:PQVLw:Z[ r D/] g XTcLVLZ[_8NQM\V5W]8DmN,+K?D/KIaPHIaw?VLZ[M[^`_:D=cLV5M\PUK:Z[_JSdZ[VLM[Wt®:p°¯LRp5PUV5_:PPQXTSRVL_]8PRDEDK:PQVLw:Z[ r D]|g Xc5VLZ[_8NRM[VLW*M[ZUSRDPHIM[W5NQXPQDTSQSRILW`N,] g _8NQM\Z[MzS1DTPZ[DTS=cILZ\DT_:PHS,DN&noV5WJENQM\V5WJSKJPRV5K:PRDeS
(λ, u, p)]8_*K:PQVLwJZ\ r DSR_:M[cILW`N
∇p− µ∆u = λu]:ILWJS
Ωdivu = 0
]:ILWJSΩ
u = 0SR_:P
∂Ω,
¦p°¯a¹5VQU
µ > 0DTS1N/ZzIcMzSREV5SRMONQXLf
uZ[Ic`M\NRDeSRSRD&DN
pZzIK:PQDTSQS1M[VLWy]:_J_:Mz]8DLp-:mWuSR_:M[cjIaW`NBZzI r XNQG:V8]8DILK:K:Z[M[^`_:XTDEM\h¦]8DeSRSR_JS,IL_qIaK:ZzILEM\DTW f8Z\DZ[DTENRD_JP&KV5_:PQPQIvPRXeS1V5_J]8PQDZkg Db8DPHEMzEDS1_JM\cjIaW`Nep
`j¨ « ¨5OGH Ω
"7A")< <4)@ G@%K"7+ < #() /(_+ >)$R$& C1/(
RN E G<!#" +( $;; "() $%"< ;dG $&$.>7-;
(λk)k≥1/ @-<+ $*?<($P . $ !#" ;</#)<G$ +, N %M"() >)$&
+ <<. <7)(uk)k≥1
/" $&#"$d -$ ?> /(L2(Ω)N /(%$5 . #$ / )<GK < I("7+,+ (N;<+,+
!#")d&>!#"uk
>7?7?><. < H1
0 (Ω)N N7 Q + $&;2"7> +,+ :/ ?<-$&$ )$pk ∈ L2(Ω)
!#"<)@< <
∇pk − µ∆uk = λkuk? ? /(>7)$ Ω
divuk = 0? ? /7>7)$ Ω
uk = 0? ? $%"( ∂Ω.qdDPQXTSR_:ZONHIjN]8DPQX5_:Z[ILPRM\NRXSR_:PlZ\DeS&noV5WJENQM\V5WJS&KJPRV5K:PRDeS=]8DvZ[I PmPQVLK?V5SRMONQM\V5Wp 9:p ºySg XNQDWJ]IL_JSQS1MnILEM\Z[D r DW`NFIa_tETILS!]8DTS/XT^`_JIaNRM[VLWJS!]8D`NRV LDTSlk8p ¯L¹5pQP2IaPFEV5W`NRPQD&Z\DWkFG:XTVLPQ r D8p 9Jp\e¹SR_:P&ZzIySRM r K:Z[MzEM\NRX]8DZ[I$KJPRD r M[PQDcjIaZ[D_:P=K:PRV5K:PQDDN=ZzI$K?V5SRMONQM\cM\NRX]8DZ[I$KJPRD r M[PQDnoVLWENRM[VLWKJPRV5K:PRDlDTS1N,nIa_:buDW*LXTW:XPHIaZ2®EV rr DDTS1NFnIa_8bZ\DK:PQM[WJEM[KD]:_ r IabM r _ r p
`j¨ « ¨5O ( W%)$P / G=+ 2? (+ >7"e>7+ <"7$ ?<G?<G%$M?<#"( + @-!#")>. # /(02&7/ )K <Q> "(?<<; -. + !#")>/# >. !")
V (x) = Ax · x A
-$;"() > $ @ !")/(@ ?< $P . )1 / + 2/( + ($O +,+ >; "(1>< #< !#" 6−∆u+ V u = λu
/(>7)$RN .
¦p°¯8e
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5 O^ < ( ¯L¯Lº( A/(@ < 0+ %$'%$%?)>%%$
H = L2(RN )%
V =
v ∈ H1(RN ); +7!#" |x|v(x) ∈ L2(RN )
.
#-G<O!#"V
-$; "7 -$ ?> =/ + << ?<"7 + ?</#"< Q$O->7+ >
〈u, v〉V =
∫
RN
∇u(x) · ∇v(x) dx +
∫
RN
|x|2u(x)v(x) dx, Q!#") +, " . 2/
V/7>7$
H-$;N% ?>; =/(@-/#" =!#"!, + ( $;; "7:$%"< ;WG $R$R>7;
(λk)k≥1/(:@-<+ $Q?< $ . $ !" ; )/# $'+, %L"7))>)$Rf< + < . ()M/
L2(RN ) (uk)k≥1!" #$& + %$%>7+ <"7$2?<#?<G%$M :+ -$o . )$:?<G?<G%$*/( 1 3 % 6% >7+ "7+ <O 7?7+ ; <'$&%$)>7+ "(G$M . #$0?<G?<G%$:1#_&)<1&)< >uk
$&#"$ + ># pk(x) exp(−Ax · x/2)
pk%$;0"( ?<+ 7 /(2/(%KG@k − 1 6% -; 4?<G@ ; ?2 )$P !#" <*+ -$:@-$ "7+ &>;$
`j¨ « ¨LO Ω
"( " < @ G@%K#"(+ L/ RN ( ! #$ / + ? (+ /
<< >. )$ ?<"7 + @!">. 2/(%$:?7+ >!")%$'>) # #)/ . 2>7" + ;-$Q/) )->)$;G -
∆ (∆u) = λu/(>7)$
Ω∂u∂n = u = 0
$ "7∂Ω.
#-G<#!#"!, +- $&; "()'$ " ;< $&$R>7;(λk)k≥1
/(o)>7+ "(G$Q?<#?<G%$L?< $P . )%$ !#"<);</d $+ <-% "() >)$&/ + <<. <7) /(>7)$
L2(Ω)/ p . #$J?<G? -$
(uk)k≥1!" >7?7?)> . ()<
H2
0 (Ω)
,+ % " t,+-/.&.10 £@8E`> 7 C|6:@TE NU68@19B4 HN7 C(F,CLMGd4/6OEl4@TEW~cjI EV5WJSRM[]8XTPRDTP_:W:DyILK:K:PQVbM r IjNQM\V5WM[W5NQDPQW:Dy]8D$ZzInoVLP r _:ZzIjNQM\V5WcjILPRMzIjNQM\V5W:W:DZ[Z[DM[W8hNRPQV]:_:MONQD=YZzIV5_JSAhiSRDTENQM\V5W$8p 9Jp\5p 2NHIaW`N!]8V5W:W:X&_JWyS1V5_JSAh¦DTSRKJI5ED
Vh]8D&Zkg DTSRKJILED&]8D&7&M[Z[wDTP1N
VfJ]8D]8M r DWJSRM[VLW´JW:M[DLf:V5WEHGJDPHEHG:DZ[DTS=S1V5Z\_8NQM\V5W (λh, uh) ∈ R× Vh
]:Da(uh, vh) = λh〈uh, vh〉H ∀ vh ∈ Vh.
¦p°¯L¯5k+K:Mz^`_:D r DW`Nef Vh
DeSANB_JWyDTSRKJILED=]|g XZ[X r DW`NHS2´JW:MzSBEV rr D=EDT_8b$M\W`NQPRV8]8_:M\NQSBKJIaPBZ\DeS/xX´JWJMOhNRM[VLWJS Jp 9Jp .=DN Jp 9Jp ¯Q.8fjDNHDeSANmZkg DTSRKJILED
L2(Ω)p5qdIlPQXTSRVLZ[_8NRM[VLW]8D,Zkg ILK:K:PQVb8M r IjNQM\V5WM[W`NRDPQW:Dkp°¯L¯5/DTS1N!nI5EM[Z\DEV rr DZ\D r VLW`NQPRDlZ\DZ[D rr DSR_:M\cjILW5Nep
9rO4 J ^G3%698M7;=3G.> Ge8M3Gv 6.a G3[GCE>HA=.B Y 3 I7I I 8M.>B=GW83[G L 8M32B=G69BO.=69BO3G_EF3 I .B[Y 3N243WG[201V35;BO.0UG=G 3\S0<3
0 < λ1 ≤ · · · ≤ λndl
L 3; ndl = dimVh,3%018 3OT70UG[V3C293=)G3aEF3
Vh i .>B[T.9.>B[YZ>8M3aE7HG H i (uk,h)1≤k≤ndl
EF3 L 3=;V32B=Gg69BO.=69BO3G)G=G. ;0<A[G i ;R 3G= ` E>01BO3Z3
uk,h ∈ Vh,3a(uk,h, vh) = λk〈uk,h, vh〉H ∀ vh ∈ Vh.
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¯ 9L¹ ! "$#%&' ()*
+ 6 ?§ ª j ªe« ?§ C/D Z[D rr D KDT_8N NRPQD EVLWS1Mz]8XPQX EV rr D _:WJD cILPRMzIaW`NQDXTcM[]8DTW`NRD]8_kFGJXVLPQ r Dp 9:p°¯u®YZ[I]8MSR XPQDWJEDlKJPReS,^5_dg DTW]8M r DWJSRM[VLWt´JW:M[DM\Z Db8M[S1NRD_:W*W:V r w:PQD=´W:Md]8DcjILZ\DT_:PQSK:PQVLK:PQDTSHp0&XeIaW r V5M\WJSUW:V5_JSUDW]8VLW:WJVLWJS_:W:D,]8X r V5WJS1NRPHIjNRM[VLWv]8M R XPQDW`NQDLfK:_:PQD r DW`NILZ\5Xw:PQMz^5_JDLfa^`_:MJEVLPQPRDeS1K?VLWJ]vK:Z[_JSBYZzI]8X r ILPQEHGJDFSR_:M[cM\D,DTW$K:PQIaNRMz^`_:DLp58VLM\N (φi)1≤i≤ndl
_:W:DwILSRD]8DVhoKJILPDb8D r K:Z[DLf Z[DTSlnoVLWENRM[VLWSl]:DwILSRD]|g _:W:D r XNQG:V8]8D$]|g XZ[X r DW`NQS´JWJM[STf|cLV5M\PZzI PmPQVLK?V5SRMONQM\V5W Jp 9Jp°LpW*EHG:DPHEHG:D
uhSRVLZ[_8NRM[VLW ]8D$¦p°¯L¯L!SRVL_JSFZzIvnoVLP r D
uh(x) =
ndl∑
i=1
Uhi φi(x).
ViW`NQPRV8]8_:MzSQIaW`N,Z[I ª « ¨L ¢U :ee Mh]8X´JWJM\DKJILP
(Mh)ij = 〈φi, φj〉H 1 ≤ i, j ≤ ndl,DN&Z[I ª « ¨5 ¢* « D « ¢ «oª56 Kh]8X´W:M\DKJILP
(Kh)ij = a(φi, φj) 1 ≤ i, j ≤ ndl,Z[DK:PQVLw:Z[ r Dk8p ¯5¯L/DTS1N&XT^`_:M[cjIaZ[DW`N&YNQPRV5_:cLDTP (λh, Uh) ∈ R× RndlSRVLZ[_8NRM[VLW*]8D
KhUh = λhMhUh.¦p°¯ 95
qdDTSvIaK:K?DZ[Z[IaNRM[VLWJSl² r IaNRPQM[EDTS]8D r I5SRSRDyDN]8DyPQM[LMz]8MONQX³yK:PQVjcM\DTW:W:DW`N]8DeSILK:K:Z[M[ETIjNRM[VLWSDWr XeEIaWJM[^`_:D]:DTSlS1V5Z\Mz]8DTSTp ,D r IaPH^`_:VLWS&^`_:DLf ]:IaWS=Z[DvETILS&VQU VhDeSANl_:WDeS1KJI5ED] g XTZ\X r DTW`NQS´W:M[STf?Z[I r IaNRPQM[EDv]8DvPRM[LMz]8M\NRX Kh
DTS1NlDb:ILENQD r DW`N=ZzI rTr D r IjNQPRMzED^`_:DEDTZ\Z[DvPQDWEVLW`NQPRXTDIL_C/GJIaKJMONQPRD ]:IaWSuZkg ILK:K:Z[M[ETIjNQM\V5W£]8DZzI r XNQG:V8]8D]8DeSuXTZ\X r DW`NQSu´JW:MzSuIL_8b K:PRV5w:Z[ r DTSDTZ\Z[M[K8NRMz^`_:DTSTpW~cLXTPRM\´JD$M rr Xe]8MzIjNRD r DW`N^`_:DyZ[DTS r IjNQPRMzEDeS MhDN Kh
S1V5W`NST+ r XNQPRMz^`_:DTSDNF]:X´JW:M[DTSFK?V5SRM\NRM[cLDTSTpq DST+8S1NR r Dk8p ¯ 95BDTS1N!_:WK:PQVLw:Z[ r D r IaNRPQM[EM\DTZ|Ia_8bycjIaZ[D_JPQS/K:PQVLK:PQDTS²1LXTW:XPHIaZ[MzS1X³Hpjq D&NRG:XTVLPQ r D&]8D&PQXT]8_ENRM[VLWySRM r _:Z\NQILW:XDcLVLM[PmKJILPmDb8D r KJZ\D&Z[D,NQG:XV5PR r D=¯:p 9Jp ]JIaWJSW'°¯)(oFIj±$P r D^`_ g M\Z|Db8M[S1NRD_:WJD r IaNRPQM[EDlM[WcLDTPQSRM\wJZ\D PhNRDTZ\Z[D^5_JD
Mh = PhP∗h ,DN Kh = Ph diag(λk)P ∗
h .
P2ILPEV5WJS1Xe^`_:DW`NTfdZ\DeSSRVLZ[_8NRM[VLWJS]8Dk8p ¯ 95SRVLW`NZ[DTScjIaZ[D_JPQSKJPRV5K:PRDeS(λk)DNZ[DTSc5DTENRD_JPQSKJPRV5K:PRDeS
(Uk,h)1≤k≤ndl
^`_:M=SRVLW`N$Z[DTS$c5DTENQD_:PHS$EV5Z\V5W:W:DeS$]8DPhp!C/DeS$cLDeENRDT_:PHS$EVLZ[VLWJW:DTSnoV5P r DTW5N]8V5WJE_JW:DwJILSRDLf VLPRNRGJVLLV5WJIaZ[DK?VL_:P Kh
DNVLPRNRGJVLW:V5P r IaZ[DK?VL_JP MhW:VL_JSM[WJ]8M\h^`_:DTPRV5WJSw:PRM[c5D r DW`NY&ZzI0,D r IaPH^5_JD!p =Jp 9=EV rr DW`NETIaZzE_:Z[DPUEDN1NRDFwJI5S1Dp UM\WJILZ\D r DW`NTfZ[DTSc5DTENRD_JPQS
Uk,hSRVLW`N=SRM r K:Z[D r DTW5NlZ\DeS,cLDeENQD_:PHS&]8DeS=EVV5PQ]8V5W:W:XTDTS,]:ILWJS&ZzIwJILSRD (φi)1≤i≤ndl]:DTS,noV5WJENQM\V5WJSuk,h^`_:MdnoVLP r DW`N&_:W:DwILSRDVLPRNRG:V5W:VLP r IaZ[Dl]:D Vh
KV5_:P=Z[DK:PQV8]8_:M\NSQEIaZzIaM[PQD]:DHp
8 ? MrEGF·xILWJS Z[DBqdD rr D!p =Jp[2VLWI,PQDK:PQMzS|Z\DeS G-+KVLNRG:eS1DeS ]8_kFG:XTVLPQ r D/p 9:p°¯ DTW£KJILP1NQM[E_:Z\M[DPef!Z[I~noV5P r Dw:M\Z[M[W:XTILM\PQD a(u, v) DeSAN*S1_:KJKV`S1XTD 6`ª « ? U?pFW£c5VLM\Ntw:M[DWZkg M r K?VLPRNQIaWEDy]8DuEDNRNRDuG-+K?VaNRGJTSRDy]:IaWJSZ[I ]:X r VLWJS1NRPHIjNQM\V5W p1:mW¡D R DNTfUS1M/DZ[Z\DuW g XNQILMONKILSS8+ r XNRPQMz^5_JDLf?VLW W:DSRIL_:PHIaM\N=KJILSlSRMUZ[DvS8+8SANQ r Dt¦p°¯ 9`,DTS1Nl]:M[ILLVLWIaZ[M[SQIaw:Z[DLfEag DTS1N1hiYjhi]8M\PQDSTg M[ZDb8M[S1NRD]8DeS&S1V5Z\_:NRM[VLWJSF]8_*K:PQVLw:Z[ r DIa_:bucjIaZ[D_:PHSFK:PQVLK:PQDTSk8p ¯5¯Lp •
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5 O^ < ( ¯ 9JqBg ILK:K:Z[M[ETIjNQM\V5W ]8_ q D rr D$p =Jp[YyZ®g IaK:KJPRVb8M r IaNRM[VLWcjIaPQM[IaNRM[VLW:W:DTZ\Z[DKJILP=XZ[X r DTW5NHS,´JWJM[S]8_KJPRV5w:Z\ r D]8D$xM\PQMzEHG:Z\DNykp[L&DeSANM rr Xe]8MzIjNRD5pUWKJPRDTWJ] V = H1
0 (Ω)fH = L2(Ω)
fDN&Zkg DeS1KJI5ED]8MzSREPRDNV0h]8DZzIxX´W:MONQM\V5W :p 9:p/.uPQILK:K?DZ[VLWJSF^`_:D
V0hEVLW`NRM[DW`N,ZzIEVLWJ]:MONQM\V5WIa_8buZ[M r M\NRDeS&]8DxM\PQMzEHG:Z\DNHp
`j¨ « ¨LMrE ( 5 #$ / + ?<G 7+ =>7"5>7+ <"7$ ?<G?<G%$' / $ N = 1
−u′′k = λkuk?<"7
0 < x < 1uk(0) = uk(1) = 0.
( $R2?<G?<($& /(d->7+ "7+ <E+ > > %0/( >)$R$&=?<#"( + > @%/( / -$ @ + @ -;$ $ P1 (
?< )/ + -$*)<&>. )$L/(2+ >A . 5# %# #-G<!#"A+ > > :/ >)$&$R Mh%$&L/(7@-
?)>
Mh = h
2/3 1/6 01/6 2/3 1/6
1/6 2/3 1/60 1/6 2/3
,
Q!#")=$&%$>7+ <"7$:? #?<-$L$R
λk(Mh) =h
3(2 + cos(kπh))
?<#"(1 ≤ k ≤ n.
#-G<!#"7N-$:". + $&0+ >o "(+ '/(Q!#")>/# >"(G:1 5# 76 N)>7+ G$O'G"'!#" Mh = h Id >7$% 2/(< <&%>)$NH%>7+ <"(+ + %$>7+ <"7$:?<G? -$Q/#" ?<G 7+ $ ?< >7+/ $O<% 8 ? UI9rOU PVL_:P,ETIaZzE_:Z[DP!Z\DeS!cILZ\DT_:PHSBDN!c5DTENQD_:PHS/K:PRV5K:PQDTS!]8_uK:PQVLwJZ\ r DSRKDeENRPHIaZr IaNRPQM[EM\DTZ¦p°¯ 95?M[Z5nIa_8NefDWLXWJXPHIaZkfEV rr DTWJEDTP KJIaPUEILZ[E_:Z\DTP ZzIFnILENRVLPQMzSRIaNRM[VLW]8D!C/G:V5Z\DeS -+]8DZ[I r IjNQPRMzED]8D r ILSQSRD Mh = LhL∗h
f:K?VL_JP&S1DPHI r DW:DTP&Ia_*EI5S,EZzILSQS1Mz^`_:DKhUh = λhUh
Ic5DTE Kh = L−1h Kh(L∗h)−1 DN Uh = L∗
hUh,KV5_:PZ\De^`_:DZ2V5W¸]8MzSRKV`S1D$]|g IaZ[LV5PRM\NRG r DeSl]:DyEIaZzE_JZm]8DcILZ\DT_:PHSDNcLDeENRDT_:PHSKJPRV5K:PRDeSp 0=VL_JSPRDTW`c5V +LV5WJSYlZzIDTENRM[VLW* 9Jp ¯=K?VL_:PmKJZ\_JSB]8D&]8XNHIaM[Z[S2SR_:PBEDTS2ILZ\5VLPQMONQG r DTS'L]8MzSRVLWJS2SRD_:Z[D r DTW5N^5_JDELg DeSAN&Zkg XNQILKDZzIK:Z[_JS&EV8NRDT_JSRDlDTWuNQD r KJS&]:DEIaZzE_JZ®pWvKDT_8N2XcM\NRDP]8DFEV5WJSANQPR_JM\PQDmZzI r IjNQPRMzED KhDNnILM\PQDBZkg XeEV5W:V r M[D/]8D!Z[I=nILENRV5PRMzSRIaNRM[VLW]8DC/G:VLZ[DTS -+l]8D Mh
SRMLV5W_8NQM\Z[M[SRDB_JW:DmnoV5P r _:Z\D!]8D/^`_JI5]8PHIjNR_JPRD2K?VL_:PUXcjILZ\_:DTPdZ[DTSEVD±yEM[DW`NHS]8DZ[I r IaNRPQM[ED Mh^`_:M`Z[I=PRDTWJ]8DF¢ « D?§U:?paC/DBK:PQV8EXT]:X/]|g M\W`NQXLPHIjNQM\V5WW_ r XPQM[^`_:D/DTS1NILK:KDTZ\X¨L?§¢U§e ªT« §¡¢:eTtoV5_² r I5SRSBZ\_ r KJM\W:a³=DWuILW:LZzIaMzSHDN!DTS1NBnoPQXT^`_:D rr DW`NF_8NRM[Z[M[SRXLp
PILPFDb8D r K:Z[DLfJSRM|VLW_8NQM\Z[M[SRDZ[IvnoV5P r _:Z\D]8D^`_JI5]8PQIaNR_:PQD :p = .5=®^`_:M _8NQM\Z[M[SRD_:W:Mz^`_:D r DTW5N,Z[DTScILZ\DT_:PHS2Ia_8bW:VD_J]:S/]|g _:W:DFnoVLWENRM[VLW$K?VL_JPBEILZ[E_:Z\DTP2_:W:D,M[W`NRX5PQILZ\DfaV5WcLVLM\N2nILEM\Z[D r DTW5NB^`_:DZ[I r IjNQPRMzED]8D r ILSQSRD MhILM\WS1M|V5w8NRDTW`_JDlDeSAN=]:M[ILLVLWIaZ[DcLVLM[PFZ®g :2b8DPHEMzED8p =p\p •
0=VL_JSvcLDTPRPQVLWS]:IaWSZzISRVL_JS1hiS1DeENRM[VLWsSR_:M[cILW`NRDt^`_:DTU® U « ` :Jdjjj,]8MzSREPRNRDTSλk,hoZ[DTSKJZ\_JSK?DNQMONQDTSHUS1V5W5N2]8DeS2IaK:K:PQVb8M r IaNRM[VLWJSEVLPQPQDTENQDTSU]8DTScjILZ\DT_:PQS
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¯ 95¯ ! "$#%&' ()*
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SRVLM\N,w:M[DW K:Z[_JS5PQILWJ]8D^`_:D r M\Z[Z\D5p0=VL_JSM[Z[Z\_JS1NRPQVLWSlEDNRNRDySRVL_JS1h¦SRDTENRM[VLWKJIaPZ[DyEIaZzE_JZm]8DTS K:PQD r M[DPHS r V8]8DeSK:PQVLK:PQDTS]8DcM[w:PHIjNRM[VLW]|g _:WNHI r wV5_:P/ r V8]8XTZ\MzSRX/EV rr DB_:W:D r D r w:PHIaW:D/EM[PQE_:Z[ILM\PQD´:b8XD/S1_:PUS1V5WwV5PQ]pWtPRXeS1V5_8N,]8VLWJElZ[DKJPRV5w:Z\ r Dl]8DxM\PQMzEHG:Z\DNkp[LB]JIaWJS!_:W*]8MzSR^`_:D]8DPQI+5VLW 5pJWt_8NQM\Z[MzS1D_JW:D r XNRG:V8]8D]|g XZ[X r DTW5NHS/´JW:MzS P1
p:q DeS/PQXTSR_:Z\NQIaNQSFS1V5W`N!K:PQXTSRDW`NQXTSF]:IaWS/ZzI M[L_JPRDp°¯DN,Z\DkUIawJZ\DeIa_¸8p =p\p W PRD r IaPH^5_JD^`_:DvZzIyKJPRD r M[PQDcjILZ\DT_:PKJPRV5K:PRDvDTS1NlSRM r K:Z[DvNQIaW]8M[S^`_:DvZ[I]:D_8b8M[ r DDN,ZzINRPQVLMzS1M[ r DS1V5W5NB²R]8VL_Jw:Z\DeSi³p
&ILW:$]8_ r V8]8DKJPRV5K:PRD ¯ 9 = .
IaZ[D_:P,KJPRV5K:PRD .8p j =Jp Lº =p 5º ¯ :p =`¯ ¯ :p =`¯ 9L¹Jp . 9 p°¯ ILZ\DT_:PQS!K:PQVLKJPRDeS!EVLPQPRDeS1K?VLW]:IaW`N,Ia_8b r V8]8DeSFK:PRV5K:PQDTSF]8DZzI UM[L_:PQDp°¯8p-!.&.@A 9B4 G 7Gd4=> G G|6 GE568@8LMN6:@19B4&E~; G 7:7OGd<=7xlIaWJSEDN1NQDtSRVL_JS1hiS1DeENRM[VLWdfdW:V5_JSWJVL_JSEV5W5NQDW`NRV5WJS] g XTW:VLWJEDPv_:W¡PQXTSR_:Z\NQIjN]8DtEVLWcLDTP1h5DWJED]8DvZ[I r XNRG:V8]8D]8DTS=XZ[X r DW`NHS&´JWJM[S=NRPQM[ILW:L_JZ[ILM\PQDTS Pk
K?VL_:PlZ\DEILZ[E_:Z]8DeS=cjILZ\DT_:PQS=DNc5DTENRD_JPQSK:PQVLK:PQDTSl]8_KJPRV5w:Z\ r D]8DxM\PQM[EHGJZ\DN¦p[ap3ViZ2DeSANw:M[DW¸EZzIaM[Pl^`_:D$EDPQXTSR_:ZONHIjNS1D5XW:XTPQILZ\MzSRDlILM[SRX r DW`N&Y]|g Ia_8NQPRDeSFK:PRV5w:Z[ r DTS!DN=Y]|g Ia_:NRPQDTS!N,+KDeS&]|g XZ[X r DW`NHS!´JW:MzSpb 4(6 MrEMr ,4.>01
(Th)h>0
293 G[201 3PE73 YZ>01818M 3[GPDB[0< F28M>01BO3[G!BOA >28:0<3B=G!EF3Ω I,/.01
V0h8M3PG.> G 3[GD6F;=3XE73
H10 (Ω) i E7A<\p0N6H>BP8M Y AaH.EF3nEF3G^A8MAYZ3?Ge\p0UG
Pk i E73E>01YZ3HG[0<.>ndl I ,4.01 (λi, ui) ∈ R×H1
0 (Ω) i 6H.>2B i ≥ 1 i 8M3G L >8M32B=G_3 L 3;V32B=G69B.69B3[G .>B[T.9.>B[YZAGEF>HGL2(Ω) Ed69B. 8Y 3eEF3 01B[0<;" 8M3 FI i BO!7AG&6B.>BOEB3;BO.0UG=G
0 < λ1 ≤ λ2 ≤ ... ≤ λi ≤ λi+1...
,/.018M3G L 832B=G69B.69B3[G
0 < λ1,h ≤ λ2,h ≤ ... ≤ λndl,h,
E738OR 6F64BO.T701Y D0<. L B[0<D0<.93818M3 I E7HG V0h I .>2BW .>2 i ≥ 1\ T A i .K
limh→0|λi − λi,h| = 0.
k8p ¯ =`f 8&3OT70UG[V3#293 YC01818M3ZEF3 L 3=;V32B=GS69B.69B3[G (ui,h)1≤i≤ndl
E73 I E7HG V0h 3818M3_3 i G=0
λi3[G[p243 L 8M32B69B.69B3G[01Y698M3 i .>K
limh→0‖ui − ui,h‖H1(Ω) = 0.
k8p ¯Q.L
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5 O^ < ( ¯ 9Q9
p°¯ uqdDTS K:PQD r M\DTPQS r V8]8DeSFK:PRV5K:PQDTSF]|g _:WNQI r wV5_:Pep
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¯ 9 = ! "$#%&' ()*
C3 698h G i G[0_8M3nG.> G 3GD6H7;=3l3!739E>BOA 6B(u1, ..., ui)
3G[ 019;8: GKE7HGHk+1(Ω)
3_G[0k + 1 > N/2 i 8.B=G_. 8 R 3G[?01Y D0<.@EQR 3B[BO32B
|λi − λi,h| ≤ Cih2k,
¦p°¯ 5.
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h i 3NG[0 λi3[G[293 L 832B64BO.64BO35G=01Y6983 i .>K
‖ui − ui,h‖H1(Ω) ≤ Cihk.
¦p°¯5a 8 ? MrE ·qdIEV5WJSANHIaW`NRD
Ci]:IaWSl¦p°¯ `mV5_k8p ¯`amNRDTWJ]:SFcLDTPQS
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DTS1N r _:Z\NRM[K:Z\D5fdZzI*S1_:M\NRDy]:DTSc5DTENRD_JPQSK:PQVLKJPRDeSlILK:K:PQV8EHG:XTSui,hK?D_:NW:D$KJI5SEV5W`c5DPQLDTPDNIL] r DNRNRPQD$K:Z[_JS1M[D_JPQSK?VLM[W`NQS] g I5EE_ r _:ZzIjNQM\V5W ^`_:M=SRVLW`Ny]8MSR?XTPRDTW`NRDTS$EV r w:M\WIaMzS1V5WJSZ\M[W:XTILM\PQDTS$]:DTS$c5DTENQD_:PHSK:PQVLK:PQDTS$]8_SRVL_SAh¦DTSRKJILED=K:PQVLKJPRDILSQSRVEM\XY
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∂u∂t −∆u = f
]:ILWJSΩ× R+
∗u = 0
SR_:P∂Ω× R+
∗u(x, 0) = u0(x)
K?VL_:Px ∈ Ω.
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p!qdIs]8MzS1NRPQM\w:_:NRM[VLW£]8DNQD r K?XPHIjNR_JPRDM[W:MONQM[ILZ\D5fY t = 0f:DTS1N=]8VLWJW:XDKJILP,Z[InoV5WJENQM\V5W
u0p_:P&Z[DwV5PQ]
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S1_:P∂Ω× R+
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x ∈ ΩDNt ∈ R+
AV5_:DW`N]8DeSP5Z\DeS=NQPReS]8M R XPQDW`NHS]:IaWJS$®:p[e=K:_:MzSR^`_ g M[ZBSg Ia5MON] g _:WJDlXe^`_JIjNQM\V5WuIa_8bu]:XPQM\c5XDTS/KJILP1NQM\DTZ\Z[DTSF]8_K:PQD r M\DTP/V5PQ]:PRDDW t DN&]8_]8D_8b8M[ r DVLPH]8PQD=DWxZ\DqIaK:ZzILEM\DTWtWJDlK?VLPRNRD^`_:DSR_:P,ZzIvcjILPRMzIaw:Z[DS1KIjNRMzIaZ[DepViWJ]:M[^`_:V5WJS=^`_ g M\ZUDb8MzSANQD]|g Ia_8NQPRDeS&V5PRM[LM[W:DTSFKJG +8SRM[^`_:DeS=]8_ S8+8SANQ r Du®:p[ep3P2IaP=Db8D r K:Z[DLf®:p[e r V8]8XTZ\MzS1DtIa_JSQS1M/Z[I]8M R _JSRM\V5W¡]|g _:W:DEVLWEDW`NQPQIaNRM[VLW u ]:ILWJSZ[Dt]:V r IaM[W:D Ω
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∂u∂t +∇p− µ∆u = f
]JIaWJSΩ× R+
∗divu = 0
]:IaWJSΩ× R+
∗u = 0
S1_JP∂Ω× R+
∗u(x, t = 0) = u0(x)
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µ > 0DTS1NmZ[IcMzSREV5SRMONQXF]:_ _:M[]:DLp &ILK:K?DZ[VLWJSm^`_:D,ZzIEV5WJ]8M\NRM[VLW$Ia_8bvZ\M r MONQDTSB]:D&x=M[PRMzEHG:Z[DNG:V r VL5W:D r V8]8XZ[MzS1DZkg I5]8G:XTPRDTWJED]8_ _:M[]:DYZ[IKJILPRV5M ]8D Ω
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PBPRXeS1DTW5NQVLWJS=PQILK:Mz]8D r DW`NlZ\DeS=]8DT_8b K:PRM[WJEM\KJIL_8b r V8]8TZ\DeS=G-+K?DPQwV5Z\Mz^`_:DTS^5_JDW:VL_S=XNR_8h]8M\DTPRV5WJSm]:ILWJSBED=EHGJILK:MONQPRD5pLq D=K:PRD r M\DTP r V]:Z[D&DTS1N/,> 6-? ªe« ?§s¢?§¢U&]8V5W5N/Z®g VLPQM\5M\WJDK:G-+SRMz^5_JDFI]:X 1YXNQX&]8MzSRE_8NRXTD&Ia_uC/GJIaKJMONQPRD5p5VLM\NΩ_:WVL_:c5DPRNw?VLPQW:X&]8D
RN]8DFnoPRV5W`NRM[PQD
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∂2u∂t2 −∆u = f
]:ILWJSΩ× R+
∗u = 0
SR_:P∂Ω× R+
∗u(t = 0) = u0(x)
]JIaWJSΩ
∂u∂t (t = 0) = u1(x)
]JIaWJSΩ.
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DTS1N!]8XeS1V5P r IaMzS2_:WJD&noVLWENRM[VLWyYcILZ\DT_:PHSmcLDeENQVLPQM\DTZ\Z[DTSm]JIaWJSRN p PmZ[_JS/K:PRXeEMzS1X r DW`NTf`S1MVLWyW:VLNRD
f(t, x)ZzIPRXeS1_JZONHIaW`NRDcLDeENRV5PRM[DZ[Z[Dem]:DTS2noV5PQEDTSBDbNRXTPRM[D_:PQDTSTfLZ[D]:XK:ZzILED r DW`N u(t, x)DTS1N=S1V5Z\_8NQM\V5W]:D
ρ∂2u∂t2 − div (2µe(u) + λ tr(e(u)) Id) = f
]:ILWJSΩ× R+
∗u = 0
SR_:P∂Ω× R+
∗u(t = 0) = u0(x)
]JIaWJSΩ
∂u∂t (t = 0) = u1(x)
]JIaWJSΩ,
:p =`
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¯ 9L O )1 )#& *<
VQUu0DeSANZ[Dy]8XKJZ[I5ED r DTW5NM[W:M\NRMzIaZkf u1
Z[IcM\NRDTSQSRD$M\W:M\NRMzIaZ[DLfdDNe(u) = (∇u + (∇u)t)/2
Z[DNQDWJSRD_JP/]8DeSB]8XnoVLP r IaNRM[VLWJSTp`8_:K:K?V5SQIaW`NBG:V r V5LTW:D,M[SRVaNQPRV5KD=Z\D r IaNRXPQMzIa_y^`_:MJV8ETE_:K?D Ω fSRI]:DWJSRMONQXDeSAN&EVLWJS1NQILW5NQDρ > 0f8]8D rr D^5_JDlSRDTS r V8]8_:Z[DTS,]8DqdI r X^`_:M cLXTPRM\´JDW`N µ > 0
DN2µ+Nλ > 0
p 8 ? O £WuK?D_8N,IAV5_8NRDTP!IL_8b$XT^`_JIaNRM[VLWJSlJp 9`2DNJp =2_:WuNRDTP r D]8_uK:PQD r M[DPV5PQ]:PRDlDWNRD r KJSfED^`_:Md]8V5W:W:D
∂2u
∂t2+ η
∂u
∂t−∆u = f
]:ILWJSΩ× R+
∗ .qdVLPHSR^`_:DlZ[DEVD±yEM[DW`NηDTS1N&KV`S1M\NRM\nAfJEDNQDP r D]:_ K:PRD r M\DTP,VLPH]8PRDEV5PRPQDTSRK?VLWJ]tY_:W:DnoVLPHED]:DnoPQDM[WJIa5DK:PQVLK?VLPRNRM[VLW:W:DTZ\Z[DYuZ[I$cM\NRDeSRSRDLp W]8M\NIL_JSQS1MU^5_dg M[ZUSTg IaLM\Nl] g _:W NRDP r Dv]|g I r V5P1hNQM[SQSRD r DW`NTpW*KJILPRZ[DIaZ[VLPHSF¢> 6 ? U ªe« ?§ ¢U§U¢: ªT« `5p •
8 ? O GF ViZdDbMzS1NRD]|g Ia_8NQPRDeS r V8]8Z[DTS=K:G-+SRMz^5_JDTS,]8V5W:WJIaW`N&Z[M\DT_ Y$]8DTS=XT^`_JIaNRM[VLWJSIL_8b$]8XPQM\c5XDeSKJIaPRNRM[DZ[Z\DeSmG-+KDTPRw?VLZ[Mz^5_JDTSTp`C/DK?DWJ]JIaW`NTfLNRVL_:N r V8]8Z[D&G-+KDTPRw?VLZ[M[^`_:D,Wdg DeSANBKILSWJXTEDeSRSQIaM[PQD r DW`Nv_JW¶K:PQVLwJZ\ r D*]|g Xc5VLZ[_8NRM[VLW ]|g V5PQ]:PRD]8D_8b|p/Cg DTS1N$Z\D EI5SW:VaNHI rr DW`N]:DTSXe^`_JIjNQM\V5WJS=]|g :m_:Z[DPlZ[M\W:XeIaPQM[SRXDeS,DWILEV5_JS1NRMz^5_JDLf:V5_]8DTS=XT^`_JIaNRM[VLWJS]8D Iab /DTZ\Z DTW XZ[DTENRPQVahr ILLW:XNRMzS r DLf^5_JMBS1V5W`N]:DTSS8+8SANQ r DTS] g Xe^5_IjNRM[VLWSG +K?DPQwV5Z\Mz^`_:DTS]|g V5PQ]:PRD_:WSRD_:Z[D r DW`NDTW¸NRD r KJSTpqdDTSMz]8XDeSEV5W`NRDW_:DeS]:IaWSEDyEHGJILK:M\NRPQDSTg XNRDTWJ]8DTW5NY EDTSKJPRV5w:Z\ r DTSTf r IaMzSZ[I]:M R XPQDWJED]|g V5PQ]:PRDDWtNRD r KJS,DW EHGJILW:LDlZzIK:PRXeS1DTW`NQIjNQM\V5W p •
#y+ , $/ !t$/H t+") $!+ '*#") t0=VL_JS2ILZ\Z[VLWJSSR_:M[c`PQD!_:W:D&]8X r ILPQEHGJD/SRM r M[Z[ILM\PQD,]:IaWJSUZkg DeS1KJPRM\NmYEDZ[Z\D,^`_:M8W:V5_JS2IlL_:Mz]8XDFIa_C/GIaK:M\NRPQD.lK?VL_:P/XNQILw:Z[M\P/Z®g Db8MzSANQDWJED&DNBZkg _:W:MzEM\NRX]:D&Z[IS1V5Z\_8NQM\V5W$]|g _JWyK:PQVLw:Z[ r D,DZ[Z[M\K8NQM[^`_:D5pC/DN1NQD]8X r ILPQEHGJD&S1D=]:XTEV r KV`S1D,DTW$NRPQVLMzS2XNHIaK?DTS 5K:PQD r M[PQD r DW`NefLXNHIaw:Z[M[Pm_:W:D=noVLP r _:ZzIjNQM\V5WcjILPRMzIjNQM\V5W:W:DZ[Z[D*kVL_SAhiS1DeENQM\V5W~Jp ¯:p\f ]8DT_8b8M\ r D r DW`NTfU]8X r V5W5NQPRDTPZ®g Db8MzSANQDWJEDyDNZkg _:W:MzEM\NRX]:DyZ[I SRVLZ[_8NQM\V5W]8DuEDN1NQD$noVLP r _:ZzIjNQM\V5W~cILPRMzIjNQM\V5W:W:DTZ\Z[DDWs_8NQM\Z[M[SQIaW`N_JW:DywJI5S1D$G:M[Z[wDTP1NQM\DTW:W:D]:DnoV5WJENQM\V5WJSlK:PRV5K:PQDTS®V5_JSAhiSRDTENQM\V5W:p°¯8p°¯LfJNQPRV5M[SRM[ r D r DW`Nef r V5W5NQPRDTPl^`_:DEDN1NQDSRVLZ[_8NRM[VLWc5XPQMO´DlwJM\DTWtZ[DK:PQVLw:Z[ r DIa_8btZ\M r MONQDTS,XNR_J]8M[Xy®8VL_JS1h¦SRDTENRM[VLW:p°¯8p 95p
[email protected] ,9m7 L <&F,NU6:@A9/4 N7J@,NU6:@A9B44GF8F,GqBg Mz]8XDuDTS1Nv]|g XTEPQM[PRD_:W:DynoVLP r _:ZzIjNQM\V5WcjIaPQMzIjNRM[VLWJW:DZ[Z\Dy^`_:M/PRDeSRSRD r w:Z[DY*_JW:D 6-? ªT« ?§¢ « 6 j§ ªT« ¸¢ « §U « ]8_~K:PRD r M\DTPlV5PQ]:PRD5f SRM r M[ZzIaM[PRDY¦p 95p PUV5_:PEDTZ[ItW:V5_JS r _:Z\NRM\hKJZ\M[VLWJS&Z®g XT^`_JIaNRM[VLW*]8DZ[I$EHGIaZ[D_:P®:p[e/KJIaP=_:W:DnoVLWENRM[VLWNQDTS1N
v(x)^`_:M W:D]:XK?DWJ]*KJI5S&]8_NQD r KJS t pQ2£ETIa_JSRD&]8D,ZzIEV5WJ]8M\NRM[VLW$Ia_8bZ\M r MONQDTSBW:VL_JSBIaZ[Z\V5WJSm]8D r ILWJ]8DTPBYED=^`_:D v Sg IaWJW`_JZ\DSR_:PZ[Dyw?VLPH]s]8DyZkg V5_:cLDTP1N
ΩfUEDu^`_:M/cjI W:VL_SKDTP r DN1NRPQDu]|g D R DTENR_:DTP_:WJDyM\W`NQXLPHIjNQM\V5W~KJIaPKIaPRNRM[D&S1M r K:Z\D®SRILWJSNQDP r D]8D,wV5PQ]p PV5_:PmZkg M[WJS1NQIaW`NBED=EILZ[E_:Z:DTS1NmK:PQM\WEM[KJIaZ[D r DW`N2noV5P r DZkp
0=VL_JSFV5w8NRDTW:VLWS,]8VLWJE∫
Ω
∂u
∂t(x, t)v(x) dx +
∫
Ω
∇u(x, t) · ∇v(x) dx =
∫
Ω
f(x, t)v(x) dx.Jp .5
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% O "&T1< < < & J &) ¯ 95ºC/V rr DW:M Ω W:M v(x) WJDcjIaPQM\DTW5NlIcLDTElZ[DNRD r KJS t f?VLW*K?D_8NlPRXTXTEPQM[PRDEDN1NQDXT^`_JIaNRM[VLW SRVL_JSZ[IvnoVLP r D
d
dt
∫
Ω
u(x, t)v(x) dx +
∫
Ω
∇u(x, t) · ∇v(x) dx =
∫
Ω
f(x, t)v(x) dx.
:2b8K:Z\V5MONHIaW`NmZ[DFnIaM\NB^5_JD,Z\DeS2cjIaPQM[ILw:Z\DeSxDNt
AVL_JDW`NB]8DTSmP5Z\DeSUNRPQTS2]8MSR?XTPRDTW`NQSTfaW:VL_SmS1XTKJIaPQVLWSEDTSdcjIaPQM[ILw:Z\DeS|DWEVLWJSRMz]8XPHIaW`Nd]8XeS1V5P r IaMzS|ZzI&SRVLZ[_8NRM[VLW u(t, x) EV rr DB_:W:DmnoV5WJENQM\V5W]8_NQD r KJSt ::¢U:§U§~:¨5y¢$¤¦?§¨ ªT« §U,]8X´W:M\DeSmSR_:P
Ω rTr D&EHG:V5SRDFKV5_:P f(t, x)
pPmZ\_SFK:PRXeEMzS1X r DW`NTf8SRM|Z®g VLWS1D]8V5W:W:Dl_:WuNQD r KJSF´WJIaZ T > 0
oXc5DW`NR_JDZ[Z\D r DTW`NFXT5ILZ Y +∞ fVLW*EVLWS1Mz]8PQD^`_:DuDTS1N=]8X´JWJM\DKJILP
u : ]0, T [ → H10 (Ω)
t → u(t),DNvWJVL_JSvEVLW`NQM\W_:DTPRV5WJSY W:VLNRDTPu(x, t)
ZzI cjIaZ[D_:Pu(t) (x)
p2q DuEHG:V5MObs]8DtZ®g DTSRKJILEDH1
0 (Ω)DTS1NXTcM[]8D rr DTW`N$]:M[ENRX KJILPZzI¸WJIjNQ_:PQD*]:_¶K:PQVLwJZ\ r D*DNK?D_:NycILPRM[DP$]|g _:W r V8]8Z[D Y_JWIa_8NQPRD5pO:mW LXTW:XPHIaZ|M[ZSTg IaLM\N&]8DZkg DeS1KILED^`_:MUEVLWcM[DW`N&KV5_:P&ZzInoVLP r _:ZzIjNQM\V5WtcjILPRMzIjNQM\V5W:W:DZ[Z[D]8_K:PRV5w:Z[ r D,SANHIjNRM[VLWJWJIaM[PRD,I5SRSRV8EM[XLpaxD rTr DLfaZ[DFNRDTP r D&SRVL_:PHED f DeSANB]:XTSRVLP r ILM[SEVLWJSRM[]:XPQXEV rr D_:W:DnoV5WJENQM\V5W]8D t YcjIaZ[D_:PHS,]:IaWS L2(Ω)pWM[W5NQPRV8]8_JMONBILZ\V5PQSZ\DFK:PQV8]8_:M\NBSRETIaZzIaM[PRD!]8D
L2(Ω)DN2ZzI=noV5P r DFw:M[Z\M[W:XeIaM[PRD a(w, v) ]8X´JW:MzSKJIaP
〈w, v〉L2(Ω) =
∫
Ω
w(x)v(x) dxDN
a(w, v) =
∫
Ω
∇w(x) · ∇v(x) dx.:mWEHG:V5M[SRMzSRSQIaW`NZzI$noV5WJENQM\V5W NQDTS1N]:ILWJSZ®g DTSRKJILED
H10 (Ω)f|V5W K?D_8NIaZ[VLPHS r DN1NRPQD*:p/.LSRVL_JSZ[InoVLP r D ]|g _:W:D SRVLPRNRD ¢> 6-? ªe« ?§"¢ « 6 j§ ªT« ® ¢ « §U « DW t p!W¬VLw8NQM\DTW`N$ILM\WS1MZ[InoV5P r _:ZzIjNRM[VLW cjILPRMzIjNQM\V5W:W:DZ[Z[DS1_:M[cjIaW`NRD JNQPRV5_:cLDTP u(t) noVLWJENRM[VLW]:D ]0, T [
Y$cjIaZ[D_:PHS&]:ILWJSH1
0 (Ω)NQDZ[Z\D^`_:D
d
dt〈u(t), v〉L2(Ω) + a (u(t), v) = 〈f(t), v〉L2(Ω) ∀ v ∈ H1
0 (Ω), 0 < t < T,
u(t = 0) = u0.
:p 5ViZ/PQDTS1NRDyKJZ\_JSRM[D_:PHSK?VLM[W`NQSY K:PQXTEMzSRDP]:ILWJSZzI noV5P r _:ZzIjNRM[VLW~cjIaPQMzIjNRM[VLWJW:DZ[Z\D:p 5KV5_:PvZ[_:M]8VLW:WJDP,_:W SRDWJS r IaNRG:X r IjNRMz^`_:DK:PQXTEM[S:?^5_JDZ[Z\DDeSAN&ZzIPRXTL_:ZzIaPQM\NRXlDW*NRD r KJS&]8D f DN]:D u fDNl^`_:DTZUSRDWS]8V5W:W:DTPYyZzIu]:XPQM\c5XDDTW NRD r KS :mWKIaPRNRMzE_:Z[M\DTPTfM\ZnIa_]8PQItIawJSRVLZ[_ r DW`N^`_:Du(t)SRVLM\N&EV5W5NQM\W_:DDTW
t = 0K?VL_:P=]:VLW:W:DTP,_:W*S1DTWJS&EV5PRPQDTENFYZzI]8VLW:WJXDM\WJMONQM[ILZ\D
u0p
PV5_:PtEDZzI~W:VL_JSuIcLVLWSwDeS1V5M\W£]|g M[W5NQPRV8]8_JM\PQD _:W:D nI r M\Z[Z\D]|g DeS1KILEDeSnoVLWENRM[VLWJW:DZzSu]8DnoVLWJENRM[VLWJS,]8DtYcjIaZ[D_:PHSF]:IaWJS,]:DTS,DeS1KJI5EDeS!]:DnoVLWJENRM[VLWJS,]8D
xp
+ 6 ;§ «ªT« ?§ F94-,/.01X
2 3[GD6F;=3 EF3 #018=3B[ i .> 648: G 7A9ABO>8M3Y 3 i 2 3[GD6F;=3 EF3 >9F;" EFA?\S04G=2B
Ω 690<3Y 3 i X = L2(Ω) i H1
0 (Ω) i .> C(Ω) FI ,/.01 2 V3Y6GQ\S480 < T ≤ +∞ I .2BC2l3?0<3B
k ≥ 0 i [email protected]([0, T ];X)
8 R 3[GD6F;=3 EF3G .4;D0<.Gk.0UG ;=.D012Y 3gEFAB[0 L 83[G!E73
[0, T ]E7G
X I ,[email protected] ‖v‖X 8MP4.B[Y 3aEF>HGX i 018
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¯ =5¹ O )1 )#& *<
3G[;8M)G=G[0<3 L .01B 3 Ck([0, T ];X)3[G[p2K3[GD6F;=3ZEF3 >9F;"Z6H.>2B#8M 4.B[Y 3
‖v‖Ck([0,T ];X) =k∑
m=0
(
sup0≤t≤T
∥
∥
∥
∥
dmv
dtm(t)
∥
∥
∥
∥
X
)
.
J 9.V3L2(]0, T [;X)
8 R 3G?6F;=3 E73[G .>9;?0<.>HG E73]0, T [
E7GX
3818M3G 3a8M .>9;?0<.>t→ ‖v(t)‖X
G.01YZ3G[2BO 8M3C3oEF3C;=B[BOA501VA >BO 83 i ;R 3G= ` E>01BO3Z3
‖v‖L2(]0,T [;X) =
√
∫ T
0
‖v(t)‖2Xdt < +∞.
20/E73;=3?V3%4.B[Y 3L2(]0, T [;X)
3[G[ > GG[042 3G?6F;=3eEF3 >97;* I _3 698: G i G[0 X 3G[&23GD6H7;=3_E73 018=3B[ i 8.B=G L2(]0, T [;X)
3[G[2n3[GD6F;=3CEF3#018=3B[/6.2B#8M3%69B. E>201pG;=8M>01BO3
〈u, v〉L2(]0,T [;X) =
∫ T
0
〈u(t), v(t)〉Xdt.
8 ? OGFGFMXDTS1NdZ®g DTSRKJI5ED
L2(Ω)feILZ\V5PQS
L2(]0, T [;L2(Ω))STg Mz]8DW`NQMO´JD!Y,Zkg DeS1KJI5ED
L2(]0, T [×Ω)K:_:MzSQ^5_JDLf:KJILPFZ\DNQG:XV5PR r D]8DJ_:w:M[W:Mkf:VLW*I
‖v‖2L2(]0,T [;L2(Ω)) =
∫ T
0
(∫
Ω
|v(t)|2(x) dx)
dt =
∫ T
0
∫
Ω
|v(x, t)|2dx dt = ‖v‖2L2(]0,T [×Ω).
PV5_:P1 ≤ p < +∞ f:V5WtK?D_8NLXWJXPHIaZ[M[SRDPFZzIxX´W:MONQM\V5W :p°¯8p[lDTW*M\W`NRPQV8]8_:MzSRILW`NFZkg DTSRKJILED]:D ILWJILEHGLp(]0, T [;X)
]8DTSnoV5WJENQM\V5WJSm]8D]0, T []:ILWJS
XNRDTZ\Z[DTS/^5_JD,Z[IlnoV5WJENQM\V5W
t→ ‖v(t)‖XSRVLM\N r DTSR_:PQILw:Z[DlDN=]8DK:_:MzSRSQIaWED p h¦ r DM\W`NQXLPHIawJZ\D5p •xlIaWJSZzIuSR_:MONQDV5W K:PQDWJ]8PHI$Z[DNQDP r DS1V5_:PHED f ]:ILWJS&Zkg DTSRKJILED L2(]0, T [;L2(Ω))f DNlVLWEHGJDPHEHG:DPHIlZ[IvS1V5Z\_8NQM\V5W
u]:ILWJSBT>\TU:¨5y¢> 6 §`D «
L2(]0, T [;H10 (Ω))∩C([0, T ];L2(Ω))
pC/DlEHG:VLM\b$KDT_8N,KJIaPHI)zNRPQD=ILPRw:M\NRPHIaM[PQDLf r IaMzS/M\Z|SRDPHIe ªT« ; 6 f`W:VLWtS1DT_:Z\D r DW`N!KIaPBZzI]8X r VLWSAhNQPQIaNRM[VLW¬]8D Zkg Db8M[S1NRDTWJED ]|g _:W:D SRVLZ[_8NQM\V5W K?VL_JPyZ[InoV5P r _:Z[IaNRM[VLW cjIaPQM[IaNRM[VLW:W:DTZ\Z[DsJp `f r IaMzSIL_JSQS1M?U:e§¶ « `§¡L?`¨¢U`v ªe« ªe« ?§¢> 6 §U5D « oc5VLM[PBZkg :2b8DTPQEM[ED:p°¯8p[epQ0=VaNRV5WJS]:X 1Yy^`_:D5fJZ[DTS,noV5WJENQM\V5WJS&]8DvEDNlDTSRKJILEDXNHIaW`NEV5W`NRM[W`_JDTS,DTW*NRD r KJSY$cjIaZ[D_:PHS&]:ILWJS L2(Ω)fZzIEV5WJ]8M\NRM[VLWM\W:M\NRMzIaZ[DIw:M[DW*_:W S1DTWJSTp
UM[WJIaZ[D r DW`Nef:Z[I$]:XPQM\c5XDDWNRD r KJS&]:ILWJS,ZzInoV5P r _:ZzIjNRM[VLWcjIaPQM[IaNRM[VLW:WJDZ[Z\D$Jp `!]8VLM\N NQPRDKJPRMzS1DIa_¡SRDWSnILM\w:Z[DK:_:MzSR^`_ g I K:PQM[VLPQMBZzI noVLWENRM[VLWt → 〈u(t), v〉L2(Ω)
W g ILK:KJILP1NQM\DTW5N^5_dg YL2(0, T )
cLVLM[P2Z[D&q D rr D =Jp =Jp[e¯K?VL_:PB_:W:D&]8X´JW:M\NRM[VLWK:PQXTEMzSRD&]8D&EDN1NRD=W:VaNQM\V5W]8D]8XPQM\c5XDpJVLPRN!GJD_:PQD_JSRD r DTW5Nef`STg M\Z Db8MzSANQD_JW:DS1V5Z\_8NQM\V5Wu]8DJp `f8IaZ[VLPHSBZkg X5ILZ\M\NRXl]:IaWSl®:p 5BW:VL_S/]8M\N^`_:D,EDN1NQD&]8XPQM[cLXD!DWNQD r KJS2DeSANNRV5_8NBED,^`_ g M\Z +I]8DFK:Z\_SmEZzILSQS1Mz^`_:D!K:_:MzSR^`_ g DZ[Z\D&ILK:KJILP1NQM\DTW5NYL2(]0, T [)
p .@AF.@A £4 7OCE<Fi6%NU6 Cd4Cd7ON F
PVL_JP]8X r V5W`NRPQDP,Zkg DbMzS1NRDWEDDN=Z®g _:W:MzEM\NRX]8DZzIySRVLZ[_8NRM[VLW ]8DZzInoV5P r _:Z[IaNRM[VLWcjIaPQM[IaNRM[VLW8hWJDZ[Z\D$Jp `f8W:V5_JSFPQDcLDTW:VLWS/IL_ETIL]8PQD5XW:XTPQILZ M\W`NRPQV8]8_:M\N&]:IaWJSFZzI$DTENRM[VLW p 9:p%0&V5_JS,IaZ[Z[VLWJS
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% O "&T1< < < & J &) ¯ =KV5_:cLV5M\PIaM[WJS1M,²R]8M[ILLV5WJIaZ[M[SRDP1³Zkg VLK?XPHIjNRDT_:PtqIaK:ZzILEM\DTW DN*W:VL_SyPHI r DW:DTPtY¡Z[I¡PRXeS1V5Z\_8NQM\V5W]|g _JW:DnI r M[Z\Z[D]8DvS1M r K:Z\DeS&Xe^5_IjNRM[VLWS&]8MSR?XTPRDTW5NQM\DTZ\Z[DTS=VLPH]8M[WJIaM[PRDeSF]8_ K:PQD r M[DP&V5PQ]8PQDLpW M[W8hNRPQV]:_:MON=]:VLWJE]8DT_8btDeS1KJI5EDeS!]:D'7&M[Z\w?DPRN
VDNHNRDTZ[S&^`_:D
V ⊂ H Ic5DTEM\WADeENQM\V5W*]:DWJSRDDNEV r KJI5ENQDcLV5M\P&¦p[LeDNmZ®g Db8K:Z[M[ETIjNQM\V5W^`_:MJSTg DW$S1_JMONpFk+K:M[^`_:D r DW`NmVLWIa_:PHI V = H10 (Ω)DN
H = L2(Ω)p
b 4(6 j`¡LOGF,4.>0<3V
3H
EF3FT 3[GD6F;=3[GPEF3#018=3B[W 38 G!3V ⊂ H
L 3=; 01 j3;D0<.n;=.Y6F; 3_3
V3[G[NE73G3_EF>HG
H I ,4.01 a(u, v) 243.B[Y 3 018:019A=>01BO35G Y A?B[0<3;.>D013C3%;=. 3BO;0 L 3 E7G V I ,/.012XV3Y6Gp\p98 T > 0 i 293_EF.9A=3_0101D0<8M3 u0 ∈ H i3N2^ 3B[Y 3#G.2B;3
f ∈ L2(]0, T [;H) I 8M.>B=GW83o64BO. 8YZ3
d
dt〈u(t), v〉H + a (u(t), v) = 〈f(t), v〉H ∀ v ∈ V, 0 < t < T,
u(t = 0) = u0,
:p a .@8 R A=>?0<.> E73 I X8:0<3 G3G 0 8M3 EF>HG
]0, T [ 293 20<3aG.>8:2D0<.u ∈
L2(]0, T [;V )∩C([0, T ];H) I C3S698: G i 01843T70UG=V3e293W;=.G[ >V3 C > 0 20 93#E7A 634E 3
EF3Ω V3818M3_3
‖u‖L2(]0,T [;V ) + ‖u‖C([0,T ];H) ≤ C(
‖u0‖H + ‖f‖L2(]0,T [;H)
)
.:p 5
8 ? U F9r >[ ªe« ªe« ?§ ¢> 6 §U5D « Jp `$K:PQVL_:c5D^`_:D¸Z[I¶S1V5Z\_8NQM\V5W·]8D :p a]8XK?DWJ]*EVLW`NRM[W r DW`N&]8DTS,]8V5W:W:XTDTSTf`DN=]8VLWEl^`_:DZ[DK:PRV5w:Z[ r DKJIaPHIaw?VLZ[M[^`_:DJp°LmDeSAN&wJM\DTWKV`S1XIa_*SRDWJS&]:D7IL]:I r IaPH]|p • 8 ? U F9 xlIaWJS!Z[DdkFG:XTVLPQ r DJp ¯:p 9V5WuK?D_8N,I R|IaM[w:Z[M\P!Zkg G-+KVLNRG:eS1Dl]8DEVDPHEM[cMONQX]8DZzInoVLP r DwJM\Z[M\WJXTIaM[PQDST+ r XNQPRMz^`_:D a(u, v) EV rr DVLW*Z®g Iy]8X 1Y$K:PQVLK?V5SRX]:IaWJS&Z®g :2b8DPHEMzEDp 9:p[ep2W¡V5w8NRM[DW`NvZ[DTS rr DTSEVLWJEZ\_JSRM[VLWJSDW¡SR_:K:K?V5SQIaW`NSRD_JZ\D r DTW`N^`_ g M\ZFDb8MzSANQDt]8DT_8bEVLWSANHIaW`NRDeS/K?V5SRM\NRM[cLDTS
ν > 0DNη > 0NQDZ[Z\DeS&^`_:D
a(v, v) + η‖v‖2H ≥ ν‖v‖2VKV5_:PFNRV5_8N
v ∈ V.:mWDR?DNTf SRMZkg V5WDR?DeENR_JDvZ[DEHGIaW:5D r DW`N]8DvnoVLWENRM[VLWM\WEVLWJW`_JD u(t) = eηtw(t)
f|V5W cLV5MON^5_JD®:p a!DeSAN,XT^`_:M[cjIaZ[DW`N=Y
d
dt〈w(t), v〉H + a (w(t), v) + η〈w(t), v〉H = 〈f(t), v〉H ∀ v ∈ V, 0 < t < T,
w(t = 0) = u0,V U¡ZzInoVLP r Dw:M[Z\M[W:XeIaM[PRD a(w, v) + η〈w, v〉HDeSANw:M[DW¬EVDPHEM[cLDS1_:P
Vp/C/DN1NQDtG-+KVLNRG:eS1DI R|IaM[w:Z\M[DDTS1N&_8NRM[Z[DLf:KJILP,Db8D r K:Z[DLf:]JIaWJSFZzIPRXeS1V5Z\_:NRM[VLWt]:DZ®g :2b8DPHEMzED:p°¯8p =Jp •
+ 6 ?§ ª j ªe« ?§ !qdI]8X r V5WJSANQPQIaNRM[VLW¸DTS1N]8M\cMzS1XTDDW~]8D_8bXNQILKDeSp xlIaWS_:W:D$K:PQD r M[PQDXNQILKD5fDWtSR_:K:K?V5SQIaW`N!Z®g Db8MzSANQDWJED]|g _:W:DS1V5Z\_:NRM[VLWufW:VL_JS/VLw:NRDWJVLWJS/_:W:DnoVLP r _:Z[D=Db8K:Z\MzEM\NRDKV5_:P
uSRVL_JS&Z[InoV5P r D]|g _:W:DSRXPQM[DVLw8NQDW_:DKJILP=]8XTEV r K?V5SRMONQM\V5W*SRK?DTENQPQILZ\D]8DeS&DeS1KJI5EDeS H
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¯ =`¯ O )1 )#& *<
DNVp :mWtKJIaPRNRMzE_:Z[M[DPefLEDN1NRD=noVLP r _:Z[D&K:PQVL_JcLD,Zkg _:WJM[EMONQXl]8D=Z[IvS1V5Z\_8NQM\V5W p5xlIaWSm_:W:Dl]8D_:bM[ r DXNQILKD5f!WJVL_JS]8X r V5W5NQPRV5WJSu^`_:D~EDN1NQD¸SRXPQM\D¸EV5W`c5DPQLD ]:ILWJStZ\DeStDeS1KILEDeS L2(]0, T [;V )
DNC([0, T ];H)
f8DN=^`_:DZzISRV rr DDTS1N&w:M[DW*_:W:DSRVLZ[_8NRM[VLW*]8D$Jp°Lp ª :B B_:KJKV`S1V5WJS^`_:D
u ∈ L2(]0, T [;V ) ∩ C([0, T ];H)DeSANS1V5Z\_8NQM\V5Ws]8D®:p apq DTSG-+K?VaNRGJTSRDTSlKDTP r DN1NRDTW`N] g ILK:K:Z[M[^`_:DTPZ\D kFG:XTVLPQ r D8p 9Jp ¯tS1_JPlZzIPRXeS1V5Z\_:NRM[VLW]8_~K:PQVLwJZ\ r DIL_8b*cILZ\DT_:PHS,K:PQVLK:PQDTS&I5SRSRV8EM[XY$ZzInoVLP r Dw:M\Z[M[W:XTILM\PQDS8+ r XNRPQM[^`_:D a(u, v) p3P2IaPEVLWJSRXT^`_:DTW`NTfM[Z DbMzS1NRD_:W:DwJI5S1DG:M[Z[wDTP1NQM\DTW:W:D
(uk)k≥1]8D
HEV r KV`S1XTD]:DcLDTENRDT_:PQS!K:PQVLKJPRDeS!]:Dykp[e¯5
uk ∈ V,DNa(uk, v) = λk〈uk, v〉H ∀ v ∈ V.W*]8X´JW:M\N
αk(t) = 〈u(t), uk〉H , α0k = 〈u0, uk〉H , βk(t) = 〈f(t), uk〉H .
PB_:M[SQ^`_:Du ∈ L2(]0, T [;V )∩C([0, T ];H)
DNf ∈ L2(]0, T [;H)
faV5W$DWy]8Xe]8_:M\N/^`_:Dαk(t) ∈
C([0, T ])DNβk(t) ∈ L2(]0, T [)
pC/V rr D (uk)k≥1DTS1N&_:W:DwJI5S1DG:M[Z\w?DPRNRM[DWJW:D]8D
Hf:V5WI
u(t) =
+∞∑
k=1
αk(t)uk,
DN=EHG:VLMzSRM[SQSRILW`Nv = uk
]:ILWJSJp°L/VLWVLw8NQM\DTW5N
dαk
dt+ λkαk = βk
]:IaWS]0, T [
αk(t = 0) = α0k.
®:p º5WcLXTPRM\´JDM rr Xe]8M[IaNRD r DTW`N=^5_JDlZkg _:W:Mz^5_JDS1V5Z\_8NQM\V5W]:Dy:p º5/DeSAN
αk(t) = α0ke
−λkt +
∫ t
0
βk(s)e−λk(t−s)dsK?VL_:P
t > 0,
ED^5_JMd]8VLWJW:D_:W:DnoVLP r _JZ\DDb8K:Z[MzEM\NRDK?VL_:P,ZzIS1V5Z\_8NQM\V5W u ®^`_:M DTS1N=]8V5WJEl_:W:Mz^5_JDep ª :BIF9 0=VL_JS&ILZ\Z[VLWS!]:X r VLW`NRPQDP&^`_:DZzIS1XTPRM[D+∞∑
j=1
(
α0je
−λjt +
∫ t
0
βj(s)e−λj(t−s)ds
)
uj®:p[T¹5
EVLWcLDTPR5D&]:IaWSL2(]0, T [;V )∩C([0, T ];H)
DN,^5_JDlSQISRV rr DLf`W:VaNQXD u(t) DeSAN,S1V5Z\_:NRM[VLWu]8D®:p apC/VLWJSRM[]:XPQVLWJS!ZzISRV rr DKJILP1NQM\DTZ\Z[DYvZkg VLPH]8PRD k ]8DEDN1NRDSRXPQM\Dwk(t) =
k∑
j=1
(
α0je
−λjt +
∫ t
0
βj(s)e−λj (t−s)ds
)
uj .®:p[Le
C/ZzIaM[PQD r DW`N wkIaK:KIaPRNRM[DW`N&Y
C([0, T ];H)KJ_:M[SQ^`_:DEHGJI5^5_JD
αj(t)DeSAN=EVLW`NRM[W_ p V5W5NQPRV5WJS^`_:DZzIS1_:M\NRD
wk DeSAN=]:DC!IL_JEHG-+y]:IaWS C([0, T ];H)p%PVL_:P
l > kf8DW*_8NQM\Z[M[SQIaW`N,Z[DEIaPHILENRTPRD
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% O "&T1< < < & J &) ¯ = 9VLPRNRG:V5W:VLP r Xl]8DeS!noVLWJENRM[VLWJSFKJPRV5K:PRDeS uj
f:V5WI‖wl(t)− wk(t)‖H ≤
∥
∥
∥
∥
∥
∥
l∑
j=k+1
α0je
−λjtuj
∥
∥
∥
∥
∥
∥
H
+
∥
∥
∥
∥
∥
∥
l∑
j=k+1
∫ t
0
βj(s)e−λj(t−s)ds uj
∥
∥
∥
∥
∥
∥
H
≤
l∑
j=k+1
|α0j |2e−2λjt
1/2
+
l∑
j=k+1
(∫ t
0
βj(s)e−λj(t−s)ds
)2
1/2
≤
l∑
j=k+1
|α0j |2
1/2
+
l∑
j=k+1
1
2λj
∫ T
0
|βj(s)|2ds
1/2
≤
l∑
j=k+1
|α0j |2
1/2
+1√2λ1
l∑
j=k+1
∫ T
0
|βj(s)|2ds
1/2
,
K:_:MzSR^`_:DFZzISR_:M\NRD,]8DTScjILZ\DT_:PQSUK:PQVLKJPRDeS(λj)DeSANmEPQVLMzSQSRILW5NQDBDN2S1NRPQMzENRD r DW`N2KV`S1M\NRM[cLD5paC/V rr D
u0 ∈ HDNf ∈ L2(]0, T [;H)
VLW*I‖u0‖2H =
+∞∑
j=1
|α0j |2 < +∞, ‖f‖2L2(]0,T [;H) =
+∞∑
j=1
∫ T
0
|βj(s)|2ds < +∞,
EDt^`_:MFDTW5NQPQIzWJDu^`_:DuZzISR_:M\NRDwk(t)DTS1N]:D C!IL_JEHG-+¸]:ILWJS
Hp(PBZ\_JSK:PQXTEM[SRX r DTW5NefV5WDW]8XT]8_JMON=^`_:DZzIS1_:M\NRD
wkc5XPQMO´D
limk,l→+∞
(
sup0≤t≤T
‖wl − wk‖H)
= 0,
Eag DTS1N1hiYjhi]8M\PQD^5_dg DTZ\Z[DDTS1N=]8DvC!Ia_JEHG-+y]:ILWJSC([0, T ];H)
p VLW`NRPQVLWJSF^`_:DZzIS1_:M\NRD
wk DeSAN&Ia_SRSRM ]8DC!Ia_JEHG-+y]:ILWJS L2(]0, T [;V )pW r _:WJMON V ]8_K:PRV8]8_JMONSRETIaZzIaM[PRD
a(u, v)XT^`_:M[cILZ\DTW`NIa_ KJPRV8]8_:M\NlSQEILZ[ILM\PQD_JSR_:DZUYuETIa_JSRD]8DZzIuEVDPHEM[c`M\NRX]8D
apOPVL_JP
l > kV5WI
‖wl(t)− wk(t)‖2V = a(wl(t)− wk(t), wl(t)− wk(t)) =
l∑
j=k+1
λj |αj(t)|2
≤ 2
l∑
j=k+1
λj |α0j |2e−2λjt + 2
l∑
j=k+1
λj
(∫ t
0
βj(s)e−λj (t−s)ds
)2
.
PTf8KJILP&IaK:KJZ\MzEIaNRM[VLWt]:DZ®g M\WJX5ILZ\M\NRX]8DvC!IL_JEHG-+`hi8EHG FIaP.S&VLW*I(∫ t
0
βj(s)e−λj(t−s)ds
)2
≤(∫ t
0
|βj(s)|2e−λj(t−s)ds
)(∫ t
0
e−λj(t−s)ds
)
≤ 1
λj
(∫ t
0
|βj(s)|2e−λj(t−s)ds
)
.
Ecol
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¯ = = O )1 )#& *<
P2ILP&IaM[Z\Z[D_:PHSTf`DTW*cLDPRNR_*]8_NRGJXVLPQ r D]8DJ_:w:M[W:M∫ T
0
(∫ t
0
|βj(s)|2e−λj(t−s)ds
)
dt =
∫ T
0
|βj(s)|2(
∫ T
s
e−λj(t−s)dt
)
ds
≤ 1
λj
∫ T
0
|βj(s)|2ds.
P2ILP&EVLWS1Xe^5_JDW`NTf8V5WtDTW]8Xe]8_:M\N^`_:D∫ T
0
‖wl(t)− wk(t)‖2V dt ≤l∑
j=k+1
|α0j |2 +
l∑
j=k+1
2
λj
∫ T
0
|βj(s)|2ds,
ED^5_JM M r K:Z[Mz^5_JD^5_JDlZzI$SR_:MONQD wkcLXTPRM\´JD
limk,l→+∞
∫ T
0
‖wl(t)− wk(t)‖2V dt = 0,
ELg DeSANRh¦Yah¦]:M\PQDl^`_ g DZ[Z[DlDeSAN=]:DvC!Ia_JEHG-+y]:ILWJSL2(]0, T [;V )
pC/V rr DtZ\DeSv]8DT_8bsDTSRKJILEDTS C([0, T ];H)DNL2(]0, T [;V )
SRVLW`NvEV r K:Z[DNQSTfUZ[IS1_:M\NRD]8DC!IL_JEHG-+wkEV5WcLDPQLDDN&V5WtK?D_8N=]:X´JW:M[P=SQIZ\M r MONQD u
limk→+∞
wk = u]:ILWJS
C([0, T ];H) ∩ L2(]0, T [;V ).
:BW KJIaPRNRMzE_:Z[M[DPefBEV rr D wk(0)EVLWcLDTPR5Dtc5DPHS
u0]:IaWS
HfmV5W DTW£]8XT]:_:MONtZ[IsEVLWJ]8M\NRM[VLWM[W:M\NRMzIaZ[DBc5VL_:Z[_:DLf
u(0) = u0^`_:M`DTS1N_:WJDBX`IaZ[MONQXBDTW5NQPRDmnoVLWJENRM[VLWJSU]8D
Hpxg Ia_8NQPRD/KJIaPRNTfeM[Z5DeSANEZ[ILM\P2^`_:D
u(t)fjDWNQILW5Nm^`_:DFS1V rr DF]8DFZzISRXPQM\DJp\e¹5dcLXTPRM\´JDBZzInoVLP r _:ZzIjNQM\V5WcjIaPQM[IaNRM[VLW:WJDZ[Z\D®:p a&KV5_:PlEHGIL^`_:DnoV5WJENQM\V5W NRDeSAN
v = ukp|C/V rr D (uk/
√λk)DeSAN_:W:DvwJILSRDG:M[Z[wDTP1NQM\DTW:W:D]:D
Vfu(t)cLXTPRM\´JD,]8VLWEFZ[InoVLP r _JZ[IaNRM[VLWcILPRMzIjNQM\V5W:W:DTZ\Z[Dl®:p aUK?VL_:PNQVL_8N v ∈ V fLEag DTS1N1hiYjhi]8M[PRD^`_:D
u(t)DTS1N&w:M[DW*ZzISRVLZ[_8NQM\V5WtPQDTEHG:DTPQEHGJXD]:Dy:p ap
PVL_JP&VLw8NQDW:M[P=Zkg DTS1NRM r IaNRM[VLW ]|g XW:DTPR5M\Dy®:p 5f:M\ZUS1_:±Nl]8DPRD r IaPH^5_JDP&^`_:DZkg V5W IK:PRV5_:cLXZ[DTS r I AVLPHIjNRM[VLWS‖wl(t)− wk(t)‖H ≤ ‖u0‖H +
1√2λ1
‖f‖L2(]0,T [;H)
DN∫ T
0
‖wl(t)− wk(t)‖2V dt ≤ ‖u0‖2H +2
λ1‖f‖2L2(]0,T [;H).
:BWK:PQDWIaW`Nk = 0
DN*DWnIaMzSQIaW`NuNQDWJ]8PQDlc5DPHSuZkg M[W8´JWJM®fFVLWVLw8NQM\DTW`N*M rr XT]8MzIjNQD r DW`NZkg DTS1NRM r IaNRM[VLW*]8XTSRM[PRXTDLp
IK IPORQ O 9 *I J+ (+-(_%)("J"E )@ 9J (m)@I(2(*J! :G%)0"! :G%)@4&2( $9)(";<("$$299' + (>9'("$91(>+ :/*)@-;<"(,("$f%)(*!#E29 2:3$9 + :RG)!2+ : %)@ 3$#;: )@ :G%)@-$$2("+-+-( / 86 5 6 6f; ("9m("9'E2:"(*92:3$9/+ (("+m$9 d02(_"_0$29/+ : 9'+- %)@ 3$
u(t) + :V3$*%)@-$ t → 〈u(t), v〉H$F ("9 %qE2:39v*)@-;3:3=2+-(
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% O "&T1< < < & J &) ¯ =-.:G 9'("$29 "+ :39'9'@4&2( +o("+-+ (D:3EEF: '%)@ (*$%q9'("+-("!#("$&%?;
L2(0, T )(*%C;
C[0, T ])6# $LE?(" %q$2:G$2!#@-$9
8F$@ 9): *)@-;<"(,:Gf9'("$29HT:G@-=2+-(>DG ("!#!#( 6 6I[ 7 / #:Gf9'("$29 (*9 @-9 %')@ =%)@-$9 6 6&~+-29JE)"@-9' !#(*$% ddt〈u(t), v〉H
("9 %m82$2@W"!#!#(m2$f"+-"!#(*$% (H−1(0, T ) / (*9 %: ;= @ )(,2$2(3)!#(+-@-$2:G@ )(
*$&%)@-$ (B9'H1
0 (0, T ) 6 EF: >+R: 3)!v2+-(⟨
d
dt〈u(t), v〉H , φ(t)
⟩
H−1,H10(0,T )
= −∫ T
0
〈u(t), v〉H dφ
dt(t) dt ∀φ ∈ H1
0 (0, T ).
: 1"$9'&2("$&% @ )(R&( +' &F:G%)@-3$A( / 86 5 6 :+-@-(" :G 9'("$9T:G@ =+-(v:3$9]0, T [
("9 %,&2@-;3:3+-("$&%1; @ )(>(
−∫ T
0
〈u(t), v〉H dφ
dt(t) dt +
∫ T
0
a(u(t), v)φ(t) dt =
∫ T
0
〈f(t), v〉Hφ(t) dt
E?3%)%v ∈ V
(*% %)3%φ ∈ C∞
c (]0, T [)E2@-9"(
C∞c (]0, T [)
("9 %/(*$29'(v2:3$9H1
0 (0, T )6
*$*+ )( d:39'9' )$9+ (+-(*%)(" )+9'@ u("9 %$2(>9'+-%)@-3$ ( / 8 6 5 6 :G+-3)9 EF: +' (<:3+-@ %)! "!#(&F ("9 %
/ 86 5 6 +R:*)@-;<*( ddt〈u(t), v〉H
:3EEF: '%)@ (*$%D;L2(0, T )
(*% 3$ E?(" % 3$ @ )( &2( / 86 5 6 :+-@-("E )("9"&2(BE2:G'%)3%>2:G$29
]0, T [6 •
.@AF. DH&HJF1@R> N6:@19B4&E0=VL_JSILK:K:Z[M[^`_:V5WJS r IaM[W`NRDWIaW`NlZ[DPRXeS1_JZONHIjNIawJS1NRPHIaM\N]8_@kFG:XV5PR r D:p°¯8p 9YuZkg Xe^`_JIjNQM\V5W]8DyZzI EHGJIaZ[D_:Pef DNW:VL_SK:PQVL_:c5VLWJS^`_:DuEDNRNRDuILK:K:PQVEHGJDcjILPRMzIjNQM\V5W:W:DZ[Z[DyI*w:M\DTWsKDTP r MzS]8DPRXeS1V5_J]8PQDZkg Xe^`_JIjNQM\V5WIL_8bt]8XTPRM[cLXTDTS!KJIaPRNRM[DZ[Z[DTS,]|g VLPQM\5M\WJDLp
b 4(6 j`¡LOGFGc,4.>01Ω2 .> L 3B[ =.B[4AWBOA >28h0<3BEF3
RN I ,4.>01S2P 3Yg69G \p98 T > 0 i293 EF.>9A=3n0101D0<83u0 ∈ L2(Ω) i 3Z2 V3B[Y 3XG.>2BO;=3
f ∈ L2(]0, T [;L2(Ω)) I 8M.>B=G8 R A=>?0<.> E738M!;*8M32B
∂u
∂t−∆u = f
6 I 6 I EF>HG Ω×]0, T [
u = 06 I 6 I G[2B ∂Ω×]0, T [
u(x, 0) = u0(x)6 I 6 I EF>HG Ω.
®:p[e¯5FEY 3N243W20<3WG.8h2?0<.>
u ∈ L2(]0, T [;H10 (Ω)) ∩C([0, T ];L2(Ω)) I _3o698: G i 018 3T70UG=V32935;=.>HG[ >V3
C > 0 20 93CEFAV6H39E!3CEF3
Ω 3818M3C3 i 6H.>2B# .>2 t ∈ [0, T ] i∫
Ω
u(x, t)2dx+
∫ t
0
∫
Ω
|∇u(x, s)|2dxds ≤ C(∫
Ω
u0(x)2dx+
∫ t
0
∫
Ω
f(x, s)2dxds
)
.®:p[ 9`+ 6 ?§ ª j ªe« ?§ (0&VL_S!ILK:K:Z[M[^`_:V5WJS!Z\D#kFG:XTVLPQ r D:p°¯8p 9YZ[InoVLP r _JZ[IaNRM[VLWucjILPRMzIjNQM\V5W:W:DZ[Z[D:p 5!]8DZkg XT^`_JIjNQM\V5W]:DlZzIEHGJIaZ[D_JP:p[e¯5*S1DeSFG +K?VaNQG:TSRDTS,SRVLW`NFnILEM\Z[D r DW`N,cLXTPRM\´JXTDTSFIcLDeEH = L2(Ω)
DNV = H1
0 (Ω)oDTWKJILP1NQM[E_:Z\M[DPefdEV rr D Ω
DeSANvwV5PRW:XuZ\D kFGJXVLPQ r D =p 9Jp ¯:]8D,DTZ\Z[MzEHG·Ij±$P r D^5_JDZkg M[WADTENRM[VLW ]8DH]JIaWJS
VDTS1NEV r KJI5ENRDp*ViZPQDTS1NRD¸Y r VLW`NRPQDP^5_JDZkg _:W:Mz^5_JD*SRVLZ[_8NQM\V5W
u ∈ L2(]0, T [;H10 (Ω)) ∩ C([0, T ];L2(Ω))
]8D EDN1NQDnoVLP r _:ZzIjNQM\V5WcILPRMzIjNQM\V5W:W:DTZ\Z[DDTS1N=w:M[DW _:WJDS1V5Z\_:NRM[VLW ]8Du®:p[e¯LpkVL_8Nl]|g ILwV5PQ] f:Z[I$EVLWJ]:MONQM\V5W Ia_8b*Z\M r MONQDTS
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¯ = O )1 )#& *<
]:Dx=M[PRMzEHG:Z[DN&SRDlPQDNQPRV5_:cLDlKJILP&IaK:K:Z[MzEIjNQM\V5W]:_PkFGJXVLPQ r DJ=Jp 9:p[ 9]8DNRPHILEDY u(t) ∈ H10 (Ω)K?VL_JPK:PRDeSR^`_:DNQVL_8N
t ∈]0, T [f|DNZ[I*EV5WJ]8M\NRM[VLW¸M\WJMONQM[ILZ\DDTS1N A_JS1NRM\´JXD$KIaPZ[I EVLW`NRM[W_:M\NRX$]8D
u(t)DTW
t = 0EV rr DnoVLWJENRM[VLW*YcILZ\DT_:PHS,]:IaWJS L2(Ω)
pM8ZzIlS1V5Z\_8NQM\V5WuDTS1NmS1_:±$SQI rr DW`NPRXTL_:Z[M[PQDoKIaPDb8D r K:Z[DLfjSRM ∂u
∂t
DN∆uIaK:KJILP1NQM\DTW:W:DTW5NY
L2(]0, T [×Ω)fED!^5_JM`DeSANcPQILM5]|g IaKJPReSdZzIPmPQVLK?V5SRM\NRM[VLWJp =p `feKJIaPUM\W`NQXLPHIjNQM\V5WKJIaPUKJILP1NQM\D5fZzIvnoVLP r _:ZzIjNQM\V5WtcjILPRMzIjNQM\V5W:W:DZ[Z[D®:p 5/DTS1N&XT^`_:M[cjIaZ[DW`NRDY
∫
Ω
(
∂u
∂t−∆u− f
)
v dx = 0,Jp\ =`
K?VL_JPNRVL_:NRDnoVLWJENRM[VLWv(x) ∈ H1
0 (Ω)DNK:PRDeSR^`_:DvNRV5_8NNRD r KJS t ∈]0, T [
p P2IaPEVLWJSRXT^`_:DTW5NefV5W]:XT]8_:M\N=]8Dy®:p[=`!^`_:D∂u
∂t−∆u− f = 0
K p KdpJ]:IaWS]0, T [×Ω.
7 @+ :9'+-%)@-3$u$F ("9 %#E2:39fE+ 9v)(32+-@-4*)( &2(
u ∈ L2(]0, T [; H10 (Ω)) ∩ C([0, T ]; L2(Ω)) $=%)@-("$&%%) % (N! *!#( "(*%'%)(N(<:G+ @ %)N! :3@-9 + :0! 9 %)@ 8B: %)@ 3$ ("$ ("9 % +-(34*)("!#("$&%DE2+-29D*+ @I: %)(6
-m$3)!#"!#(*$%;v+R:("! : S&2(B86 76 f+-(9'("$29,E)"@-9 ( / 8 6 6 ("9 %
−∫ T
0
∫
Ω
uvdφ
dtdxdt +
∫ T
0
∫
Ω
∇u · ∇vφdxdt =
∫ T
0
∫
Ω
fvφ dxdt / 8 6I[ 6E?,%)3%)(R3$*%)@-$
v(x) ∈ C1c (Ω)
(*%φ(t) ∈ C1
c (]0, T [)6 $ )"9'+ %d:G%X"+ :39'9'@4&2(v% :3$2:3+ 2 9'(q$9
@ %O&2(m+' ("$9'("!=+ (, ("9 "3!=2@-$2:3@-9'$9 + @-$:3@ )("9 (1E )@-%J (m%)("+-+-("9 3$*%)@-$9v(x)φ(t)
("9 % ("$9'(2:3$9
C1c (]0, T [×Ω)
6 $ $2G%)(σ = (u,−∇u)
+ : 3$*%)@-3$0;v;3:3+-(" )91;<(_%)3)@-("+-+-("9,2:G$29RN+1 3$%
+R:v@-;<(*"(3("$*(("$ ("9'EF:"( a%)("!#E9,("9 % ∂u∂t−∆u
6WGO @ ("$&%)@ %) / 864[ 6 $291@ %1&2("(*%'%)( @-;<(*"(3("$"(:=@-("$ 2$ 9'("$9?T:G@-=2+-( / ;<3@-v+ : 8F$@-%)@-3$ 6 76 6 (_%f("9 %(<:G+ (0;+ : 3$*%)@-$f&@1:3EEF: '%)@ (*$%
;L2(]0, T [; L2(Ω)) & + (:3+-@ %) E )("9"&2(EF: '%)% 2:G$29
]0, T [×Ω6 u+:3 %0"("E?(*$F2:G$&%@:3@ )(
:G%'%)("$&%)@-$&2(q$29:;<$9 !#3$%')?&2(q+ : @ P*)(*$"( ∂u∂t−∆u
:GE2E2:G'%)@-("$&%>;L2(]0, T [; L2(Ω)) ! :3@-9EF:G9d0F:&2( %)(*)!#( @ $2@-;@ 2(*+ +-("!#(*$%6
8 ? OGF >[` ªT« ªe« ?§ ¢> 6 §U`)D « :p[ 9`!M\W]8M[^`_:Dv^5_JDZ[I$WJVLP r D]:DZ[IySRVLZ[_8hNQM\V5Wt]:ILWJS/Z®g DTSRKJILED]|g XWJDPQLM[D&DTS1NFEVLW`NRP LZ[XD=KJIaP!Z[IW:VLP r D]8DeS/]:VLW:W:XTDTSTpQViZ DeSAN,YW:VLNRDTP!^`_:DEDNRNRDW:V5P r DW:DvEVLPQPQDTSRKV5WJ]uKILSFNQVL_AV5_:PQS=YZ[I²1cPQILM\D³lXTW:DPQLM[DK:G-+SRMz^5_JDy]:ILWJS&Z[DEI5S&]8DZzI EHGJILZ\DT_:PZ®g XWJDPQLM[DNRG:DTP r Mz^5_JD$DTS1NK:PRV5KV5P1NQM\V5W:W:DTZ\Z[D$Y ∫Ω u(t, x) dx
pUqBg M[W:XT5IaZ[M\NRX Jp\595IyXNRXVLw8NQDW_:DvEV rr DEV5WJSRXT^`_:DWED]8D:p 5fED^`_:METI r VL_ DS1V5W VLPQM\5M\WJDDNlSRVLW M[W`NRDTP1hKJPRXNQIjNQM\V5W*K:G-+SRMz^5_JDLpJqdDTS&Db8DPHEMzEDTS,SR_:M[cjIaW`NQS=KDTP r DNRNRDTW5Nl]|g V5w8NRDTW:M[P]8M[PRDeENQD r DW`NJp\595FYKIaPRNRM[Pm]8D,Z®g XT^`_JIaNRM[VLW]8D,ZzIEHGJIaZ[D_:P,®:p[e¯LDTW_:NRM[Z\MzSRILW`Nm_:W:D 6 DJ «oª56 ¢> 6 §U`D « ^`_:M:W:DFnILMON^`_ g Db8K:PQM r DTP_:W~w:M[Z[ILW¸KJG +8SRM[^`_:D5p :mWsKJILP1NQM[E_:Z\M[DPef EDeSDTS1NRM r IjNRM[VLWSVL_~X`IaZ[MONQXTS]|g XW:DTPR5M\DA_SANQMO´JDTW`NFZ[DEHG:VLM\by]8DZ®g DTSRKJI5ED
L2(]0, T [;H10(Ω)) ∩C([0, T ];L2(Ω))
KV5_:P+yEHG:DPHEHG:DTP!]8DeSSRVLZ[_8NQM\V5WJSFEIaP&ELg DeSAN,K:PQXTEM[SRX r DTW5N&T>\:¨5¢> 6 §U`)D « $ELg DeSANRh¦Yah¦]:M\PQDZkg DTSRKJILEDl]8DPQX5_:Z[ILPRM\NRXr M[W:M r _ r ]:IaWJSFZ[DT^`_:DTZ Z[DTSFX`IaZ[MONQXTS,]|g XWJDPQLM[DV5W5N&_JWSRDWJSTp • `j¨ « ¨5 OGF ( 2$ "7?(?<($&=!#" + %$f7?<< $&-$Q/#"DB)@% % 3A$&#-N@ @-%$
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% O "&T1< < < & J &) ¯ =8( *$%"(?7?< $.>7-J!"):+ >*$R+,".
u/(=1 %76 %$; >)$&$R G@%K"7+ GM/(>7)$ ]0, T [×Ω
N !#"7N ?<"79;"t ∈ [0, T ]
N(# > + @%K7>7+ ;@ /) @<)<GK :$ " )>7;
1
2
∫
Ω
u(x, t)2dx+
∫ t
0
∫
Ω
|∇u(x, s)|2dx ds =1
2
∫
Ω
u0(x)2dx
+
∫ t
0
∫
Ω
f(x, s)u(x, s) dx ds.
®:p[ `
%# J@ + >:?<G?< @ ;@L$ " )>7;(N)>7?7?<<+ @%.+ Q/ G >7+,+ -$ z %$& "()N . %. (") /(
[0, T ]/7>7$
R+ ;<+,+ 2!#"
z(t) ≤ a+ b
∫ t
0
z(s) ds ∀ t ∈ [0, T ],
a, b
$RL/ " Z )$;&>7;%$0?< $ . -$'" 7"(+,+ -$-N>7+ G$
z(t) ≤ aebt ∀ t ∈ [0, T ].
# :>7?(?7+ !">7Q+ + 0/( G >7+,+>) z(t) = 12
∫
Ωu(x, t)2dx
N/(@-/#"< G0/( 1 -5(6!#"7N ?<"79;"t ∈ [0, T ]
N
1
2
∫
Ω
u(x, t)2dx+
∫ t
0
∫
Ω
|∇u(x, s)|2dx ds ≤ et
2
(∫
Ω
u0(x)2dx
+
∫ T
0
∫
Ω
f(x, s)2dx ds
)
.
®:p[L
`j¨ « ¨L OGFGF "d7"2/ A1 (6 N( A+ > #$&&>7-;C
%$;Q /(@<?<</7>7-; / TN<#d) !#"
+ ;< et ! %$&W%<&> < ?>)$:?. >7+ /7>7)$A+ > > " G>. #1 3 6 ;%$&. >. ?<<" G > @<+ G@% < > $&#(>70/ + >a> $%"< >7-;7N9> "()C)> >7; /" + / # >7+,+
( a ∈ R+%g ∈ L2(]0, T [)
;<+!#"g ≥ 0 E #-G<J!")(N#$P
z(t)%$&g . 7"2/(
[0, T ]/7>7)$
R+%p)@<
z(t) ≤ a+ 2
∫ t
0
g(s)√
z(s)ds ∀ t ∈ [0, T ],
>7+ G$z(t) ≤
(√a+
∫ t
0
g(s)ds
)2
∀ t ∈ [0, T ].
%# J@/" =/(1 -576 !")(N ?<#"(;#"- t ∈ [0, T ]N
∫
Ω
u(x, t)2dx+ 2
∫ t
0
∫
Ω
|∇u(x, s)|2dx ds ≤(
(∫
Ω
u0(x)2dx
)1/2
+
∫ t
0
ds
(∫
Ω
f(x, s)2dx
)1/2)2
.
®:p[T`
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¯ =5 O )1 )#& *<
qBg XT5IaZ[M\NRX]|g XWJDPQLM[Dy:p[ `!W g DTS1NKJI5S,Z[IuS1DT_:Z\DKV`SRSRM\wJZ\DK?VL_JP&Z®g XT^`_JIaNRM[VLW ]8DZ[IyEHGIaZ[D_:PEV rr DZ\D r VLW`NQPRDZkg DbDTPQEM[EDlSR_:M[cjIaW`NTp `j¨ « ¨5 OGF ( $ "7?(?<($& !") + %$D 7?<< $R%$:/"NB)@% % 3 $&#-g)@< @%-$-N9!#"u0 ∈ H1
0 (Ω)N9 0!#") + > $&+,".
u/( 1 %76 -$;0>)$R$& G@%K"7+ G/(>7)$
]0, T [×Ω !#")(N ?<"7;#"-
t ∈ [0, T ]N > + @-K(>7+ ;@=/ @<)<GK $ " >7-;
1
2
∫
Ω
|∇u(x, t)|2dx+
∫ t
0
∫
Ω
∣
∣
∣
∣
∂u
∂t(x, s)
∣
∣
∣
∣
2
dxds =1
2
∫
Ω
|∇u0(x)|2dx
+
∫ t
0
∫
Ω
f(x, s)∂u
∂t(x, s)dxds.
®:p[Tº5
M[DW¡S :PefdZ[D!kFG:XTVLPQ r Du]|g Db8MzSANQDWJEDu:p°¯8p S1DtLXTW:XPHIaZ[M[SRDnILEM\Z[D r DTW5NIa_¡ETILS] g IL_8NRPQDTSEVLWJ]:MONQM\V5WJSBIL_8b$Z\M r MONQDTS/VL_y]|g _:WuV5KXTPQIaNRDT_:PmDZ[Z[M\K8NQM[^`_:DLXTW:XPHIaZ?EV rr D&Z[D r VLW`NQPRD&Z[DTS/Db8DTP1hEM[EDTS,SR_:M[cILW`NTp `j¨ « ¨5 OGFMr
Ω"( ")<<4)@G@%K#"(+ */
RN "7 ; ?)$5 >7+T > 0
N"7)/(7)@% < . >7+
u0 ∈ L2(Ω)N9 A"( ; $&"71
f ∈ L2(]0, T [;L2(Ω)) + > /(/(+ > > G!#" %# !")I+ @-!#">. / + >X)>7+ "(*>) P #)/ . >7" + ;%$/ " >77
∂u∂t −∆u = f
/(>7)$Ω×]0, T [
∂u∂n = 0
$ "7∂Ω×]0, T [
u(x, 0) = u0(x)/7>()$
Ω
®:p°¯a¹5>/ % "7 "7< !#":$R+,".
u ∈ L2(]0, T [;H1(Ω)) ∩ C([0, T ];L2(Ω)) `j¨ « ¨5 OGF H
Ω"( #" )< <4)@'G@%K"7+ </
RN A(x)"7 #) . L/(
Ω/(>7)$
+ <)$& 7+ :/(%$ > %%$M$ @% !#"-$:@-<+,+ -$L; +,+ %$*!#"!, +( $&;2/ " #$&&>7-;-$β ≥ α > 0)@< >7-
β|ξ|2 ≥ A(x)ξ · ξ ≥ α|ξ|2 ∀ ξ ∈ RN ,? ? x ∈ Ω.
"7 ; ?)$5 >7+T > 0
N "7) /(7@- . >7+ u0 ∈ L2(Ω)
NJ% "7 ; $R"7U%f ∈
L2(]0, T [;L2(Ω)) O O!") + ?< (+ :>7" + ;%$
∂u∂t − div (A(x)∇u) = f
/7>7$Ω×]0, T [
u = 0$ "7
∂Ω×]0, T [u(x, 0) = u0(x)
/(>7)$Ω,
>/ % "7 "7< !#":$R+,". u ∈ L2(]0, T [;H1(Ω)) ∩ C([0, T ];L2(Ω)) W*K?D_8N&XNRDW]8PRDZ[D#kFG:XV5PR r DJp ¯:p°Ia_:buXT^`_JIaNRM[VLWJS,]8D`NQV5DTSFM[WJSANHIjNQM\V5W:WJIaM[PQDTSTp
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% O "&T1< < < & J &) ¯ =`ºb 4(6 j`¡LOGF ,4.>01
Ω2 . L 3B[ =.B[4A%BOA F28:0<3Bg;=.93T 3EF3
RN I ,/.01 2 3Yg69G4\p9>8T > 0 i 293 EF.9A=3!0101D0<8M3 u0 ∈ L2(Ω)N 3818M3P3
divu0 = 0EF>HG
Ω i 3W2 V3B[Y 3G.>2BO;=3
f ∈ L2(]0, T [;L2(Ω))N I 8M.B=G#8M3G5A=>?0<.>HGEF35,V.73[GW01HG[VD0<.901B3[G
∂u∂t +∇p− µ∆u = f
E7HGΩ×]0, T [
divu = 0E7G
Ω×]0, T [u = 0
G=2B∂Ω×]0, T [
u(x, t = 0) = u0(x)EFG
Ω
®:p°¯8FEY 3?V3p243#20<3WG.>8:2?0<.>
u ∈ L2(]0, T [;H10 (Ω))N ∩ C([0, T ];L2(Ω))N I
@K= JO 1=%)(*$2@ ,2$(3)!+ :G%)@-$D;: )@R: %)@-$2$("+-+-(( / 86 7Y[ 6 $9"!v=2@-$23$291+-("9: "(!#("$&%)9,(+ :D7 3293 r9'(*%)@-3$8 6 76I[/(*%(+ :f "!#$9 %'d:G%)@-$ .10"3)4*!#( 6 6 8 6 $ @-$&%')2@ %+-("9,("9'E2:"(*9 ( @-+-=?(*'%
V =
v ∈ H10 (Ω)N %)("+F&2(
divv = 0E 6 E 6F:3$9
Ω
,
(_%H =
v ∈ L2(Ω)N %)("+F&2(divv = 0
E 6 E 6F:3$9Ω
,&H
("9 %f2$ 9'93 r("9'EF:"((*)!# (H(div)
&F 3$ E?("%f:329'9'@182$2@ C"3!#!#(#+' :30*)("$*( (V
2:G$29L2(Ω)N / ;<3@-+ : 7293 r9'(_%)@ 3$ 6 6 7 6 6 329 = %)("$23$29#+ : G)!+R: %)@-$ ;3:G)@ :G%)@-3$2$2(*+ +-(
9'@-;:G$%)(
d
dt
∫
Ω
u(t) · v dx + µ
∫
Ω
∇u(t) · ∇v dx =
∫
Ω
f(t) · v dx ∀ v ∈ V, 0 < t < T,
u(t = 0) = u0,
/ 86 77 6
& +' 2:G%)@-$ ( / 8 6 77 6 : +-@ (* :3 9'("$9?T:G@-=2+-( :3$9]0, T [
6H $ :GE2E+ @4&2(#+-(D.102"G)4"!#(8 6 76 ;*(*%'%)(G)!2+ : %)@ 3$f;3:G)@ :G%)@-$$2("+-+-( / 8 6 737 6 / 9'("9~0Y2 E?G%)024"9'("9~9'3$% :"@-+-("!#("$&%J;<*)@ 82"("9 6 (*%~$=%)@-("$&%+' (*C @-9 %)("$*((*%>+ $2@I"@ %)/(9):9'3+-%)@-$
u ∈ L2(]0, T [; H10 (Ω))N ∩ C([0, T ]; L2(Ω))N 6
.o %)(+ :/@D"+ %)>)"9'@ ( :3$29J+ :qE )("2;(1(1"(*%'%)(>9'+-%)@-$ ( / 86 77 6 ("9 %1=@ (*$#2$2(,9'3+ %)@-$ ( / 8 6 7Y[ 6 6.G : "$2@ %)@-$ :3 CL+-@-!#@-%)(*9f($B@ )@Id02+-(*%f9'( )(*%')32;<( EF: #:GE2E2+-@I: %)@ 3$ .102"G)4"!#( 6 6I[L (D%'d:"( ;
u(t) ∈ H10 (Ω)N E?3#E )("9"&2(%)%
t ∈]0, T [ (*% + : *$F @ %)@ 3$ @-$2@ %)@ :3+-(A("9 %!u29 %)@ 8F*(EF:G>+ :"3$&%)@ $@-%)B(
u(t)("$
t = 0:G
u0 ∈ H6
m)(*%');<(*1+' &F: %)@-$ 3$E )J"4 ("3!#!#(:3$9m+ :v "!#$9 %'d:G%)@-$D .10"3)4*!#(B8 6 7&6 5&67@ + :9'3+-%)@-$
u(*9 %9'#9):3!#!#(*$%m)(+ @-4*)( $D= %)@ (*$%
∫
Ω
(
∂u
∂t− µ∆u − f
)
· v dx = 0 / 8 6 7 6E?3 E)("9"&(1%)%
t ∈]0, T [ (_%&2(*+B&2(9'3@-% v(x) ∈ C1c (Ω)N %)(*+Z&2(
divv = 02:3$9
Ω6Y-m!#!#(
E?3m+ (E)3=2+-4"!#((R7&%)<(*9m9 %d:G%)@-3$2$F:G@ )( / ;<@ m+R:7 3293 r9'(*%)@-$ &6 6 7 6 @ +B:3 %, 2@ )(( / 86 7 6+' (*C @-9 %)("$*(v 2$2()$*%)@-$p(t, x)
%)("+-+-( &2(+' &F:G%)@-3$ / 8 6 7Y[ 6 :G@-%+-@-(" 6 +%:3 % ("$"3)(/ %)@ +-@-9'(*+-(.10"3)4*!#(( ( >0F:3! 6 6 / q!#@-$9 2$2(1("(*9J;3:G)@ :G$%)(*9 6 @ :f)!#(.( +' G'%)02($2:3+2("9;(*%)(" )9.;/@-;<(_"(("$"($+ +-(,("9 %m+ ("$9'("!=+-( ("9U(Gd: @ (*$%)9)6W-m(E?@-$&%1 :3$F:G+ 29'(>("9 %1:39'9'( "+-@I:G%/ (*$"G)(vE+ 9 9'@+ :#9'+- %)@ 3$
u$F ("9 % EF:G9)(32+-@-4*)( 6 (*%$2329 :!#(_%'%')$29>9'@-!#E2+-("!#("$&%>+' (*C @ 9 %)(*$"(
2$2(%)("+-+ (E )("9'9'@-$p9):G$29,E)"@-9'(*>! "!#();C&("+ (*9'EF:*(("+-+ (/:GE2E2:G'%)@-("$&%6
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d2
dt2
∫
Ω
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∫
Ω
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∫
Ω
f(x, t)v(x) dx.:p°¯ =
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〈w, v〉L2(Ω) =
∫
Ω
w(x)v(x) dxDN
a(w, v) =
∫
Ω
∇w(x) · ∇v(x) dx.8VLM\N_:W*NQD r KJS=´JWJIaZ T > 0Xc5DW`NR_:DTZ\Z[D r DTW5NlX`IaZdY +∞ fVLW SRDv]8V5W:W:DZ[DNQDP r DS1V5_:PHED
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u0 ∈ H10 (Ω)DN
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C([0, T ];H10 (Ω)) ∩ C1([0, T ];L2(Ω))
NRDTZ\Z[D^`_:D
d2
dt2〈u(t), v〉L2(Ω) + a (u(t), v) = 〈f(t), v〉L2(Ω) ∀ v ∈ H1
0 (Ω), 0 < t < T,
u(t = 0) = u0,du
dt(t = 0) = u1.
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)
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uk ∈ V,DNa(uk, v) = λk〈uk, v〉H ∀ v ∈ V.
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k=1
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αk(t = 0) = α0k,
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dt(t = 0) = α1
k.
®:p°¯aº5@2!N1NQDW`NRM[VLW*Y_JW:DEVLW:no_JS1M[VLWK?V5SQS1M[w:Z[Dl]:ILWJS!Z\DeSFW:VaNHIjNQM\V5WJS 8Z[I]8VLWJW:XDlM\WJMONQM[ILZ\D
u1Wdg IvPRM[DWYvcLV5M\PFIcLDeE&ZzInoV5WJENQM\V5WuK:PQVLKJPRD
ukKV5_:P
k = 1p°PV`SRILW5N
ωk =√λkf`Zkg _:W:Mz^5_JDSRVLZ[_8NQM\V5Wt]8D®:p°¯aº`BDeSAN
αk(t) = α0k cos(ωkt) +
α1k
ωksin(ωkt) +
1
ωk
∫ t
0
βk(s) sin(ωk(t− s))ds, :p 9L¹`ED^5_JMd]8VLWJW:D_:W:DnoVLP r _JZ\DDb8K:Z[MzEM\NRDK?VL_:P,ZzIS1V5Z\_8NQM\V5W u ®^`_:M DTS1N=]8V5WJEl_:W:Mz^5_JDep ª :BIF9 PVL_JP&]8X r VLW`NQPRDTP&^5_JDlZzIS1XTPRM[D
+∞∑
j=1
(
α0j cos(ωjt) +
α1j
ωjsin(ωjt) +
1
ωj
∫ t
0
βj(s) sin(ωj(t− s))ds)
uj®:p 9:e
EVLWcLDTPR5D ]:IaWSC([0, T ];V ) ∩ C1([0, T ];H)
fuV5W cjI r VLW`NQPRDTP^`_:D Z[I S1_:M\NRD wk =∑k
j=1 αj(t)uj]8DTSFSRV rr DTS!KJILP1NQM\DTZ\Z[DTS!]8DEDN1NRDSRXPQM\DlDTS1NF]:DC!IL_JEHG-+Lp8xILWJS V WJVL_JSFEV5WJSRMOh]:XPQVLWJStZ\D¸K:PQV8]8_:M\NSQEIaZzIaM[PQD
a(u, v)K?VL_:PZ[DT^`_:DZZzIsnI r M[Z[Z\D (uj)
DTS1NV5P1NQG:VL5VLWJILZ\D5pP2IaPV5P1NQG:VL5VLWJILZ\M\NRX]:ILWJSHDNv]:ILWJS
V]8D
(uj)cLV5M\PZ[DakFG:XV5PR r Dy8p 9Jp ¯5f|V5W¸V5w8NRM[DW`Nef KV5_:P
l > kf:DN,K?VL_:P&NRVL_:N!NQD r KJS t fa(
wl − wk, wl − wk)
+
∥
∥
∥
∥
d
dt
(
wl − wk)
∥
∥
∥
∥
2
H
=
l∑
j=k+1
(
λj |αj(t)|2 +
∣
∣
∣
∣
dαj
dt(t)
∣
∣
∣
∣
2)
.
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%% "&T1< < < & J)1 ( ¯ . 9PTf8DTW r _:ZONQM\KJZ\MzIaW`NJp ¯Lº5/KJILP dαk/dt
DN&DW*M\W`NQXLPHIaW`NFDWNRD r KJSfJVLWVLw8NQM\DTW5N∣
∣
∣
∣
dαj
dt(t)
∣
∣
∣
∣
2
+ λj |αj(t)|2 =∣
∣α1j
∣
∣
2+ λj
∣
∣α0j
∣
∣
2+ 2
∫ t
0
βj(s)dαj
dt(s)ds.
x=DZzIvnoVLP r _:Z[D$:p 9L¹`/VLWM\W8noTPRD^`_:D∣
∣
∣
∣
dαj
dt(t)
∣
∣
∣
∣
≤ ωj
∣
∣α0j
∣
∣+∣
∣α1j
∣
∣+
∫ t
0
|βj(s)|ds.
:mW EV r w:M[WJIaW`N&EDeS,]8D_8bPRXeS1_JZONHIjNQSFV5WtDTW ]8XT]8_JMON∣
∣
∣
∣
dαj
dt(t)
∣
∣
∣
∣
2
+ λj |αj(t)|2 ≤ 2∣
∣α1j
∣
∣
2+ 2λj
∣
∣α0j
∣
∣
2+ 2t
∫ t
0
|βj(s)|2ds.®:p 95¯L
C/V rr D u0 ∈ Vfu1 ∈ H
DNf ∈ L2(]0, T [;H)
f8VLW*I‖u0‖2V = a(u0, u0) =
+∞∑
j=1
λj |α0j |2 < +∞, ‖u1‖2H =
+∞∑
j=1
|α1j |2 < +∞,
‖f‖2L2(]0,T [;H) =
+∞∑
j=1
∫ T
0
|βj(s)|2ds < +∞,
ED^`_:M M r K:Z\Mz^`_:D^5_JD=ZzISRXPQM\D5f`]8V5W`N/Z[D&NRDTP r DlLXTW:XPHIaZDTS1N/Z[D r D r w:PRDl]8Dl5Ia_EHG:D=]8DJp 9`¯LfDTS1N=EVLWc5DPQLDW`NQDLfEag DTS1N1hiYjhi]8M\PQD^5_JDlZzIS1_JMONQDwkcLXPQM\´JD
limk,l→+∞
max0≤t≤T
(
‖wl(t)− wk(t)‖2V +
∥
∥
∥
∥
d
dt(wl(t)− wk(t))
∥
∥
∥
∥
2
H
)
= 0,
Ia_8NQPRD r DTW`N!]8M\NTfDZ[Z[DDeSAN,]8DC!Ia_JEHG-+]:IaWJS C1([0, T ];H)DN,]:ILWJS
C([0, T ];V )p:C/V rr DlEDeSDTSRKJILEDTSFS1V5W`N,EV r K:Z[DNHSf8ZzIvSR_:M\NRD]8DC!IL_JEHG-+ wk EVLWc5DPQLD=DN,VLWKDT_8N=]8X´W:M\P&SQIvZ\M r MONQD u p:mWtKIaPRNRMzE_:Z[M\DTPTfEV rr D (wk(0), dwk
dt (0))EVLWcLDTPR5D&cLDTPQS
(u0, u1)]:IaWJS
V ×H fVLWuV5w8NRM[DW`NZ\DeS,EVLW]8MONQM\V5WJS!M\WJMONQM[ILZ\DeSFcLVL_JZ\_:DeSp8xvg IL_8NRPQDKJIaPRNTfM[Z|DTS1N&EZzIaM[PF^`_:Du(t)fJDWtNQIaW`N,^`_:DS1V rr D]8DZzISRXPQM\D¡®:p 9:cLXTPRM\´JD ZzI~noVLP r _JZ[IaNRM[VLW cjIaPQM[IaNRM[VLW:WJDZ[Z\D¡®:p°¯ `KV5_:PtEHGIL^`_:D noV5WJENQM\V5WNRDTS1N
v = ukp2C/V rr D (uk/
√λk)DeSAN_:WJDywJILSRDG:M[Z\w?DPRNRM[DW:WJDu]8D
Vfu(t)cLXTPRM\´JD$]8VLWEZ[InoVLP r _:ZzIjNQM\V5WscILPRMzIjNQM\V5W:W:DTZ\Z[DJp ¯ 5K?VL_JPNQVL_8N v ∈ V
fUELg DeSANRh¦Yah¦]:M\PQDu^`_:Du(t)DTS1Nvw:M[DW¡Z[IS1V5Z\_8NQM\V5WtPQDTEHGJDPHEHG:XD]8D$Jp ¯ 5p%P2ILP&IaM[Z\Z[D_:PHSTf8VLW*IvDTWnIaM\N r V5W`NRPQXl^`_:D
a(wl − wk, wl − wk) +
∥
∥
∥
∥
d
dt(wl − wk)
∥
∥
∥
∥
2
H
≤ C(
‖u0‖2V + ‖u1‖2H + T‖f‖2L2(]0,T [;H)
)
,
DNZkg DTS1NRM r IaNRM[VLW]|g XWJDPQLM[D=Jp ¯`a|STg V5w8NRM[DW`NIaZ[VLPHS nILEM\Z[D r DW`NdDTWK:PQDWJILW`N k = 0DNDWnIaMzSRILW5NNRDW]8PRD
lcLDTPQSFZkg M\W8´W:M®p
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¯Q. = O )1 )#& *<
.F. DH&HJF1@1> NU6:@A9/4&E0=VL_JSIaK:K:Z[Mz^5_JVLWJS r ILM\W`NRDTWJIaW`N Z[DmPQXTSR_:ZONHIjNdILwJSANQPQILMONd]:_kFG:XV5PR r D2Jp 9Jp\mYFZkg Xe^`_JIjNQM\V5W]:DTSV5WJ]8DeSf:DN&W:VL_S,K:PQVL_:c5VLWJSF^`_:DEDN1NQDIaK:KJPRV8EHG:DcjIaPQM[IaNRM[VLW:WJDZ[Z\DIw:M[DW*K?DP r MzS&]8DPRXeS1V5_J]8PQDZkg XT^`_JIjNQM\V5WIL_8bt]8XTPRM[cLXTDTS!KJILP1NQM\DTZ\Z[DTS,]|g VLPQM[LM[W:DLp
b 4(6 OOMr ,4.>01Ω2 .> L 3B[=.B[4AZBOA F28:0<3B EF3
RN i 3g2 3Y6G\p9>8T > 0 I
J l;=.G=0<E B3 293 E7.9A=3 0101?0<>8M3(u0, u1) ∈ H1
0 (Ω) × L2(Ω)3%2@V3B[YZ3_G.2B;3
f ∈L2(]0, T [;L2(Ω)) I 8M.B=G#8 R A=D0<[email protected][G
∂2u
∂t2−∆u = f
6 I 6 I E7G Ω×]0, T [
u = 06 I 6 I G[2B ∂Ω×]0, T [
u(x, 0) = u0(x)6 I 6 I E7G Ω
∂u
∂t(x, 0) = u1(x)
6 I 6 I E7G Ω.
®:p 9 95
7EY 3N293W20<3WG.>8:2D0<.u ∈ C([0, T ];H1
0 (Ω)) ∩C1([0, T ];L2(Ω)) I C3N698: G i 018&3TF0UG[V32435;=.G[ >V3C > 0
20 435E7A 634E 35E73Ω3EF3
T 3818M3C3 i 6H.>2B# .>2 t ∈ [0, T ] i∫
Ω
(
∣
∣
∣
∣
∂u
∂t(x, t)
∣
∣
∣
∣
2
+ |∇u(x, t)|2)
dx ≤ C
(∫
Ω
(
|u1(x)|2 + |∇u0(x)|2)
dx
+
∫ t
0
∫
Ω
|f(x, s)|2 dxds)
.
®:p 9 =`
+ 6 ?§ ª j ªe« ?§ 0=VL_JSFILK:K:Z[M[^`_:V5WJS!Z\D#kFG:XTVLPQ r DlJp 9Jp\lYZ[InoVLP r _:ZzIjNQM\V5WucjIaPQMzIjNRM[VLWJW:DZ[Z\D®:p°¯ .5]8DyZkg Xe^`_JIjNQM\V5W]8DTSvVLWJ]:DTSV5w8NRDTW_:DuY ZzIV5_JSAhiSRDTENQM\V5Ws:p 9:p[ SRDTSvG +K?VaNQG:TSRDTSSRVLW`NnI5EM[Z\D r DW`Nc5XPQMO´JXTDTSIcLDTE H = L2(Ω)DNV = H1
0 (Ω)pWViZUPQDTS1NRDY r V5W`NRPQDP^`_:DvZ®g _:W:Mz^`_:DSRVLZ[_8NQM\V5W
u ∈ C([0, T ];H10 (Ω)) ∩ C1([0, T ];L2(Ω))
]8D$EDN1NQDnoVLP r _:ZzIjNQM\V5WcjIaPQM[IaNRM[VLW:WJDZ[Z\DDeSANwJM\DTWs_:W:DuSRVLZ[_8NRM[VLWs]8D:p 9 9`pQkVL_8N]|g ILwV5PQ] f Z[I EVLWJ]:MONQM\V5W~IL_8bZ[M r M\NRDeS]8Dtx=M[PRMzEHG:Z[DNSRDPRDNRPQVL_:c5DvKIaPIaKJK:Z\MzEIaNRM[VLW~]8_@kFG:XV5PR r D =p 9Jp\59t]8DNQPQI5ED$Y u(t) ∈ H10 (Ω)KV5_:PNRV5_8N
t ∈ [0, T ]fLDN/Z[IEVLWJ]:MONQM\V5W$M\W:M\NRMzIaZ[D=DeSAN A_JSANQMO´XD=KIaPBZzIEVLW`NRM[W_:MONQX=]8D
u(t)DWt = 0EV rr DnoV5WJENQM\V5WuYcjILZ\DT_:PQS!]:ILWJS
H10 (Ω)DNF]8D
du/dt(t)DWt = 0EV rr DnoVLWENRM[VLWtYcjIaZ[D_:PHSB]JIaWJS
L2(Ω)pMZzI&SRVLZ[_8NRM[VLW
uDTS1NUSR_8±ySRI rr DTW`NdPRXTL_:Z[M[PQDLfKIaPdM[W`NRXTLPHIjNRM[VLWKJILPdKJILP1NQM\DBZ[I&noVLP r _:ZzIjNQM\V5WcjILPRMzIjNQM\V5W:W:DZ[Z[D®:p°¯ .L/DeSAN&Xe^5_JM\cjIaZ[DW`NQDY
∫
Ω
(
∂2u
∂t2−∆u− f
)
v dx = 0,
K?VL_JPNRVL_:Nv(x) ∈ C1
c (Ω)DNK:PRDeSR^`_:DvNRV5_8N
t ∈]0, T [pW¸DWs]8Xe]8_:M\N]8VLWEZ®g XT^`_JIaNRM[VLW~]8D®:p 9 9`pJM ZzI$SRVLZ[_8NRM[VLW
uW g DTS1N&KJILS,KJZ\_JS&PRXTL_:Z[M\TPRD^`_:DED^`_:M|DeSAN=]:VLW:W:XKIaP,Z[DkFG:XV5PR r DJp 9Jp\5f5V5WuV5w8NRM[DW`N!NRVL_:N,]:D rr DZkg Xe^5_IjNRM[VLWtIa_tSRDWS²1K:PQDTSQ^`_:D=KJILP1NQVL_8N1³HfDTWuPQDK:PQDWJILW`N/Z[DTSILPR5_ r DW`NQS]8DZzIy]8X r VLWJS1NRPHIjNQM\V5W ]8_XkFG:XTVLPQ r D:p°¯8p *®^5_JDW:V5_JS=WJDv]8XNQIaM[Z[Z\V5WJS&KJI5SQp|WWJVaNRD
σ = (∂u∂t ,−∇u)
Z[InoVLWJENRM[VLWYcjIaZ[D_JPQS!cLDeENQVLPQM\DTZ\Z[DTS!]:IaWSRN+1 fDN,VLWtKDT_8N r VLW`NRPQDP
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%% "&T1< < < & J)1 ( ¯ .Q.^5_dg DTZ\Z[D=IL] r DN/_:WJD=]8M[cLDPQLDTWJEDFnILM\wJZ\D&DTWv²1DTSRKJILEDhkNRD r KJSA³F^`_:M?DTS1N#A_JS1NRD r DW`N ∂2u
∂t2 −∆u^`_:MIaK:KJILP1NQM\DTW`N,]:VLWJEY
L2(]0, T [;L2(Ω))p
:BW¸Zkg IawJSRDWJED]:DvnoV5PQEDTSTff = 0f VLW¸KDT_8NI r XZ[M[VLPQDPlZ®g DTS1NRM r IjNRM[VLW~]|g XW:DTPR5M\D*Jp 9 =`=DNVLw8NQDW:M[P,_:W:DK:PQVLKJPRM[XNQXl]8D¨5§Ue5: ªe« ?§¬¢ T> 6 §U5D « ª ª :®y^`_:M DeSANFNRPQTSFM r KV5P1NHIaW`NRD]8_ K?VLM[W5N]8Dvc`_JDv]8DeSILK:K:Z[M[ETIjNRM[VLWSp?qmg XW:DTPR5M\DNQVaNHIaZ[DDTS1NMzEMZzIyS1V rr D]8Dv]:D_8b*NRDTP r DeS ]|g _JW:DKJIaPRN,Z®g XWJDPQLM[DEM\WJXNRMz^`_:D |∂u
∂t |2DN&]|g Ia_8NQPRDKIaPRNFZkg XW:DTPR5M\D r XTEILW:Mz^5_JD |∇u|2 p
B? «ªT« § U J nG3d698M7;=3CG. G58M3G# 6H.>T [G3G E> HA=.>B Y 3 I7I L 3=; f = 0 I nG.8h2?0<.> EF3 8OR AD0<. E73[G^.4EF3G I L AB[0 \3 i 6.2B .>2 t ∈ [0, T ] i 8 R A 8:01VAXE73;.>HG3B L >?0<.> E738 R A93BF0<3
∫
Ω
(
∣
∣
∣
∣
∂u
∂t(x, t)
∣
∣
∣
∣
2
+ |∇u(x, t)|2)
dx =
∫
Ω
(
|u1(x)|2 + |∇u0(x)|2)
dx .®:p 9Q.L
+ 6 ?§ ª j ªe« ?§ J:mW PQDK:PQDWIaW`N$Z[I~]8X r V5WJS1NRPHIjNRM[VLW¬]8_ kFG:XTVLPQ r D :p 9:p[*Ic5DTE f = 0fEag DTS1N1hiYjhi]8M\PQD
βk = 0f!VLW]8XT]8_JMON ]8M[PRDeENRD r DW`N]8D Jp ¯Lº5$^`_:D¸Z®g XWJDPQLM[D]8DZkg V5SQEM[Z\ZzIjNQD_:PGJIaP r VLW:Mz^`_:DDeSAN=EVLWJSRDPQcLXTDLf8Eag DTS1N1hiYjhi]8M[PRD^`_:D∣
∣
∣
∣
dαj
dt(t)
∣
∣
∣
∣
2
+ λj |αj(t)|2 =∣
∣α1j
∣
∣
2+ λj
∣
∣α0j
∣
∣
2,ED^`_:Md]8V5W:W:DZ®g X`IaZ[MONQX$oK:Z[_8NLN^`_:DZ®g M[W:X`IaZ[MONQXe
a(wl − wk , wl − wk) +
∥
∥
∥
∥
d
dt(wl − wk)
∥
∥
∥
∥
2
H
=
l∑
j=k+1
∣
∣α1j
∣
∣
2+ λj
∣
∣α0j
∣
∣
2,
DN®:p 9Q.5FSg VLw8NQM\DTW`NDW KJPRDTWJIaW`Nk = 0DNlDW nILM[SQIaW`N&NQDWJ]8PQD
lc5DPHS,Zkg M[W8´JW:Mkp MdZzIySRVLZ[_8NRM[VLW
uDeSANPRXTL_:Z[M[PQDLfJV5W K?D_8N]8X r V5W`NRPQDP=KJZ\_JS]8M[PQDTENQD r DW`N:p 9Q.5,DTW r _JZONQM\K:Z[MzIaW`NZkg Xe^`_JIjNQM\V5W]8DTSFV5WJ]8DTS®:p 9 9`BKIaP ∂u
∂t
DN&DWM\W`NRXTLPHIaW`N,KJIaPFKJILP1NQM\D$cLVLM[P,Zkg :2b8DTPQEM[EDJp 9Jp\p
0=VL_JStPQDcLDTW:VLWS r IaM[W5NQDWJILW`NSR_:PtT>\ ªe« ªT« ?§"¢> 6 §U5D « :p 9 =$]:ILWJSuZ[DETILSuLXhW:XPHIaZf 6= 0
p2qmg Db8DPHEMzEDS1_JM\cjIaW`N r VLW`NRPQDtEV rr DW`N*®:p 9 =K?D_8N NRPQDtV5w8NRDTW`_JD]8M[PQDTENQDhr DTW`NuYKIaPRNRM[P]:D~®:p 9 9`vY¸Zkg ILM[]:D*] g _:W ILPR5_ r DW`NSRM r M[ZzIaM[PRD Y~EDTZ\_:M&]:D Z[I PmPQVLK?V5SRMONQM\V5W:p 9:p/.t^`_:M/XNHIaw:Z[MONv_:W:D 6 D?: «ª 6 ¢> 6 §U`)D « ^5_JMmW:D$nILMONv^`_ g Db8K:PQM r DTP_:Wsw:M[ZzIaW~K:G-+8S1Mz^`_:DLp:mW·KJILP1NQM[E_:Z[M\DTPTfFEDeSuDeSANQM r IjNQM\V5WJStVL_£XT5IaZ[M\NRXTS]|g XW:DTPR5M\DA_JSANQMO´DW`N Z\D¸EHG:VLM\b£]8DZkg DTSRKJILEDC([0, T ];H1
0 (Ω))N ∩ C1([0, T ];L2(Ω))NKV5_:P+*EHG:DPHEHG:DP&]:DTS=SRVLZ[_8NQM\V5WJS&EILP=Eag DTS1N=K:PQXTEMOhS1X r DTW`NT>\TU:¨5¢> 6 §`D « Eag DTS1N1hiYjhi]8M[PRDZkg DeS1KJI5ED]8D$PQXL_JZ[ILPRM\NRX r M\W:M r _ r ]:ILWJSZ\De^`_:DZZ\DeSFX5ILZ\M\NRXeS,]|g XW:DTPR5M\DVLW`N,_:W SRDWJSTp
`j¨ « ¨L OO ( 2$ "7?7?< $&=!#" + -$f 7?<< $R%$'/"D@- # A$Rp)@< @-%$ ( *$%"(?7?< $.>7-J!"):+ >*$R+,".
u/(=1 76 %$; >)$&$R G@%K"7+ GM/(>7)$ ]0, T [×Ω
N !#"7N ?<"79;"t ∈ [0, T ]
N(# > + @%K7>7+ ;@ /) @<)<GK :$ " )>7;∫
Ω
∣
∣
∣
∣
∂u
∂t(x, t)
∣
∣
∣
∣
2
dx+
∫
Ω
|∇u(x, t)|2 dx =
∫
Ω
u1(x)2dx+
∫
Ω
|∇u0(x)|2 dx
+2
∫ t
0
∫
Ω
f(x, s)∂u
∂t(x, s)dxds.
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¯Q. O )1 )#& *<
%# / @/" G!#" +# $&; "() )$;&>7;C(T )
1 / @ ?<</(>7; / -$ /(7@-%$:>7"-G !")T 6 ;<+,+ 2!#"∫
Ω
∣
∣
∣
∣
∂u
∂t(x, t)
∣
∣
∣
∣
2
dx+
∫
Ω
|∇u(x, t)|2 dx ≤C(T )
(∫
Ω
u1(x)2dx+
∫
Ω
|∇u0(x)|2 dx
+
∫ t
0
∫
Ω
f(x, s)2dxds
)
.
#: #-G<#!#" +( $;; "()% #$&&>7-;C
1 )/(@<?< )/7>7;Q/(Q;";%$L+ %$ / #()@%-$&e ?< $T 6 ;<+,+ 2!#"∫
Ω
∣
∣
∣
∣
∂u
∂t(x, t)
∣
∣
∣
∣
2
dx+
∫
Ω
|∇u(x, t)|2 dx ≤ C
(∫
Ω
u1(x)2dx+
∫
Ω
|∇u0(x)|2 dx
+
(
∫ t
0
(∫
Ω
f(x, s)2dx
)1/2
ds
)2
.
ViZ Db8M[S1NRD]|g Ia_8NQPRDeS,^`_JIaW`NRM\NRXeS,EV5WJS1DTPRc5XDeS!EV rr DZ®g M\W]8M[^`_:DZkg Db8DPHEMzEDSR_:M\cjILW5Nep `j¨ « ¨5 OOGF ( $ "7?(?<($& !") + %$D 7?<< $R%$:/"NB)@% $&#-g)@< @%-$-N9!#"+ 0;< $&#"(1 %$&*("7+
f = 0 L!#"A+ >=$R+,".
u/( 1 76 -$;*G@%K#"(+ :/(>7)$
[0, T ]×Ω G<O!#"7N ?<"79;"L -.
m ≥ 1NA>
d
dt
∫
Ω
(
∣
∣
∣
∣
∂mu
∂tm
∣
∣
∣
∣
2
+
∣
∣
∣
∣
∇∂m−1u
∂tm−1
∣
∣
∣
∣
2)
dx = 0.
M[DW¡S :PefdZ[D!kFG:XTVLPQ r Du]|g Db8MzSANQDWJEDu:p 9:p = S1DtLXTW:XPHIaZ[M[SRDnILEM\Z[D r DTW5NIa_¡ETILS] g IL_8NRPQDTSEVLWJ]:MONQM\V5WJSmIa_:bZ[M r M\NRDeS=oKIaPBDb8D r K:Z[DLf5]:D&0&DT_ r IaW:Wf5V5_$Ia_yEI5Sm]|g Ia_8NQPRDeSmV5KXTPQIaNRD_JPQS2^`_:DZ[DqdILK:ZzILEM[DWEV rr D∂2u
∂t2− div (A(x)∇u) = f.Cg DTS1N&_:WDb8DPHEMzED]8D5XW:XTPQILZ\MzSRDPFEDPRXeS1_JZONHIjN&Ia_8btXT^`_JIjNQM\V5WJS,]8DZkg XTZ[I5SANQV8]%+WJI r Mz^5_JDLp
`j¨ « ¨5 OO Ω
"7*" <4)@ @-K"7+ <H #()M/ RN L"()M/ #()@%* < . >7+
(u0, u1) ∈ H10 (Ω)N ×L2(Ω)N
N( '"(0;< $R"7U% f ∈ L2(]0, T [;L2(Ω))N #-G< !" , + $&; "()"( !#":$&#+,"-. #
u ∈ C([0, T ];H10 (Ω))N ∩ C1([0, T ];L2(Ω))N
/
ρ∂2u
∂t2− div (2µe(u) + λ tr(e(u)) Id) = f
/(>7)$Ω×]0, T [,
u = 0$ "7
∂Ω×]0, T [,u(t = 0) = u0(x)
/(>7)$Ω,
∂u∂t (t = 0) = u1(x)
/(>7)$Ω.
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% O * V <&13<#(* & & ¯ .` 2$ "7?(?<($R>7Q!#"+ >A$&#+,"-. #
u%$;M>)$&$R @-K"7+ G(N E!")(N ?<#"(O;" t ∈ [0, T ]
N7> + @%K7>7+ ;@ /) @ K :$ " )>7;
ρ
2
∫
Ω
∣
∣
∣
∣
∂u
∂t
∣
∣
∣
∣
2
dx+ µ
∫
Ω
|e(u)|2 dx+λ
2
∫
Ω
(divu)2dx =ρ
2
∫
Ω
|u1|2dx
+µ
∫
Ω
|e(u0)|2 dx+λ
2
∫
Ω
(divu0)2dx+
∫ t
0
∫
Ω
f · ∂u∂tdxds.
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ΩDNlSR_:PlZ[IuST+ r XNQPRM[Dv]8DvZ[InoVLP r Dw:M[Z[M\W:XeIaM[PRD a(u, v) ]8DZ[InoV5P r _:ZzIjNRM[VLWtcjIaPQM[IaNRM[VLW:W:DTZ\Z[DLp •
"' "U + )H,+ t+ ) $!+ '*#"U ) 0=VL_JSlDb8I r M\W:V5WJS r ILM\W`NQDWJILW5NZ[DTSK:PRM[WJEM\KJILZ\DeS=KJPRV5K:PRM[XNQXTSl^`_JIaZ[MONHIjNQM\c5DTSl]8DvZzIuSRVLZ[_8NQM\V5W]8DtZkg XT^`_JIjNQM\V5W¡]8DZ[I¸EHGJILZ\DT_:PTfW:VaNHI rr DW`NZ\DeSvKJPRV5K:PRM[XNQXTSv]8DtPQX5_:ZzIaPQMONQXLfEV r K?VLPRNRD r DTW`NILS8+ r K8NQVaNQM[^`_:DeSFKV5_:P,Z\DeSFLPHIaWJ]:DTS!cjIaZ[D_:PHS,]8D t DN&K:PQM\WJEM\K?D]8_ r Ijb8M r _ r p
.&.10 9LMH9m7:6-G(L G4B6 N E LMH&68926:@ F<G0=VL_JSmXNR_]8M\V5WJSZ[D&EV r K?VLPRNRD r DTW`N2]:D,Z[ISRVLZ[_8NRM[VLW]8D,Zkg Xe^5_IjNRM[VLW]8DFZzIlEHGIaZ[D_:P2DTWvNQD r KJSZ\V5W:Jf ELg DeSANRh¦Yah¦]8M[PQDvZ[VLPHSR^`_:D
tNRDW]c5DPHS
+∞ p0=VL_JSILZ\Z[VLWJSc5XPQMO´DP^`_:DLfdEVLW8noV5P r X r DTW`NYZ®g M\W`NQ_:MONQM\V5W~K:G-+SRMz^5_JDLf|S1MBZ\D$SRDTEV5WJ] r D r w:PQD f(x)DeSANM[WJ]8XK?DW]:IaW`N]8_¸NRD r KJS t fdIaZ[VLPHSZ[IS1V5Z\_8NQM\V5Wt]8DZkg XT^`_JIjNQM\V5Wt]8DZzIEHGJIaZ[D_:P!NQDWJ]*ILS8+ r K8NRVLNRMz^5_JD r DW`NFcLDPHS!Z[ISRVLZ[_8NRM[VLW¸S1NQIaNRM[VLW8hWJIaM[PRDd]:_qIaK:ZzILEM\DTW p 0&V5_JSEV rr DWVLWSKIaP2Db:I r M[W:DTPUZ[D,EI5SU]:D!Zkg XT^`_JIjNQM\V5Wv]8DFZzIEHGJIaZ[D_:PG:V r VL5W:D5p
B? «ªT« § rO4 ,/.01Ω2@. L 3B[ =.B[4ACBOA >28h0<3B_E73
RN I ,4.>01 u0 ∈ L2(Ω)3u8M
G.>8:2D0<.KE>C69BO. 8 Y 3
∂u∂t −∆u = 0
E7HG]0,+∞[×Ω
u(x, t) = 0G[2B
]0,+∞[×∂Ωu(x, 0) = u0(x)
EFGΩ.
®:p 9`L 8M.B=G i u(t) ;=. L 3B 3 L 3B=GABO. E7G L2(Ω)
8M.>B=G3t 39E L 3B=G +∞
limt→+∞
‖u(t)‖L2(Ω) = 0.®:p 9L5
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¯Q.a O )1 )#& *<
+ 6 ?§ ª j ªe« ?§ |WPQDKJPRDTWJ]Z[I]8X r VLWJS1NRPHIjNQM\V5W]8_akFGJXVLPQ r D,:p°¯8p 9]:ILWJS2Z[D&EILS f = 0fELg DeSANRh¦Yah¦]:M\PQD
βk = 0p?WVLw:NRM[DW`N,nILEM\Z[D r DW`N&^`_:DZzISRV rr DKJIaPRNRM[DZ[Z[Dlc5XPQMO´D
‖wl(t)− wk(t)‖2H =
l∑
j=k+1
|α0j |2e−2λjt,
Ic5DTEH = L2(Ω)
f?ED^`_:MdEVLWJ]8_JMONef8DW*K:PQDWJILW5Nk = 0DNl = +∞ f8DN&DW r I AVLPHIaW`NTf8Y
‖u(t)‖2H ≤ ‖u0‖2He−2λ1t
^`_:M NRDW]tc5DPHSQSXPQVZ\V5PQSQ^`_:DtNRDTWJ]cLDTPQSFZkg M[W8´JWJM K:_:MzSR^`_:D
λ1 > 0p
qdD&EILS2] g _:W$SRDTEVLWJ] r D r w:PRDFW:V5WW`_JZ®faM[WJ]8XTKDTWJ]:ILW5NB]:_vNQD r KJSTfaDTS1Nm_:W$SRM r K:Z[D,Db8DPHEMzED^`_:DW:V5_JS,ZzIaMzSRSRVLWS!IL_tZ[DTENRDT_:PTp `j¨ « ¨5 OMrE ( G<?<G</I+ %$#(?< $&-$ / + > OG?<($P . ( f(x) ∈ L2(Ω)%u(t, x)
+ >2$R+,". 2/(
∂u∂t −∆u = f
/7>7$]0,+∞[×Ω
u(x, t) = 0$ "7
]0,+∞[×∂Ωu(x, 0) = u0(x)
/7>7$Ω.
v(x) ∈ H1
0 (Ω)+ >2$&#+,"-. #2/
−∆v = f/(>7)$
Ωv = 0
$ "7∂Ω.
G<O!#"limt→+∞ ‖u(x, t)− v(x)‖L2(Ω) = 0 WsK?D_8NDW~nILMONK:PRXeEMzS1DTPZ[IEVLWEZ[_JS1M[VLWs]8DuZ[IPBPRV5KV`S1M\NRM[VLWs:p =Jp[yEV rr DuZ\D r V5W5NQPRDZkg Db8DPHEMzEDSR_:M\cjILW5N&]8V5W`NFZkg M\W`NRDTPRKJPRXNQIjNQM\V5WtDeSAN,ZzI$S1_JM\cjIaW`NQDu(t, x) ≈
(∫
Ω
u0u1 dx
)
e−λ1tu1(x)Z\V5PQSQ^`_:D
t→ +∞,
VQUu1(x)
DeSANtZzIsK:PQD r M[PQD noVLWENRM[VLW KJPRV5K:PRD¶oW:V5P r IaZ[MzS1XTDe$]8_qdIaKJZ[I5EM[DWp[8p2=S8+ r K8hNQVaNQM[^`_:D r DTW`NTfNRV5_8NRDeSZ\DeS$S1V5Z\_:NRM[VLWJS$]8D*Zkg Xe^`_JIjNQM\V5W¶]8D ZzIEHGIaZ[D_:PG:V r V5LTW:D]:XTEPQVLMzSQS1DTW5NDb8KV5W:DW`NRM[DZ[Z[D r DW`N=DTW*NRD r KJSlIcLDeEZ\D rTr DK:PQVa´JZUS1KJIaNRMzIaZU^5_JMdDTS1N]8V5W:W:XKIaP u1®^5_JDZ[Z\D^`_:DSRVLM\N&Z[I]8V5W:W:XTDlM[W:M\NRMzIaZ[Dep
`j¨ « ¨5 OMrEGF ( ?<G</ + %$v 7?<< $R%$M/ + > EG?< $ . ( O!" , + ( $;;"7)W%)$;&>7; ?< $P . )
C; +,+ =!")
‖u(t)− α01e
−λ1tu1‖L2(Ω) ≤ C e−λ2t ∀ t > 1,>)
α01 =
∫
Ω
u0u1 dx,Jp 95º5
λk
/(@%$ K) + >k o)>7+ "(? #?<*/#" $<>7?7+ > 0>% / . *>7" + ;%$ /( (+ %
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Ω2X.> L 3B[ =.B[4AWBOA F28:0<3BE73
RN i 3S2 V3Yg69G \S48 T > 0 I,4.01u0 ∈ L2(Ω) i f ∈ L2(]0, T [;L2(Ω)) i 3 u ∈ C([0, T ];L2(Ω)) ∩ L2(]0, T [;H1
0 (Ω))8 R 20<3eG.8:2D0<.XE73 I FI ,0 f ≥ 069BO3G3N6B[V.2EFG
]0, T [×Ω3u0 ≥ 0
64BO3G36H>B[ .>2EF>HG
Ω i >8M.B=G u ≥ 069BO3G3%6B[V.2E7G
]0, T [×Ω I+ 6 ?§ ª j ªe« ?§ V5MON
u− = min(u, 0)^`_:MILK:KJILP1NQM\DTW5N$w:M[DW£Y
L2(]0, T [;H10 (Ω))
DWcLDPRNR_*]8_ q D rr D'.:p ¯:p ¯ =vDN=^5_JM cLXTPRM\´JDLfK?VL_:P 0 < t < Tf
∫
Ω
∇u(t) · ∇u−(t)dx =
∫
Ω
|∇u−(t)|2dx. ®:p =5¹5 WsPHIaMzS1V5W:W:D r DTW5NSRM r M[ZzIaM[PRDyY EDZ[_:M/^`_:M!I KDTP r MzS]8Du]:X r VLW`NRPQDPt:p =5¹` r VLW`NRPQDy^`_:DLfUS1M∂u∂t ∈ L2(]0, T [;L2(Ω))
fJIaZ[VLPHS∫
Ω
∂u
∂t(t)u−(t)dx =
1
2
d
dt
(∫
Ω
|u−(t)|2dx)
.®:p =J
0&VL_SI5] r DNRNRPQVLWJS^`_:DZkg M[]8DTW`NRM\NRX Jp =euPRDeSANQDcPQILM\D rr DsS1M ∂u∂t
W g IaK:KJILP1NQM\DTW`NKJI5StYL2(]0, T [;L2(Ω))
p PILPlEVLWJSRXT^`_:DTW`NTf?DW K:PQDWIaW`Nv = u−
]:IaWJSZzI$noVLP r _JZ[IaNRM[VLW cjIaPQM[IaNRM[VLW8hW:DZ[Z\Dy®:p 5!]8DZkg Xe^`_JIjNQM\V5W]8DZzI$EHGJIaZ[D_JP!V5WtV5w8NRM[DW`N1
2
d
dt
(∫
Ω
|u−|2dx)
+
∫
Ω
|∇u−|2dx =
∫
Ω
fu− dx,
ED^`_:Md]8V5W:W:DKJIaP,M[W`NRXTLPHIjNRM[VLWtDWNRD r KJS1
2
∫
Ω
|u−(t)|2dx+
∫ t
0
∫
Ω
|∇u−|2dx ds =
∫ t
0
∫
Ω
fu− dx ds+1
2
∫
Ω
|u−(0)|2dx.
C/V rr D u−(0) = (u0)− = 0
V5WtDTW ]8XT]8_JMON1
2
∫
Ω
|u−(t)|2dx+
∫ t
0
∫
Ω
|∇u−|2dx ds ≤ 0,
Eag DTS1N1hiYjhi]8M\PQD^5_JDu− = 0
K:PQDTSQ^`_:DKJIaPRNRV5_8NF]JIaWJS]0, T [×Ω
pC/V rr Dv]:IaWJS=Z\DvEILS=DZ[Z\M[K8NQM[^`_:D5fJZ[DK:PQM\WEM[KD]:_ r Ijb8M r _ r noV5_:PQW:MdKJILP=ZzI PmPQVLK?V5SRMONQM\V5W:p =Jp°¯&DeSANmEV5W8noVLP r D!YZkg M[W`NR_:M\NRM[VLWK:G-+8S1Mz^`_:DLpjxlIaWJSZ[D,EI5]8PRDF]8_ r V]:Z[DF]:D/NQG:DP r M[^`_:D,]8XeEPQMONIa_tC/GJILK:MONQPRDLf5SRMJZzINQD r KXTPQIaNR_:PQD,M[W:MONQM[ILZ\D u0(x)
DTS1NBDWNQVL_8N!KV5M\W`N/SR_:K?XPQM\DT_:PQD=YZzIcjIaZ[D_:P0YZ[I5^5_JDZ[Z\DVLW r ILM\W`NQM\DTW5NZzI NQD r KXTPQIaNR_:PQDtSR_:PZzIKJILPRV5M ∂Ω
DNSRMFZ\DtNRDTP r D*S1V5_:PQEDuDeSANKV`S1M\NRM\n®EV5PRPQDTSRKV5WJ]:ILW5NY_:W¸D R DN]8DyEHGJIL_%R|IaLDf|IaZ[VLPHSlM\ZBDTS1NEZ[ILM\P^`_:DZzItNRD r K?XPHIjNQ_:PRDDTS1NUK?V5SRMONQM\c5DBDTWNRV5_8NKV5M\W`NDN2Y,NRV5_8NM\WJS1NQILW`NTp 0&V5_JSdWdg Ic5VLWJS nIaM\N^`_:D!cLXTPRM\´JDPU^`_:D/Z[D r V8]8TZ\Dr IaNRG:X r IjNRMz^`_:DPRDTK:PQV]:_:MON,w:M[DW EDN1NQDK:PQVLK:PQM\XNRXM[W5NQ_:M\NRM[cLDLp
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¯ L¹ O )1 )#& *<
ViZ|DeSANFwV5W]:DPQXTILZ\MzS1DTP!^`_:DEDeS!PRXeS1_:Z\NQIaNQS/KDT_:cLDTW`N!XT5IaZ[D r DTW5N,STg XTW:VLWJEDPFS1V5_JS/K:Z\_S1M[D_:PHSnoV5P r DeSvXe^`_:M\cjILZ\DTW5NQDTSTp2W¶KDT_8NTf2KIaPDb8D r K:Z[DLf2EV r KJIaPQDP]8DT_8b¡S1V5Z\_:NRM[VLWJS]8D~Jp\¯L SRMu0 ≤ u0
]:IaWSΩDNf ≤ f
]:IaWS]0, T [×Ω
f|DNS1MuDNu]8XeS1M[LW:DTW`NPRDeS1K?DTENRM[cLD r DTW`NZ\DeSSRVLZ[_8NQM\V5WJSB]8DJp\¯L2EVLPQPRDeS1K?VLWJ]JIaW`NBIa_8by]8V5W:W:XTDTS
(u0, f)DN
(u0, f)PQDTSRK?DTENQM\c5D r DW`Nef5IaZ[VLPHSV5WI
u ≤ u ]:IaWS ]0, T [×ΩpqdDTS&]8DT_8buDbDTPQEM[EDTSFSR_:M\cjILW5NHS,M[Z\Z[_JS1NRPQDW`N&^5_JDZz^5_JDTS,IaKJK:Z\MzEIaNRM[VLWJSFM[W`NRXTPRDeSRSQIaW`NRDeS/]:_KJPRM[W8hEM\K?D]8_ r Ijb8M r _ r p
`j¨ « ¨5 OMrE Ω
"7 #" )< < )@0G@%K#"(+ /(RN ( )<;
u1+ > ?<G o #) . ? #?<=/#"$<>7?(+ >) < /7>7$
Ω>W / . 2/ &7+ %N
λ1+ >5)>7+ "( ?<G? =>)$&$R2 @% (
>7?7?<<+,+ :!") + # ?< "-o&) $ u1 > 0
/(>7)$Ω
1 ) + fB)@% =/( G) " >7:3 (6%E# >/ % G>0!") + # >0>7")$&$∂u1/∂n > 0
$%"(∂Ω f = 0
Nu0 ∈ L2(Ω)
%u+ "( !#"
$R+,". 1$%"(?7?< $R@% G@%K"7+ G 6 /(1 %76 ε > 0 9 G<O!#"+ ?< "-L#" )< "7) )$;&>7;A?< $P . )
K;<+,+ =!")
−Ku1(x) ≤ u(x, ε) ≤ Ku1(x) ∀x ∈ Ω,Jp =¯L
%L A/ @/" G:!#" +7( $;;"7# #$&&>7-;A?< $ . C
;<+,+ =!#"
maxx∈Ω|u(x, t)| ≤ Ce−λ1t ∀ t > ε.
Jp = 95 `j¨ « ¨5 OMrEMr
Ω"( #" )< <4)@'G@%K"7+ <7/
RN u0 ∈ L∞(Ω)Nf ∈ L∞(R+×
Ω)N(%
u ∈ C([0, T ];L2(Ω))∩L2(]0, T [;H10 (Ω))
+ "7< !#"0$&#+,"-. #:/( 1 %(6% #-G<9!")
‖u‖L∞(R+×Ω) ≤ ‖u0‖L∞(Ω) +D2
2N‖f‖L∞(R+×Ω),
®:p = =`
D = supx,y∈Ω |x − y| %$; + I/ > G/(Ω ( ?<"74 > / > <G/ )$P /(@<G< + ^%>)$
?7+,")$>) + f ≡ 0
NE?(" $-N /7>7$ + C->)$0K(@<)@< >7+ NO". + $& L+ > . ψ ∈ H1
0 (Ω);<+,+ !")
−∆ψ = 1/7>7$
Ω .&. 7:9H&N NU6:@A9B4 &@A6-G(E E Gµ@14 l4=@TG
0=VL_JS2Ic5VLWJSU]8X 1YlXTcLV8^`_:X!YZzI &D r IaPH^`_:D5p ¯:p ºEDN1NRD,KJPRV5K:PRM[XNQX/SR_:PQK:PRDTWJIaW`NQD!]8DFZkg Xe^`_JIjhNQM\V5W]8D$ZzI EHGJIaZ[D_JP dZzI EHGJIaZ[D_:PvS1D$K:PQVLKIaLD$Y _JW:D$cMONQDTSQS1DyM[W8´JWJM\D2C/D$PRXeS1_JZONHIjN]8XeEVL_JZ\D] g _:W « §U¨ « B¢U£: « ¸ ¤¦ ª ^`_:DW:V5_JS&XTW:VLWVLWS r IaM[W5NQDWJILW`NSRILWJS&]8X r VLWJS1NRPHIjhNQM\V5W p(0=VL_JSZ\Dyc5XPQMO´DPQVLWJSK:Z[_JSnILEM[Z[D r DW`NZ[VLPHSR^`_:DZ[Du]8V r ILM\WJD ΩDeSANvZ®g DTSRKJILED
RN NQVL_8NDTW`NRM[DPoc5VLM[P,Z[I$V5_JS1h¦SRDTENRM[VLWtJp =p .5p jB? «ªT« ?§ rOUc,/.01
Ω2m. L 3B[ =.B[4AKBOA >28h0<3BnE73@;8)G=G3 C2 E73
RN I ,4.01 2V3Yg69Go\p98T > 0 I ,4.>01 u0 ∈ L2(Ω)
3u8PG.8:2D0<. 20<3aEF>HG
C([0, T ];L2(Ω)) ∩L2(]0, T [;H1
0(Ω))EC69B. 8Y 3
∂u∂t −∆u = 0
EF>HG]0, T [×Ω
u(x, t) = 0G[2B
]0, T [×∂Ωu(x, 0) = u0(x)
EF>HGΩ.
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re.
% O * V <&13<#(* & & ¯ J
J XG[6F6.)G3ZEF3g698: G_3 u0(x) ≥ 069BO3G3g6B[V.2%EF>HG
Ω3o3
u0/R 3[G[Q6HG0<EF3D0
3YZ3N2818M3 I 8M.>B=G i 6.2B#V.2N 3Y6G ε > 0 i .
u(x, ε) > 0 ∀x ∈ Ω.Jp =-.L
Cg DTS1NFZ®g M\WJX5ILZ\M\NRXl ª « ¨ ª $]8D:p = .52^`_:M DTS1N!PQD r ILPQ^`_JILw:Z\DvoVLWtIcILMONF]8X 1Yv_:W:DM\WJX5ILZ\M\NRXZ[ILPR5DBKJILPZ[DFK:PRM[WJEM\K?D!]:_ r Ijb8M r _ r ]:D!ZzIPmPRV5KV`S1M\NRM[VLWv:p =Jp°¯Lp:mWD R DNefaS1M u0I_:WS1_JK:KV5P1NEV r KJI5ENv]JIaWJS Ω
DNS1M/VLW¡SRDuK:ZzILEDyDW¡_:WK?VLM[W5Nx ∈ Ω
DTW¡]8DGJVLPHS]8_¡SR_:K:K?VLPRN]:Du0fVLW¶NRPQVL_:c5Dt^`_:D
u(x, ε) > 0wJM\DTW¬^`_ g M[W:MONQM[ILZ\D r DW`N u0(x) = 0
p2&_8NQPRD r DW`Ny]8MONefm]:IaWSZ\DEIL]:PRD*]8D ZzI r V8]8XTZ\MzSRIaNRM[VLW¬]8D Zkg XTcLVLZ[_8NQM\V5W ]8D*ZzINRD r K?XPHIjNQ_:PRD5f rr D S1M=Z\D*K?VLM[W5N x DeSANM\W:M\NRMzIaZ[D r DTW5N/noPRV5M[] u0(x) = 02DN/NQPReSBZ[VLM[Wu]8DKJIaPRNRM[DEHGJIL_J]8D&M[W:M\NRMzIaZ[DZ\DlS1_JK:KV5P1NF]8D
u0fM\Z!]8DcM[DW`NM[WJSANHIaW`NQILW:X r DTW5NEHGIa_J]~K:_JM[SQ^`_:DyK?VL_:PNQVL_8NNRD r KJS t = ε
rr D$NQPReSK?DNRM\NHfVLWIu(x, ε) > 0
pW2=M\WJSRM2¨ 4 : TUD «oª eT « §(;§ « uK:_JM[SQ^`_:DvSRVLW DR?DNDTS1NM rr Xe]8MzIjN rr DYLPHIaW]8Dt]8MzS1NQIaWEDViZ,Sg Ia5MON$EZzIaM[PQD r DW`N] g _:W¬]8XnIa_8N$]8_ r V8]8TZ\Dr IaNRG:X r IjNRMz^`_:DK:_:MzSQ^5_JDZ®g VLW SQIaM\N=^`_:DPRM[DW WJDKDT_8NSRDK:PQVLKJILLDPFK:Z[_JS,cM\NRD^`_:DZzIcMONQDTSQS1D]8DZ[IvZ\_ r M[PQDoDWtnIaM\N&Eag DTS1N!ZzIZ\V5M|]8D :VL_JPRM[DPA5p 9`B^`_:M|DeSAN,DW]8XnIL_8NHpCg DeSAN,_:W r V8]8Z[DLf^5_IaZ[MONHIjNRM[cLD r DW`NBDN!^5_IaW`NRM\NQIaNRM[cLD r DTW5N!EVLPQPQDTENBYw:M\DTWy]8DTS/X`IaPH]:STfaEV rr D=Zkg VLW`N/]8X r V5W`NRPQX]|g ILM\Z[Z[D_:PHS K:Z[_JSRM\DT_:PHS PQXTSR_:ZONHIjNHS K:PQXTEXT]8DTW5NHSfTEVLW:noVLP r DeSdY,Zkg M\W`NR_JMONQM\V5WK:G-+8S1Mz^`_:DLf r ILM[SED/W g DeSAN^5_dg _:W r V8]8Z[DM[]:XTIaZ[MzS1X]8DZzIPRXeIaZ[MONQXLp 8 ? U rOr·qdI rr DtKJPRV5K:PRM[XNQXu]8D$²AKJPRV5KJIa`IjNRM[VLWsYc`M\NRDeSRSRDuM[W8´JWJM\D³uKDT_8NyIa_JSQS1M NRPQD,VLwS1DTPRc5XDFKV5_:PmZ[DTS2Xe^`_JIjNQM\V5WJSm]8DNRV5DTS,®:p°¯Lp`xILWJSmED&EIL]:PRD5faVLWDTS1N/]|g IaM[Z[Z\DT_:PQSK?D_:N1h NRPQD r VLM[WJS,S1_JPRK:PQMzSFKJIaP,EDKIaPHIL]8Vb8DSRM|VLW IPQXTIaZ[MzS1X^`_:DZ®g G-+`K?VaNQG:TSRD]|g M\WJEV r K:PQDTSQS1M[w:M[Z\M\NRX]|g _JW _:M[]:D$PRDTc`M[DW`NY ]8M[PRD$^`_:DZ[IcM\NRDTSQSRD]:_~SRVLW + DeSANM[W8´JW:M[DLp2=_8NRPQD r DTW5N]:MONef DW~_8NQMOhZ\MzSRILW`N!Zkg ILK:K:PQVb8M r IjNQM\V5Wu]|g M\WEV r K:PQDTSQSRM\w:M[Z[MONQXLf8VLWM[W5NQPRV8]8_JMON,M r KJZ\MzEM\NRD r DTW`N,]JIaWJSFZ[D r V8]8TZ\DZ[IK?V5SQS1M[w:M[Z\M\NRX]|g _:W:DKJPRV5KJIa`IjNRM[VLWt]8DZ®g M[W8noVLP r IjNRM[VLW*YcMONQDTSQS1DM[W8´JW:M[DLp •
.&. BDC B<F,N7J@i6%C G|6G'G|6¶7OC B<F,N7J@TE N24/6xlIaWSZ\DFEI5SdDTZ\Z[M\K:NRMz^5_JD!WJVL_JSIcLV5WJSc`_^`_:D/ZzIPRXTL_:ZzIaPQMONQX/]8D!ZzISRVLZ[_8NRM[VLWvDTS1N]:M\PQDTENRD r DTW5NZ\M[XDYuEDTZ\Z[Dv]:DTSl]8VLW:WJXDTSTpxlIaWS=Z[DETILS=KJIaPHIaw?VLZ[M[^`_:D5fJZzIuSRMONQ_JIjNQM\V5W DTS1N]:M R XPQDW`NRDEILPTf?S1MUZ\DNRDP r DSRVL_:PHED=DTS1N/W_:Z f = 0
f5M[Z?Db8M[S1NRD_:Wt ª 6 DU: « TJ§ ª ]8DZ[IvEV5WJ]8M\NRM[VLW$M[W:MONQM[ILZ\D ]8D r IaWJM\TPRDBSR_:PQK:PQDWJILW5NQDLf rTr D/SRM5ZzI&]8V5W:W:XD/M\WJMONQM[ILZ\D u0DTS1NdNRPQTSdKDT_PQXL_JZ\M[PQDLfTZ[I&SRVLZ[_8NQM\V5W]8DcM[DW`N,M\WJS1NQILW`NQIaWJX r DW`N,NRPQTSFPQX5_:Z[M\TPRD5p
B? «ªT« § rOm,/.01Ω2@. L 3B[ .>B[9A_BOA F28:0<3BZEF3a;8MGG3 C∞ E73
RN i 3oG.>01%2 3Yg69G\S48T > 0 I ,4.>01 u0 ∈ L2(Ω) i 3 u 8 R 20<3 G.8:2D0<. EF>HG
C([0, T ];L2(Ω)) ∩L2(]0, T [;H1
0(Ω))EF3
∂u∂t −∆u = 0
EF>HG]0, T [×Ω
u(x, t) = 0G[2B
]0, T [×∂Ωu(x, 0) = u0(x)
E7HGΩ.
®:p = ` 8M.B=G i 6.2B#V.2 ε > 0 i u 3[G[%EF3C;8)G=G3
C∞ 3x3tEF>HG
Ω× [ε, T ] I
Ecol
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re.
¯ 5¯ O )1 )#& *<
¢ 6 *¢*¢ 6 §U ª j ªT« §LOPmZ[_8N aN=^`_ g _:W:D]8X r VLWSANQPQIaNRM[VLWyPQM\5VL_:PQD_S1D r IaMzSFILSQS1D-S&NRDeEHG8hWJM[^`_:D¸DN*KDT_·KJILPRZzIaW`NQDLf!cLVLM[PZ®g :2b8DPHEMzED¸:p =Jp/.L$WJVL_JSK:PQVLK?V5SRVLWJSu_:WETIaZzE_:ZnoVLP r DZ^`_:Mr V5W5NQPRDvw:M[DWZkg M[]8XTDDeSRSRDW`NRM[DZ[Z[Dv]8DTPRPQM\TPRDvEDPQXTSR_:Z\NQIaNl]8DPRXTL_:ZzIaPQM\NRXuZ[I]8X r V5WJSANQPQIaNRM[VLW DeSANKJZ\_JS,nILEM[Z[D]:IaWJSFZ[DEI5SΩ = RN f:cLV5M\P,Zkg :mbDTPQEM[ED:p =Jp º5p%PV5_:P k ≥ 1
VLW*W:VLNRDv = ∂ku
∂tk
DNV5W ]8XPQM[cLDuonoVLP r DTZ\Z[D r DTW5N k noV5M[S=Z®g XT^`_JIaNRM[VLW ]8DvZ[IyEHGIaZ[D_:P®:p = 5,KJIaPPQILK:KV5P1N=IL_ NRD r KJSK?VL_JP,VLw8NQDW:M[P
∂v∂t −∆v = 0
]:ILWJS]0, T [×Ω
v(x, t) = 0SR_:P
]0, T [×∂Ω
v(x, 0) = ∂ku∂tk (0, x)
]JIaWJSΩ,
®:p =a^`_:M8DeSAN2DTWJEV5PRD/_:W:DFXT^`_JIaNRM[VLWv]:D!ZzIEHGJIaZ[D_JPTpjM ∂ku
∂tk (0, x)ILK:KJIaPRNRM[DW`N2Y
L2(Ω)faV5WvIaKJK:Z\Mz^`_:DZ[D kFG:XTVLPQ r Dy:p°¯8p ]|g Db8MzSANQDWJEDuDN]|g _JW:M[EMONQXtY®:p =L^5_JM/W:V5_JS]8M\N^`_:D v ILK:KJIaPRNRM[DW`NY
L2(]0, T [;H10(Ω)) ∩ C([0, T ];L2(Ω))
p :mWKJILP1NQM[E_:Z[M\DTPTfuDeSANPRXTL_:Z[M[DP=DTW NRD r KJSp|xvg IL_8NRPQDKIaPRNTfKJIaP$XT5ILZ\M\NRX5f
v = (∆)kuILK:KJILP1NQM\DTW5N$Ia_ rr D DTSRKJILEDLpP2IaPPQXL_JZ[ILPRM\NRXtDZ[Z\M[K8NRMz^`_:DcLV5M\P&Z[D5kFG:XTVLPQ r D'.8p°¯8p°¯ `!VLW*DW ]8Xe]8_:M\Nl^`_:D u DTS1N=PRXTL_:Z[M[DP,DW DeS1KILED5pJq DKV5M\W`N=Z[DK:Z[_JS]:XZ[M[ETIjNK?VL_:P]:VLW:W:DTP_:WsSRDWJSY ED$PQILM[SRVLWJW:D r DW`NnoVLP r DZmDeSAN^`_:D$ZzI ]8V5W:W:XD$M[W:M\NRMzIaZ[Du]8D®:p =LlW g DeSANKILSI5SRSRD%SPQXL_JZ\M[PQDLpCg DTS1NK?VL_:PEDNRNRD$PHIaMzS1V5W¸^`_:D$ZzI*PRXTL_:ZzIaPQM\NRX$]8D
uWdg DeSANcjILZ[ILw:Z\D^`_:DK?VL_:P&Z[DTS!NQD r KJS t > ε > 0
p
:mW¬K:PQXTSRDWEDt]:DuNQDP r DeSS1V5_:PQEDTSTfZ[D rr DtPHIaMzSRVLW:W:D r DW`N^`_:D EDTZ\_JMF]:DZzI¸]8X r VLWSAhNQPQIaNRM[VLW ]8D ZzI PmPQVLK?V5SRMONQM\V5W :p =Jp/.K?DP r DNt]:D PRDNRPQVL_:c5DP$_:W PQXTSR_:Z\NQIaNu]8D PQXL_JZ[ILPRM\NRX*K:Z[_JSEZ[I5SRSRMz^5_JDv^`_:M2W g DTS1NKJILSSQIaWJSlPHIaK:K?VLPRNlIc5DTEvZ®g X`IaZ[MONQX]|g XWJDPQLM[D®:p[Tº5oDTWDR?DNTf Zkg DeS1KILEDIL_J^`_:DZucjI$ILK:KJIaPRNRDTW:M[P!DeSAN&EDZ[_:Md^`_:Md]8V5W:W:D_:W SRDWJS,Y*Jp\eº5Rp
jB? «ªT« ?§ rO c,/.01Ω2X. L 3B[ =.B[9ABOA >28h0<3BEF3
RN i 3S2 V3Y6G \S48 T > 0 I .2B 2 V3B[YZ3^G.2BO;=3
f ∈ L2(]0, T [;L2(Ω))3C293KEF.>4A3X0101D0<83KBOA >28:0BO3
u0 ∈H1
0 (Ω) i .> ;=.G[0<EBO3P8MXG.8:2D0<. 20<3u ∈ L2(]0, T [;H1
0 (Ω)) ∩ C([0, T ];L2(Ω))E73
8OR AD0<.E73K8M ;"H>8M32B I FI 8.B=G i ;=3?V3XG.>8:2D0<.c3[G[d698: G^BA F28:0BO3 G3HGn.∂u∂t ∈ L2(]0, T [;L2(Ω))
3u ∈ L2(]0, T [;H2(Ω)) ∩ C([0, T ];H1
0 (Ω)) I IK IPOoOM $AE?("% =2@-("$9 !#$&%)(* ("$A)(32+ :G)@ %)f(_%B= %)("$2@ )(v+ :#9'+- %)@ 3$
u(
+ &2:G%)@-$(>+ : d0F:G+-(" / 8 6I[ 7 6 ("9 %:G29'9'@F)(32+-@-4*)()&(>+' $ ;<(*% E?3);$&2(,+-("91 $2$"("9 u0(*%
f+-(19'3@-("$&%m:G29'9'@ / ;<@ 5 6 6 -m("E?("$F:3$&% 9'@+' $v;<(" %O(m+ : 9'3+ %)@-$ u
9'3@ %J)(2+-@-4*)(, 4"9 +' @-$9 %d:3$&%@ $@ %)@R:G+ @ +WT:G% &(+ (*913$2$"("9 u0
(_%f;<*)@ 8F("$&% (*9."3$F@ %)@-$9m (1"!#E2:G%)@-=2@-+-@ %)6 @-$9'@ 2:3$9 + :~)3E?9'@ %)@-$ 86 6 v@-+?("9 %1("! :G$F;/+ : "3$F @-%)@-3$@-$@-%)@ :G+ (
u0 (;<_)@I82(*1+ :>"$2@ %)@-$D:3 C#+-@ !#@ %)("9
(>B@ )@I_0+ (_% / "( &2@ $F *%d:G@-%EF:G9 $"("9'9):G@-)(qE? +' (*C @-9 %)("$*( $2(9'3+ %)@-$N2:G$29>+ (v.102*3)4"!#(8 6 76 5 6 6 G ("9v:3 %')("9?*$F @ %)@ 3$29v(D"3!#EF: %)@ =@-+ @ %) 9 = %)@ (*$2$2(*$%q("$)("! :GS&F:G$&% (#+-("9 *)@-;<"("99'"(*9'9'@ ;("9(
uEF: d:3EE?3'% :3A%)("!#E9
t9'3$&%:39'9'@9'+- %)@ 3$29 &F:G%)@-3$29B (v+ :Dd0F:3+-(" :;<(
"$2@ %)@-$29:3CB+ @-!#@ %)("9 (.B@ )@Id02+-(*%6G :G(*C ("!#E2+-( + :"$2@ %)@-$q@ $@ %)@R:G+-(1E? + : *)@-;<"(JE)("!#@-4*)(("9 % ∂u
∂t(0) = f(0) + ∆u0
6&2( ∂u∂t
9'3@ %v)(32+-@-(* @-+U:3 %f $@&2(D*(*%'%)( 3$2$2*(#@ $@ %)@R:G+-(;<*)@ 8F(+R:?"$2@ %)@-$ :GC +-@-!#@ %)("9,() @-)@Id02+-(*%
f(0) + ∆u0 = 09'
∂Ω "(R&@?("9 %,2$("3$F@ %)@-$()"3!#EF: %)@ =@-+ @ %)("$&%')(u0
(_%f6 •
IW I OoO I ! #"$&% ')(*+-,."/ / 102 342 5-2(- !67% '189:+<;4 .'-=>, !?@- A
m ≥ 0=B%DC :":,'67
W 2m(Ω) = v ∈ H2m(Ω), v = ∆v = · · ·∆m−1v = 0": !
∂Ω, / 8 6 <8 6
Ecol
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% O * V <&13<#(* & & ¯ Q9 &% C - !. ;!&% ' #< ‖v‖2W2m(Ω) =
∫
Ω|(∆)mv|2dx
=; -:' <C % % :" . '#9% $ % ' #! &;!
H2m(Ω)2 ,;4' % ' ;!-#" :'/ - ;4 :! 0-2 -2 :'!
% ' " . $ (wk);!:""@- :" ,'./ % % :" :" ;! '< 6 ;!'<#"
C`([ε, T ], W 2m(Ω))2
.&. F<NU6:@A9B4 ;G FTN > NFTGd<&7¬;N24Es689B<,6 F GE HN> G %)(*)!#@-$(*$"(*%'%)(A9'(_%)@ 3$ $29#@-$F@4&23$29 =)@-4";("!#("$&%D"3!#!#("$&%)"9'3F )(A+' &F: %)@-$
(+ :$_02:3+-("BE?9'"(2:3$9 %) %+' ("9'EF:"(RN 6 :GE2E?("+-$9)&2(v+' :GE2E )J_0(9'E?(*%'d:G+-(f9'2@-;@-(f2:G$29
*(D_02:3E2@ %')(f(*9 %v+-@-!#@ %)"( :3 :39q(*9q32;<(_'%)9=?3)$2*9 / *(*%'%)(#+-@ !#@ %d: %)@ 3$L("9 %q("$ T:3@ %:G'%)@ 8B*@ (*+ +-( (*%:G=29'3+ !#("$&%>EF:39$2*("9'9):3@ )(qE?*%d:G=2+-@ B+' (*C @ 9 %)(*$"(q(*%+ $2@I"@ %) (/+R:f9'3+-%)@-$N% $2(/&F: %)@ 3$E2:Gd:G=?+-@4( 6 6-m3$29'@ _)$29+' 2:G%)@-$N (+R:Cd0F:G+-("023!#9(4*$2(q2:G$29%) % +' ("9'EF:"(
RN ,!$2@-(
2$2(B$$2"( @ $@ %)@R:G+-(
∂u∂t−∆u = 0
:3$9]0, +∞[×RN
u(x, 0) = u0(x):3$29
RN ./ 86 6
G (q)*9'2+ %d:G%"+ :39'9'@4&2(9'@-;:G$%!#$&%')( &2(/+ :9'3+-%)@-$AE )=2+-4"!#( / 86 6 (*9 %3$2$"(v(_CE+-@4*@I %)(*!#("$&%1"!#!#(+ (/E )2@ %("$;<3+ %)@-$A (+R:# $$2"(@-$2@ %)@ :3+-(u0
:;<($2(B:39'9'@-("$2$(/$&%+' : '%: a% 2 E?(R*) %"3!#!#( √
t6
YBKI O O y1\ ^)[ `<` X ^*]0N*[2] u0 ∈ L2(RN ) x h-X jd^ h ]/`Fj X<i"h_w ] l x p bN[\?]^ Xh [ ZTYaX \[\ Y N*[2] u ∈ C(R+, L2(RN )) ∩ C1(R+
∗ , L2(RN )), e X \\Pk)]>` b3ju(x, t) =
1
(4πt)N/2
∫
RN
u0(y)e−|x−y|2
4t dy. / 86 \ 6@K= JO
t ≥ 0,$291@-$&%')2@-9'3$291+ :%'d:3$2933)!#"((XA23)@-(*, (
u(t) / ;@ 6 (*9 %: ;= @ )(/(B+ :$_%)@ 3$x 7→ u(x, t) 82$2@-(EF:
u(k, t) =1
(2π)N/2
∫
RN
u(x, t)eik·xdx,
E?3k ∈ RN .
7 @u ∈ C(R+, L2(RN )) ∩ C1(R+
∗ ; L2(RN ));<*)@ 8F( / 8 6 6 $E?(" %B:GE2E+ @4&2(_ + :
%'d:G$2933)! :G%)@-3$A(>A23)@-(*:3 CD(" C &F:G%)@-3$29 / 8 6 6 E?3,3=%)("$@-
u ∈ C(R+, L2(RN )) ∩ C1(R+∗ , L2(RN )) ,
∂u∂t
+ |k|2u = 0E?
k ∈ RN , t > 0 ,u(k, 0) = u0(k)
E? k ∈ RN ,
/ 86 Y[ 6
&u0(k) = 1
(2π)N/2
∫
RN u0(x)eik·xdx(*9 %B+ :f%'d:3$2933)!#"(q (?A)@-(*B(
u06G (/9329 %)4*!#( / 86 Y[ 69'()"9' %>(:5"3$ "+-"!#("$&%d:G@-)( E22@-9"&F 3$ :$2(&F:G%)@-3$ @ P_)("$&%)@ (*+ +-(E?3 d0F:( ;3:3+-(" >(
k6B $ = %)@-("$&%
u(k, t) = u0(k)e−|k|2t E? (k, t) ∈ R
N ×R+,(_%m@-+?("9 %OT:"@-+ (B% ("$ "2@ )( / 8 6 \ 6 E2:GJ%'d:3$2933)! : %)@ 3$(XA2 )@ (_m@-$;<(*)9'( / E2@ 9"&( "(*%'%)(,%'d:3$293 3)! : %)@ 3$0"3$ ;(*'%)@ %,2$ E )@-%> (R"$;<3+-%)@-$D("$ E) 2@ %,9'@-!#E2+-( 6 6
#IK I O O GO %)@-+-@-9):G%)@-$ (f+ :D%'d:3$93G)! :G%)@-$ (DA23)@-(*q: E?(_)!#@ 9B(u@ :($2:3+-@-9'(* +' &F: %)@-$N(/+ :C_02:3+-(" / 8 6 6 (_%B (q9'(/d:3!#(*$2(*X;#+ :#)*9'+-%)@-$N $2(9'@-!#E2+-(/&F: %)@-$@ P )(*$%)@-("+-+-(3d @-$F:3@ )( / 86 J[ 6 6Z-m(*%'%)(B!#*%)0( (*9 % $ %')4"99'("!v=2+ :3=+ ( :3$9+ ("9'E )@ % ;v+' :3EE)Jd02(
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¯ = O )1 )#& *<
9'E?(*%'d:3+-(/E)"(*!#!#("$&%1%)@-+-@-9'"(/(*%&2@?)("E?39'(v:G29'9'@ 9'>2$ :G"(32!#("$&%>(B@ :(3$F:G+-@ 9): %)@-$ 6~$% :3 %')("91%)(*)!#("9 |k|2 (_%
eik·x 9 @-$%)(_)E)4*%)("$&%1"3!#!#( ("919'G'%)("9>( ;3:3+-(" )9(*%$_%)@ 3$29,E)3E)("9 Go:GE2+ :"@-("$ 2:G$29
RN 6 >("! :GS&23$29(vE+-29 +-(f!# ( (A)@-(* |k| ("9 %R(Gd:3$F E2+-29 +R: *)@-9'9):3$"(/(*C E?$("$&%)@ (*+ +-(B(*$D%)("!#E9(
u(k, t)("9 %,d:GE2@ (+"(*%>:G!#3'%)@-9'9'("!#("$&%E+-29d:3E@R (BE?
+ (*91E?(*%)@ %)("91+-$((")9m% $2( / k (Gd:3$2 6 (*9 %+-@ ;+ (P(_%1)(+ :G)@-9):3$&%>(>+' &F:G%)@-3$ (+ :>_02:3+-("/ ;<@ >+' C (*""@I"(/86 6 $*@I ("9'9'9 6 6 •
IK IPOoOTSU G ()"9'2+ %d:G% / 86 \ 6 E?(" %9 @-$%)(_)E)*%)(*,(*$D%)(*)!#( (X$*%)@-$A ( )("("$ 6~$E?9):3$&%
G(x, t) = 1
(2πt)N/2 e−|x|24t 3$ *)@ % / 8 6 \ 6 9'29,+ :?3)!#(
u(t) = G(t) ∗ u0 9 ("9 %: <; @-)(
u(x, t) =
∫
RN
G(x− y, t)u0(y) dy.
Go:?$_%)@ 3$()("(*$ ("9 %>+ :f9'+-%)@-$A*+ *!#("$&%d:3@ )(/(+' &F: %)@ 3$A(+ :Cd0F:G+ (*2:G$29RN × R+
∗"(>&2@P9'@I($2@ 82(>(B+' $ E?(" %>;<*)@ 8F(*&(
∂G∂t−∆G = 0
2:G$29]0, +∞[×RN
G(x, 0) = δ0(x)2:G$29
RN .
:39'("$29, ("9,@-9 %')@-=2 %)@ 3$29 9& δ0(*9 %,+R:q! :39'9'( ()B@ d:);v+' 3)@I(3@ $(6-m( E?@-$&%,(; ( E?(" %,:39'9'@
*%')(>";<(*+ 3E2E?>2:G$29J2$# ! :G@ $(,=?3)$2,(*%U"3$F@ %;B$2(!#*%)0(>(1)"9'3+-%)@-$(*9m&F: %)@-$29EF:Gd:G=?+-@4&2("9>@ P*)(*$%)(/ (B+' :3E2E )J_0( 9'E?(*%'d:3+-( B9'2@-;@ ( @4*@a6 •
GO (*C (*""@I"(/9'@ ;3:3$&%,E?(*)!#(_%,(B)"9'3F)(+ &2:G%)@-$A (B+ :C_02:3+-(" ,$2$ 02!#(4"$(6 IW I OoO (- !
u0 ∈ L2(RN )
t > 0=-1#$
S(t)u0% '-6/ ; !7 ,'. % "$76:-;
;! 0-2 5:2 S(t):" ! -, '$ !% 7'. 6:-/! ;!
L2(RN );!'<#"
L2(RN )2B
,."$'<S(0) = Id
$% C ;!/ @ ;!L2(RN ))
= 4 (S(t))t≥0" !"@ 9 + !,. ; C -, '$ !"
.<;!,.;! 6:-/!;! t=-6.C :" 9#9 ;- <C D% " 4
S(t + t′) = S(t)S(t′), !
t, t′ ≥ 02
< f ∈ C1(R+; L2(RN ))
2 -: % ,!%
∂u∂t−∆u = f
;!'<#"]0, +∞[×RN
u(x, 0) = u0(x);!'<#"
RN .
';4 ! !. " % A u ∈ C(R+; L2(RN )) ∩ C1(R+∗ ; L2(RN ))
=; !7 ,'.
u(t) = S(t)u0 +
∫ t
0
S(t− s)f(s) ds,
6.C :" 99 ;-
u(x, t) =
∫
RN
u0(y)e−|x−y|2
4tdy
(2πt)N/2+
∫ t
0
∫
RN
f(y, s)e−
|x−y|24(t−s)
dy ds
(2π(t− s))N/2.
Go:DG)!2+-(v(*C E2+-@I"@ %)( / 8 6 \ 6 E?(*)!#(*% (q)(*%');<(* 9):3$29B@D"+-%) E?B+ (vE)3=2+-4"!#( / 8 6 6E?9'/2:G$291%)3%,+' ("9'EF:"( +-("9E )E )@ _%)"92:3+-@ %d:G%)@-;<("9,*%)2@-"("9E )" ("!#!#("$&%6W-X ("9 %>+' =Y! (*% ("9(*C (*""@I"("9#9'@ ;3:3$&%)9f& +' $ $2G%)(*d:
u+R:9'+- %)@ 3$ / 86 \ 6 E )=2+-4"!#( / 8 6 6 : ;( + : $$2"(
@ $@ %)@R:G+-(u0 ∈ L2(RN )
6
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% * V <&13<#(* & )1 ( ¯ . IY 3I POoOM B IYZI : .=B,.- !B@ T > 0
=
1
2
∫
RN
u(x, T )2dx +
∫ T
0
∫
RN
|∇u(x, t)|2dxdt =1
2
∫
RN
u0(x)2dx.
IY 3I POoOF I K 4FTK K .="/ u0 ∈ L∞(RN )='<% <+"
u(t) ∈L∞(RN )
‖u(t)‖L∞(RN ) ≤ ‖u0‖L∞(RN ) ∀ t > 0.
=-"A u0 ≥ 0,.:" ,'.+ ;!'<#"
RN =4'!% !+"u ≥ 0
;!'!#"RN × R+ 2
IY 3I POoO uI IY= ZZ u ∈ C∞(RN × R+∗ )
2
IY 3I POoOTSU 9PK B="IKIZD B oKB I
lim|x|→+∞
u(x, t) = 0 ∀ t > 0,/
limt→+∞
u(x, t) = 0 ∀x ∈ RN .
IY 3I POoOTSFS *I"I IF B OI -: .= "A u0 ≥ 0/
u0 6≡ 0,'!% !+"u(x, t) > 0
;!'!#"RN × R+
∗2
"' "U + )H,+ t+ ) $!+ 'o" ) .F.10 BDC G 73E`@TI@8F1@i6%C Gd4 6%GLMHJE
0=VL_JSlDb8I r M\W:V5WJS r ILM\W`NQDWJILW5NZ[DTSK:PRM[WJEM\KJILZ\DeS=KJPRV5K:PRM[XNQXTSl^`_JIaZ[MONHIjNQM\c5DTSl]8DvZzIuSRVLZ[_8NQM\V5W]8DZ®g XT^`_JIaNRM[VLWt]:DTS!VLWJ]:DTSTf^`_:M SRVLW`N!NRPQTSF]8MSR?XTPRDTW`NRDTS,]:DlEDZ[Z\DeS,]8DZzIS1V5Z\_:NRM[VLWt]:DZkg Xe^`_JIjNQM\V5W]8DZzI¡EHGJILZ\DT_:Pep/qdI¡K:PQVLK:PQM\XNRX ZzIKJZ\_JSunoPHIaK:KJILW`NRDLf!]8X 1Y¶Xc5V8^5_JXDIL_ C/GJIaKJMONQPRD5f/DeSANZ[I 6 ?5j « « «oª 6 § ª s]8DEDN1NRDXe^`_JIjNQM\V5W p B? «ªT« § 94 ,/.01
Ω2 .> L 3B[=.>B[9A!BOA >28h0<3B E73
RN i 3d2 3Yg69G\S48T >
0 I ,/.01 (v0, v1) ∈ H10 (Ω) × L2(Ω) i 3#2 V3B[Y 3!G.2BO;=3
f ∈ L2(]0, T [;L2(Ω)) I 8M.>B=G8 R A=>?0<.>nEF3G#.>9E73[G! 6`ª DJ¢*§ ª sU 01VA >BA353 BO3Y .V 8M3#V3Yg69GW`6B[D01BEF3T
∂2v∂t2 −∆v = f
6 I 6 I E7G Ω×]0, T [v = 0
6 I 6 I G=2B ∂Ω×]0, T [v(x, T ) = v0(x)
6 I 6 I EFG Ω∂v∂t (x, T ) = v1(x)
6 I 6 I EFG Ω
®:p/.L¯5FEY 3 243g20<3G.8h2?0<.>
v ∈ C([0, T ];H10 (Ω))∩C1([0, T ];L2(Ω)) I C3S698: G i G[0 u(t, x)3[G[58 G.8:2D0<.mEF3X8 R A=>?0<.> E73[GX.4EF3G I 3G[0
v0(x) = u(x, T )EF>HG
H10 (Ω)
3v1(x) = ∂u
∂t (x, T )EF>HG
L2(Ω) i 8M.>B=G.>K v(t, x) = u(t, x) I+ 6 ?§ ª j ªe« ?§ &WnIaM\NZ\DyEHGIaW:5D r DW`N]|g M\WJEVLW:W_:D w(x, t) = v(x, T − t) DNtJp .5¯L]8DcM[DW`Nv_:WJDuXT^`_JIaNRM[VLW¡]:DTSvVLWJ]8DeS²1K:PQVLLPQDTSQSRM\c5D³Ic5DTE$]8V5W:W:XTDuM[W:MONQM[ILZ\DtDW
t = 0EV rr DZ®g XT^`_JIaNRM[VLW&²A_S1_:DTZ\Z[D³=®:p 95EV rr DmZzI&]8XPQM[cLXD2DTWNRD r KSdDeSAN] g V5PQ]8PQDB¯8fM\Z5W g +IFKJI5Sd]8D/EHGJIaW8hLD r DTW5N]:DyS1M[LW:Dy]JIaWJSZkg XT^`_JIjNQM\V5W¸ILK:PQTSEDtEHGJIaW:5D r DW`N]|g M\WEVLWJW`_JDepPILPIaKJK:Z\MzEIaNRM[VLW f
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¯ O )1 )#& *<
]:_!kFG:XTVLPQ r DJp 9Jp =:p/.L¯5mI5] r DN,]8VLWJE=w:M[DWu_JW:D_JW:M[^`_:DSRVLZ[_8NRM[VLW p:M v0(x) = u(x, T )DN
v1(x) = ∂u∂t (x, T )
fJZzIyS1V5Z\_:NRM[VLWu(t, x)
]8Du®:p 95FDeSANlIa_JSQS1MS1V5Z\_8NQM\V5W ]8Du®:p/.L¯LpOP2IaP&_JW:M[EMONQXV5WtDTW ]8XT]8_JMONv(t, x) = u(t, x)
pqdDvEILPQI5ENQPQDPRXTcLDTPQSRM\wJZ\DDTW NRD r KSl]:DZkg Xe^5_IjNRM[VLW]8DTSVLWJ]:DTS=It]8DWJV r wJPRDT_JS1DeS=EVLWJSRXh^`_:DTWJEDeSp`qdIK:Z[_JSBM r K?VLPRNQIaW`NQD&DTS1N/^`_ g M\ZW g +IIL_JE_:Wy ª 6 DU8 « T:§ ª KV5_:PmZkg Xe^5_IjNRM[VLW]:DTSV5WJ]8DeSIa_EVLW`NRPHIaM[PQD]:DtEDt^5_JM/SRDyKJILSQSQIaM\NKV5_:PZkg XT^`_JIjNQM\V5W]8DyZzI EHGJILZ\DT_:PTp:mWD R DNefSRMELg XNQILMONlZ\DEILSTf:DTWEHGJIaWJLDTILW`N&Z\DS1DTWJS=]8_ NRD r KJSEV rr D]:IaWJS=Z[I PmPRV5KV`S1M\NRM[VLW Jp .:p\5f:VLWV5w8NRM[DW]8PQILMON,_:WD R DN/²R]8XTPRXTL_:ZzIaPQM[SQIaW`N1³&EVLW`NRPHIL]:M[ENRVLM[PQDvoZzIS1V5Z\_:NRM[VLWtSRDPHIaM\N r VLM[WJS!PRXTL_:Z[M\TPRD^`_:DZzI ]8VLWJW:XD´JWJILZ\DDTW
Tf ED$^`_:MmW g DTS1NKJILSK?V5SQSRM\w:Z[DK:_:MzSR^`_:DZkg DR?DNPQXL_JZ[ILPRMzSRILW`Nl]8V5MONIL_JSQS1MSg IaKJK:Z\Mz^`_:DPp P2IaPEVLWJSRXT^`_:DTW5Nef « =§> · § « D? « §µ§ « B` ª s¢U¸ 6 D?U8 «oª56 KV5_:PZzI*S1V5Z\_:NRM[VLW]:DZkg XT^`_JIjNQM\V5W]8DeSVLWJ]:DTSlKJIaPPHIaKJKV5P1NIa_8b]:VLW:W:XTDTSlM[W:MONQM[ILZ\DeSpW¸K?D_8NNQVL_8NIL_¶KJZ\_JSuIj±$P r DTPTf2EV rr D ]JIaWJSZ[D EILSDTZ\Z[M[K8NRMz^`_:DLfB^`_:D ZzI¸PRXTL_:ZzIaPQM\NRX*]8D Z[I~SRVLZ[_8NRM[VLW¬]8DZkg XT^`_JIjNQM\V5W]:DTSFVLW]8DTSFDeSAN=]8M[PQDTENQD r DW`N,Z[M\XTDYEDZ[Z\D]:DTS,]8V5W:W:XDeSp
)HB O ORQ VFX3YZΩ [\ X [ n ]_j Z,i)X jd\Pk/j)k3[ h Y ]_jqe&] RN x V?X3YZ [\ Z ] w `F^~z~\?b h T > 0 |~[ \?]
e X \2\?k)] Y \ YZTY b h ] u0 ∈ H10 (Ω) ∩H2(Ω) ] Z u1 ∈ H1
0 (Ω) |J[ \ Z ]_j w ]v^ X [j'cd] f ∈ L2(]0, T [; L2(Ω)) |] Z u ∈ C([0, T ]; H1
0 (Ω))∩C1([0, T ]; L2(Ω))h bv^ Xh [ ZTYaX \A[ \ Y N*[2]e&] h L k:N*[2b ZTYaX \e&]_^ X \?e<]_^ l x p x h-X j_^)| u b_`<` b3j ZTY ]_\ Z Q C([0, T ]; H1
0 (Ω) ∩H2(Ω)) ∩ C1([0, T ]; H10 (Ω)) ∩ C2([0, T ]; L2(Ω)) x
329f:3!#(*%'%)3$29q+ : )E?39'@-%)@-3$ 86 &6 7&2@ ("9 %f9'@-!#@-+ :3@ )(0;A+ : )E?39'@ %)@ 3$ 86 6 &2( $29: ;$29J:G29'9'@?:3!#@-9'(6Y $E?("%J=2@-("$9 u!#3$&%)(* m(*$)(+ :G)@ %);BE2:G'%)@ m( "()"9'+-%d: %m(_%m3=%)("$@-(+ :/9'+-%)@-3$
u (+' &F: %)@-$D("913$F ("9 / 8 6 6 ("9 %,:G29'9'@F)(32+-@-4*)(R&2(>+' $ ;("% E?3);$&2(+ (*9f3$2$"("9
u0 u1
(*%f+-( 9'@-("$&%f:329'9'@ / : ;( ";<("$&%)2(*+ +-("9 "3$F@ %)@-$9f ("3!#EF: %)@ =@-+ @ %)D("9
$$2"(*9 ;<3@ + : >("! : S(B86 6 5 6 6
.F.@A 9LMH9m786%GLMGd4/6 N E LMH&68926:@!<&G G|6 CF<=@TH&N786:@i6:@19B4 ;G F Cd4&G 7B@,G
U§> uUJ¢U « §U¨ « /¢Us8 « U K?VL_:P2Zkg Xe^`_JIjNQM\V5W]8DeS2VLW]8DTSTp :mW$Zkg ILwJS1DTWJED]:DNRDP r DS1V5_:PHED$ f = 0f rTr DSRM ZzIc`M\NRDeSRSRDM[W:MONQM[ILZ\DDeSAN&W_:Z[Z[Dy u1 = 0
!DNSRM ZzI]8VLW:WJXDM[W:M\NRMzIaZ[DDTS1N=K?V5SRMONQM\c5Dyu0 ≥ 0
fZ[I$SRVLZ[_8NRM[VLWuK?D_8NEHGJIaWJLDP,]8DvS1M[LW:DIL_ EVL_JPQS,]8_*NQD r KJSTpC/DN1NQD!ILwJS1DTWJED/]8D/KJPRM[WJEM[K?D/]8_ r Iab8M r _ r DTS1NEVLW8noV5P r DBY=Z®g M\W`NQ_:MONQM\V5WK:G-+8SRM[^`_:D5p V r Ia5M\WJVLWJS_JW:D,EVLPH]8D/VL__:W:D r D r w:PHIaWJD/XTZ[I5SANQM[^`_:D aS1M8V5WZzI]:XnoVLP r D/M[W:M\NRMzIaZ[D r DW`Nm]:ILWJSU_:W:DFKV`S1M\NRM[VLWIL_ ]8DTSQSR_JS=]8DvS1V5W K:Z[ILW ]8DPQDK?V5STf:DTZ\Z[DcjIycM[w:PRDTP&DNKJILSQS1DTP=IaZ\NRDTPRWIjNRM[cLD r DW`N&DW]:DTSQS1_JS=DNIL_£]8DTSQSRVL_JSt]8DEDK:ZzIaW ®Ia_8NQPRD r DTW`N]8M\N u EHGJIaW:5D]8D¸S1M[LW:Dp IjNQG:X r IjNQM[^`_:D r DW`NTfFEDEVLW`NRPQDh¦Db8D r KJZ\DFK?D_8N/STg XTEPQM[PRD,SRM r K:Z[D r DW`N/SRVL_SZ[IlnoV5P r D&SR_:M\cjILW5NQDLp58VLM\N w(x)
ZzIK:PRD r M\TPRDnoV5WJENQM\V5W K:PQVLK:PQD]8_¸qdILK:Z[I5EM[DW ]:ILWJS_:W]8V r ILM\WJDw?VLPQW:XvEVLW:W:Db8D Ω IcLDeEEV5WJ]8M\NRM[VLWIL_8bZ[M r M\NRDeS]8D/xM\PQMzEHG:Z\DNTpxg IaKJPReSdZ[DNkFG:XTVLPQ r DF8p 9Jp\e¹:fV5WK?D_8NW:VLP r IaZ[M[SRDP w ]8DBNRDTZ\Z[D r ILW:M\TPRD^`_:Dw(x) ≥ 0
]:IaWSΩp-:mWtW:VaNHIaW`N
λ = ω2ZzIK:PQD r M[PQD=cjIaZ[D_:P!K:PQVLK:PQD=I5SRSRV8EM[XD=Y w fM\Z DeSANnI5EM[Z\D$]8D$c5XPQMO´DP^5_JD
u(t, x) = cos(ωt)w(x)EHGJIaW:5D]:DyS1M[LW:DIa_~EV5_:PHSl]:_NQD r KJSNQVL_8NDTW$XNHIaW`NmZ[ISRVLZ[_8NQM\V5W_JW:M[^`_:D]:IaWJS
C([0, T ];H10 (Ω))∩C1([0, T ];L2(Ω))
]8D&Zkg Xe^5_IjNRM[VLW$]8DeSV5WJ]8DeS:p 9 9`!SRILWJS/NRDP r DS1V5_:PHEDlDN=IcLDeE=Z[DTS=]8VLW:WJXDTSFM[W:M\NRMzIaZ[DTSu(x, 0) = w(x) ,
∂u
∂t(x, 0) = 0
]:ILWJSΩ.
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% * V <&13<#(* & )1 ( ¯ m§> s ¢?§¨yU:v§?§sU¢Ut¨5s/J ª § ª : s ª ªe« ? UDTW$NRD r KS/Z[VLWJKV5_:P/Zkg XT^`_JIjNQM\V5Wu]8DeSBV5WJ]8DTS/DWt]8V r IaM[W:DwV5PRW:X5p 2=_8NRPQD r DTW5N,]8M\NTf rr DSRM Z\D=NRDTP r DSRVL_JPQED
fW:D]8XK?DW]uKJI5S,]8_tNRD r KJSTf8ZzIvSRVLZ[_8NRM[VLW u WJDEVLWcLDTPR5D=KJI5S!cLDPHS/_:W:DZ\M r MONQDSANHIjNRM[VLWJWJIaM[PRDZ\V5PQSQ^`_:D,Z\D=NRD r KS t NRDTWJ]$c5DPHS2Z®g M[W8´JW:MkpQ:BWyKJIaPRNRMzE_JZ\M[DPef5SRM f = 0
f5Z®g M\W J_:DWED]:DTSBEVLWJ]:MONQM\V5WJSM\W:M\NRMzIaZ[DTSDTS1NZzI rr DyYtNRV5_8NNRD r KSK:_:MzSR^`_:DZkg XTW:DPQLM[DDTS1NEV5WJS1DTPRc5XDDNW:D$]8XeEPQVzNKJILSocLV5M\PtZ®g :2b8DPHEMzED~:p 9:p[ep/qdD rr D~EV5W5NQPRDhkDb8D r K:Z[D u(t, x) = cos(ωt)w(x)K?DP r DN*]:DcLVLM[P^5_dg M[Z,W g +¡IKILS]8D*Z\M r MONQD SANHIjNRM[VLWJWJIaM[PRD r IaMzS]:DTSV5SQEM[Z[Z[IaNRM[VLWJS^`_:M,KDTPQ]:_:PRDTW`NSQIaWJSI r VLPRNRMzSRSRD r DTW5NepC/DTZ[IW g DTS1NBXcMz]8D rr DW`N/KILSmZ[DETILSBKV5_:PBZkg Xe^`_JIjNQM\V5Wy]8DTS/VLWJ]:DTSBI r VLPRNRM[D:p/. 9`2EV rr DZ\D r VLW`NQPRDlZ®g Db8DTPQEM[EDSR_:M[cjIaW`NTp
`j¨ « ¨L O η > 0 ( _%)$P / G + @-!#">. /(%$'/ -$Q> .
∂2u∂t2 + η ∂u
∂t −∆u = f/7>7$
Ω× R+∗
u = 0$ "7
∂Ω× R+∗
u(x, 0) = u0(x)/7>7$
Ω∂u∂t (x, 0) = u1(x)
/7>7$Ω.
®:p/. 9`
( $ "7?7?< $&M!#"u%$;Q"7M$R+,". *$ "$R> - @-K"7+ G*/ =1 (6 %J!#") f %$&L7"(+>7?< $
"(=; ?$ #-G< N + > /( /) "( + =/( G >7+,+1< + <1 % %#6 N!#") u %∂u∂t
/ @G $R$&< 7?<# -. +,+ -S)<G$ @ + G$R!#"A+ :; ?$t;</C $:+, qI PmPRV5KV`S1M\NRM[VLW:p 9:p/.IXNHIaw:Z[M\NF_JW:DK:PRV5K:PQM\XNRX]8DlEVLWJSRDPQcjIjNQM\V5Wt]8DZkg XTW:DPQLM[D&NQVaNQILZ\DDTWZ®g IawJSRDWEDF]8D!NRDP r D,S1V5_:PQEDLp WPQXTSR_:Z\NQIjNK:Z[_JSK:PQXTEM[STfa]8M\N2¢> 6-? « 8 ªT«ªT« ?§¸¢,> 6 §`D « ?fIj±$P r D^`_:DvZkg XW:DTPR5M\DvNRVLNQIaZ[DS1DKJILP1NHIaLDvXT^`_:M\NQILw:Z\D r DW`NlDTWXTW:DPQLM[DvEM\WJXNRMz^`_:DDNXWJDPQLM[Dr XeEILW:M[^`_:D5f:ILS8+ r K:NRVaNQM[^`_:D r DW`N,KV5_:P,Z\DeSFNRD r KJS,LPHIaW]:Sp
`j¨ « ¨L O GFH u(t, x)
+ > $&#+,"-. #N$%"(?7?< $R@%=$%"$R> < G@%K"7+ G(N/ + @!">. / -$Q#)/(%$:1 (6% + > )$& )%=/ ;< =$R"7U%(N #-G<O!#"
limt→+∞
1
t
∫ t
0
∫
Ω
∣
∣
∣
∣
∂u
∂t
∣
∣
∣
∣
2
dx = limt→+∞
1
t
∫ t
0
∫
Ω
|∇u|2 dx =1
2E0,
>E0
+ @ K < . >7+
E0 =
∫
Ω
|u1(x)|2 dx +
∫
Ω
|∇u0(x)|2 dx."7& + >2 "(+ . ?7+ < > + @-!#")>. # 1 76 ?)> u Q ;@%K# G> ?)> ?><. .F. ·@i6%GE-E Gµ;G H=7:9HN NU68@19B4 l4@,G
W:Dv]:DPQW:M\TPRDK:PQVLKJPRM[XNQXM r K?VLPRNQIaW`NQD]8DZkg Xe^5_IjNRM[VLW ]:DTS,V5WJ]8DTS&DTS1NEDTZ\Z[DLf?]8MONQD]8DUj:©UD? ªe« ?§ «ª Te ;U§ « ?p30=VL_JSlIcLV5WJS&]8X 1Yyc_ Ia_~C/GJIaK:M\NRPQDutoDW]8M r DTWJS1M[VLW N = 1DNl]JIaWJS&NRVL_:NZkg DeS1KJI5EDΩ = R
&^`_ g M\ZUDb8MzSANQD_:W¸E LWJD]8DZ[_ r M[PQDuoV5_]8V r ILM\W:Dv]8Dv]8XTKDTW8h]:IaWJEDeB^5_JMDTW:LZ[VLw?D&NQVL_8NQD=Zkg M[W8noVLP r IjNQM\V5WuS1_JP/ZzISRVLZ[_8NRM[VLWy]:DZkg Xe^`_JIjNQM\V5Wy]8DTS/VLWJ]:DTSlocLV5M\P/Z[I
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¯ L O )1 )#& *<
UM[L_:PQDLp 95p PBZ\_JSK:PRXeEMzS1X r DW`NTf|ZzItSRVLZ[_8NQM\V5W u DTW (t, x)W:Dy]:XK?DWJ]~^`_:D$]8DTScjILZ\DT_:PQS]8DeS]:VLW:W:XTDTSBM\W:M\NRMzIaZ[DTS
u0DNu1S1_:P/Z\DlS1DT r DW`N [x− t, x+ t]
pW$DTWu]8Xe]8_:M\NF^`_:D=SRM?Z\DeSB]8V5W:W:XDeSM[W:M\NRMzIaZ[DTS/S1V5W5NBYSR_:K:K?VLPRN/EV r KJILEN K = [kinf , ksup] ⊂ Rf`IaZ[VLPHSZ[IS1V5Z\_8NQM\V5W$Ia_NRD r KJS t DTS1NYSR_:K:K?VLPRNEV r KJILENl]:ILWJS [kinf − t, ksup + t]
p :mWNRDTP r DeSKJG +8SRM[^`_:DeSEDZzIuc5D_8N]8M[PRD^`_:D]:DTSlKDTP1NQ_:PQwJIjNQM\V5WJS=M[W:M\NRMzIaZ[DTSSRDvK:PQVLKIaLDTW`NYucM\NRDeSRSRD´JWJM\DwV5PRWJXDKJIaPLpdC/DNRNRDS1M\NR_IjNRM[VLWDeSANDWJEVLPQDu_:WJDynoVLMzSw:M[DW¡]8MSR XPQDW`NRD*]8D*EDt^`_:MFSRDtKJI5SRSRDyK?VL_:PZkg Xe^5_IjNRM[VLW¶]8DtZzIEHGIaZ[D_:P®EV r KIaPQDPIcLDTEvZzI PBPRV5KV`S1M\NRM[VLW¸:p =Jp 95p|qBg DbDTPQEM[EDSR_:M[cjIaW`NKDTP r DN]:D5XW:XTPQILZ\MzSRDPEDNRNRD]:M[SQE_JSQSRM\V5WIL_8bt]8M r DWJSRM\V5WJS N = 2, 3p
`j¨ « ¨5 O ( 5 #$ / + @-!#")>. #=/(%$M#)/(%$'/7>7$Q;" + %$%?)>%RN
∂2u∂t2 −∆u = 0
/(>7)$RN × R∗
+
u(x, 0) = u0(x)/7>()$
x ∈ RN
∂u∂t (x, 0) = u1(x)
/7>()$x ∈ RN ,
®:p/. =`>) "7:/ #()@%2 . >7+
(u0, u1)@-K"7+ G =$ "7?7?< N% ?> 9!#") + > $&+,".
u(t, x)?<<"'$& G=$&#"$0+ >5
u(x, t) = (Mu1)(x, t) +
(
∂(Mu0)
∂t
)
(x, t),
M
-$; "(2#?<@< >;<"7E/ )-<7:/ @ <9?)>
$PN = 1, (Mv)(x, t) =
1
2
∫ +t
−t
v(x− ξ)dξ,
$PN = 2, (Mv)(x, t) =
1
2π
∫
|ξ|<t
v(x− ξ)√
t2 − |ξ|2dξ,
$PN = 3, (Mv)(x, t) =
1
4πt
∫
|ξ|=t
v(x− ξ)ds(ξ),
ds(ξ)
%$;:+ > %$%"(G2$%"(> !#"2/(+ > $ ?2 A/(@-/#"< G2!#" + >A$&#+,"-. #u<
(t, x))
/(@<?< )/ !#" / -$g>7+ <"7$ / -$0/(7@-%$ . >7+ %$u0
u1
$ "7Q+ > <"7+ |x| ≤ t 1 "7 $.> % <'#" )0+ %$ ?<-$&$ )$& / -$&$%"$ / 0+ #?<@< >;<"7 M N )")$ G<))%)$E>7"g&>7? /( 5; 6 `j¨ « ¨5 O Mr ( P%)$P / G + @-!#")>. # /(%$ #)/(%$ 1 6 /7>7$ "( / > ) Ω ⊂ RN>) /(%$o / . )$ >7" + ;-$Q/( (+ % "2/ <" >77 1d K )%$ 6 N %*"7:/ #()@% . >7+
(u0, u1)@-K"7+ G %I$ "7?(?< ?)> A/(>7)$
Ω * @< <:!" + ( $;; "7 ; ?)$T > 0
;<+J!") $ "7=+ ;<)>7+,+ [0, T ]
+ > $R+,". %$&A< G /(7)@%?)> =+ %$_ "(+ -$ /(+ <1 qBg DbDTPQEM[ED,S1_:M[cjIaW`N r DNmDTWPQDZ[M[DnZzI]8MSR?XTPRDTWJEDFDeSRSRDW`NRM[DZ[Z[D!DTW`NRPQD,Z[I]8M r DWJSRM\V5W$]|g DeS1KILEDN = 2
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% ^ < ( 1 &&&)1 ( ¯ 5º `j¨ « ¨L O A8U « ¨5 ªT« § ¸U « ¨L:® >/ &>72!") + $R $R ?<G?>)K $R<+ + @!">. / -$0#)/(%$N !" +O! %$&2?)>)$A?< $R$P7+ A/) @ #"-; /( + > ")$P !#" 14>7"/7+ 6/(>7)$'"( /(*/ / $ $%?)>. >7+ N = 2
N >7+ G$ !") %$&Q1< )<"7G<")$& < 6 ?< $&$(+ < / <)$P # N = 3
,+ % " t,+ tt+ ) $!+ '*#") txlIaWSEDNRNRDySRDTENRM[VLW f W:V5_JS r VLW`NRPQVLWJSlEV rr DTW`NZzI r XNRGJV]:D]:DTSXZ[X r DTW5NHS´JW:MzS$oK:PQXhS1DTW5NQXDlIa_C/GJILK:M\NRPQD 5/STg I5]:IaK8NQDYZ[IPRXeS1V5Z\_8NQM\V5WyW_ r XPQM[^`_:Dl]8D=Zkg XT^`_JIjNQM\V5Wu]8DlZ[IvEHGJIaZ[D_JP0VLW_8NQM\Z[MzS1D*]8DTSXTZ\X r DTW`NQSv´JW:MzSKV5_:PvZzI]:M[SQEPQXNQM[SQIjNQM\V5W¡S1KIjNRMzIaZ[DLfUDN$]8DeS]:M R XPQDWJEDTS´JW:M[DTSKV5_:P,Z[I]8MzSQEPQXNRMzSQIjNRM[VLWtNRD r KV5PRDTZ\Z[DLp
.F.10 *G(L @ ;@TE> 7OC|6:@TE NU6:@A9B4 Gd4 GE H&N2> GViZ/Sg Ia5MON]8D$]8MzSREPRXNRMzS1DTPl§ :¨5TU§ ª ZzItnoVLP r _:ZzIjNQM\V5WcjIaPQM[IaNRM[VLW:WJDZ[Z\D:p 5]8DvZkg XT^`_JIjNQM\V5W]8DZzItEHGIaZ[D_:PJp\p PVL_:PEDTZ[IJf|EV rr D]:IaWJSlZ[D$EILSl]8DeSK:PRV5w:Z[ r DTSDZ[Z\M[K8hNRMz^5_JDTSTfLW:V5_JSmEV5WJS1NRPQ_:M[SRVLWS_:W:D&IaKJK:PRVb8M r IjNRM[VLWcjIaPQM[IaNRM[VLW:WJDZ[Z\DFM[W5NQDPQW:D,DW$M[W`NRPQV8]8_:MzSRILW5Nm_JWS1V5_JSAh¦DTSRKJI5ED
V0h]8DH1
0 (Ω)f:]8D]8M r DTWJS1M[VLWt´JW:M[DLp2k+K:M[^`_:D r DW`NTf V0h
S1DTPQI_:WSRVL_SAh¦DTSRKJILED]|g XTZ\X r DW`NQS´JW:MzS PkVL_
QkSR_:Pv_:W r IaM[Z[Z[ILLD$NQPRMzIaW:5_:ZzIaM[PRD VL_¡PQDTENQILW:L_:ZzIaM[PQDeEV rr DK:PRXeEMzS1XIL_8btxX´JW:M\NRM[VLWS :p 9:p/.DN :p 9:p°¯ .:pqItSRD r M\h¦]:M[SQEPQXNQM[SQIjNQM\V5W ]8D®:p 5=DTS1N]:VLWJEZ®g IaK:KJPRVb8M r IaNRM[VLW cjIaPQMzIjNRM[VLWJW:DZ[Z\DS1_JM\cjIaW`NQD NRPQVL_:c5DP
uh(t)noV5WJENQM\V5W]8D
]0, T [YcILZ\DT_:PHS,]:IaWJS
V0hNRDTZ\Z[D^`_:D
d
dt〈uh(t), vh〉L2(Ω) + a (uh(t), vh) = 〈f(t), vh〉L2(Ω) ∀ vh ∈ V0h, 0 < t < T,
uh(t = 0) = u0,h
®:p/. .5V U
u0,h ∈ V0hDeSAN_:W:DyILK:K:PQVb8M r IjNQM\V5W]8DZzI]:VLW:W:XTDM\W:M\NRMzIaZ[D u0
pUC/DNRNRD r XNQG:V8]8D$]|g IaK8hK:PRVb8M r IjNRM[VLWtDTS1N&Ia_JSQS1M EVLW:W_:DSRVL_JSFZ[DW:V r ]8D=² r XNRG:V8]8D]8DeS,Z[M\5W:DTSA³HpW K?D_8NyIL]JIaK8NQDPZ\D*EI5]8PQDILwJSANQPQILMON$]8_ kFG:XV5PR r Dt:p°¯8p 9KV5_:P r V5W`NRPQDP^5_JD~®:p/. .5IL] r DN/_:W:D&_:WJM[^`_:DlS1V5Z\_8NQM\V5W f r IaMzSmM[ZJDeSAN/wDeIa_JEVL_:K$K:Z[_JS/S1M r K:Z\D=DNBKJILPRZzIaW`NB]8D=cLXPQM\´JDPB^`_:D:p/. .5!DTS1N=DW*nILMON_:W ST+8S1NR r Dv¢> 6 ? U ªe« ?§$¢ « 6 j§ ªT« `®Jj¢ « §U « jY$EV`D±yEM[DW`NQSEVLWSANHIaW`NQSy]:VLW`NyVLW£ETIaZzE_:Z[D*nILEM\Z[D r DW`NuZkg _:W:Mz^5_JDS1V5Z\_:NRM[VLW pBxD r IaWJM\TPRD K:PHIjNQM[^`_:D5fmK?VL_:PPRXeS1V5_J]8PQD*®:p/. .5=V5W~M\W`NRPQV8]8_:M\N_:W:DuwJILSRD(φi)1≤i≤ndl
]8DV0h N,+K:Mz^`_:D r DW`Nef Z[I*wILSRDy]8DTSXZ[X r DW`NQS!´W:M[SHfJDN&VLW*EHG:DTPQEHG:D uh(t)
S1V5_JS!Z[InoV5P r Duh(t) =
ndl∑
i=1
Uhi (t)φi,
®:p/. `IcLDTE
Uh = (Uhi )1≤i≤ndl
Z\DcLDeENQD_:P]8DTSEVVLPH]8VLW:WJXDTSl]8Duhp ViZ2DTS1NM r K?VLPRNQIaW`N]8DW:VLNRDP^5_JD]:IaWJS®:p/. `BZ[DTS,noVLWJENRM[VLWJS,]8DwILSRD
φiWJD]8XK?DWJ]:DW`N=KJI5S,]8_NRD r KJS&DN=^`_:DSRD_:Z[DTSFZ[DTSEVVLPH]8VLWJW:XDeS
Uhi (t)SRVLW`N&]8DeS!noVLWJENRM[VLWJS,]8_NQD r KJS t pJxD rr DLfJVLWKV`S1D
u0,h =
ndl∑
i=1
U0,hi φi,
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¯`j¹ O )1 )#& *<
DNJp .Q.L!]8DcM[DW`NTf8K?VL_JPFNRVL_:N1 ≤ i ≤ ndl
f
ndl∑
j=1
〈φj , φi〉L2(Ω)dUh
i (t)
dt+
ndl∑
j=1
a(φj , φi)Uhi (t) = 〈f(t), φi〉L2(Ω)
Uhi (t = 0) = U0,h
i
ViW`NQPRV8]8_:MzSQIaW`N®EV rr D]:ILWJS!Z[I]8X r V5WJSANQPQIaNRM[VLW]8_qdD rr Dp =Jp[eBZ[I ª « ¨L*¢U :eTMh]:X´JW:M[DKJILP
(Mh)ij = 〈φi, φj〉L2(Ω) 1 ≤ i, j ≤ ndl,DN&Z[I ª « ¨5 ¢* « D « ¢ «oª56 Kh]8X´W:M\DKJILP
(Kh)ij = a(φi, φj) 1 ≤ i, j ≤ ndl,Zkg IaK:K:PQVb8M r IaNRM[VLWcjILPRMzIjNQM\V5W:W:DZ[Z[D®:p/. .5dDeSANmXe^`_:M\cjILZ\DTW5NQD,Ia_$ST+8S1NR r D,Z[M[W:XTILM\PQD,¢> 6 ? U ªe« ?§¢ « 6 j§ ªT« vJj¢ « §U « jYEVD±yEM[DW`NHS,EVLWSANHIaW`NQS
MhdUh
dt(t) +KhU
h(t) = bh(t), 0 < t < T,
Uh(t = 0) = U0,h
®:p/.5aIc5DTE
bhi (t) = 〈f(t), φi〉L2(Ω)
p8qmg Db8MzSANQDWJEDDN!Zkg _JW:M[EMONQXLf8IaM[WJSRM|^5_dg _:WJD=noV5P r _:Z\DDb8K:Z[M[EMONQDLf8]8DZzIS1V5Z\_:NRM[VLW]8DJp .`aUSg VLw:NRM[DW:W:DTW`N2EZ[I5SRSRMz^5_JD r DW`N2KJIaP2SRM r K:Z[D&]8MzIa5VLWJILZ\MzSRIaNRM[VLWSRM r _:Z\NQILW:XD]:D MhDN KhocLV5M\P,ZzI$VL_SAhiS1DeENQM\V5W*p =Jp[YEDS1_ADNHp?C/V rr DM\Z|DeSAN=]:MO±yEM[Z[DDN&EV8NRDT_8b]:D]8MzIa5VLWJILZ\MzS1DTP=Jp .`af5DTWuK:PHIjNQM[^`_:D&V5WyPQXTSRVL_8N!W`_ r XTPRMz^`_:D r DTW5NJp .`aKJILP!]:M[SQEPQXNQM[SQIjNQM\V5WDN r IaPHEHG:D,DTW$NRD r KSp ViZJDbMzS1NRD=]8DW:V r w:PQD_JSRDTS r XNRG:V8]8DeS/EZ[I5SRSRM[^`_:DeSm]8D=ETIaZzE_:ZW_ r XPQM[^`_:D]:DTSS1V5Z\_8NQM\V5WJS]|g Xe^`_JIjNQM\V5WJSt]:M R XPQDW`NRM[DZ[Z[DTSVLPH]8M[WJIaM[PRDeSp*0&V5_JSuDTW·c5DPQPRV5WJSu^`_:DTZ[^`_:DeSu_:WJDTS]JIaWJSZ[ISRVL_JS1h¦SRDTENRM[VLW SR_:M[cILW`NRDLp 2cjIaW`NEDZzI:f?W:VL_SXTW:VLWVLWS=_:WPRXeS1_JZONHIjNl]:DEVLWcLDTPR5DWJED]:DTS,SRVLZ[_8NRM[VLWJSm²RSRD r MOhi]8MzSREPRNRDeSi³l]:D®:p/. .5Bc5DPHS!Z[I$SRVLZ[_8NRM[VLWDb:I5ENRD]8D$®:p 5p
)HB O OTS VFX3YZ(Th)h>0 [\P]m^)[ YZ ]e&] w b Yhh b &]_^ Z j Y b3\ 3[ h b Y j']_^mj'k 3[ h Y ]_jd^me&] Ω x V?X3YZ V0hh ]^ X [ ^ s ]_^a` b<cd]e&] H1
0 (Ω) |~e&kz~\ Y ` bGj h b w k Z P X e&]/e&]_^Bk h k w ]_\ Z ^zJ\ Y ^ Pk |~e&]e Yw ]_\2^ YaX \ ndl x V?X3YZf(t) ∈ L2(]0, T [; L2(Ω)) | u0 ∈ H1
0 (Ω) | ] Z u ∈ L2(]0, T [; H10 (Ω))∩C([0, T ]; L2(Ω)) | h b^ Xh [ ZTYaX \
[ \ Y N*[2] e&] h L k"N*[b ZTYTX \ e<] h b cSP b h ]_[j l x p x V?XGYZ uhh b ^ Xh [ ZTYaX \ [\ Y N*[2] e&] h L b_`<`Fj X M Yw b ZTYaX \n b3j Y b ZTYaX \\P] hh ] l x p eb3\2^ V0h x V2Y limh→0 ‖u0,h − u0‖L2(Ω) = 0 | b h-X j_^ X \ b
limh→0‖u− uh‖L2(]0,T [;H1
0(Ω)) = lim
h→0‖u − uh‖C([0,T ];L2(Ω)) = 0.
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MdU
dt(t) +KU(t) = b(t)
U(t = 0) = U0
®:p/.a`PUV5_:PS1M r K:Z[MO´JDTPZ®g IaWIaZ +SRDVLW¸cI*SR_:K:K?V5SRDP^`_:D b(t) DTS1NEV5W5NQM\W_~S1_JP [0, T ]
pdW~]8XTEVL_:K?DZ®g M\W`NQDPQcILZ\Z[D]8DNRD r KJS [0, T ]DTW
n0M[W`NRDTPRcjIaZ[Z[DTSFVL_KJI5S,]8DNRD r KS ∆t = T/n0
DN&VLW*K?V5SRDtn = n∆t 0 ≤ n ≤ n0.W W:VaNQD
UnZkg IaK:K:PQVb8M r IaNRM[VLW ]8D U(tn)
ETIaZzE_:Z[XKIaP=_JWSREHGJX r I:p3PUV5_:PEIaZzE_JZ\DTP=W_ r XPQMOh^5_JD r DW`N&]8DTS,SRVLZ[_8NQM\V5WJSFIaK:K:PQV8EHG:XDeSF]8D$:p/.a`mZ[DSREHGJX r IZ[DK:Z[_JS,SRM r K:Z[DDN,Z\DK:Z[_JS,_:NRM[Z\MzS1XDTS1N&Z\Dθ©HT¨ 4<6 ¡®]8X 1Yc_ f:c5VLM[P®¯8p/.LRMUn+1 − Un
∆t+K
(
θUn+1 + (1− θ)Un)
= θb(tn+1) + (1− θ)b(tn).®:p/.aº`
q VLPHSQ^5_JDθ = 0fJV5WIaK:K?DZ[Z[DuJp .Lº5&e¨ 4(6 ~`| « ¨ «oª fZ\V5PQSQ^`_:D
θ = 1fe¨ 4(6 « s « ©¨ «oª ?fJDNlKV5_:P
θ = 1/2fJT¨ 4<6 ~¢ j:§ © « ¨ 4 ?e§/p W K?D_:NPQXXTEPRM[PQD®:p/.aº`,S1V5_JSZ[IvnoVLP r D
(M+ θ∆tK)Un+1 = (M− (1− θ)∆tK)Un + ∆t(θb(tn+1) + (1− θ)b(tn)).®:p L¹`
,D r IaPH^`_:VLWS2^5_dg DTW5XW:XTPQILZ8Z[I r IjNQPRMzED M W g DTS1NmKJILSB]8M[ILLV5WJIaZ[DLfDN!]8VLWJE,^`_:D5f rr D,K?VL_:PZ\DBSQEHG:X r I&Db8K:Z[M[EMONQDLfM[Z`DTS1NdW:XTEDTSQSRILM\PQDm]8DmPQXTSRVL_]8PRD2_JWST+8S1NR r DmZ[M[W:XTILM\PQDmK?VL_:PUEIaZzE_JZ\DTP Un+1DW$noV5WJENQM\V5Wy]8DUn DN!]8_tS1DeEVLW] r D r w:PQD b ®SRIL_8ndS1M?VLW$_:NRM[Z\MzS1D_:W:D=noVLP r _:Z[D=]|g M\W`NQXLPHIjNQM\V5WW`_ r XTPRMz^`_:Dt^`_:M!PQDWJ]8D M ]8MzIaLV5WJIaZ[DLfdcLVLM[PZzII&D r IaPH^`_:Dp =Jp 95p mcMz]8D rr DW`NefUV5WK?D_8NEVLWSANQPR_:M[PQD_JW:D$noVL_JZ\Du]:DuSREHGJX r ILSDTWSg M[WJS1KJM\PHIaW`N]8DyED_8bs]8_¶C/GJILK:MONQPRDu¯:pC/M\NRV5WJS A_JS1NRDKV5_:P r X r VLM[PRD_:W*DbD r K:Z\D]8DSQEHG:X r IY!9W:M[cLDTIL_8bt]8DlNQD r KJSTf:Z\De¨ 4(6 ¢8f
M3Un+1 − 4Un + Un−1
2∆t+ KUn+1 = b(tn+1).
®:p :C/DTS,SQEHG:X r ILS!SRVLW`N,w:M\DTWS :P,EV5WJSRM[S1NQILW5NHSocLV5M\PFZzIx=X´JW:M\NRM[VLW ¯8p°¯8p =`BDN,VLWKDT_8N,nILEM\Z[D r DTW5NIaWJILZS+8SRDPFZ\DT_:P,K:PQXTEM[SRM\V5W¸_:W:Mz^`_:D r DW`N&KJILPFPQILK:KV5P1N&YZzIvcjILPRMzIaw:Z[D]8DNRD r KJSQp `j¨ « ¨L OO #-G<!#"0+ '$O@ >*/( G>7 &)+ $&#L% +,"</ E-> $&#-E/) G/#% 14<2; ?)$ 6 N7&>7/ $'!#")+
θ $&)@ > ?<"7
θ 6= 1/2%$&Q/) /# (
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
¯`a¯ O )1 )#& *<
8 ? OOGF ViZ2DTS1NKV`SRSRM\wJZ\D]8D r DN1NQPRDDTWVD_Jc`PQD_:W:D r XNRGJV]:D] g XTZ\X r DTW`NQSl´JW:MzS§¬:¨5 ª § ª sLf r IaMzS,EDZzIvW:DK:PQXTSRDW`NRDlKJILSF]|g M[W5NQXP N,KJILP1NQM[E_:Z\M[DP,SQIa_8n]:IaWSFZ\DETILSFVQUtZ[D]8V r ILM\W:D Ω(t)cjILPRM[DDWtnoVLWJENRM[VLW ]8_NRD r KSp •
0=VL_JS/]8VLW:WJVLWJS r ILM\W`NQDWJILW5N/_:W:D=]8X´JW:M\NRM[VLWu]:D&Z[ISANHIaw:M[Z\M\NRXl]8D=EDTSBSQEHG:X r I5Sm^`_:MJDeSAN!_:W:DcjILPRMzIaW`NRD]8DZzIx=X´JW:M\NRM[VLW ¯8p°¯8p :p+ 6 ;U§ «oªe« ?§ U ] ^G;"HAY !>FT!E>0 AB34;3\p0<3Gp6.2B I 3G[oE>01pG[V 8M3#G[0
MUn · Un ≤ C 6H.>2B# .>20 ≤ n ≤ n0 = T/∆t,
. 8M ;.>HG[VV3C > 0
3G[019E7A 639EF>V3^EF3∆t
3_EF3P8@E01Y 3G=0<.> E GG[ Y 3ndl E7.9;CE_6)G5E Y 01818 3
h )i YZ>0UG6H32E7A 634EBO3ZE738M!EF.>9A=30101D0<8M3 U0 i E G3;=.>9EY 3YBO3b i 3%EF3 T I
8 ? OOMr q DlEHG:VLM\b]8DZ[IW:VLP r D √MU · U ]:IaWJSBZ[Ix=X´JW:M\NRM[VLWuJp Jp 9STg Db8K:Z\Mz^`_:DKIaPBZ[D,nILMONF^5_JD MU ·U =∫
Ω |u|2dxIcLDeE
u ∈ V0hZzInoV5WJENQM\V5Wu]8D=EV`V5PQ]:VLW:W:XTDTS
U]JIaWJSBZ[IwILSRDvEHG:V5M[SRM\Dv]8D
V0hM DTS1Nw:M[DW¸]8X´W:M\DvKV`S1M\NRM[cLDp &IaKJKDTZ\V5WJS^`_:D5f?]JIaWJSZ[Itx=X´JW:M\NRM[VLW¯:p ¯:p ¡]8DZzI¡S1NQILw:M\Z[M\NRXIL_£S1DTWJS]8DTS]8MSR?XTPRDTWJEDeSy´JWJM\DeSf!VLW K?VLWJ]:XPHIaM\NtKJILP
∆xZzIWJVLP r DDT_JEZ[Mz]8M\DTW:W:D]8D
UK?VL_:P,PQDNQPRV5_:cLDTP!IL_JSRSRM|Zkg ILWJIaZ[VL5M\DIcLDeE=ZzIW:VLP r D]:D u ]:IaWJS L2(Ω)
p • L ,0
1/2 ≤ θ ≤ 1 i 83 θ G;"HAYZ I 3[G[d019;=.4E01D0<.438183Y 3eG=V 8M3 iV4E0UG53 i G[0 0 ≤ θ < 1/2 i 018 3G[pG[V 8M3G.> G#8M ;=.4E01D0<.
maxiλi ∆t ≤ 2
1− 2θ,
®:p 5¯L. 8M3G
λiG.>8M3[G L 8M32B=G69BO.=69BO3G5EF3 KU = λMU
L .01B I FI 8 ? OOµW$W:D&PQDTEVLW:WJI[NmKILS2M rr Xe]8MzIjNRD r DW`N!]:ILWJS&®:p 5¯LUZzIEVLWJ]8M\NRM[VLWtC q¦C/VL_:PHIaW`NRh JPRM[DT]8PQMzEHGJSAhiq D +:&_JSR_:DTZ\Z[D
∆t ≤ Ch2K?VL_JPZkg Xe^5_IjNRM[VLW]:DyZ[IEHGJIaZ[D_:PtocLV5M\PZ[I8VL_JS1h¦SRDTENRM[VLW*¯8p°¯8p 95p-:mWnILMONef:VLWKDT_8N r VLW`NRPQDP,^`_:D5f:S1M|Z[D r ILM\Z[ZzIaLD Th
DeSAN,_:W:M\noVLP r X r DW`NPQX5_:Z[M\DTPIL_¸SRDWSV UNRVL_:NRD r IaM[Z\Z[D$EV5W5NQM\DTW`N_JW:D$wV5_:Z[Dy]8DPQI+5VLW Ch ®IcLDeE C > 0M[WJ]8XhK?DW]:IaW`N$]8DtZzI r ILM\Z[Z[DefmILZ\V5PQSVLW IDR?DeENRM[cLD r DW`N maxi λi = O(h−2)
pmW¶K:PQVLK?V5SRDtIa_Z[DTENRDT_:Pl]:Dvc5XPQMO´JDTPEDnIaM\NDW]8M r DWJSRM[VLW N = 1Z\V5PQS]8DZ®g :2b8DPHEMzED:p :p =uEM\h¦]:DTSQS1V5_JSp3:mWKJPQIaNRMz^5_JD=VLWtW g _8NRM[Z\MzSRDlKILS/Z\D
θhiSREHGJX r IK?VL_:P θ < 1/2
EILPFZ[IvEVLW]8MONQM\V5Wt]8DS1NQIawJM\Z[MONQX$:p 5¯5DeSAN&w?DTIL_JEV5_:KNRPQVLK SRXcLTPRD JDTZ\Z[DVLwJZ\M[LDZkg _JSQIa5D]8DNRPQTS&KDNRM\NQS,KJI5S&]8DNQD r KJS=^`_:MdPRDTWJ]8DW`NZ[DEILZ[E_:Z|w?DTIa_EVL_JKuNQPRV5KEV 8NQD_8b|p •+ 6 ?§ ª j ªe« ?§ UW*PQXXeEPQMON,Z[DSQEHG:X r I*®:p L¹5!]:ILWJSFZzIwJILSRDV5P1NQG:VLWJVLP r ILZ\DlK?VL_:P M DN]:M[ILLVLWIaZ[DK?VL_:P K oc5VLM[P,Z[I]8X r V5WJS1NRPHIjNRM[VLW]8_ q D rr Dv8p =p\
( Id + θ∆t diag(λi))Un+1 = ( Id− (1− θ)∆t diag(λi))U
n + ∆tbn,®:p 95
Ic5DTE M = PP ∗ f K = P diag(λi)P∗ f Un = P ∗Un
fDNbn = P−1(θb(tn+1)+(1−θ)b(tn))
pW*]8Xe]8_:M\N=]8DyJp Q95!^`_:DZ\DeS&EV r K?V5SQIaW`NQDTS Uni
]:DUnc5XPQMO´DW`N
Uni = (ρi)
nU0
i +∆t
1 + θ∆tλi
n∑
k=1
(ρi)k−1bn−k
i .®:p =`
Ecol
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lyte
chni
que.
Ne
pas
repr
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re.
% ^ < ( 1 &&&)1 ( ¯5 9IcLDTE
ρi =1− (1− θ)∆tλi
1 + θ∆tλi.xILWJSEDN1NRDwJI5S1DZzIEVLWJ]:MONQM\V5W¸]8D$S1NQILw:M\Z[M\NRXDTS1N ‖Un‖M = ‖Un‖ ≤ C
pPILPEVLWS1Xe^5_JDW`NTf_:W:DEVLW]8MONQM\V5WuW:XeEDeSRSQIaM[PRDDN&SR_8±ySRILW5NQD]8DS1NQILw:M\Z[M\NRXlDTS1N |ρi| ≤ 1K?VL_:P!NQVL_8N
ifEDl^`_:M|Wdg DeSANPRM[DWs]|g Ia_8NQPRD$^`_:D$ZzI EV5WJ]8M\NRM[VLW Jp `¯LlS1M
0 ≤ θ < 1/2f|DN^5_JMmDTS1NNRV5_AVL_:PHSSQIjNQM[S1nIaM\NRD$SRM
θ ≥ 1/2p
8 ? U DViZBDeSANEZ[ILM\Pv]:IaWJSZ®g DTS1NRM r IjNRM[VLW ®:p =`l^`_:DyKJZ\_JS θ DTS1NLPHIaWJ]|f|KJZ\_JSZ\DEVD±yEM[DW`N]8DcjIaW`NZ\DuNRDP r D bn−ki
DTS1NvK?DNQMONep:mWnILMONefUEDN1NQDtKJPRV5K:PRM[XNQXuEV5PRPQDTSRK?VLWJ]~Y _JWI r VLPRNRMzSRSRD r DTW5N,DbK?VLWJDW`NRM[DZKJIaP&Z[DSQEHG:X r I$]8DeS=EVLW`NRPQM\wJ_8NRM[VLWJS&KJILSQS1XTDTS,]8_*NQDP r DS1V5_:PHEDLpPILPtEVLWJSRXT^`_:DTW5Nef rr D¸S1MlKV5_:PtNRVL_:NRDcjILZ\DT_:P 1/2 < θ ≤ 1
Z\Dθh¦SQEHG:X r IDTS1NS1NQIawJZ\D5fS1V5W r IabM r _ r ]8DSANHIaw:M[Z\M\NRX DeSANIjN1NQDM[W`NuKV5_:P θ = 1
oZ[DTSuDPQPRDT_:PHSW`_ r XTPRMz^`_:DTSyKJILSQS1XTDTSSg I r VLPRNRMzSQS1DTW5Np?Cg DTS1N=K?VL_:PH^`_:VLMdZ\DSQEHG:X r IM r K:Z[M[EMONQDDTS1N=K:Z[_JS&PQVLwJ_JSANQDDNlS1V5_:cLDTW5N&_:NRM[Z\MzS1XKV5_:Pu]8DeSyK:PQVLw:Z[ r DeS²APHIaMz]8DTSA³wJM\DTW£^5_dg M[ZSRVLM\N r VLM[WJS$K:PQXTEM[Sy^`_:D Z[DSQEHG:X r Is]8DC/PHIaW 5h0&MzEHG:VLZzS1V5W p • 8 ? U £q D&S8+8SANQ r D&]|g XT^`_JIaNRM[VLWJSm]:M R XPQDW`NRM[DZ[Z[DTS2V5PQ]8M[WJILM\PQDTS,®:p/.a5DeSAN/]:MON²1PQILMOh]8D³,EIaPefjS1DeSS1V5Z\_8NQM\V5WJSnIaMzSRILW`NM[W`NRDTPRc5DW:M[PU]:DTSUNRDTP r DeS2]8_N,+K?D exp(−λit)
fjXc5VLZ[_:DW`NmSR_:P2]8DTSXTEHG:DTZ\Z[DTS ]:D2NRD r KS|NRPQTSd]8M R XPQDW`NQDTSBPQILK:K?DZ[VLWJS ^`_:D mini λi = O(1)DN
maxi λi = O(h−2)p
ViZdDb8MzSANQD]8DeS r XNQG:V8]8DTS=W`_ r XTPRMz^`_:DTS=]|g M[W`NRXTLPHIjNRM[VLW ]|g XT^`_JIaNRM[VLWJS=]8M R XPQDW`NQM\DTZ\Z[DTS=VLPH]8M\WIaM[PRDeSPQILM[]8DeSK:Z[_JSvD±yEI5EDeSDNK:Z[_JSvEV r K:Z[M[^`_:XTDTSv^5_JDuZ[D θ hiSREHGJX r I:p2C/MONQVLWJSKV5_:P r X r V5M\PQDyZ[DTSr XNRG:V8]8DeS,]8D=,_:W:5Dh _8N1NHItcLV5M\PefKJIaPFDb8D r KJZ\D5f8Z\DEHGJILK:MONQPRD ]:DC'[( pO0&XTILW r VLM[WJSTf8M\Z+I*_:WsEV r K:PRV r M[SYNQPRV5_:cLDTPDW`NRPQDZ®g _8NRM[Z[M[SQIjNQM\V5W]|g _JW:D r XNRGJV]:DyW_ r XPQM[^`_:D$]|g M\W`NQXLPHIjhNRM[VLW$DWNQD r KJSmPQVLwJ_JSANQDLf r ILM[SmEHGJPQDLfaDNmZ[INHIaM[Z\Z[DFNRPQTSmM r K?VLPRNQIaW`NQD&]8DTS/ST+8S1NR r DeS2YPRXeS1V5_J]8PQDoPHIaK:K?DZ[VLWJS,^`_:DZzIvNQIaM[Z[Z\DDeSAN&K:PQVLK?VLPRNRM[VLWJW:DZ[Z\DIa_W:V r w:PQD]8D r ILM\Z[Z[DTSHp •W$K?D_:N/_8NQM\Z[M[SRDPBZ\D=ETIaPHILENRPQDFcjIaPQM[IaNRM[VLW:W:DTZ:]8D&ZzI]8MzSREPRXNRMzSRIaNRM[VLWKJIaPBXZ[X r DW`NQSm´JW:MzSB^`_:MIEVLWJ]:_:MON=Y*®:p/.a5BK?VL_:P r VLW`NQPRDTPFIL_8NRPQD r DTW5N,ZzISANHIaw:M[Z\M\NRXM[WJEVLWJ]8M\NRM[VLWJW:DZ[Z\D]8_ θ hiSQEHG:X r I:p `j¨ « ¨L OOGF ( _%)$P / GA+
θ $&)@ > 1 7(6 > 1/2 ≤ θ ≤ 1 ( )<; ‖U‖M =√
MU · U J@ #-G< + @-!#"< >7+ <Q/ $O<%L$%"< >7-M/(A+, )@%K(>7+ ;@:/) @ K 1 3 6
‖Un0‖2M + ∆t
n0∑
n=0
KUn · Un ≤ C(
‖U0‖2M +
∫ T
0
‖f(t)‖2L2(Ω)dt+O(1)
)
.
"7& + >N ?<G</ > + A? /#"< Q$O%>7+ > G=/ 1 F76 >) Un = θUn+1 + (1− θ)Un `j¨ « ¨L OO #-G<O!#")A+ =$&)@ >A/ E>< 1 5 )6 -$;* )%/ . #()<+,+ < $;&> 7+ `j¨ « ¨L OOMr ( G@%$R"L?> @<+ @ <;$ $ P1
$O@ >0?(+ ) ;*<0; ?)$'+ @!">. / + >C&>7+ <"7 1 %(6 A/ <)$P N = 1 ( "-. + $R"()W "(+ 2/(2!#")>/#G>"7G2!" G</+ > > % M / >)K >7+ 1 ) Q+ > > G!#") 3 =+ ) U) 3 )6 ( >7?7?<<+,+ A!#") + > > % K %$&M/(7)@%?><Q1 5# %76 *!#"! > %>7+ "7+ @=$&-$%>7+ <"7$=? #?<-$=+ G$*/ + <1 #( # O!")=/(>7)$g #->)$0+ >C #)/ . $ 1 5 %76 %$;< A/#"=?7?<
∆t ≤ Ch2
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
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re.
¯`= O )1 )#& *<
A@-$F:3+-("!#(*$% $2329>E?2;$29,"$$"(*2$ )"9'+ %d:G% (R"$;<(_"(("$"(q(>"(*%'%)(/!#*%)0( (v @-93 *)*%)@-9):G%)@-$ &($29,$291(<:Gd (*)$9=@-("$A(B"!#3$%')(_ / +R:# "!#$9 %'d:G%)@-$ (B+ :# )E?9'@ %)@-$8 6 6 f("9 % 2:G$29+' ("9'E)@ % ("9>*!#$29 %'d: %)@-$29E )" ("$&%)("9 6 62>E+ 9>( *%d:3@-+-9 :G@ $9'@#&2( E?("9,("9 %)@-! :G%)@-3$29>% (*')(" )9,(*%>("9,;@ %)("9'9'("9>(R*$;<(*"(3("$"( (*C E2+-@I"@ %)("9 $29)("$;<2&3$29 ; a6)HB O O VFX3YZ
uh b ^ Xh [ ZTYaX \^d[ ^*b ww ]_\ Z j'k 3[ h Y j']fe<] h L k:N*[b ZTYaX \ e<] h bcSPb h ]_[jl x p x VFX3YZ (Th)h>0 [ \?]^)[ YZ ] e&] w b YThh b &]_^ Z j Y bG\ 3[ h b Y j']_^j'k3[ h Y ]_j_^e&] Ω x V?XGYZ V0h
h ]#^ X [ ^ s]_^r` bc_]ve&] H1
0 (Ω) | e&kz~\ Y ` b3j h b w k Z P X e&]ve&]_^/k h k w ]_\ Z ^Jz~\ Y ^ Pk x V?XGYZ (∆t) [\P]/^)[ YZ ]qe&],` b3^e&]Z ] w `F^N*[ YoZ ]_\?e n ]_jd^Gk_j X x V?XGYZ unh ∈ V0h
h b X \Fc ZTYaX \ e X \ ZJh ]_^c XX j'e X \2\?kd]_^ Un e<bG\2^ h b i b3^*]e&]_^k h k w ]_\ Z ^Jz~\ Y ^Be&] V0h ^ X \ Z c)b h c*[ h k)]_^1` b3j h ] θ
s ^_cSP2k w b&x V2Y limh→0 u0h = u0 e<bG\2^ L2(Ω) |1] Z ^ Y h
] Z ∆tZ ]_\Fe&]_\ Zn ]_jd^]_\j)]_^a`]'c Z b3\ Z,h bc X \?e YZTYaX \e&]/^ Z b i*Yh YZ k l x p |1b h-X j_^ X \Nb
limh→0, ∆t→0
max0≤n≤n0
‖u(tn)− unh‖L2(Ω) = 0.
PVL_JP$NRDTP r M[W:DP$W:V5_JS$M[Z\Z[_JS1NRPQVLWJSWJVaNRPQD*K:PQVLK?V5SDW¬PQDK:PQDWIaW`N$Z®g Db8D r K:Z[D ]8D ZzIVL_SAhSRDTENRM[VLW :p 9tSR_:PZ[I]8MSR?_S1M[VLW~]:IaWSZ®g IjN r V`S1KJG:PQDv]|g _:W¸KV5Z\Z[_JILW5NX r MzSlKJIaP_:WJDSRVL_JPQEDvZ[VahETIaZ[M[SRXDtocLV5M\PlZ[D r IaM[Z\ZzIa5DvYuZ[I M[L_JPRD Jp°$DNZ\DS1DeEV5WJ] r D r w:PRD f YyZzIUM[L_:PQD Jp\ =`p?qdI]:VLW:W:XTD&M[W:MONQM[ILZ\Du0DTS1N/K:PQMzS1D&W_:Z[Z[D]:ILWJSmZ[D]8V r IaM[W:DLpqdIUM[L_:PQD=Jp\&KJPRXeS1DTW5NQD&Z[DTS/PRXeS1_:Z\NQIaNQSK?VL_JP<=lNQD r KJS2]8MSR?XTPRDTW`NQSTpaC/V rr DFZ[D!NRDP r D!SRVL_JPQED/DeSAN2M[WJ]8XK?DW]:IaW`Nm]8_vNRD r KJSTfjZzISRVLZ[_8NRM[VLWEVLWcLDTPR5D=w:M[DW*c5DPHS/_JWPQX5M r DS1NQIjNQM\V5W:WJILM\PQDZ[VLPHSQ^5_JDZ[DNRD r KSFNRDTWJ]cLDPHS!Z®g M[W8´JW:Mkp
Jp\ ViWJS1NQIaW`NHIaW:XeS,Ia_8byNQD r KJS t = 1f2f3f5]8DZzIEV5WJEDTW5NQPQIaNRM[VLW p
H ,+ *% " t,+ +") $!+ 'o" ) qdDTS r XNRG:V8]8DeSBW_ r XTPRMz^`_:DTSBKV5_:PBPQXTSRVL_J]:PRD,Zkg Xe^`_JIjNQM\V5Wy]8DTS/VLWJ]:DTSBSRVLW`N/NRPQTSBSRD r w:ZzIaw:Z[DTS®]:IaWSZ\DK:PQM[WJEM[KD5f r IaMzSKJI5SlNRVL_ AVL_:PHS]:IaWSZ[I*K:PHIjNQM[^`_:DlY EDZ[Z[DTS^`_:D$W:V5_JSIcLV5WJSlc`_JDTSK?VL_JP,Z®g XT^`_JIaNRM[VLW*]8DZ[IEHGJILZ\DT_:Pep
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% 5^ < ( 1 & ¯5 . .¦-/.10 *G(L @ ;@TE> 7OC|6:@TE NU6:@A9B4 Gd4 GE H&N2> GW]8MzSREPRXNRMzS1D!`§:¨Le`U§ ª ZzI&noV5P r _:ZzIjNRM[VLWcjILPRMzIjNQM\V5W:W:DZ[Z[DJp ¯Q.Ld]8D/Zkg XT^`_JIjhNRM[VLWu]8DeSBV5WJ]8DeS=Jp 9`p PVL_:P!EDZzI:f5VLWuEVLWJS1NRPQ_:M\N/_:WJDIaKJK:PRVb8M r IjNRM[VLWcjIaPQMzIjNRM[VLWJW:DZ[Z\D,M[W`NRDTPRW:DDWsM[W5NQPRV8]8_JM[SQIaW`N_:W¡SRVL_SAh¦DTSRKJILED
V0h]8D
H10 (Ω)fU]:Du]8M r DTWJSRM\V5W~´JW:M[D N,+K:Mz^`_:D r DW`Nef_:WS1V5_JSAh¦DTSRKJI5EDv] g XTZ\X r DTW`NQS´JW:MzSQpdqdISRD r MOhi]8MzSREPRXNRMzSRIaNRM[VLW]:D*®:p°¯ .5&DeSAN]8V5WJEZkg IaK:K:PQVb8M r IahNRM[VLW cjIaPQM[IaNRM[VLW:WJDZ[Z\DSR_:M[cILW`NRD :NQPRV5_:cLDTP
uh(t)noVLWJENRM[VLW ]8D
]0, T [Y$cjIaZ[D_JPQS&]JIaWJS
V0hNQDZ[Z\D^5_JD
d2
dt2〈uh(t), vh〉L2(Ω) + a (uh(t), vh) = 〈f(t), vh〉L2(Ω) ∀ vh ∈ V0h, 0 < t < T,
uh(t = 0) = u0,h,∂uh
∂t(t = 0) = u1,h
®:p Q.5V U
u0,h ∈ V0hDNu1,h ∈ V0h
S1V5W`N,]:DTS&ILK:K:PQVbM r IjNQM\V5WJS/]:DTS,]8V5W:W:XDeSFM\WJMONQM[ILZ\DeS u0DNu1p
PV5_:P r V5W5NQPRDTPt^`_:D :p Q.5$IL] r DNt_:WJD_:W:Mz^`_:DSRVLZ[_8NQM\V5W£DNZ[I¶EIaZzE_JZ\DTPt]8D r ILW:M[PQDK:PQIaNRMz^`_:DLf:V5WtM[W`NRPQV8]8_:M\N=_:W:DwJI5S1D(φi)1≤i≤ndl
]:DV0h®^5_JMdW:D]8XTKDTWJ]*KJI5S,]8_*NRD r KSQfJDNVLW*EHG:DPHEHG:D
uh(t)SRVL_JSFZzIvnoVLP r D
uh(t) =
ndl∑
i=1
Uhi (t)φi,
IcLDTEUh = (Uh
i )1≤i≤ndl
Z[DcLDeENRDT_:P,]8DTS=EVVLPH]8VLWJW:XDeS/]8Duhp%:mW KV`SRILW`N
u0,h =
ndl∑
i=1
U0,hi φi, u1,h =
ndl∑
i=1
U1,hi φi, bhi (t) = 〈f(t), φi〉L2(Ω), 1 ≤ i ≤ ndl,
Z®g IaK:KJPRVb8M r IaNRM[VLWvcILPRMzIjNQM\V5W:W:DTZ\Z[D:p Q.5DTS1NmXT^`_:M[cILZ\DTW`NRD,Ia_$S8+S1NR r D,Z\M[W:XTILM\PQD,¢> 6-? ªe« ?§¢ «6 § ªe« ®`vJj¢ « §U « j]|g VLPH]8PQD¯Y$EVD±$EM\DTW`NQS,EV5WJS1NQIaW`NHS
Mhd2Uh
dt2(t) +KhU
h(t) = bh(t), 0 < t < T,
Uh(t = 0) = U0,h,dUh
dt(t = 0) = U1,h,
®:p `V U¡V5WPQDNRPQVL_JcLDyZ[DTS rr DTS ª « ¨5`¸¢U :eT Mh
ª ¢U¬ « D « ¢ «ª 6 Kh^5_JDuK?VL_:PZ®g XT^`_JIaNRM[VLW*]8DZ[IEHGJILZ\DT_:P
(Mh)ij = 〈φi, φj〉L2(Ω), (Kh)ij = a(φi, φj) 1 ≤ i, j ≤ ndl.qmg Db8M[S1NRDTWJEDDNZkg _JW:M[EMONQXLfdILM\WS1M2^`_ g _:W:DnoVLP r _:Z[DDb8K:Z[M[EMONQDLf ]8DZzItSRVLZ[_8NRM[VLW¸]8D Jp 5Sg VLw8hNRM[DW:W:DTW`NnI5EM[Z\D r DW`NKIaPS1M r K:Z\D*]8MzIa5VLWJILZ\MzSRIaNRM[VLWSRM r _:Z\NQIaWJXDt]:DTS r IjNRPQMzEDTS MhDN KhpC/V rr DM\Z DeSANl]8MO±yEM\Z[DDNEV 8NQD_8b*]8D]8MzIa5VLWJILZ\MzS1DTP:p `f:DTW*K:PQIaNRMz^`_:DVLW*PQXTSRVL_8N,W_ r XhPRMz^`_:D r DW`NJp 5/KJILP&]8M[SQEPQXNQM[SQIjNQM\V5WuDN r ILPQEHGJDDTWtNQD r KJSTp
`j¨ « ¨L OG2 T+ $ $& T+ )@-> G/ @-!#")>. #$/ @ -. +,+ %$G/ > -$ -; ("?)> $& / $O<@%. $R>. #=/( + @!">. 2/(%$M#)/(%$'> . 1 76
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¯` O )1 )#& *<
.¦-/.@A £@TE> 7OC 68@8E N6:@19B4 68926%N F,G Gd4$GE H&N2> G e6-G(L HJEWs_8NQM\Z[MzS1Dy_JW:D r XNQG:V8]8Du]8Dt¢ «6 §U¨L;§ « § ª sK?VL_:PvPRXeS1V5_J]8PQDZ[DuS8+8SAhNQ r D$]|g XT^`_JIjNQM\V5WJS]8MSR?XTPRDTW`NRM[DZ[Z\DeSV5PQ]:M\WJILM\PQDTS®:p 5p PUV5_:PS1M r K:Z[MO´JDTPZ\DeSW:VaNHIjNQM\V5WJSf W:V5_JSPQXXeEPQM\c5VLWJS/Z[DST+8S1NR r D$Jp 5!SQIaWJS r DW`NQM\V5W:W:DP,ZzI]8XK?DW]:IaWJEDS1KJIaNRMzIaZ[DDW h
Md2U
dt2(t) +KU(t) = b(t)
U(t = 0) = U0,dU
dt(t = 0) = U1,
®:p `aVQU*VLW S1_JK:KV`S1D^`_:D
b(t)DTS1NEVLW`NRM[W_*SR_:P
[0, T ]p?W ]8XeEV5_:KDZkg M\W`NRDTPRcjILZ\Z[D]8DNQD r KJS [0, T ]DTW
n0KJILS]8DtNRD r KJS ∆t = T/n0
f2VLWK?V5SRDtn = n∆t 0 ≤ n ≤ n0
f2DNVLW¶W:VaNQDUnZkg IaK:K:PQVb8M r IaNRM[VLWs]8D U(tn)
EIaZzE_JZ\XtKJIaPv_:W¶SQEHG:X r IJpPVL_:P 0 ≤ θ ≤ 1/2V5WK:PQVLK?V5SRDyZ[D
θ©Qe¨ 4<6
MUn+1 − 2Un + Un−1
(∆t)2+K
(
θUn+1 + (1− 2θ)Un + θUn−1)
= θb(tn+1) + (1− 2θ)b(tn) + θb(tn−1).
®:p L5qdVLPHSR^`_:D
θ = 0faVLWIaKJKDTZ\Z[DJp 55Ue¨ 4<6 u`| « ¨ «oª yoM[ZJWdg DeSANBDTWnIaM\Nmc`PHIaM r DW`NmDb8K:Z[MzEM\NRD^`_:D!S1M`ZzI r IjNQPRMzED!]8D r ILSQSRD M DTS1NU]:M[ILLVLWIaZ[Dep PVL_:P]8X r ILPRPQDP Z[D/SQEHG:X r I=M\Z5nIa_:NEVLW:WJI[NQPRD
U0 DN U1 fED^`_ g VLWVLw8NQM\DTW5N,5P 5EDIa_8btEVLWJ]:MONQM\V5WJSFM\WJMONQM[ILZ\DeSU0 = U0
DN U1 − U0
∆t= U1.
WSREHGJX r IFK:Z[_JS noPQXT^`_:D rr DTW`Nd_8NRM[Z[M[SRXBEILP K:Z[_JSLXWJXPHIaZjDTS1NdZ[D/e¨ 4(6 v¢ $8 2pPV5_:P,PRXeS1V5_J]8PQDZ[DST+8S1NR r D=²RI r V5P1NQM ³
Md2U
dt2(t) + C dU
dt(t) +KU(t) = b(t)
V5WILK:K:PQV8EHG:DU(t), dU/dt(t), d2U/dt2(t)
KJILPFNRPQVLMzS!SR_:M\NRDUn, Un, Un
MUn+1 + CUn+1 +KUn+1 = b(tn+1)
Un+1 = Un + ∆t(δUn+1 + (1− δ)Un)
Un+1 = Un + ∆tUn +(∆t)2
2
(
2θUn+1 + (1− 2θ)Un)
®:p Lº5
Ic5DTE0 ≤ δ ≤ 1
DN0 ≤ θ ≤ 1/2
p8qdVLPHSR^`_:DZ[I r IjNQPRMzED]|g I r VLPRNRMzSRSRD r DTW5NFDTS1N,W_:Z\Z[D$ C = 0fV5WtK?D_:N=XZ[M r M[W:DP,Z[DTS,SR_:M\NRDeS Un, Un
DN®:p Lº`BDeSAN,Xe^5_JM\cjIaZ[DW`N=YMUn+1 − 2Un + Un−1
(∆t)2+K
(
θUn+1 + (1
2+ δ − 2θ)Un + (
1
2− δ + θ)Un−1
)
= θb(tn+1) + (1
2+ δ − 2θ)b(tn) + (
1
2− δ + θ)b(tn−1).®:p j¹5
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% 5^ < ( 1 & ¯55,D r IaPH^`_:VLWS,^5_JDKV5_:P δ = 1/2
Z[DSQEHG:X r I]8D 0=D r IaP uPQDT]:VLW:W:DZ[D θ hiSREHGJX r I:pO:BW K:PQIahNRMz^5_JDLfaK:Z[_JSδDeSANm5PQILWJ]|fjK:Z[_JS2Z[D&SREHGJX r IlDTS1NB]8M[SQSRM\KJIaNRM\n|DNmPQVLw:_JS1NRDZ\DeS2DPQPRDT_:PHSW`_ r XTPRMz^`_:DTSKJILSQS1XTDTSFSTg I r V5P1NQM[SQSRDW`N!KJZ\_JS&c`M\NRDf rTr DSTg M\Z|DeSAN r VLM[WJSFK:PQXTEM[STp
`j¨ « ¨L OGGF #-G<:!") + $O@ > / E > -$; /) /# 1 ; ?$ 6 ?<#"(δ 6= 1/2
N(/) /# % ?<"7δ = 1/2
%θ 6= 1/12
N %L/ /G A$δ = 1/2
θ = 1/12
1#=$&+ ;< > + @!">. =$.>7$M> . $&$R - 6%W~XNR_J]8M[D$ZzI SANHIaw:M[Z\M\NRX$]8DyEDTSSQEHG:X r I5SIa_sS1DTWJS]8D$ZzI*x=X´JW:M\NRM[VLW:p :p 9:p PVL_JPXcMONQDP]8DTS/EIaZzE_JZ[SmNRPQVLK$Z[VL_:PH]:STfaVLWuSRD=EVLW`NRDTW5NQD&]|g XNR_]8M\DTP/ZzIEVLWJ]8M\NRM[VLW$WJXTEDeSRSQIaM[PQD&]8D=S1NQIawJM\Z[MONQX]8D VLW 0=D_ r ILW:WcLVLM[P!Z[I ,D r ILPQ^`_:D¯8p°¯8p°¯ =`p`qdDPQXTSR_:Z\NQIaN!SR_:M\cjILW5NFDTS1N,]:IaWJS!Z[D rTr DlDTSRK:PQMON^5_JDlZ[Dq D rr D¯:p 9Jp Jp L 94 J ;=.G=0<E B3%8M3G;"HAYZ5EF3_3 Y B I >I ,90 δ < 1/2 i 018H3[G[QV.[j.>2B=G01HG[ 83 I ,676H.G.>HG E7A[G.>B[YZ>0UG 3 δ ≥ 1/2 I @;.>9E>01?0<.> 4A=;3G=G01BO3^E73!G[ 018:01VA^E73 ._32Y n3G=V.[j.2B=G L AB[0 \A35G[0 δ ≤ 2θ ≤ 1 i >9E>0UG_3 i G[0 0 ≤ 2θ < δ
3818M35/R 3[G[G>?0UG >01 3Z3#G. GW8M ;=.9E>01?0<.>
maxiλi (∆t)2 <
2
δ − 2θ,
®:p . 8M3G
λiG.>8M3G L 832B=G69B.69B3[G_E73 KU = λMU
L .01B I FI 8 ? U 9 F·W$W:D&PQDTEVLW:WI)zNmKJILS2M rr XT]8MzIjNQD r DW`N!]JIaWJS&®:p UZ[IEVLWJ]:MONQM\V5WtC UqkC/VL_JPQILW5NRh JPRM[DT]:PRMzEHGJS1h¦qdD +: _JSR_:DZ[Z[D
∆t ≤ Ch K?VL_:PZkg Xe^`_JIjNQM\V5W]8DeSVLWJ]:DTSFoc5VLM[PUZ[D,q D rr D¯8p 9:p 5pW:mWnIaM\NTf|V5WK?D_8N r VLW`NRPQDP^`_:DLf|SRMZ\D r IaM[Z\ZzIa5D ThDTS1N_:W:M\noVLP r X r DW`NlPQX5_:Z[M\DTPlIL_S1DTWJS&VQU*NRVL_:NRD r IaM[Z[Z\DEVLW`NRM[DW`N=_JW:Dw?VL_:Z[D]8DvPQI+5VLW Ch ®IcLDTE C > 0
M\WJ]:XK?DWJ]:ILW`Nl]8DZzIr ILM\Z[Z[DefUIaZ[VLPHSVLWI D R DTENQM\c5D r DW`N maxi λi = O(h−2)pq DuZ\DeENRDT_:PvI K:_cLXTPRM\´JDTPEDynILMONDW]8M r DWJSRM\V5W N = 1
Z\V5PQS]8DuZkg :mbDTPQEM[EDu:p :p =JpC/VLW`NRPHIaM[PRD r DW`NIa_ETILSKJILPQILwV5Z\Mz^`_:DLfdZ[IEVLW]8MONQM\V5WC Uq ®:p !W g DeSAN&KILSFNQPRV5K*SRXc5PQDK:_:MzSR^`_ g VLW*K?D_8N=KJPRDTWJ]8PQD]:DTS,KJI5S&]8DNQD r KJS∆t]:DBZkg V5PQ]8PQDm]8_KILSU]|g DeS1KILED
hpjC/DK?DWJ]:ILW`NTf^5_JMONRNRDFY,M\Wc5DPHS1DTP _:WS8+8SANQ r DmZ[M\W:XeIaM[PRDoK?VL_:PKV5_:cLV5M\PBETIaZzE_:Z[DP
Un+1DTWnoVLWJENRM[VLWy]8D
Un, Un−1DN/]8_ySRDTEV5WJ] r D r w:PRDf`Ia_8NHIaW`Nm_8NRM[Z[M[SRDPZ\D&SQEHG:X r I]8D0=D r IaP K?VL_:PB]:DTS2cjIaZ[D_JPQS2]8D δ DN θ NQDZ[Z\DeSm^`_ g M[ZJSRVLM\NmM\WEVLW]8MONQM\V5W:W:DTZ\Z[D r DW`NSANHIaw:Z[DLp|q D$SRD_:ZmETILSM\W`NQXPQDTSQSRILW5NK?VL_:P_:W~SREHGJX r ItS1NQILw:Z[DSRVL_JSEVLWJ]8M\NRM[VLWsC q¶DeSANZ[D$EILSV U$M[ZJDTS1N/`| « ¨ «oª f5Eag DTS1N1hiYjhi]8M\PQD&^`_ g M\ZW g +IKJI5SB]8DST+8S1NR r D&Z[M[W:XTILM\PQD&YPQXTSRVL_J]8PQD,YEHGJI5^5_JDKJILS/]8D,NRD r KJSp :mWyDR?DNTf`_:WuSQEHG:X r IDbKJZ\MzEM\NRDW:XTEDTSQS1M\NRD,NQPReS2K?D_u]|g VLK?XPHIjNQM\V5WJS2KJILPBKJI5SB]8DNRD r KJS&DN&EV5WJ]8_:M\N=]8VLWElY]8DeS,EILZ[E_:Z[SFK?D_ EV 8NQD_8b|p:qdISRD_:Z[DKV`SRSRM[w:M\Z[M\NRXKV5_:P,Z[DSREHG:X r I]8D 0&D r IaP Jp°a¹5F]|g NQPRDDb8K:Z[M[EMONQDDTS1N^`_:D θ = 0DN^`_:DZzI r IjNRPQMzED]8D r I5SRSRD M SRVLM\N]8M[ILLV5WJIaZ[DLPLEDvYt_:W:DvnoVLP r _:Z[Dv] g M[W`NRX5PQIaNRM[VLWILK:K:PQV8EHG:XDocLV5M\PlZzI ,D r IaPH^5_JDp =Jp 95p|C/DSREHG:X r IvDb8K:Z\MzEM\NRDDeSAN=SRVL_:c5DW`NF_8NRM[Z\MzSRXDW*K:PHIjNRMz^`_:DIcLDeE δ = 1/2
p •+ 6 ?§ ª j ªe« ?§ :mZ\Z[DDTS1NFNRPQTS,SRD r wJZ[ILw:Z\DYEDTZ\Z[D]8_qdD rr D:p :p/. r IaMzSFIcLDeE]8DeS,EV r hK:Z\MzEIaNRM[VLWJS=NRDeEHG:W:Mz^`_:DTSl^`_:MPQILK:KDTZ\Z[DW`NZ\Dq D rr D$¯8p 9:p :p W]8XeEV r K?V5SRD Un
DNZ\DS1DeEV5WJ]r D r w:PQDt]:D¸®:p j¹5v]:ILWJSZzIwJILSRDuV5P1NQG:VLW:V5P r ILZ\DyK?VL_JP M DNV5P1NQG:VL5VLWJILZ\D$K?VL_JP K p P2IaP
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¯`j O )1 )#& *<
EVLWJSRXT^`_:DTW`NTf Jp°a¹5BDTS1N&XT^`_:M[cjIaZ[DW`NTf:EV r KV`SRILW`NRDKJIaP,EV r KV`SRILW`NRDLf:YUn+1
i − 2Uni + Un−1
i
(∆t)2+λi
(
θUn+1i + (
1
2+ δ − 2θ)Un
i + (1
2− δ + θ)Un−1
i
)
= bni ,®:p a¯L
Ic5DTE ]8DeSyW:VLNQIaNRM[VLWJS$XTcM[]8DTW`NRDTSoZ[DTSλiS1V5W5NtZ\DeSycjIaZ[D_:PHS$K:PQVLKJPRDeSy]8_·S8+8SANQ r D r IjNRPQMzEM[DZ
KVi = λiMVip!C/V rr D*Z\D SQEHG:X r I :p a¯5DeSANyYNRPQVLMzSW:M[cLDTIL_8b·oc5VLM[PZ[I~VL_SAhiS1DeENQM\V5W¯:p ¯:p .5f8V5WtM[W`NRPQV]:_:MON&_JW:D r IaNRPQM[ED]|g M\NRXTPQIaNRM[VLWJS Ai
NRDTZ\Z[D^`_:D(
Un+1i
Uni
)
= Ai
(
Uni
Un−1i
)
+(∆t)2
1 + θλi(∆t)2
(
bni0
) IcLDeEAi =
(
a11 a12
1 0
)
,
a11 =2− λi(∆t)
2( 12 + δ − 2θ)
1 + θλi(∆t)2, a12 = −1 + λi(∆t)
2( 12 − δ + θ)
1 + θλi(∆t)2.
WDW ]8Xe]8_:M\N=^`_:D(
Un+1i
Uni
)
= Ani
(
U1i
U0i
)
U0i +
(∆t)2
1 + θλi(∆t)2
n−1∑
p=0
Api
(
bn−pi
0
)
.®:p 95
qIEVLW]8MONQM\V5W*W:XTEDTSQSRILM\PQDl]:DSANHIaw:M[Z\M\NRX]:D VLW0&DT_ r IaW:W*DTS1N ρ(Ai) ≤ 1pW EILZ[E_:Z\D]8V5WJEZ[DTSFcjILZ\DT_:PQSFKJPRV5K:PRDeSF]8D
Ai^`_:MdSRVLW`N,Z[DTSFPHILEM[W:DeS,]8_tK?VLZ +W r DDTW µ S1_JM\cjIaW`N
µ2 − a11µ− a12 = 0]:VLW`N,Z\D]8MzSQEPQM r M[WJIaW`N,DTS1N∆ =
−4λi(∆t)2 + λ2
i (∆t)4(
( 12 + δ)2 − 4θ
)
(1 + θλi(∆t)2)2.
WcLXPQM\´JDIaMzS1X r DW`Nl^`_:DZ[DTSPHILEM[W:DeS=]8DEDK?VLZ +W r DvSRVLW`N]:D r V8]8_:Z[DM[W8noXPQM\DT_:P=V5_ X`IaZYulSRM?DN&S1DT_:Z[D r DW`N!SRM VLWtDTS1N!]:ILWJS!Z®g _:Wt]:DTSF]8D_:b$ETILSFSR_:M\cjILW5NHS 8SRVLM\N ∆ ≤ 0DNa12 ≥ −1
fSRVLM\N∆ > 0
DN1−a12 ≥ |a11|
p WEILZ[E_:ZnILS1NRMz]8M[D_8b r ILM[S2SRM r KJZ\D,]:ILWJSUZ\D,K:PQM[WJEM[KD,EVLWJ]8_JMONILZ\V5PQSY*ZzI EVLWJ]8M\NRM[VLW ®:p ecLV5M\PSRMmW:XeEDeSRSQIaM[PRD$Z[DNRGJXVLPQ r D JfSRDTENQM\V5W 9:fdEHGJILK:M\NRPQD ]JIaWJSW'[ (op
`j¨ « ¨5 OG ( n #$ / T+ ^->)$ + ; /"&$ 3 N %$;' / Gδ = 1/2
λi (∆t)2 = 4
1−4θ #!") + '$O&)@ > /( > '%$& )$;&> 7+ Q/7>7)$ %g%>)$J<%)@< >7-
!#")
Ai =
(
−2 −11 0
)
, An
i = (−1)n
(
n+ 1 n−n 1− n
)
.
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cos(ωit)fIcLDTE
ω2i = λi
pC/V rr D$Z[DTSSQEHG:X r ILSlW_ r XPQMz^5_JDTS]|g M\W`NQXLPHIjNQM\V5W]8D ®:p `aV5W5NNQDWJ]:ILWJEDYI r V5P1NQM\P_:W¡K?D_ EDeSV`SREM\Z[Z[IaNRM[VLWJSoV5W¡]8M\Ny^5_dg M[Z[SSRVLW`N]8MSR _JS1M\nSVL_¶]8M[SQSRM\KJIaNRM\nS cLV5M\PvZ[I]8M[SQE_SRSRM\V5W IaK:PQTS=Z[Itx=X´JW:M\NRM[VLW¯:p 9Jp .5fVLW K?D_8NZ[D_:PlK:PQXnoXPQDP_:W:DIa_8NQPRD r XNRG:V8]8D]8DPQXhS1V5Z\_8NQM\V5Wu]8DJp awJI5S1XTDSR_:P!_:W:D]8XTEV r KV`S1M\NRM[VLWt]8D=ZzIvSRVLZ[_8NQM\V5WuS1_JP/Z[DTS/cLDeENRDT_:PHSmK:PQVLK:PQDTSVi]:_ ST+8S1NR r D r IjNQPRMzEM[DZ KVi = λiMVi
p W EHG:DPHEHG:DZzI$S1V5Z\_8NQM\V5WU(t)]8Dy®:p `aFSRVL_SFZ[InoVLP r D
U(t) =∑
i∈I
xi(t)Vi,
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dt2(t) + λixi(t) = bi(t)^5_dg V5WK?D_:NBM\W`NQXLPQDPnI5EM[Z\D r DW`NTfaPHIaKJM[]8D r DW`NTfaDNmK:PQXTEMzSRX r DW`NmKJILP2Z[I r XNQG:V8]8D,W_ r XPQM[^`_:D]8DlSRVLWtEHG:VLM\b|p8C/DN1NRD r XNQG:V8]8DLf:]8M\NRD]8DTB`B? «ªT« ?§U¢ ¢ULf5DTS1NBNRPQTS!_8NQM\Z[MzS1XTDDTWr XeEILW:M[^`_:DcM[w:PQIaNRV5M\PQDLpW:mZ[Z\DyIZkg IcjIaW`NHIaLD]|g NRPQDKDT_s]8M[SQSRM\KJIaNRM[cLD5f Eag DTS1N1hiYjhi]8M\PQDv] g I r V5P1NQM\PNRPQTSK?D_¸Z\DeSV5SQEM[Z\ZzIjNQM\V5WJSp M[DW~S:Pef|ZzI^`_JILZ\M\NRXeS]8DTSPQXTSR_:Z\NQIaNQS]8XK?DW]K?VL_JP_:W:DLPHIaWJ]:DKJIaPRNB]8_uEHG:V5MOb]8DTS r V8]8DeSmDTW5NQPQILW`NB]:IaWJSBZ[I]:XTEV r KV`S1M\NRM[VLW f`EHG:VLM\bSRVL_JcLDW`N r VaNQM\c5X,KJIaP/]8DTSEVLWS1Mz]8XPHIjNQM\V5WJS!K:G-+8S1Mz^`_:DTSTp •
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uh b ^ Xh [ ZTYaX \ T^)[ ^_b wvw ]_\ Z j'k3[ h Y j)] e&] h L k:N*[b ZTYaX \ e&]_^ X \?e<]_^l x p x VFX3YZ (Th)h>0 [\?]^)[ YZ ] e&] w b YThh b &]_^ Z j Y b3\ 3[ h b Y j)]_^ j'k 3[ h Y ]_jd^e&] Ω x V?X3YZ V0h
h ] ^ X [^ s]_^a` b<cd]e<] H1
0 (Ω) | e&kz~\ Y ` b3j h b w k Z P X e<]ve&]_^qk h k w ]_\ Z ^Jz~\ Y ^ Pk x V?X3YZ (∆t) [\?]^)[ YZ ]ve<],` b3^/e<]Z ] w `F^$N*[ Y,Z ]_\?e n ]_j_^ k_j X x VFX3YZ unh ∈ V0h
h b X \?c ZTYaX \ e X \ ZBh ]_^fc XX j'e X \2\Pk)]_^ Un e<b3\^ h b i bG^"]e<]_^/k h k w ]_\ Z ^~zJ\ Y ^ e&] V0h ^ X \ Z c)b h c*[ h kd]_^m` b3j h ] ^_cSP2k w be&]q] w bGj<x V2Y limh→0 u0
h = u0 e<b3\^L2(Ω) | limh→0 u1
h = u1 e<bG\2^ L2(Ω) |] Z ^ Y h ] Z ∆tZ ]_\Fe&]_\ Z,n ]_jd^ ]_\ j']_^a`])c Z bG\ Zh b c X \?e YZTYaX \
e<]/^ Z b i*YTh YZ k l x p |mb h-X jd^ X \blim
h→0, ∆t→0max
0≤n≤n0
‖u(tn)− unh‖L2(Ω) = 0.
PV5_:PmNQDP r M[W:DTP/W:V5_JSmM[Z[Z\_JS1NRPQVLWSmW:VLNRPQD,K:PRV5KV`SmDTWuS1M r _JZ[ILW5N/Z[IKJPRV5KJIa`IjNRM[VLW]|g _:W:DVLWJ]8DS1K:GJXPQM[^`_:D]:ILWJSF_:W:DETIc`M\NRXETIaPQPRXTDlSR_:P,Z[DTS,KJILPRV5M[SF]8DZzIL^`_:DTZ\Z[DDZ[Z\DSRDPRX JXTEHGJMONepWPQXTSRVL_:N]8VLWJE/Zkg Xe^5_IjNRM[VLW]8DTSV5WJ]8DeSU]JIaWJSΩ =]0, 1[2
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x ∈ Rn fmM[ZSTg IaLM\Ny¢>[J ªT« « e ªe« ?§ § :8 « 8¨5§ ªe« §|ULfNHIaWJ]8MzS,^`_:DS1Mx ∈ Zn VLWtII R ILM\PQDY]8D,>[J ªT« « e ªe« ?§s¨L? « §U ª « jVL_DTW$cjIaPQM[ILw:Z\DeS2]8MzSREPRNRDTSTp ILZ\5PRXFZ[DTSmILK:KJILPRDTWJEDeSfjZkg V5K8NRM r M[SQIjNQM\V5WDW$cjIaPQM[ILw:Z[DTS2EV5W`NRM[W`_JDTS2DeSANSRVL_JcLDW`NK:Z[_JS&²inI5EM[Z\D³^`_:DZkg V5K8NRM r M[SQIjNQM\V5WDTWcjIaPQMzIaw:Z[DTS]8MzSREPRNRDeSlETIaPVLW¸K?D_8N_8NQM\Z[MzS1DTPZ[IWJVaNRM[VLW ]:D]8XPQM[cLXD^`_:MDTS1N&noVLPRN=_8NQM\Z[DNHIaW`N]8_ K?VLM[W`N]8Dc_:DNRG:XTVLPQM[^`_:D^`_ g IaZ[LV5PRM\NRG r Mz^`_:DLpqBg V5K8NRM r M[SQIjNQM\V5WEV r w:M\WIjNRV5M\PQDDTS1N&WJIjNQ_:PQDZ[Z\DDNDTSQS1DTW5NQM\DTZ\Z[D]:IaWS&]8DW:V r w:PQD_8btK:PRV5w:Z[ r DTS]:DyZ[I*PQDTEHGJDPHEHG:DVLK?XPHIjNRM[VLWJW:DZ[Z\D5pUCg DTS1N_:W]8V r ILM\WJD$V U fUY EVaNQXu]8DyPQXTSR_:Z\NQIaNQSNQG:XV5PRMz^`_:DTSPQM[LVL_JPRDT_8b|f JDT_:PRMzSQS1DTW5N2]8D!W:V r w:PQD_JSRDTS ²AGJD_:PQM[S1NRMz^`_:DTSA³BDTSQSRDW`NRM[DZ[Z\DeSdK?VL_JPUV5w8NRDTW:M[P]:D/w?VLW:WJDTSK?DPRnoVLP r IaWJEDTSFIaZ[LV5PRM\NRG r Mz^`_:DTSTp
PVL_JP´JWJM\PEDNRNRDtw:PRTcLDyM[W`NRPQV8]8_JENQM\V5WsW:VL_SM[WJ]8Mz^5_JVLWJSvZ\DuK:Z[ILW¡]8DtZ[I SR_:M\NRD]8_¡EV5_:PHSpC/D EHGIaK:M\NRPQD cjI~K:PRM[WJEM\KJILZ\D r DW`NyKV5P1NQDPuSR_:PyZzIs^`_:DTS1NRM[VLW ]8DZ®g Db8MzSANQDWJED DNt]8D Zkg _:W:MzEM\NRXDTW¶V5K8NRM r M[SQIjNQM\V5W EVLW`NRM[W_:DLf2^`_:D ED SRVLM\N$DW¬]8M r DTWJS1M[VLW¶´JW:M[D VL_¶M\W:´JW:M[DLp:mW¬KJILP1NQM[E_:Z[M\DTPTfWJVL_JSvcLDTPRPQVLWJSZ[DP5Z\DuEPR_EMzIaZ/]:DuZzI¨L?§ ` «ª 6 K?VL_:PvVLw8NQDW:M[P]:DTSvPRXeS1_:Z\NQIaNQSv]|g Db8M[S1NRDTWJEDDTW¶]8M r DWJSRM[VLW¡M[W8´JWJM\D5pqdD C/GIaK:M\NRPQDe¹]8XTcLDZ[VLKJKDTPQI Z[DTSEVLWJ]8M\NRM[VLWSv]|g VLK:NRM r ILZ\M\NRXDNZ[DTSILZ\5VLPQMONQG r DTS!W_ r XPQM[^`_:DeS&^`_:M DTW*]:XTEV5_:Z[DW`NTpJqdDvC/GJIaKJMONQPRD$5EVLWJS1NRM\NR_JD_:W:DM[W5NQPRV8]8_ENRM[VLWIL_8b r XNQG:V8]8DTS]8D$Z[I*PQDTEHGJDPHEHG:DvV5KXTPQIaNRM[VLW:WJDZ[Z\D5f|M\WJEZ\_IaW`NZ[IK:PQVL5PQI rr IaNRM[VLWZ\M[W:XTILM\PQDDN]:DTS r XNRGJV]:DTSEV r w:M[WJIjNQVLM[PRDeSpWPVL_:PK:Z[_JS]8D]8XNHIaM[Z[SSR_:PlZ®g VLK:NRM r MzSQIjNRM[VLWW:VL_SPQDWc5V +LVLWSZ[DZ[DTENQD_:P&IL_8buV5_:cPQILLDTSd'/.)(¦f/' º>(kfQ'[e¯)(kfQ'[=>(kf/'°¯a¹>(kf/'°¯L¯(¦fQ' 9 9)(¦p .102.@A G(L HJF,GE
P2ILSQSRVLWJS=DW PQDc_:D^`_:DTZ[^`_:DeS&K:PQVLwJZ\ r DeS,N,+K:Mz^`_:DTS]|g V5K8NRM r M[SQIjNQM\V5W f ]|g M r KV5P1NHIaWJEDK:PHIjhNQM[^`_:DV5_yNRGJXVLPQMz^5_JDM[W:X`IaZ[DLf r IaMzSF^5_JM|KDTP r DNRNRDTW5N&]8DlnILM\PQDZ[DNRVL_JPF]:DTS,]8MSR?XTPRDTW`NRDTS2²1w:PHIaW8hEHGJDTSA³]8DZkg V5K8NRM r M[SQIjNQM\V5W pC/V rr DTW V5WJS=KIaP^`_:DZz^`_:DTSDb8D r K:Z[DTSlDTWj¨ 4 `j¨ 4 6 j ªT« §U§®?fELg DeSANRh¦Yah¦]8M[PQDDTW*VLK8NQM r MzSRIaNRM[VLW*]8DZ[I5DTS1NRM[VLWuV5_ ]8DZ[IK:PQVL5PQI rr IaNRM[VLWt]:DTSFPQDTSQS1V5_:PQEDTSTp
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)<#-< "&1* ¯aQ9 sL 44 AjU®*¢ ª j:§TB ª ViZlSTg IaLM\Nt]|g _:W£Db8D r K:Z[D]8DK:PRV5LPHI rr DZ\M[W:XTILM\PQD*oVL_K:PQVLLPHI rr IjNQM\V5W Z\M[W:XeIaM[PRDp|q Dw:_8NDeSAN]|g VLK8NQM r MzS1DTPZ[ItZ\M[cPQILM[SRVLW]|g _JW:D r ILP1hEHGJIaWJ]:M[SRD o_:W~K:PQVLw:Z[ r DEZzILSQS1Mz^`_:DDW¸Z\V5LMzSANQM[^`_:DpW~]8MzS1K?V5SRD$]8D M DTW5NQPRDTK LNQSTf|M\W]8M[EXTSKJIaP
1 ≤ i ≤M f5]8MzS1K?V5SQIaW`N/EHGILE_JWu]|g _:WyW:M[cLDeIa_y]8DlSANQVE 8S sip-ViZnIa_8N!Z\M[cPRDTP
NEZ[M[DW`NQSTf5M[W8h]8M[EXTSBKJIaP
1 ≤ j ≤ N f5^`_:MJV5W`NBEV rr IaW]8X=EHGJI5E_:W_:W:D^5_IaW`NRM\NRX rj p`q D=EV 8N/]:D,NRPHIaWJSRK?VLPRN_:W:M\NQIaM[PQD$DW`NRPQD$Z®g DW`NQPRDTK LNiDNZ[DyEZ[M[DW`N
jDeSAN]:VLW:W:X$KIaP
cijpdqdDTScjIaPQMzIaw:Z[DTS]8Dy]:XTEMzSRM\V5WS1V5W5NFZ[DTS,^`_JIaW`NQMONQXTS
vij]8D r IaPHEHGJIaWJ]:M[SRD=KJILP1NHIaW`N,]8DZ®g DW`NRPQDK aN i c5DPHSBZ[DEZ[M\DTW5N j pWuc5D_8Nr M[W:M r M[SRDP$Z[D EV8Ny]8_ NRPHIaWJSRKV5P1NNQVL_8N$DW SQIjNQM[S1nIaMzSRILW`NZ[DTSyEV rr IaWJ]:DTS]8DTSuEZ[M\DTW5NHS oV5WS1_:KJKV`S1D^`_:D ∑M
i=1 si ≥∑N
j=1 rjp 2=_8NRPQD r DW`N=]8M\NTfJVLWcLD_:N&PRXeS1V5_J]8PQD
inf(vij )
M∑
i=1
N∑
j=1
cijvij
S1V5_JSFZ\DeS,EV5W5NQPQILM\W`NQDTSF]8DZ[M r M\NRDTS=]8DTS,S1NRV8E 8SFDN&]8DSQIjNQM[S1nILENRM[VLW*]8DTS,EZ\M[DW`NQSvij ≥ 0,
N∑
j=1
vij ≤ si,
M∑
i=1
vij = rjK?VL_JP
1 ≤ i ≤M, 1 ≤ j ≤ N.
C/DlKJPRV5w:Z\ r DlDeSAN&_JW*ETILSFKJILP1NQM[E_:Z[M\DTPF]:DK:PRV5w:Z[ r D]8DNRPHIaWJSRK?VLPRNTp • sL 4 F AjU®*¢>[ ¨ ª ªT« § ViZSTg ILLM\N]|g _:W Db8D r KJZ\D¡] g V5K8NRM r M[SQIjNQM\V5WEV r w:M[WJIjNQVLM[PRD&V5_uDTWucjIaPQM[ILw:Z[DTSmDTW`NRM[PQDTSTpQV r Ia5M\WJD%S=c5VL_JS!YZzIN NQDl] g _:WJDlILLDWED r IaNRPQM r VahW:M[ILZ\D5p\p[p8VLM\N
NnoD rr DeSf`M\W]8M[EXDeSBKJILP 1 ≤ i ≤ N f`DN N G:V rr DTSTf5M[WJ]8MzEXeSmKJILP 1 ≤ j ≤ N pMZ[I=noD rr D i DNUZkg G:V rr D j SRVLW`N2]|g ILEEVLPH]KV5_:P2S1D r IaPQM\DTPdZ[D_:PcjIaPQMzIaw:Z[D/]|g ILETEV5PQ] aij
cjIa_8NT]:ILWJS Z[DBETILSdEVLW`NRPHIaM[PQDUDTZ\Z[DBcjIL_8N¹:p,DTS1NRV5WJSdEZ[I5SRSRMz^5_JDTS S1DT_:ZzS Z[DTS r ILPRMzIa5DTS G:XNRXPQV5SRDb8_:DTZ[SS1V5W5NIa_8NQVLPQM[SRXTSDNZzI=K?VLZ +5I r M[DBW g DTS1NKJILSUIL] r MzS1D5p\p[p8kWvcLDPQPHI&]:IaWSdZzIl8VL_JS1h¦SRDTENRM[VLWy5Lp 9:p ^5_JDEDNRNRDG-+KVLNRG:eS1D]8D r V5W:VL`I r M\DW g DTS1N&KJI5SFW:XTEDTSQSRILM\PQD°/qdDw:_8N=]8_ ADT_ DTS1N=]8D r Ijb8MOhr MzSRDP=Z[DW:V r w:PQD]8D r ILPRMzIa5DTSFDW`NQPRDeS=EDeS N noD rr DTSDN N G:V rr DTSTp32=_8NRPQD r DW`N]:MONefV5WEHG:DPHEHG:Dy_:WJDtK?DP r _8NHIjNQM\V5W σ ]:IaWSZkg DTWJSRD r wJZ\D]8DTSvKDTP r _8NQIaNRM[VLWJS SN]8D 1, ..., N ^`_:MPRXeIaZ[M[SRDZ[D r Ijb8M r _ r ]:D
maxσ∈SN
N∑
i=1
aiσ(i).
W:D&cjILPRMzIaW`NRD,EVLWJSRM[S1NRD&YIL_8NRV5PRMzSRDPm]8DeS2cjIaZ[D_:PHSm]8DaijDTW`NRPQD&¹DN=5p`C/D,N,+KD&]:D&K:PQVLw:Z[ r DTSDTS1NuIaK:K?DZ[X K:PQVLw:Z[ r D ]|g I R DTENQIaNRM[VLW oM[Z=M[W`NRDPQcM\DTW`Nu]:IaWJSu]8DTSyEVLW`NRDbNRDTSyM\WJ]:_JSANQPRM[DZzS$K:Z[_JSS1XTPRM[D_8bEV rr DBZkg I R?DeENQIaNRM[VLW]8DTSXT^`_:M[KJIa5DTSDN2]8DTSUIcM\V5WJSU]:IaWJS_:W:DFEV r KJIa5W:M\D!IaXTPRM[DW:WJDep
M[DW^`_:DEDlW:DlSRVLM\NFKJILS/noVLPHEX r DW`N!ZzI r DTM\Z[Z[D_:PQD r ILW:M\TPRD]8DlKV`S1DTPFZ\DlK:PQVLw:Z[ r DLfV5WuK?D_8NZ®g XTEPRM[PRD!S1V5_JS_:W:D/noVLP r D!cLV5M[SRM[W:D/]8D!Zkg :2b8D r KJZ\DFº:p[Lp[Lpq DeScjILPRMzIaw:Z[DTS]8D,]8XTEM[SRM[VLWSRVLW`NWJVaNRXTDTSvij^`_:M!cjIa_8NuSTg M[Z+sI r ILPRMzIa5D$DW`NRPQDuZzI*noD rr D i DNvZkg G:V rr D j DN¹S1M[W:V5W pWc5D_8Nr Iab8M r MzS1DTP
sup(vij )
N∑
i=1
N∑
j=1
aijvij
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¯L = ) < *<,T%<
SRVL_SFZ\DeS,EVLW`NQPQILM\W`NRDeSvij = 0
V5_1,
N∑
j=1
vij ≤ 1,
M∑
i=1
vij ≤ 1K?VL_JP
1 ≤ i, j ≤ N.
WK?VL_JPRPHIaM\NEPQVLM[PQD^`_:D!EDmKJPRV5w:Z\ r DB]|g I R DTENHIjNQM\V5WDTS1NUSRM r K:Z[DmK:_:MzSQ^5_dg M[ZQ+I,_:WW:V r w:PRDm´JW:M]:D&KV`SRSRM[w:M\Z[M\NRXTS/^`_ g M[Z`²RSR_8±NA³&]|g XTW_ r XPQDPmK?VL_:PBNRPQVL_:c5DP2Zkg V5K8NRM r _ r p ViZSTg IaLM\Nmw:M[DWyS :PB]|g _:WZ[D_JPRPQDEILP=ZzIyEILPQI5ENRXTPRMzS1NRMz^5_JD]:DTS&KJPRV5w:Z\ r DTS=EV r w:M\WIjNRV5M\PQDTSFDeSANlZ\DT_:P&NQPReS&5PQILWJ]W:V r w:PRD]:D EV r w:M[WJIaMzS1V5WJSKV`SRSRM[w:Z\DeSy^`_:M&D r K EHGJD*NRVL_:NRD XW_ r XTPQIaNRM[VLW¡Db8GJIa_JS1NRM[cLD*DTW KJPQIaNRMz^5_JDLp0=XTILW r VLM[WJSlKV5_:PEDK:PQVLw:Z[ r DM\ZBDb8M[S1NRD]8DeSlNRDTEHGJW:M[^`_:DeS]8DPRXeS1V5Z\_:NRM[VLWD±yEILEDTSoc5VLM[PZ[I8VL_JS1h¦SRDTENRM[VLWLLp 9:p ap • sL 4U AjU®`¡¢U e:¨j©
©Q¢U WyK:PQVLwJZ\ r DEZzILSQS1Mz^`_:D&DTS1N/EDZ[_:M?]:_uSRI5EhYah¦]:V5STpV5MON
nVLw ADNQS¸]8D KV5M[]JSPQDTSRK?DTENQMOnS
p1, . . . , pn ∈ RfDN~]|g _8NQM\Z[MONQXTS~PQDTSRKDeENQM\c5DTS
u1, . . . , un ∈ Rf|DN
P ∈ R_:W¸K?VLMz]:S r Ijb8M r ILZm^5_JDZkg VLW¸DTS1N]:M[SRKV`S1X$YK?VLPRNRDPepdWK?V5SRD
xi = 1S1M V5W r DNFZ\D i hk r D=VLw ADN,]:IaWJS/Z[DlSQILEh¦Yah¦]8V`SfLDN xi = 0
SRM[W:VLW p:Wuc5D_8N r Ijb8M r MzSRDPZkg _8NRM[Z\M\NRX]:_*SQILEY]8V`S,S1V5_JS,EV5W5NQPQILM\W`NQDl]8DK?VLMz]:S:
maxx ∈ 0, 1n
∑
1≤i≤nxipi ≤ P
∑
1≤i≤n
xiui .
:BWJEV5PRD2_:WJD2noVLMzS|ZzI,]8MO±yE_:ZONQXmcM\DTW`N]8DmED/^5_JD2Z\DeS cjIaPQM[ILw:Z[DTS|]|g VLK8NQM r MzSRIaNRM[VLW xiS1V5W`Nd]8M[SQEPQNQDTScLV5M\PFZkg :2b8DTPQEM[ED$LLp =Jp/.K?VL_JP&_:W:D r XNQG:V8]8D]8DPQXTSRVLZ[_8NRM[VLWp •
º:p[ kVL_:PQW:XTDlK?VL_JP,_:W*cLV +5ILLDT_:P/]8DEV rr DPHED]:IaWSFZ®g :2b8D r K:Z[Dº:p[Lp =Jp sL 4r ª ?dj§ 6 ¢¬? D?d*¢ ¨5¡`¨5 W Db8D r K:Z[D EXZ[w:PQD DWV5K8NRM r M[SQIjNQM\V5WvEV r wJM\WJIaNRV5M\PQD/DeSAN2Z[DFK:PRV5w:Z[ r D!]:_vc5V +5Ia5D_:PU]8D,EV rr DTPQEDLp WPQDK:PQXTSRDW`NQILW`NVL_ P/&]:VLM\N=cM[SRMONQDP
ncM[Z\Z[DTSS1_EEDeSRSRM[cLD r DTW5N,DNPQDcLDTW:M[P&YyS1V5W*KV5M\W`Nl]:D]8XKIaPRN&DW _:WNQD r KJS r M[W:M r _ r pWtWJVaNRD tij Z[DNQD r KJSFK?VL_:PFPQD AV5M\WJ]:PRDlZ[IvcM\Z[Z[D i ]:DlZzIvc`M[Z[Z\D j oXTcLDTW5NQ_:DZ\hZ[D r DTW5N,]8MSR?XTPRDTW`N!]8D tjipJW$NRPHILEDIaZ[VLPHSmZ[D=5PQILK:G:D&V5PRM[DW`NQX]8DeS
ncM[Z\Z[DTS/PQDZ[M\XTDTSmDTW`NRPQD=DZ[Z[DTS
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re.
)<#-< "&1* ¯a .KJIaP=]:DTSlIaPHES,K?VLWJ]:XPQXTS,KJILP=Z[DTS
(tij)oc5VLM[P=ZzI UM\5_:PQDº:p[ep3ViZdnIL_8NILZ\V5PQSFNQPRV5_:cLDTP&_:W E+EZ\D]:IaWJSED5PQILK:G:D^`_:MdKILSQS1D_:WJDDNl_:W:DSRD_:Z[DnoV5M[S,KIaP&NQVL_8NQDTS=Z\DeS,cM\Z[Z\DeSp3ViZdDeSAN=K?V5SQSRM\w:Z[D]8Dr DN1NQPRDEDK:PQVLw:Z[ r DSRVL_JS&_:W:DnoV5P r D]8DN,+K?DK:PQVL5PQI rr DlZ\M[W:XeIaM[PRDDTW cILPRMzIawJZ\DeS,DW`NRM[PQDTSocLV5M\PFZzIyVL_SAhiS1DeENQM\V5WL5p Jp\lK?VL_:P,_:WJD r XNRG:V8]8D]:DlPQXTSRVLZ[_8NQM\V5Wp •
sL 4 1¨ 4 « § ¢U ¨5 ª « § « ¸uV5MON&_JWt5PQILK:G:DlVLPQM\DTW5NQX G = (N ,A)fV U N DTS1NdZkg DTWJSRD r wJZ\DB]:DTS W_J]:S DN A ⊂ N×N DTS1Nd_:WDTWJSRD r wJZ\DB] g ILPQETS PQDZ[M[ILW5NdEDTSdW_J]:STpWILSQS1V8EM\D!Y=EHGJI5^5_JD/IaPHE
(k,m) ∈ A _:WvEV 8N w(k,m)p5W´:b8D!_:WW_]]|g VLPQM\5M\WJD
iDN_:WW_J] ]8D]8DTS1NRM[WJIjNQM\V5W
jpq DK:PRV5w:Z[ r D]8_ EHG:D r M\W]:DEV8N r M\WJM r _ r EVLWS1MzSANQDYNQPRV5_:cLDTP_:WsEHG:D r M[W¸]8_~5PQILK:G:DIaZ[Z[ILW5N]8D i Y j DTW_JW¸WJV r wJPRD$^`_:DZzEV5WJ^`_:D]|g XNQILKDeS^5_JMBS1V5MON]8DEV :N2NRVaNHIaZ r M[W:M r _ r p 2=_8NRPQD r DW`NB]8M\NTf5VLW$EHG:DTPQEHG:DF_:WJD&S1_:M\NRD&]:D,W_J]:S i = `0, . . . , `T = jNRDZ[Z[D^`_:D
(`r, `r+1) ∈ AKV5_:PlNRV5_8N
r = 0, 1, ..., T − 1DNw(`0, `1) + · · · + w(`T−1, `T )DTS1N r M[W:M r IaZkpqdVLPHSR^`_:DZ[DTS=W_J]:SS1V5W`N]8DTS=cM\Z[Z\DeSf?^`_:D A DTS1NZkg DTWJSRD r wJZ\D]:DTS&PQVL_:NRDTS=]8M\PQDTENRDeS=]|g _:W:Dc`M[Z[Z\D¸YZkg IL_8NRPQDLf/DN ^5_JD
w(k,m)DeSANZ[I¶]8M[S1NQILWJEDPRV5_8NRM[PQD DW`NQPRDZ\DeSucM[Z\Z[DTS
kDNmf!VLWPRDNRPQVL_:c5D!Z[D&K:PQVLwJZ\ r D&EZ[I5SRSRMz^5_JDF]:_$K:Z\_SBEV5_:PRNBEHG:D r M[W p ILZ\5PRX,SRVLW$ILSRK?DTENBEV r w:M\WIjNRV5M\PQDLfEDK:PQVLw:Z[ r DDTS1N,nILEM\Z[DYvPQXTSRVL_]8PRDJ+tEV r K:PQM[SFK?VL_JP&]8DTSFM[WJS1NQIaWEDTS,]:DLPHIaWJ]8DNQILM\Z[Z[DLp •
V5M[EM _:W DbD r K:Z\DILZ\5Xw:PQM[^`_:DlNQPReS,S1M r K:Z\D^`_:MdK:PRVjcM[DW`NTf8KIaP,Db8D r KJZ\D5fJ]8DZzI$]8M[SQEPQXNQMOhSRIaNRM[VLWKJIaP,XTZ\X r DTW`NQS!´JW:MzS&]8DeSFXT^`_JIaNRM[VLWJS,]8D`NQV5DTSoc5VLM[P,Z[I$V5_JS1h¦SRDTENRM[VLW Jp 9Jp =p sL 4 1J ªT« « e ªe« ?§ ? U:¢dj ªe« ? U ¨L?§ ª j « § ª $ « § 6 « j` 8VLM\N A_:W:D r IjNRPQMzEDEIaPQPQXD] g V5PQ]8PQD n f:S8+ r XNRPQMz^5_JDl]8X´JW:M[DK?V5SRM\NRM[cLDLpJ8VLM\N B _:W:D r IjNQPRMzEDlPRDeENHIaW8hL_:ZzIaM[PRD]8DNHIaM[Z\Z[D
m× n pJ8VLM\N b _:W*c5DTENQD_:P&]:D Rn pW*cLD_:NFPQXTSRVL_]8PRDlZ[DK:PRV5w:Z[ r Dinf
x∈KerB
J(x) =1
2Ax · x− b · x
.
qdIyEVLW`NRPHIaM[W`NRD]|g IaK:KIaPRNRDWIaWJEDYKerBPRDTWJ] EDN1NQD r M\W:M r M[SQIjNQM\V5W W:V5W XcM[]:DW`NRDocLV5M\P=Z[IVL_JS1hiS1DeENRM[VLWT¹Jp ¯:p ¯KV5_:P&SQIvPQXTSRVLZ[_8NQM\V5Wp •
W Ia_8NQPRDDb8D r KJZ\DIaZ[LXTw:PQM[^`_:DlS1M r K:Z\DDeSAN&EDZ[_:M ]8_*^5_JVaNRM[DW`N&]8D2&I+Z[DM[LGu^`_:MdKDTP r DN]8DEILZ[E_:Z\DTPFZ\DeS,cjIaZ[D_:PHS!DN&cLDeENQD_:PHSFK:PQVLK:PQDTSF]|g _:W:D r IjNQPRMzEDS8+ r XNRPQM[^`_:D5p sL 4 Aj « 5j::®`yUjJUj ¸V5MON
A_:W:D r IjNRPQMzED*EIaPQPQXDu]|g VLPH]8PQD n fST+ r XNQPRMz^`_:DLpWcLDT_8N=EILPQI5ENRXTPRMzSRDP!DN&EILZ[E_:Z\DTP,Z\DeS,S1V5Z\_:NRM[VLWJS,]8D
infx∈Rn,‖x‖=1
Ax · x,
V U ‖x‖ DeSANZ[IW:V5P r DDT_JEZ[M[]:M\DTW:W:D]8D x p30=VL_JS&cLDPQPQVLWJS,^`_ g M\ZUSg Ia5MON&wJM\DTW*S :Pl]8DTS,c5DTENRD_JPQSK:PRV5K:PQDTSF]8DAILSQSRVEM\XeSFYSQIK:Z[_JSFKDNRM\NRDcjILZ\DT_:P,K:PQVLK:PQD®EnAp:ZzIyV5_JSAhiSRDTENQM\V5We¹:p°¯8p°¯Lp •C/V5WJSRM[]8XTPRV5WJS!_:W*Db8D r K:Z[DEZzILSQSRM[^`_:DlDW*XTEVLW:V r M\D5p
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¯L ) < *<,T%<
sL 4 1¨L?§e¡ ªT« §¶¢ 6 §UD W EVLWS1Mz]8PQDµ_:W r XWJILLD ^`_:MK?D_:N2EVLWJSRV rr DP n N,+KDeS]:D r IaPHEHGJILWJ]8MzS1D!]8VLW`NZ[DTSUK:PQM\blnoV5P r DTW5N_:WvcLDTENRDT_:P p ∈ Rn+
paVLWPQDc5DW_uY]8XK?DWS1DTP!DeSANF_:WPRXTDZb > 0fDNFSRDTSFEHG:V5MObu]8DEV5WJSRV rr IjNRM[VLWtS1V5W5N,SR_:K:K?V5SRXTS NRPQDr V8]8XTZ\MzS1XeSlKJIaP_:W:DnoV5WJENQM\V5W]|g _8NRM[Z[MONQX u(x) ]8D Rn
+
]:ILWJSREPRV5M[SQSQIaW`NRDvDNEV5WJEIc5Def ^`_:Mr DeS1_:PQD,Z[D&wXTW:X´?ED=^`_:D&Z[D r XWIaLDFNQM\PQD=]8D=Z[IEVLWJSRV rr IjNQM\V5W]:D&Z[I^5_IaW`NRM\NRX x ]8DTS n r ILP1hEHGIaWJ]8MzSRDTSTp:qdI$EV5WJSRV rr IjNRM[VLW ]8_ r XWIaLDSRDPHIZ\DcLDTENRDT_:P x∗ ^`_:M PQXTIaZ[MzS1DTPQIZ[D r Iab8M r _ r]:D
maxx∈Rn
+,x·p≤b
u(x),ELg DeSANRh¦Yah¦]:M\PQD ^`_:M r Ijb8M r M[SRD Zkg _8NQM\Z[M\NRXSRVL_S$_:W:D¸EVLW`NQPQILM\W`NRD ]:D w:_J]85DN r IabM r IaZocLV5M\PuZ[I8VL_JS1h¦SRDTENRM[VLWT¹:p 9:p°¯KV5_:P,Z[IPQXTSRVLZ[_8NRM[VLW?p •P2ILSQSRVLWJStYs_:W£DbD r K:Z\D¸]|g VLK8NQM r MzSRIaNRM[VLW£]|g _:WS8+8SANQ r D r V8]8XZ[MzS1XKJIaPt_:W:DXe^`_JIjNQM\V5W]:M R XPQDW`NRM[DZ[Z[DVLPH]8M\WIaM[PRD5f`ELg DeSANRh¦Yah¦]8M[PQDlY_:WK:PQVLw:Z[ r D]8D¨5¡:§¢UJ ªT« :?p
sL 4 1¨L?:§U¢* ªe« J W EV5WJSRM[]8TPRDs_JW ST+8S1NR r D¡]:M R XPQDW`NRM[DZvZ\M\hWJXTIaM[PQD!Ic5DTE!EPQMONQPQDF^`_JI5]8PHIjNRMz^`_:DLpjqdD!wJ_8NmDTS1NB]8DFL_:Mz]8DTP2_:WPRV5wVLNFVL__:WDW:5M\WSRKJIjNQM[ILZ®fj_:Wc5XG:MzE_JZ\D5fjDNQELp Ia´JW^5_dg M[Z:S1_JM\c5D2²RIa_K:Z[_JSK:PQTSA³/_:W:DFNQPQI ADTENRVLM[PQD/K:PQXT]:X´JW:M[DLpaqBg XNQIaN2]:_PQVLw?VaNYZkg M\WJS1NQILW`NtDeSAN=PQDKJPRXeS1DTW5NQXKJIaP,_:WJDnoVLWJENRM[VLW
y(t)YcjIaZ[D_:PHS&]:ILWJS
RN N,+K:Mz^5_JD r DW`NTfJZ[IK?V5SRM\NRM[VLWDNBZzIcMONQDTSQS1Dp`W$ILLM\NBS1_:PBZ\D&PQVLw?VaN2KJILPmZkg M[W`NRDP r XT]8MzIaM[PQDF] g _:WJD=EV rr ILWJ]8D v(t) YcjILZ\DT_:PQS,]JIaWJSRM N,+K:Mz^5_JD r DW`NTfJZ[IK:_:MzSQSRILWJED]8_ r VaNQD_:Pef8ZzI$]8M\PQDTENRM[VLW*]8DTS&PRV5_:DTSTfDNQELp p
:BWsK:PRXeS1DTWJEDy]:D$noVLPHEDeSf(t) ∈ RN Z[DTSZ\V5M[S]8DyZzI r XeEILW:M[^`_:DyEVLWJ]:_:M[SRDW`NvY*_JWST+8S1NR r D] g Xe^5_IjNRM[VLWS,]8M R XPQDW`NQM\DTZ\Z[DTSFV5PQ]8M[WJILM\PQDTSSR_:K:K?V5SRXDeSFZ\M[W:XeIaM[PRDeS/K?VL_JP&S1M r K:Z\M\´JDTPH
dy
dt= Ay +Bv + f
KV5_:P0 ≤ t ≤ T
y(0) = y0
®º:p[eVQU
y0 ∈ RNDTS1NZkg XNQIjNvM\W:M\NRMzIaZ/]8_ST+8S1NR r D5f A DN B S1V5W`N]8D_8b r IjNQPRMzEDeSEVLWSANHIaW`NRDeS]8D]:M r DWS1M[VLWJSPQDTSRKDeENRM[cLDeS N ×N DN N ×M p|W WJVaNRD z(t) _:W:DvNRPHIADeENQVLM[PRD²REM[w:Z[D³DN
zT_JW:DKV`S1M\NRM[VLW´JWJILZ\D²1EM\wJZ\D³Hp PVL_JPILK:K:PQV8EHG:DPIa_ r M[D_:bEDeSEM[w:Z\DeSDNK?VL_:P r M\WJM r MzS1DTPZ\DEV 8NF]8_uEVLW`NRP LZ[DLf5VLWyM[W`NRPQV]:_:MON/NRPQVLMzS r IjNQPRMzEDeSBS8+ r XNRPQM[^`_:DeSBK?V5SRM\NRM[cLDTS R,Q,D ]:VLW`N!SRD_:Z[DRDTS1N=S1_:KJKV`S1XTDlDTW*K:Z\_S NRPQD]8X´JW:M[DlK?V5SRM\NRM[cLDLp W]8X´JW:M\N=IaZ[VLPHS!_:W EPQMONQPQD^`_JI5]8PHIjNRMz^`_:D
J(v) =
∫ T
0
Rv(t) ·v(t)dt+∫ T
0
Q(y−z)(t) · (y−z)(t)dt+D (y(T )− zT ) · (y(T )− zT ) .
&D r IaPH^`_:VLWJS=^5_JDZ[I$noVLWJENRM[VLW y(t) ]8XTKDTWJ] ]8DZ[I$cjILPRMzIaw:Z[D v YNRPHIcLDPHSºJp\p|C/V rr DZ\DeSEV rr IaWJ]:DTSI5] r M[SQSRM\w:Z[DTSvS1V5W`NvXc5DW`NR_JDZ[Z\D r DTW`NZ\M r MONQXDeSZ[IK:_:MzSRSQIaWEDu]|g _:W r VLNRD_JPvDeSANSRVL_JcLDW`N=w?VLPQW:XTDLp[p\p°fVLW M[W`NRPQV]:_:MON_:WEV5WcLDb8DnoDTP r XWJVLW cM[]:D K ]8D RM ^5_JMdPQDK:PQXTSRDW`NRDZkg DWJSRD r w:Z\D]8DeS&EV rr ILWJ]8DeS!I5] r M[SQSRM\w:Z[DTSTp:q DKJPRV5w:Z\ r DlDeSAN&]8V5WJE]8DPQXTSRVL_J]8PQDinf
v(t)∈K, 0≤t≤TJ(v).
ViZmnIa_J]8PHI:f w:M[DWsS :Pef|K:PRXeEMzS1DTP]:IaWJS^`_:DTZ[SDeS1KILEDeSlnoVLWJENRM[VLW:WJDZzSVLW r M[W:M r M[SRD J(v)DNVLW]:X´JW:M\N&ZzISRVLZ[_8NQM\V5W
y]8D$º:p[ecLV5M\PFZzI8VL_JS1h¦SRDTENRM[VLW¸T¹Jp =p ¯KV5_:P&ZzIvPQXTSRVLZ[_8NQM\V5Wp •
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)<#-< "&1* ¯a sL 44 A « § « « T ªe« ?§s¢>[§U 6 §`D « 6 ¨LJ§ «@? ViZ STg ILLM\N!]8D r M[W:M r MOhS1DTPdZkg XTW:DPQLM[D r XTETIaW:Mz^`_:DB]|g _:W:D r D r w:PQILW:DmV5_w:M[DWZkg XW:DTPR5M\DmXTZ\DeENQPRV`SANHIjNRMz^`_:DB]|g _:WEVLW]8_JEhNRD_JPTp%0&V5_JS/PRDTW`c5V +LV5WJSmIa_:buC/GIaK:M\NRPQDTS=DN .KV5_:PFK:Z\_S!]:Dl]8XNQILM\ZzS/SR_:P!Z[I r V8]8XZ[M[SQIjNQM\V5WyDNS1_:PZ[DTSUW:VLNQIjNQM\V5WJS r IjNRGJX r IjNQM[^`_:DeSpaV5MON Ω _:WVL_JcLDPRNw?VLPQW:XF]8D RN
DNf ∈ L2(Ω)
pLxvg IaK:PQTSZ[I PmPQVLK?V5SRMONQM\V5W .8p°¯8p f`KV5_:P!PRXeS1V5_J]8PQD=Z[DK:PQVLwJZ\ r Dl]8Dx=M[PRMzEHG:Z[DNFKV5_:PFZ\DqdILK:ZzILEM[DWdf5M[Z nIa_8Nr M[W:M r M[SRDP,Zkg XTW:DPQLM[D J(v)]:X´JW:M[DK?VL_:P
v ∈ H10 (Ω)KJILP
J(v) =1
2
∫
Ω
|∇v|2dx−∫
Ω
fv dx.
2&_8NQPRD r DTW`N=]8MONef:VLWcLDT_8N&PQXTSRVL_J]8PQDinf
v∈H10 (Ω)
J(v).
W£KDT_8N S1D¸K?V5SRDPuZ[D rTr DK:PQVLwJZ\ r DKV5_:P ]8DTStXT^`_JIaNRM[VLWJStK:Z[_JS*EV r KJZ\Mz^`_:XDeSt^`_:D¸Z\DqdIaKJZ[I5EM[DWEV rr DuZ\DeSXT^`_JIaNRM[VLWJSv]8DNRV5DTSTpUDTZ\V5WsZ®g :2b8DPHEMzED.8p 9:p[T¹:fdZ[IPRXeS1V5Z\_8NQM\V5Ws]8DTSXT^`_JIjNQM\V5WJS,]8D`NQVLDeSFDTS1N&XT^`_:M[cjIaZ[DW`NRDYZ[I r M[W:M r MzSRIaNRM[VLWinf
v∈H10 (Ω)N
%)(*+&2(divv=0
J(v) =µ
2
∫
Ω
|∇v|2dx−∫
Ω
f · v dx
.
,D r IaPH^`_:VLWS2^5_dg M[Z +I_:W:DEVLW`NQPQILM\W`NRDm²R]|g M\WJEV r K:PQDTSQS1M[w:M[Z\M\NRX³,]JIaWJS/EDNRNRD r M[W:M r MzSQIjNRM[VLWdfLDN^5_JDZ[IK:PQDTSQS1M[VLWtDTS1N=IawJSRDW`NQD]:DZ®g XW:DTPR5M\D5p%0&VL_S,cLDTPRPQVLWJS!^`_:DEDeS,]8D_:bunIaM\NQS=S1V5W5N,XNRPQVLM\NRDhr DTW`N&Z\M[XTSTp •xVLWJW:VLWJS r ILM\W`NQDWJILW5N&_JWDb8D r K:Z[DM[SQS1_ ]8_ ETIaZzE_:Zd]:DTS,cjIaPQMzIjNRM[VLWSp%7&MzS1NRVLPQMz^5_JD r DW`NTfJM\ZSg Ia5MON2]|g _:W]:DTSK:Z[_JScM\DT_8bK:PQVLw:Z[ r DeSU] g V5K8NRM r M[SQIjNQM\V5W fPRXeS1V5Z\_vKJIaP?XTW:V8]8VLPQDBDTW`cM[PRV5Wv]8DT_8bS1M[TEZ\DeS=IcjIaW`N=W:VLNRPQDPQDDNl]8V5W5NZ[Iy]:X r VLWJS1NRPHIjNQM\V5W EV r KJZ\NRDDeSAN]8_:DvYDTM\DTPQS1NRPHILSQS,cLDTPQSZ[Iv´JW ]8_ ]8M\bhkW:DT_:cM\ r DSRM[TEZ[DLp sL 44 iUjJ*¢U + « ¢?§ M\PQLM[Z[DPHILEVLW`NRD]:IaWS,Z®g mW:XTM[]:D^5_JDlZ[VLPHSQ^5_JDZ[I*PQDM[W:DyxM[]8V5W¸noVLWJ]:I*ZzI*cM\Z[Z[Dy]8DC!IaPRNRGIaLD5f M\Z/W:D$Z\_JMmno_8NvILZ\Z[VL_JXEV rr DyS1_JKDTP1´EM\Dy^`_:D²1ED^`_:DK?VL_:PQPQILMON=EVLW`NRDTW:M[P&_:W:DvKDeIa_ ]8Dw?VD_8nA³HpW:mZ[Z\D]8XTEVL_:KJI$ILZ\V5PQS=EDNRNRDK?DTIL_ DW*´W:DTSZ[ILW:M\TPRDeSDN$DWJEDPHEZzI]8D EDN1NQD EVLPH]8DTZ\DN1NRD*SQI no_8NQ_:PQDcM[Z\Z[D*S1M\NR_:X IL_¶w?VLPH]¡]8D*ZzI r DTPTp2qdI^5_JDTS1NRM[VLW XNQILMON]8VLWJE*]8D*NQPRV5_:cLDTP$Z[I¸K:Z[_JS$5PQILWJ]8D SR_:P1nI5ED*K?V5SQS1M[w:Z[D Sg IaK:KJ_O+5ILW5N$SR_:Py_:W:D]8PRV5MONQDoZ[D=PQM[cILLDDNF]8D&noPQVLW`NRM[PQD,NQDPQPRDeSANQPRD=]:D=Z[VLW:5_:D_:P/]8VLW:WJXDLpqdIPRXTKV5WJS1D=DTS1N/w:M[DWuS :P_:Wu]8D r M ]8MzSR^`_:DvocLV5M\P/Z®g :2b8DPHEMzEDT¹:p°¯8p[T¹`p :mWuNRDTP r DTS r IjNRGJX r IjNQM[^`_:DeSm_:WyK?D_tS1M r K:Z\M\´JXeSfZ\DKJPRV5w:Z\ r DDTS1N]8DNRPQVL_:c5DP,ZzIyEV5_:PRw?DK:ZzIaWJD]8DZ[VLW:5_:D_JP&´:b8XD l ≥ 0
^`_:MDWJEZ\VLNIcLDeEZ\DS1DT r DW`N,PRDTZ\MzIaW`N&SRDTS,]8DT_8buDbNQPRX r M\NRXeSFZ®g IaM[PQD r IabM r _ r p32&_8NQPRD r DTW`N,]:MONefJVLWPQXTSRVL_8Nsup
∫ ξ
0
y(x) dx,
IcLDTEZ[DTS,EV5W`NRPHIaM[W5NQDTSξ ≥ 0, y(0) = 0,
∫ ξ
0
√
1 + y′(x)2dx = l,
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¯LL ) < *<,T%<
VQUξDTS1NFZkg DbNRPQX r M\NRX]8_SRD r DW`N!DN y(x) ZzIK?V5SRMONQM\V5Wt]8DZzIvEVL_:PQw?DlIL_]8DeSRSR_JSF]8_KV5M\W`N x]:_*SRD r DW`NTp •
DTW:VLWJSDTWYZkg V5K8NRM r M[SQIjNQM\V5W]|g _:W ª v¢ « ª « 6 fjEag DTS1N1hiYjhi]8M[PRD r V8]8XTZ\MzSRX!KIaP_:W:DXe^`_JIjNQM\V5WIL_8bt]8XTPRM[cLXTDTS!KJIaPRNRM[DZ[Z[DTSTp sL 44F A¨5§ ª ?¢>[§U Uj:§U W EVLWS1Mz]8PQD_:W:D r D r w:PHIaW:DXTZ[I5SAhNQM[^`_:D5f`´:b8XDSR_:PFS1V5WuEV5W`NRVL_JPTf5DN!SRD]:XnoVLP r IaW`N!SRVL_SBZkg I5ENQM\V5Wu]|g _:W:D&noV5PQED f p:C/V rr D=W:V5_JSZkg IcLVLWS/c_*YZ[I$V5_JS1h¦SRDTENRM[VLW5p 9Jp 9Jf8EDK:PQVLw:Z[ r DDTS1N r V]:XZ[M[SRXKJIaP
−∆u = f + v]:ILWJS
Ωu = 0
SR_:P∂Ω,VQU
uDeSANBZ[D=]8XTK:ZzILED r DW`NBcLDTP1NQM[ETIaZJ]:D=ZzI r D r wJPQILW:D,DN v DTS1NB_:W:D=noVLPHED&]8DlEV5W5NQP5Z\D&YW:VaNQPRD]:M[SRKV`S1M\NRM[VLWdpC/DEVLW`NRP LZ[DvDeSANN,+K:Mz^`_:D r DW`N_:W~ILENRM[VLW:W:DT_:PK:M[X%SVLhkXTZ\DeENQPRMz^`_:Dv^`_:MmILLM\NSR_:P_JW:DKIaPRNRM[D
ω]8_]8V r IaM[W:D Ω IcLDTE&_:W:DlM\W`NRDTWJSRMONQXZ[M r M\NRXTDLpJWt]8X´JW:M\NF]:VLWJEZ®g DWS1D r w:Z[D]8DeSEVLW`NRP LZ[DTSFI5] r M[SQS1M[w:Z[DTS
K = v(x) NQDZd^`_:D vmin(x) ≤ v(x) ≤ vmax(x)]:ILWJS
ωDNv = 0]:ILWJS
Ω \ ω ,VQUvminDNvmaxSRVLW`Nu]8DT_8b¶noV5WJENQM\V5WJSu]:VLW:W:XTDTSTpFW£EHG:DTPQEHGJD Z\DEVLW`NRP LZ[D ^`_:M=PQDW]¬Z\D]:XK:ZzILED r DW`N u IL_JSQS1M K:PQV8EHG:Dl^`_:DlK?V5SQS1M[w:Z[Dl]|g _:W*]8XKJZ[I5ED r DTW5N&]8XeS1M[PQX u0
f8DN&^`_:M|SRVLM\NF]|g _:WEV 8N r V8]8XPQXLpW ]8X´JW:M\N=]8VLWE_JW*EPRM\NRTPRDJ(v) =
1
2
∫
Ω
(
|u− u0|2 + c|v|2)
dx,
Ic5DTEc > 0p:qdDK:PRV5w:Z[ r D]8DEV5W5NQP5Z\DSTg XeEPQMON
infv∈K
J(v).
ViZ`PQDTS1NRDmw:M[DWSJPY&KJPRXeEMzS1DTPdZ[D/EHG:V5MOb]8DTSDTSRKJI5EDTS noVLWJENRM[VLW:WJDZzS K?VL_:PvDNZ[DTSUIa_8NQPRDeSd]8V5W:W:XTDTS]:DEDK:PQVLw:Z[ r D$oc5VLM[P!ZzIyV5_JSAhiSRDTENQM\V5We¹:p =Jp 9K?VL_:P,ZzIPRXeS1V5Z\_:NRM[VLWp •
.102. C4=@i6:@19B4&E G|6¬4&926%NU6:@A9B4EqBg V5K8NRM r M[SQIjNQM\V5W¡I _:W¶cLV8EILw:_:ZzIaM[PRD$KJILP1NQM[E_:Z[M\DTP UM\W`NRPQV8]8_:MzS1V5WJS^`_:DTZ[^`_:DeSW:VLNQIaNRM[VLWJSvDN]:X´JW:M\NRM[VLWSdEZzILSQSRM[^`_:DeSp0=VL_JSEV5WJS1Mz]8XTPRV5WJS K:PQM\WJEM\KIaZ[D r DW`N]8DeS K:PQVLwJZ\ r DeS ]8D r M[W:M r MzSQIjNRM[VLW®SRI5EHGJIaW`Nm^`_ g M\ZJSR_8±N/]|g DW$EHGJILW:LDTPZ\D=SRM[LW:DFK?VL_:PmV5w8NRDTW:M[P2_:W$K:PQVLw:Z[ r D,]8D r Ijb8M r M[SQIjNQM\V5WpkVL_8Nl]|g ILwV5PQ] f8Z®g DTSRKJILED]:IaWS,Z\De^`_:DZ DeSAN&K?V5SRXZ[DK:PQVLw:Z[ r DLf8WJVaNRX V f:DeSAN=SR_:K:K?V5SRX NQPRD_JWsDTSRKJILED$cLDTENRV5PRM[DZ2W:V5P r X5fdEag DTS1N1hiYjhi]8M[PRD r _JW:M/]|g _:W:DuW:VLP r D$W:VaNQXD ‖v‖ pdxlIaWSZ[IVL_JS1hSRDTENRM[VLW$º:p[Lp =
VSRDPHIlZ®g DTSRKJILED
RNfLNQILWJ]8MzSB^`_:D=]:ILWJS2ZzIS1DeENQM\V5WyS1_:M[cjIaW`NRD
VS1DTPQI_:W$DeS1KILED]:D7=M\Z[w?DPRN&PRXTDZmoV5WtK?VL_JPRPHIaM\N!XT5ILZ\D r DTW`N,EVLWJSRMz]8XPQDP!Z\DETILSTfK:Z\_S,LXTW:XPHIaZkf8]|g _JWtDeS1KJI5ED]8D
ILWJILEHG f?Eag DTS1N1hiYjhi]8M[PRD_JWDTSRKJI5EDc5DTENQVLPQM\DTZdW:V5P r XEV r K:Z[DNpdWSRDv]8V5W:W:DvX5ILZ\D r DW`N_:WSRVL_SAh¦DWJSRD r w:Z[D K ⊂ V V UZ®g VLW$cjIEHG:DTPQEHGJDP2ZzIS1V5Z\_8NQM\V5W aVLW$]8M\N/^`_:D K DTS1NBZkg DTWJSRD r wJZ\D&]8DeSXTZ\X r DW`NQSmJ¢ « T « !]8_K:PQVLw:Z[ r DLfLVL_w:M[DWy^`_:D K ]8X´JW:M\NBZ\DeSm¨L?§ ª j « § ª =STg Db8DP ILW5N
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
)<#-< "&1* ¯a5ºS1_:PvZ\DtK:PQVLw:Z[ r DuEVLWJSRM[]:XPQXLp1:mW:´JW fZ\D¨a «oª `?fUVL_sZzI¤¦?§¨ ªT« ?§ ¨5 ª fUVL_¡Z[I ¤¦?§¨ ªT« §J e`¨ ªT« ¤RfUY r M[W:M r M[SRDPefdW:VLNRX J f DeSAN_JW:DnoVLWJENRM[VLWs]8X´JW:M[DuS1_JP K Y*cILZ\DT_:PHS]:IaWS R
pdq DK:PRV5w:Z[ r DXNR_]8M\XSRDPHI]8VLWJElW:VLNRXinf
v∈K⊂VJ(v).
®º:p°¯Lq VLPHSQ^5_JDvZkg V5W_:NRM[Z\MzS1D$ZzIW:VaNHIjNQM\V5W
infK?VL_:P_:W~K:PQVLwJZ\ r D$]8D r M[W:M r MzSRIaNRM[VLW fEDZzIM\W]8M[^`_:D^5_JDyZ®g VLWsWJDuSRILMONvKJILSTfdI K:PQM[VLPQM®fdSRMBZzI cjIaZ[D_JP]8_ r M\WJM r _ r DTS1NvIjNRNRDM[W`NRD5fEag DTS1N1hiYjhi]8M[PRDySTg M\ZDb8M[S1NRD
v ∈ K NRDTZd^`_:DJ(v) = inf
v∈K⊂VJ(v).MdZkg V5W cLDT_8N=M[WJ]8Mz^5_JDP^`_:DZ[I$cjILZ\DT_:P=]8_ r M[W:M r _ r DTS1NIjNRNRDTM\W`NRD5fJV5W _8NRM[Z\MzSRDv]8DKJPRXnoXPQDWJEDZ[IW:VLNQIjNQM\V5W
minv∈K⊂V
J(v),
r ILM[SM[ZBWJDySg Ia5MONKJI5S]|g _JW:DuEV5WcLDW`NQM\V5W_:W:M[cLDTPQSRDZ[Z[D*®^`_:VLMz^`_:DnoVLPRNPRXTKJIaWJ]:_:Dep1PVL_:PZ[DTSK:PRV5w:Z[ r DTS&]:D r Ijb8M r MzSQIjNRM[VLWdf:Z\DeS&W:VLNQIaNRM[VLWJS supDN
maxPRD r KJZ[I5EDW`N inf
DNminf?PQDTSRKDeEhNRM[cLD r DTW5NepOPmPQXTEM[SRVLWJSF^`_:DTZ[^`_:DeS,]8X´JWJMONQM\V5WJS&]8DwJI5S1D5p
+ 6 ;§ «ªT« ?§ 44 J nE>01%3 u 3[G[2^YC0101YC2Y . 2Z6.01oE73YC0101YC2Y 8. ;=8 EF3JG[2B
KG[0p3pG328M3Y 3SG[0
u ∈ K 3 ∃δ > 0 , ∀v ∈ K , ‖v − u‖ < δ =⇒ J(v) ≥ J(u) .
J E0153u3[G[#2 YC0101YC2Y .l2X6.015E73 Y_0101Y_2Y F8M. =8gE73
JG[2B
KG[0W3
G328M3YZ3pG[0u ∈ K 3
J(v) ≥ J(u) ∀v ∈ K .
+ 6 ;§ «ªT« ?§ 4 F J 676H3818M3 01 \pY_2Y E73JG[2B
K . i 698: G@;.>2BOYCY 3 i L >8M32BY_0101Y_2Y )i 3P8 R .> EFAG[0>43C6BP8M 4.VD0<.
7I i 8 .>B[93 G=6AB[0<32BO3XEFGR
EF3G;.>HG[VV3Ge20 YC019.B3
JG[2B
K I ,90 J 4R 3G[96HGdYC019.>BOA=3eG[2BK i 8.B=GW8 R 01 \pY_2Y L 2
−∞ I ,0 K 3[G[ L 0<EF3 i 6B_;=.> L 3?0<.>X8 R 01\SYC2Y 3G=+∞ I
] 93G[201 3Y_0101Y_0UG 3CEF3 J EF>HGK
3G[p293#G[201 3(un)n∈N
V3818M3Z3
un ∈ K ∀n 3lim
n→+∞J(un) = inf
v∈KJ(v).
P2ILPZzI ]8X´JWJMONQM\V5W rr D~]8D~Z®g M[W8´ r _ r ]8D J S1_:P K M\ZDbMzS1NRDNRVL_ AVL_:PHS*]8DTS*SR_:MONQDTSr M[W:M r M[SQIaW`NRDeSp .102. H&6:@TL¥@8EQNU6:@A9/4 Gd4 ;@TLMGd4&E@19B4 4=@TG
ViW`NQXPQDTSQS1V5WJSlW:VL_JS r IaM[W`NRDTWJIaW`NYuZzI*^`_:DTS1NRM[VLW~]8DZ®g ` « ª §U¨Ls¢U « § « ¡tK?VL_JP]8DTSK:PRV5w:Z[ r DTS2]|g VLK:NRM r MzSQIjNRM[VLWK?V5SRXTSDW]8M r DWJSRM\V5W´JW:M[DLp 0=VL_JS2SR_:K:K?V5SRDPQVLWJS]:ILWJS2EDN1NQD&S1V5_JSAhS1DeENRM[VLWtSQIaWJSKDTP1NQD/]8D/LXTW:XPHIaZ[MONQXed^5_JDV = RN
^`_:D/Z®g VLW r _:W:M\N]:_K:PQV8]8_:M\NUSQEIaZzIaM[PQDm_JSR_:DZu · v =
∑Ni=1 uivi
DN=]8DZzIW:VLP r DlDT_JEZ[Mz]8M\DTW:W:D ‖u‖ =√u · u p
W PRXeS1_JZONHIjN&ILSQS1D-S5XW:XTPQILZ 5IaPHIaW`NQM[SQSRILW`N/Zkg DbMzS1NRDWED]|g _:W r M[W:M r _ r DTS1N,Z\DSR_:M[cjIaW`NTp
Ecol
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lyte
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¯LºL¹ ) < *<,T%<
b 4(6 O « ª §U¨L ¢>\U§ « § « ¸(§¬¢ « § « ?§ ;§ « ,4.>01K
23HG3Y8M3 3B[Y A59. L 0<E73ZEF3
RN i 3 J 293 .4;D0<. ;.>D013#G=2BK
` L 8M32B=G5EFGRL AB[0 \NN8e69B.69B[0<AVA i E01V3 01\S0<3C` 8OR 01 \S0 i
∀(un)n≥0G[201 3_EFG
K , limn→+∞
‖un‖ = +∞ =⇒ limn→+∞
J(un) = +∞ .ºJp 9`
8.B=G#018 3TF0UG[V35> Y .01G#2Z6H.>01%EF35Y_0101Y_2Y EF3JG=2B
K I C3648: G i .a6H323T7?BO>01BO3E73 V.2V3CG[201 3 Y_0101Y_0UG 3!EF3JG[2B
K243CG. G G[201 3 ;.> L 3B 73= L 3B=GZ2!6H.>01eE73
YC0101YC2Y G[2BK I
+ 6 ?§ ª j ªe« ?§ 8VLM\N(un)_:W:D*S1_:M\NRD r M\W:M r M[SQIaW`NQD]8D J S1_JP K pUqIEV5WJ]8M\NRM[VLW·®º:p 95DTW`NRPHI) W:D^`_:D
unDTS1N$wV5PRW:XTDtKJ_:M[SQ^`_:D
J(un)DeSAN_:W:D SR_:MONQD ]8DtPQXDTZ[S r IAV5PRXTDLpx=V5WJEaf2M[ZDb8M[S1NRD_:W:DSRVL_SAhiS1_JMONQD
(unk)^`_:MdEV5WcLDPQLDcLDTPQS!_:W K?VLM[W`N
u]8D
RNp IaMzS
u ∈ K K:_:MzSR^`_:DKDeSANnoDP r XLf DN J(unk )
EVLWc5DPQLDc5DPHSJ(u)KJILPEV5W5NQM\W_:M\NRX5f ]|g V U
J(u) = infv∈K J(v)] g ILK:PReSFZ[IxX´JWJMONQM\V5W*ºJp\5p ¯:p
8 ? O Mr 0&VLNRVLWS^`_:DvZ[I$KJPRV5K:PRM[XNQXuºJp 9`f ^`_:MUI5SRSR_:PQD^`_:DNQVL_8NQDvSR_:MONQD r M[W:M r M\hSQIaW`NQD]8DJS1_:P
KDTS1NwV5PRWJXDLf|DeSANIL_8NRV r IaNRMz^`_:D r DW`NcLXTPRM\´JXTDSRM K DeSANw?VLPQW:X5p qdVLPHSR^`_:DZkg DWJSRD r w:Z\D K W g DTS1NKILSwV5PRW:X5fEDN1NQDEVLWJ]:MONQM\V5W Db8K:PQM r D^5_JDLf ]:IaWJS K f J DTS1N « §;§ « T> « §;§ « p •
`j¨ « ¨5 O #-G< ?)> M/ -$ ?(+ -$ !")+ > A!#"K
%$&# < @ #" !#")J
%$&%. (")*-$; <:K @ @ G>7+)@ %%$R$R> G2?<"7 + $&;< 0/) "( " J#()< "(: ?(+ / #) . #-. 7") % )G@%=/
R/(>7)$
R >/ % &>7-0?)>)$M/( < " $ "7
R `j¨ « ¨5 O GF #-G<!#")=+ #2?<<"' ?7+ > <E+ >A?<#?< @ ;@ & A+ < :1 76 ?)> + >_ #)/ . ?7+,"$> 7+
infv∈K
J(v) < limR→+∞
(
inf‖v‖≥R
J(v))
.
`j¨ « ¨5 O #-G< !#") + # ?< "- G ?7+ > I+ >m #-. 7"< ;@ /(J
?)> I+ > $R %. (" ;@2 @ "(G2/ J
/ @ ?><
∀(un)n≥0$ " ;:/7>7)$
K , limn→+∞
un = u =⇒ lim infn→+∞
J(un) ≥ J(u) .
`j¨ « ¨5 O Mr #-G<!" , + ( $;;*"7 < " ?<#"( + %$ ) ?7+ %$ 7 N 7 5 % #( 3 `j¨ « ¨5 O H
a%b/ " @-<+ $O>)
0 < a < bN %J?<"7
n ∈ N∗ N%$R Pn+ <)$& (+
/(%$*?<+ 7 %$ P /( /(%KG@ @< <"7 #"0@-K(>7+ n; + $ !#"
P (0) = 1 #"( P ∈ PnN( )<;
‖P‖ = maxx∈[a,b] |P (x)| (: #-G< !") + A? (+
infP∈Pn
‖P‖ ®º:p =`> "7) $R+,".
Ecol
e Po
lyte
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O "&T1< < , < , % < ¯aºJ%# ( >7?7?<<+,+ 2!#" + %$ ?<#+ ( -$M/ #4&) &)
Tn(X)$&#-'/ @ $ ?)> + %$ G<+ >. #$
T0(X) = 1 , T1(X) = X , Tn+1(X) = 2XTn(X)− Tn−1(X) .
'!#")+ /(%K#@ / Tn
%$&2@%K(>7+ n
%A!#"I?<"7Q;#"-θ ∈ R, Tn(cos θ) =
cos(nθ) A/(@-/#" + ( $;;< =/(n+ 1
@-<+ $ξn0 = 1 > ξn
1 > ξn2 > · · · > ξn
n = −1; + $'!#")Tn(ξn
k ) = (−1)k ?<"70 ≤ k ≤ n M!#"
max−1≤x≤1 |Tn(x)| = 1 #= O!") + "7 !"):$&+,". 2/(1 # 6 -$;0+ A?<#+ (
P (X) =1
Tn
(
b+ a
b− a
)Tn
b+ a
2−X
b− a2
.
#$+|, $/ % t% % , % ,+d [email protected] GLMHF,G(E;Gµ4&9B4 5G&@8EL6%Gd4> GC/DN1NQDtSRVL_JS1h¦SRDTENRM[VLWDTS1NvEV5WJSRI5EPQXD$Y ]8DT_8b¸Db8D r K:Z[DTS r VLW`NQPQILW5N^`_:DyZkg Db8M[S1NRDTWJEDy] g _:Wr M[W:M r _ r DWv]:M r DWS1M[VLWvM\W8´W:M\D!W g DTS1Nm8Ue?U§ ª : D:J§ ªT« KJIaP]8DTSEVLWJ]8M\NRM[VLWSU]:_N,+`K?DB]8D/EDZ[Z[DTS|_8NQM\Z[MzS1XTDTSd]JIaWJS|Zkg XW:V5WJEXm]8_kFGJXVLPQ r DmºJp\5p 9JpTC/DN1NQDB]8M\±yE_:Z\NRX/DTS1N M\W`NRM r D r DW`NZ\M[XDIL_unILMON=^`_ g DW*]8M r DTWJSRM\V5WM[W8´JWJM\DZ[DTS!noDTP r XTSFw?VLPQW:XeSFW:DS1V5W`NFKILS,EV r KJILENQS C/V rr DTW V5WJS&KIaP]8VLW:WJDP=_JW Db8D r K:Z[DvILwJS1NRPHIaM\N^`_:MDb8K:Z\Mz^`_:Dw:M[DWZ\D r XeEILW:M[S r Dv]8D²ino_:M\NRDYZkg M[W8´JWJM ³^`_:M D r K EHG:DZkg Db8M[S1NRDTWJED]|g _:W r M\WJM r _ r p
sL F94¬8VLM\NZkg DeS1KILED*]8D 7&M[Z\w?DPRN ]8D ]8M r DWJSRM[VLW¡M[W8´JWJM\Dv]:DTSS1_JMONQDTS]8D EILPRPQXS1V rr ILw:Z\D]:ILWJSR
`2(R) =
x = (xi)i≥1NRDTZd^`_:D +∞
∑
i=1
x2i < +∞
,
r _:W:Mm]:_K:PQV8]8_:M\NSRETIaZzIaM[PRD 〈x, y〉 =∑+∞
i=1 xiyipW~EV5WJS1Mz]8TPRDvZ[ItnoVLWENRM[VLW
J]8X´JW:M[DSR_:P
`2(R)KJILP
J(x) =(
‖x‖2 − 1)2
+
+∞∑
i=1
x2i
i.
PmPRDTWJIaW`NK = `2(R)
fJVLW EV5WJS1Mz]8TPRDlZ\DKJPRV5w:Z\ r Dinf
x∈`2(R)J(x) ,
®º:p/.LKV5_:PZ\De^5_JDZ!W:VL_JSILZ\Z[VLWJS r VLW`NRPQDP^`_ g M\ZFW g Db8MzSANQDuKJI5S]:DtK?VLM[W`N]8D r M\WJM r _ r p XPQM\´JVLWJSNRVL_:N=]|g ILwV5PQ]^`_:D
(
infx∈`2(R)
J(x)
)
= 0.
Ecol
e Po
lyte
chni
que.
Ne
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¯Lº5¯ ) < *<,T%<
ViW`NQPRV8]8_:MzSRVLWJS=Z[ItS1_:M\NRDxn ]JIaWJS `2(R)
]8X´JW:M[DKJILPxn
i = δinK?VL_:PNRV5_8N
i ≥ 1p W cLXTPRM\´JDILM[SRX r DTW5N&^`_:D
J(xn) =1
n→ 0^`_JILWJ]
n→ +∞.C/V rr D J DeSAN/KV`S1M\NRM[cLD5fLVLW$DTWu]8Xe]8_:M\N!^`_:D xn DTS1N/_:W:DS1_:M\NRD r M\W:M r M[SQIaW`NQD&DNF^5_JD&Z[IcjIaZ[D_:P]:_ r M[W:M r _ r DTS1NmW`_JZ\Z[DLp`C/DTKDTWJ]:ILW5NefaM[Z:DTS1N2XcM[]:DW`NB^`_ g M[Z:W g Db8MzSANQD,Ia_JE_:W x ∈ `2(R)NRDTZJ^`_:D
J(x) = 0p P2IaPEV5WJS1Xe^`_:DW`NTf M[Z2W g Db8M[S1NRDKJI5S]8DKV5M\W`N]8D r M\W:M r _ r K?VL_:P$ºJp .5pWcLVLM\N]JIaWJSEDNDb8D r KJZ\D^`_:DvZ[ItS1_JMONQD r M\WJM r MzSRILW5NQD xn
²AKIaPRNYyZkg M[W8´JWJM ³DNW g DTS1NKJI5SEV r KJI5ENRD]JIaWJS`2(R)ow:M[DW ^`_ g DZ[Z\DSRVLM\N,wV5PRWJXDep •
V5M[EM r IaM[W`NRDTWJIaW`Nt_JW£Db8D r K:Z[D r V8]8TZ\D¸^5_JMW g DTS1N*KJILStSRILWJSyPQDTSQS1D r w:ZzIaWEDIc5DTE*Z[DTSKJPRV5w:Z\ r DTS=]:D r M\WJM r MzSRIaNRM[VLW]|g XWJDPQLM[D^`_:DW:V5_JSIc5VLWJS&PRDTWJEV5W5NQPRXeS,Z[VLPHS=]8DvZ[I$PQXTSRVLZ[_8NQM\V5W] g Xe^5_IjNRM[VLWSlIL_8b]8XTPRM[cLXTDTSKJIaPRNRM[DZ[Z[DTSoc5VLM[PTf KJILPlDb8D r K:Z[DLf|ZzI PBPRV5KV`S1M\NRM[VLW .8p°¯8p ap ILZ\5PRXSRVLW EILPQI5ENRTPRDtS1M r K:Z\M\´JX5fmEDN$DbD r K:Z\DDTS1NNRPQTSPQDK:PQXTSRDW`NQIaNRM\n=]8D*K:PQVLw:Z[ r DTSvPQXTIaZ[MzSANQDTSDNKJPQIaNRMz^5_JDTSU]8D r M[W:M r M[SQIjNQM\V5Wv]|g XW:DTPR5M\DeS]:DBNQPQILWJS1M\NRM[VLW]8D!K:GJILSRDTSUDTWvSQEM[DWJED!]8DeS r IaNRXTPRMzIa_8b|p sL F9 FWEVLWJSRM[]:PQDZkg DTSRKJILEDy]8DtV5wV5Z\DTc
V = H1(0, 1) r _:WJM/]8DyZzI W:V5P r D‖v‖ =
(
∫ 1
0
(
v′(x)2 + v(x)2)
dx)1/2 ^`_:M?DTS1NF_:WuDeS1KJI5ED=]:DJ7=M\Z[wDTP1N,]8Dl]8M r DTWJS1M[VLWuM\W8´W:M\D5fc5VLM[PFZ\DvC/GJILK:M\NRPQDJ=pW*K?V5SRDK = V
DNTf8K?VL_JP1 ≥ h > 0
f:VLW*EV5WJSRM[]8TPRDJh(v) =
∫ 1
0
(
(|v′(x)| − h)2 + v(x)2)
dx .
qBg ILK:K:Z[M[ETIjNQM\V5WJDeSAN=EVLW`NRM[W_:DS1_JP
Vf:DNFZzI$EV5WJ]8M\NRM[VLW~ºJp 9`BDeSAN&c5XPQMO´XDlK:_:MzSR^`_:D
Jh(v) = ‖v‖2 − 2h
∫ 1
0
|v′(x)|dx + h2 ≥ ‖v‖2 − 1
2
∫ 1
0
v′(x)2dx− h2 ≥ ‖v‖2
2− h2 .
VLW`NRPQVLWJSF^`_:Dinfv∈V
Jh(v) = 0 ,ºJp `
ED^5_JM|M r K:Z[M[^`_:DTPQI^`_ g M\Z|W g Db8MzSANQDKJILS,]8D r M[W:M r _ r ]8D JhSR_:P
V8DWD R DNefJSRM®º:p 5!IZ\M[D_DNSRM
uXNQILMON_:W r M[W:M r _ r ]8D Jh
SR_:PVfdVLW¶]8DcPHIaM\NIcLVLM[P
Jh(u) = 0fU]|g V U
u ≡ 0DN
|u′| ≡ h > 0K:PRDeSR^`_:DKJILP1NQVL_8NFS1_JP
(0, 1)f:ED^`_:MdDTS1NFM r KV`SRSRM[w:Z\D5p
0 1k/n
h/n
º:p°¯ t_JMONQD r M[W:M r M[SQIaW`NRD un K?VL_JP,Z®g :2b8D r K:Z[Dº:p°¯8p°¯8p
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O "&T1< < , < , % < ¯aºQ9PUV5_:P,VLw:NRDWJM\P®º:p 5f8V5W EVLWSANQPR_:M\N,_:W:DSR_:M\NRD r M\W:M r M[SQIaW`NQD (un)
]8X´JWJM\DK?VL_:Pn ≥ 1KJILP
un(x) =
h(x− k
n)
SRM kn≤ x ≤ 2k + 1
2n,
h(k + 1
n− x) SRM 2k + 1
2n≤ x ≤ k + 1
n,
K?VL_JP0 ≤ k ≤ n− 1 ,
EV rr D~Z\D r VLW`NQPRD~ZzI UM[L_:PQDsº:p°¯8p=Wc5VLM\N*nILEM[Z[D r DW`N ^`_:D un ∈ VDN^`_:D~ZzI¬]8XPQMOhcLXD
(un)′(x)W:DK:PQDWJ]µ^`_:D¡]:D_8bµcjIaZ[D_:PHS
+hDN −h p&P2IaPEVLWJSRXT^`_:DTW`NTf Jh(un) =
∫ 1
0 un(x)2dx = h2
4n
fFED¸^`_:M=K:PQVL_JcLD¡®º:p 5f!Eag DTS1N1hiYjhi]8M[PRD ^`_:DJhWdg I5] r DNtKILSu]:DK?VLM[W5N]8D r M\WJM r _ r S1_:P V p :mNK?VL_:PRNQILW5NefJSRM h = 0
fJM[ZdDeSANEZ[ILM\P^5_JDJ0I5] r DN=_JW _:W:Mz^`_:DK?VLM[W5N]8D r M\W:M r _ r v ≡ 0
• `j¨ « ¨L OGF / < + >e%)$; ") . A?<@ @/ -;=?<#"( #-G<9!" , +7 ( $;;2?>)$*)?(+,")$M/ " /(
J$%"(
C1[0, 1] 2 ZzI¸Z\_ r M[PQD]:DEDTS$EV5W`NRPQDh¦Db8D r K:Z[DTSTfdDb8I r M\W:V5WJSZ[I¸]8M\±yE_:Z\NRX ^`_:M&SRDtK:PQXTSRDW`NQDtDTW]8M r DTWJS1M[VLWuM\W8´W:M\DDNFS1V5_JSB^`_:DTZ\Z[DTS/G-+`K?VaNQG:TSRDTSBW:VL_SBK?VL_:c5VLWJSBDTSRKXTPRDTPBV5w8NRDTW:M\P/_:WyPQXTSR_:Z\NQIjN]|g Db8M[S1NRDTWJED K?VL_:P$_JW KJPRV5w:Z\ r D ]8D r M\W:M r M[SQIjNQM\V5W¶K?V5SRX ]:ILWJS_:W¬DTSRKJILED ]8D 7&M[Z\w?DPRNt]8D]8M r DTWJS1M[VLWM[W8´JW:M[DLp8VLM\N
V_:WDTSRKJILED!c5DTENRVLPQM[DZ:]8DFW:V5P r D ‖v‖ pLV5MON J _:W:DFnoVLWENRM[VLW$]8X´JW:M[D&S1_JP2_:W:D,KJILP1NQM\D
K]8D
VY$cjIaZ[D_:PHS=]JIaWJS
Rf?cLXTPRM\´ILW5N&ZzIuEVLWJ]:MONQM\V5Wº:p 95oM[W8´JW:M[DYyZ®g M\W:´JW:M p 2&Z[VLPHSfJNRV5_8NRDS1_:M\NRD r M[W:M r MzSQIaW`NRD (un)]8_*K:PQVLw:Z[ r D
infv∈K
J(v)®º:p a
DTS1NywV5PRW:XTDLp :mW£]:M r DWS1M[VLW¶´JW:M[DSRM V = RNfBVLW¬EVLWEZ[_8NuIaMzS1X r DW`NyEV rr D ]:ILWJSZ[IVL_JS1hiS1DeENRM[VLWº:p[Lp =DW$_8NQM\Z[MzSRILW5NmZzIEV r KJILEMONQX,]8DTSmnoDP r XeS2w?VLPQW:XTS&DNBDTW$S1_:KJKV`SRILW5NB^`_:D KDTS1N&noDP r XDN^5_JD J DeSANEV5W5NQM\W_:DV5_ S1D r MOhiEV5W5NQM\W_:DM\W8noXTPRM[D_JPRD r DTW`NHp) IaZ[G:DT_:PRDT_JSRD r DW`NTf_:WNRDZPRXeS1_JZONHIjN,DTS1N,nIa_8bDW ]8M r DWJSRM\V5W*M\W8´W:M\D5fEV rr DW:VL_JSFc5DW:V5WJS,]8DZ[DEV5WJS1NQIjNQDPep:x=Dr ILW:M[PQD&LXTW:XPHIaZ[DLfaV5W$KDT_8N!EVLWJEZ\_JPRD&SRMZ[D,NQPRM[K:Z[DN (V,K, J)
cLXPQM\´JD,Z[IEVLW]8MONQM\V5WuSR_:M\cjILW5NQD2KV5_:PFNRV5_8NRDSR_:M\NRD(un)n≥1
]:ILWJSKNRDZ[Z[D^5_JD
supn∈N ‖un‖ < +∞ VLW*Ilim
n→+∞J(un) = ` < +∞ =⇒ ∃u ∈ K NRDZ^5_JD J(u) ≤ ` . º:p 5
2&M[WJS1MkfJSRVL_JSFZ[DTS,EVLWJ]:MONQM\V5WJSºJp 9`BDNºJp `f8Z[DK:PRV5w:Z[ r D$º:p a!I5] r DN&_:W:DSRVLZ[_8NQM\V5W pIaZ[G:D_:PQD_S1D r DTW`NTfjZzIlEVLWJ]:MONQM\V5W®º:p 5DTS1N2M\W_8NQM\Z[M[SQIawJZ\D,EILPM\WcLXTPRM\´ILw:Z\D!DWLXTW:XPHIaZ LWKDT_8NEDTKDTWJ]:ILW5N&ZzIcLXTPRM\´JDP,K?VL_JP&_:W:DEZ[I5SRSRDKJIaPRNRMzE_:Z[M[PQD]:DK:PRV5w:Z[ r DTSTfNRPQTS,M r KV5P1NHIaW`NQSDWNRG:XTVLPQM\D,EV rr D!DTWK:PQIaNRMz^`_:D:aZ\DeSK:PQVLw:Z[ r DTS2]:D r M[W:M r MzSQIjNRM[VLW$¨5§ ?5 ?p8C/V rr D!WJVL_JSZ\DcLDTPRPQVLWJSl]:ILWJS=ZzI*VL_JS1hiS1DeENRM[VLW¸º:p°¯8p 9:f S1M
VDTS1N_:WDeS1KJI5ED]8D!7&M[Z\w?DPRNTf
K_:W~¨L?§ ` noDP r X]8D V f!DN^5_JD J DTS1Nu_:W:D noV5WJENQM\V5W·¨L?§ ` DN*EV5W5NQM\W_:D SR_:P K f/ILZ\V5PQS ®º:p 5$IZ\M[D_ DNuZ\DK:PRV5w:Z[ r DºJp°LIL] r DNt_:W:DSRVLZ[_8NRM[VLWdpBq DeS r VaNQM\cjIjNQM\V5WJS$K?VL_:PuM\W`NRPQV8]8_:M[PRD EDTSEVLW]8MONQM\V5WJSS1V5W`NTf ]|g _JW:DKJIaPRNTf ^`_:DZ\DeS=G-+KVLNRG:eS1DeS]8DEVLWcLDb8MONQXvSRVLW`NSRVL_JcLDW`NlWJIjNQ_:PRDTZ\Z[DTS]:IaWJSw?DTIa_EVL_JK]|g IaK:K:Z[MzEIjNQM\V5WJSTfDN]|g Ia_:NRPQDKJILP1Nef ^`_ g M[ZUSTg ILLM\Nl]|g _:W:D]8DeSPHIaPQDTSEZzILSQS1DeS=]8DK:PRV5w:Z[ r DTSK?VL_:P2Z[DTSQ^`_:DZzSZzI=NQG:XV5PRM[DFDTS1NBS1_:±$SQI rr DW`N2LXTW:XPHIaZ[D!DNBEV r K:Z\NRD5p IaMzS2EDeEM:W:D
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¯Lº = ) < *<,T%<
SRM[LW:M\´JDlKJILSF^`_:DEDeS!EVLWJ]:MONQM\V5WJS,S1V5W`N!Z[DTSFSRD_:Z[DTSF^`_:M ILSQSR_:PRDTW`N!Zkg Db8M[S1NRDTWJED]|g _:W r M[W:M r _ r 0=XTILW r VLM[WJSfTDW]:DG:V5PQS ]:_EI5]8PRDmEVLWcLDb8Dm]8Xc5DZ[VLK:K?XB]:ILWJS Z[DTSUS1V5_JS1h¦SRDTENRM[VLWJS|SR_:M[cILW`NRDTSTfT]:DTS]:MO±yE_JZONQXTS=]:_*N,+`K?D]8DvEDZ[Z[DTS,PQDWJEVLW`NRPQXDeS,]:IaWS,Z\DeS=EVLW`NRPQDh¦Db8D r K:Z[DTSFKJPRXeEXT]:DW`NQS,K?D_JcLDW`NSR_:PQcLDTW:M\Pep .@AF.@A 4N F E G > 9B4 G GxlIaWJSmNRV5_8N/ED^5_JMSR_:MONefLW:V5_JSmSR_:K:K?V5SRDPQVLWJSB^5_JD
VDTS1NB_:W$DeS1KJI5ED&]8D7&M[Z\w?DPRN r _JW:M]|g _:WKJPRV8]8_:M\N&SRETIaZzIaM[PRD 〈u, v〉 DN&]|g _:W:DW:V5P r DILSQS1V8EM[XD ‖v‖ p&ILK:KDTZ\V5WJS,^`_ g _:WtDTWJSRD r wJZ\D K DTS1NEVLWcLDb8DSTg M[Z|EVLW`NQM\DTW5N!NRV5_JS!Z\DeSFS1DT r DW`NQS/PQDZ[M[ILW5N,]8DT_8bu^`_:DZzEV5WJ^`_:DTSF]8DS1DeS!KV5M\W`NQScLVLM[P!Z[IxX´W:MONQM\V5W~e¯8p[Lp º5p8x=V5W:W:VLWS,^5_JDZz^5_JDTSFK:PQVLK:PQM[XNRXeS,]8DTS!noV5WJENQM\V5WJS,EV5WcLDb8DeSp
+ 6 ;U§ «oªe« ?§ F94 J E>01 SR 293 .4;D0<.JE7A<\p0<3G[2Bo2!3G3Y 83e;=.> L 3OT 39. L 0<E73
K ∈ V 3o` L >8M32B=GE7GR3G[;=. L 3T 3#G[2B K G[0N3SG3283Y 3SG[0
J(θu+ (1− θ)v) ≤ θJ(u) + (1− θ)J(v) ∀u, v ∈ K , ∀ θ ∈ [0, 1] .®º:p º5
C3648: G i J 3G[NE01V3#G[DB[0<;V3YZ3N;=.> L 3OT 3WG[0 8 R 014A 8h01 A 7I 3G[SG[?B[0<;V3#8M.>B=G3 u 6= v
3θ ∈]0, 1[ I 8 ? OGFGFM
JDTS1NB_:WJD&IaK:K:Z[MzEIjNQM\V5W$]8X´JWJM\D=SR_:P
KYcILZ\DT_:PHSm]:IaWS
RfLVLW$ILK:KDTZ\Z[D
6 « Dj8 4 ]8DJZkg DWJSRD r w:Z\D Epi(J) = (λ, v) ∈ R × K , λ ≥ J(v) p2&Z[VLPHS J DTS1NEVLWcLDb8DlSRM|DN=SRD_:Z[D r DTW5N=SRM Epi(J)
DeSAN,_:WJDKJIaPRNRM[DEVLWc5Db8Dl]:DR× V p •
`j¨ « ¨5 OGFGFH -J1
J2
/(<" #) . )$ % )-$ $ "7V, λ > 0
N%ϕ
"7o #) . % )S< $&$.>7-;J$ "7#"7' ;<)>7+,+ /(
R #-; )>7O+ <)$& (+
J1(V ) G<(!#"J1+J2
Nmax(J1, J2)
NλJ1
ϕ J1
$&% )-$ `j¨ « ¨5 OGF
(Li)i∈I"7)P> +,+ 1@ -")<+,+ < < 6 / ^ . )$ > )%$
$%"(V !#")
supi∈I Li%$&e # )$ "7
V @ ?<G!") -%N$& J "7)_ #) . % )W #-. 7")*$ "7
V #-G<!#"J%$& @%K7>7+ 0>7"
supLi≤J Li + %$N . )$
Li$R
> )%$ PVL_JPZ[DTSunoVLWENRM[VLWSEVLWcLDb8DTSyM[ZW g +£I¡KJI5St]:D¸]8MSR XPQDWJEDDTW5NQPRD r M\W:M r I¡Z[V8EIL_8b£DN5Z\V5wJIa_:buEV rr DZ\D r V5W5NQPRDZ[DPRXeS1_JZONHIjN,XZ[X r DW`NHIaM[PRDS1_JM\cjIaW`Nep
jB? «ªT« ?§ F9Uc,90J3[G[o293 .>9;?0<.>@;=.> L 3OT 35G[2B2 3HG3Y8M3Z;=. L 3T 3 K i V.26.01EF3ZYC0101YC2Y 8M.;=8SEF3JG=2B
K3[G[%2KY_0101Y_2Y F8M. =>8p3%8 R 3HG3Y8M3aE73[G6H.>01?G
E73YC0101YC2Y 3[G[p2K3G3Y 8M3C;=. L 3T 3 A L 3D38183Y 3 L 0<EF3 FI,90 E73N698: GJ3G[ G=DB[0<; 3YZ3p;=.> L 3OT 3 i >8M.B=GW018Q3OT70UG[ 3#>648: Gd256.01EF3#YC0101YC2Y I
+ 6 ?§ ª j ªe« ?§ 2VLM\Nu_:W r M\WJM r _ r Z\V8EILZ]8D J SR_:P K pxvg IaK:PQTS,ZzIyxX´JWJMONQM\V5Wº:p[Lp[LfWJVL_JSFK?VL_:c5VLWJSFXeEPQM\PQD
∃ δ > 0 , ∀w ∈ K , ‖w − u‖ < δ =⇒ J(w) ≥ J(u) .ºJp\e¹5
Ecol
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O "&T1< < , < , % < ¯aº .VLM\N
v ∈ K pQPVL_:P θ ∈]0, 1[S1_8±ySQI rr DW`N/K?DNQMONef wθ = θv+(1− θ)u cLXTPRM\´JD ‖wθ−u‖ < δDN
wθ ∈ KKJ_:M[SQ^`_:D
KDTS1N!EVLWcLDbD5pLxVLWJELf
J(wθ) ≥ J(u)]|g ILK:PQTS=®º:p[T¹`fLDN!ZzIEVLWcLDbM\NRX]8D
JM r KJZ\Mz^`_:D ^`_:D J(u) ≤ J(wθ) ≤ θJ(v) + (1 − θ)J(u)
fBED ^5_JM r VLW`NRPQDtwJM\DTW¬^`_:DJ(u) ≤ J(v)
fJEag DTS1N1hiYjhi]8M[PRD^`_:DuDTS1N,_:W r M\W:M r _ r 5Z\V5wJIaZ|SR_:P K pxvg Ia_8NQPRDFKJILP1NefaS1M
u1DNu2SRVLW`Nm]8D_:b r M\W:M r IlDNmS1M θ ∈ [0, 1]
fjIaZ[VLPHSw = θu1+(1−θ)u2DTS1N&_:W r M[W:M r _ r K:_:MzSR^`_:D w ∈ K DN=^`_:D
infv∈K
J(v) ≤ J(w) ≤ θJ(u1) + (1− θ)J(u2) = infv∈K
J(v) .
q D rr DPHIaMzS1V5W:W:D r DTW`N,Ic5DTE θ ∈]0, 1[ r VLW`NRPQD^`_:DLf?S1M J DTS1N=S1NRPQM[ENRD r DTW5NlEVLWc5Db8DLf:ILZ\V5PQSW:XTEDTSQSRILM\PQD r DW`N u1 = u2p
0=VL_JSUW:V5_JSUS1DTPRcM[PRV5WJS|KJILPZzI=SR_:M\NRD/] g _:WJD/W:VLNRM[VLW]:DU²AnoVLPRNRD/EVLWcLDb8MONQX³mU®& ª « ¨ ªT« ^5_JDlZzI$S1NRPQM[ENRDEV5W`c5Db8M\NRXLp+ 6 ;§ «ªT« ?§ F9r J E>01WSR 293 .9;D0<.
JE7A<\p0<3aG=2BC2 3HG3Y8M3 ;.> L 3OT 3 K 3[G[
.B[V3Y 3o;=.> L 3OT 3WG[0p3pG328M3YZ3pG=0 018 3OT70UG[ 3 α > 0 38 3
J
(
u+ v
2
)
≤ J(u) + J(v)
2− α
8‖u− v‖2 . ®º:p[L
J KE01% G=G[0pEFG5;3C;=)G53 J 3[G[α ;=.> L 3OT 3 IxlIaWS&ZzIyx=X´JW:M\NRM[VLWºJp ¯:p =f8Z[I$noVLPRNRDEV5WcLDb8M\NRX]8D
JW g DTS1N&NQDTS1NRXDv^5_JDKV5_:P]:DTS=EV r w:MOhWJIaMzS1V5WJSlEVLWc5Db8DTS&]:DvK?VLMz]:S
θ = 1/2p C/DZzIyW g DTS1NlKILS_:W:DPRDeSANQPRMzENQM\V5W KV5_:PZ\DeS&noV5WJENQM\V5WJSEVLW`NQM\W_:DeS,EV rr DZ\D r V5W5NQPRDlZkg DbDTPQEM[EDS1_:M[cjIaW`NTp
`j¨ « ¨L OGFMr #J
%$& #-. 7") %α % )7N #-G<O!#"7N ?<"79;" θ ∈ [0, 1]
N
J(θu+ (1− θ)v) ≤ θJ(u) + (1− θ)J(v) − αθ(1− θ)2
‖u− v‖2 . ®º:p[e¯5 `j¨ « ¨L OGF H
A"7 > $ @ !") / /G
N b ∈ RN "7 x ∈ RN N
T?< $&J(x) = 1
2Ax · x − b · x '#-G<'!")
J-$;5 # ) $E 2$& "(+ <2$P
A%$;
$& /(@ ?< $P . )(N 2!")J
%$&:$& ; -# # ) $9%:$R<"7+ -:$PA
-$;2/(@ < ?< $P . ) >7$% / 4 <o%>)$N G<Q!#") J -$;2>7"$R$P/ ; <# ) :#" )<0+ > ) +,+ "(G# )$;&>7; α `j¨ « ¨L OGF
Ω"7 #" )<'/(
RN H1(Ω)
+ %$ ?> =/( <#+ >)$R$&2 @1 ) + >J@ < . # #6% + ># . #
J/(@ < =$%"(
Ω?><
J(v) =1
2
∫
Ω
(
|∇v(x)|2 + v(x)2)
dx−∫
Ω
f(x)v(x) dx ,
>f ∈ L2(Ω) O G<O!#"
J-$;N ; <% )=$%"(
H1(Ω)
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¯Lº ) < *<,T%<
qdD$PRXeS1_:Z\NQIaNS1_:M[cjIaW`NSRDPHIuDeSRSRDW`NRM[DZ/]:IaWJSZkg VLw8NQDW`NRM[VLW~]|g _:WsPRXeS1_JZONHIjN]|g Db8M[S1NRDTWJED$]|g _:Wr M[W:M r _ r DTW]:M r DWS1M[VLWM\W:´JW:M[DLp :mWKJIaPRNRMzE_JZ\M[DPefM\Z8K?DP r DNm]8D,EV5WJEZ[_:PQD/^`_ g _:W:D!noVLWJENRM[VLW JnoV5P1NQD r DW`N,EV5W`c5Db8D=DNFEVLW`NRM[W_:DS1_:PF_:WtDWJSRD r w:Z[D K EV5W`c5Db8D&noDTP r XlW:VLWtcM[]8DlDTS1NB²AM[W8´JWJM\DYZkg M[W8´JWJM ³]:IaWSKf:Eag DTS1N1hiYjhi]8M[PRDlcLXTPRM\´JDZzIvKJPRV5K:PRM[XNQX$º:p 95p
jB? «ªT« ?§ F9m,90J3G=S;=. L 3T 3W;.>D013G[2Bg2 3HG3Y8M3
K;=. L 3T 3 3B[YZAe4.
L 0<EF3 i >8M.B=G018 3TF0UG[V3243 .>B[YZ3_8:014A=01BO3 ;=.?013L ∈ V ′ 3293C;=.HG[V 3 δ ∈ R
V3818M3[G3
J(v) ≥ L(v) + δ ∀ v ∈ K .®º:p[ 9`
,90dEF3648: GJ3[G[.B[V3Y 3#;.> L 3OT 3 G[2B K i >8M.B=Ga018o3OT70UG[V3 E73FT ;=.G[ >V3[G
γ > 03
δ ∈ RV3818M3[G53
J(v) ≥ γ‖v‖2 − δ ∀ v ∈ K .ºJp\ =`
+ 6 ?§ ª j ªe« ?§ %PmPQVL_JcLVLWSU] g ILwV5PQ]tº:p[ 9`pLMJDTS1N2EVLWcLDb8D!EVLW`NRM[W_:DoVL_SRM r K:Z[D r DW`NSRD r M\h¦EVLW`NRM[W_:DM[W8noXPQM[D_:PQD r DW`N=SR_:Pl_:WDTWJS1D r w:Z[D K EV5W`c5Db8DnoDTP r XW:VLW cMz]8DLf IaZ[VLPHS=SRVLWXTK:M[LPHIaK:G:D
Epi(J)]8X´JW:M/]JIaWJSZzI ,D r ILPQ^`_:D$ºJp ¯:p ¯5=DeSANEVLWcLDb8DvnoDTP r X$W:V5W~c`Mz]8D5pVLM\N
v0 ∈ KDNλ0 < J(v0)
p*Pm_:MzSR^`_:D(λ0, v0) /∈ Epi(J)
f!W:VL_JSt]8XT]:_:M[SRVLWSy]8_ kFG:XTVLPQ r D¯8p[Lp[Tº ]:D*SRXKIaPHIjNRM[VLW¶]|g _JW¶K?VLM[W`NDNy]|g _JW¬EVLWcLDb8DuZkg DbMzS1NRDWED*]8Dα, β ∈ R
DNy]|g _:W:DnoV5P r DZ[M\W:XeIaM[PRDEVLW`NRM[W_:D L ∈ V ′ NRDTZ[S,^`_:Dβλ + L(v) > α > βλ0 + L(v0) ∀ (λ, v) ∈ Epi(J) .
º:p[5.5C/V rr D5fmK?VL_:P v ´:b8XLf/VLW¬KDT_8NtKJPRDTWJ]8PQD λ IaPQw:M\NRPHIaM[PRD r DTW`N5PQILWJ]¬]:IaWJSyZ\D r D r w:PQD ]8D`Ia_JEHGJD]8D$ºJp\.Lf`M[Z|DTS1N,EZzIaM[PF^`_:D
β ≥ 08]8DK:Z[_JSTf:EV rr DlVLWtKDT_8N,K:PRDTWJ]8PQD v = v0
]:ILWJSZ[D r D r wJPRD]8D5IL_JEHG:D]:D®º:p[5.5f β WJDlK?D_:N NRPQDW`_JZ®pWIv]8V5WJE β > 0fDNFV5Wt]8Xe]8_:M\N=]8D®º:p[5.5/^`_:D
J(v) + L(v)/β > α/βK?VL_JPFNRVL_:N
v ∈ K f:ED^`_:M|K:PQVL_JcLD$ºJp\595pPmPQVL_JcLVLWS r IaM[W5NQDWJILW`NvºJp\ =`p?VLM\N=DWEVLPQD v0 ∈ K ´:b8XLp3PVL_:P,NQVL_8N v ∈ K fd®º:p[Le,DN®º:p[ 9`BM r K:Z[M[^`_:DTW5N=^`_:DJ(v)
2+J(v0)
2≥ J
(
v + v02
)
+α
8‖v − v0‖2 ≥
L(v) + L(v0)
2+α
8‖v − v0‖2 + δ .
WDW ]8Xe]8_:M\NJ(v) ≥ α
4‖v‖2 − α
2〈v, v0〉+ L(v) + C1 ,Ic5DTE
C1 = (α/4)‖v0‖2 + L(v0) − J(v0) + 2δpdxvg IaK:PQTSZkg M[W:XT5IaZ[M\NRXy]8DtC!IL_JEHG-+5hA8EHG FIaP.SILK:K:Z[M[^`_:X Y 〈v, v0〉 DN$ZzI¸EVLW`NRM[W_:M\NRX ]8D L fM®p DLp |L(v)| ≤ ‖L‖V ′‖v‖ oc5VLM[PZzI¸x=X´JW:M\NRM[VLW¯8p[Lp[LfM\Z|cM[DW`N
J(v) ≥ α
4‖v‖2 −
(
‖L‖V ′ +α‖v0‖
2
)
‖v‖+ C1 ≥α
2‖v‖2 − C ,
K?VL_JPC ∈ R
wJM\DTWEHG:V5M[SRMkp
kDP r M\W:V5WJSlEDNRNRDS1V5_JS1h¦SRDTENRM[VLW*KJIaPl_:W:DvK:PQVLK:PQM\XNRXIaLPQXTILw:Z[D]8DTS=noVLWJENRM[VLWJSlEV5WcLDb8DeS2Z[DTS!noV5WJENQM\V5WJS,EV5WcLDb8DeSU²1K:PQVLK:PQDTSA³vELg DeSANRh¦Yah¦]8M[PQDl^`_:M|W:DK:PQDWJW:DW`N,KJILSFZzIcjIaZ[D_:P+∞ !SRVLW`NEVLW`NRM[W_:DTSTp
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O "&T1< < , < , % < ¯aº `j¨ « ¨L OGFGH
v0 ∈ V%J
"7)g . #d # ) > " G@%*$%"( "() <#"(+ /(d%<v0
O!")J-$; G@% % #-. 7")0$%"( % ; <"7+
.@AF. BDCE<&FA6-NU6OE; G&@TE56%Gd4=> G0=VL_JSvK?VL_:c5VLWJS r ILM\W`NRDTWJIaW`NXTW:VLWEDP_:WK:PQD r M[DPPQXTSR_:Z\NQIaNv]|g Db8M[S1NRDTWJEDt]8D r M\WJM r _ r]:IaWJSFZ[DEI5S!KJIaPRNRMzE_:Z[M[DP,V U
JDeSAN,noV5P1NQD r DW`N&EV5WcLDb8D α h¦EVLWcLDb8Dep
b 4(6 j`¡L OGF « ª §U¨L ¢>\U§ « § « ¸U?¨5:$¤¦J ª `¡`§ ª ¨L?§ ` ,4.>01K2K;=. L 3T 3 3B[YZAC4. L 0<E73 EQR 2 018=3B[
V3J293 .4;D0<.
α ;.> L 3OT 3C;=.D013G[2B
K I 8M.>B=G i 018 3OT70UG[ 352^20<3Y_0101Y_2Y uE73JG[2B
K3o.
‖v − u‖2 ≤ 4
α[J(v)− J(u)] ∀ v ∈ K .
ºJp\ 5 6B[D0<;28h0<3B i V.2V3WG[201V3#YC0101YC0UGV3ZEF3 J G=2BW8 R 3HG3Y8M3
K;=. L 3B 3 L 3B=G u I
+ 6 ?§ ª j ªe« ?§ VLM\N(un)_:W:DSR_:M\NRD r M\WJM r MzSRILW5NQD]8D J SR_:P K p:xg IaK:PQTS®º:p[=f J DeSANr M[W:V5PRXTDSR_:P K DNef:K?VL_:P n,m ∈ NZ[IK:PQVLKJPRM[XNQX®º:p[Le!]8DnoV5P1NQDEVLWc5Db8MONQXDTW`NRPHI) W:D^5_JD
α
8‖un − um‖2 + J
(
un + um
2
)
− infv∈K
J(v) ≤ 1
2
(
J(un)− infv∈K
J(v)
)
+1
2
(
J(um)− infv∈K
J(v)
)
,EDy^`_:M r V5W5NQPRD$^`_:D$ZzI SR_:MONQD (un)DTS1N]8DC!IL_JEHG-+Lf|DNv]8VLWE$EVLWc5DPQLDcLDPHS_:W:D$Z[M r M\NRD u f^5_JMDeSANlW:XTEDTSQSRILM\PQD r DW`N=_JW r M\W:M r _ r ]8D J SR_:P K K:_:MzSR^`_:D J DeSANEV5W5NQM\W_:DvDN K noDP r XLpqmg _:W:MzEM\NRX ]8_ KV5M\W`Nu]8D r M[W:M r _ r I¸XNRX r VLW`NRPQXD*]:IaWSZ[I PmPQVLK?V5SRMONQM\V5W ºJp ¯:p 9Jp:mW8´JW f/S1M
v ∈ K f (u+ v)/2 ∈ K ETIaP K DTS1N&EV5W`c5Db8DLf8]|g V Udf8NRV5_AVL_:PHSFLPLEDlY*®º:p[Lefα
8‖u− v‖2 ≤ J(u)
2+J(v)
2− J
(
u+ v
2
)
≤ J(v)− J(u)
2,
EIaPJ
(
u+ v
2
)
≥ J(u)p
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J^`_:M?S1V5W5N!S1DT_:Z\D r DW`NBEV5W`c5Db8DTS=oDN/W:VLW$KJI5SnoVLPRNRD r DW`N/EV5WcLDb8DeSQp`C/DK?DWJ]JIaW`NTf`Ia_8NHIaW`NmZ[I]8X r VLWSANQPQIaNRM[VLW]8_KkFG:XV5PR r Dvº:p°¯8p yDeSANXZ[X r DTW5NHIaM[PRD5f?IL_8NQILW`NEDZ[Z\D]8_NQG:XV5PR r DvSR_:M\cjILW5NDTS1N]8XTZ\MzEIaNRD5p(:mZ[Z[DPQDK?V5SRDuDTWKJILP1NQM[E_:Z\M[DPSR_:PZ[IW:VaNQM\V5W¡]8D*EVLWc5DPQLDWED$nILM\w:Z[Dt^`_:DZkg V5WKDT_8N=EV5WJS1Mz]8XTPRDTPFEV rr D&²AG:V5PQS1h¦K:PRV5LPHI rr D³=]:ILWJSFZ\DETIL]8PQD]8DEDEV5_:PHSp
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2 ;=.> L 3OT 33B[Y A4. L 0<EF3ZE&R 2n3[GD6F;=3ZEF3 #018=3B[
V i 3 J 293 .>9;?0<.> ;=.> L 3OT 3C;.>D013#G=2B K i20S3[G[ 01\S0<35`Z8 R 01\S0dEF>HGK i ;R 3G[ ` E01BO3C20 L AB[0 \N3#8 ;=.4E01D0<.
7I )i `CG L .01B i∀(un)n≥0
G[201 3_EFGK , lim
n→+∞‖un‖ = +∞ =⇒ lim
n→+∞J(un) = +∞ .
8.B=G#018 3TF0UG[V32XY_0101Y_2Y EF3JG[2B
K I 8 ? OGFq DgkFG:XTVLPQ r DFº:p°¯8p =]:VLW:W:DFZkg Db8M[S1NRDTWJED,]|g _:W r M\W:M r _ r EV rr D!Z[D,K:PQXhEXT]8DTW`N kFG:XTVLPQ r D/ºJp ¯:p Jf r IaMzSdWJD/]8M\NUPQM[DWSR_:PUZ®g _:W:MzEM\NRX!W:M8S1_JPdZkg DeSANQM r IjNQM\V5Wv]|g DPQPQD_:P!º:p[ `p&D r IaPH^`_:VLWJSIL_KJI5SRSQIa5Dv^`_:D ®º:p[ 5SRDPHIynoV5P1N_8NQM\Z[D$KV5_:PZkg XNR_J]8Dy] g ILZ\5VLPQMONQG r DTSlW_ r XhPQMz^5_JDTSm]8D r M[W:M r M[SQIjNQM\V5W$K:_:MzSR^`_ g DZ[Z\D&noVL_:PQW:M\N/_:W:D=DTS1NRM r IaNRM[VLWy]8D=Z[IcM\NRDeSRSRD&]8D=EVLWcLDTPR5DWJED] g _:WJDS1_:M\NRD r M[W:M r MzSQIaW`NRD (un)
cLDTPQS!Z[DKV5M\W`N=]8D r M[W:M r _ r u p • 8 ? OGF q DkFGJXVLPQ r D,º:p°¯8p =PRDeSANQD,cPQILMJS1MJZ®g VLW$SR_:K:K?V5SRD&S1M r K:Z\D r DW`N/^`_:D V DeSAN_JWDeS1KILED]8D IaWILEHGuPQX Db8MOn,oMkp DLpJ^`_:DZ[D]8_JIaZ ]:D
V ′ DeSAN V p • 329B@-$F @4$9 =)@-4";("!#("$&%)"3!#!#("$&%>$NE?("%/ "!#$&%')(*+-(f.10"3)4"!#( 6 76 5 :3$9B+-(:G9
% 2$#("9'E2:"( ( @-+-=?(*'%19'*EF:Gd:G=2+-( / 9 ("9 %: <;= u @ )(&2@F:!#(_%m$2(>=F:39'(>02@-+-=?(*'%)@-("$$2(B "$23!= d:3=2+-( ;<@ >+ :f~)3E?9'@ %)@-$ [76I[6I[ 6 6F $ 8F$@-%1+R:f$3%)@-$ ( 9Z ?IYFI I B I :3$9V6
O ORQ OTS&U y1\Ne YZ N*[UL [ \?]/^)[ YZ ] (un) e&] V c X \ n ]_j&]"b Yai"h ] w ]_\ Z n ]_jd^ u ∈ V ^ Y∀ v ∈ V , lim
n→+∞〈un, v〉 = 〈u, v〉 .
7 3@-%(ei)i≥1
2$2(q=F:G9'(0@ +-=?(*'%)@-("$$2(v(V6F7@ $$2G%)(
uni = 〈un, ei〉 +-("9"!#E?39):3$&%)("9:3$29
"(*%'%)(>=F:39'( % 2$(,9'2@ %)(un $2@ G)!#"!#("$&%J=?3)$2*( :3$29
V @-+F(*9 %T:"@-+ ( (>;<*)@ 8F(_ &2(,+ : 8F$@I %)@ 3$ 6 76I[\f ( + :?*$;<(*"(3("$"(T:G@ =+-(B("9 %&@ ;3:3+-("$&%)(X;q+ : 9 ?IWFI 3I I "IW IW? "IW I 9PK Z B*I
(uni )n≥1
E?i ≥ 1
6-m!#!#(19'3$$! +' @ $2@4&2(1+ :*$;<(*"(3("$"(:3@-=2+-(("9 %J$2(1$3%)@-$ uE+-29#T:G@-=2+-( 1&2(m+R:X"$;<(*:
(("$"(/29'2(*+ +-(q2:G$29V E2@-9"&2( limn→+∞ ‖un − u‖ = 0
@-!#E2+-@4(R&2(un *$;<(*"(3(>:3@-=2+-("!#(*$%
;<(*)9u6 >"@-E )Y("!#("$&% (*$ @ !#(*$29'@-$@-$82$2@-(@-+ (*C @-9 %)(f ("9B9'2@ %)("9)@ "$;<(_"(("$&%:3@-=2+-("!#(*$%
! :3@-9fEF:G9#:3 9'("$9#29'2(*+ / (D+' $ :3EE?("+-+ ( EF:G:@-9 <"$;<(_"(("$"(3'%)( DEF: #EE?9'@ %)@-$ 6 6 : (*C ("!#E2+-( + : 9'@ %)(un = en "$;<(*"(3(RT:3@-=+ (*!#("$&% ;<(_)9 "_) ! :G@ 9>EF:G93'%)("!#(*$%>E2@-9"&F ("+-+-(v("9 %
(B$3)!#()"3$29 %d:G$%)((<:3+-( ;[36BGH @-$&%)* *% (B+ :C"$;<(*"(3("$*(T:G@ =+-(q;@-("$&%>)"9'2+ %d: %9'@-;:G$%6H IKLKI PORQ OTSFS ] ZrX [ Z ]m^)[ YZ ] un i)X jd\Pk)],e<b3\2^ V
X \ `]_[ Z ]:M Z j'b Y j)]1[ \?]m^ X [ ^ s ^)[ YZ ]XN*[ Y c X \ n ]_j &]"b Yai"h ] w ]_\ Z xK ~OC-m3!#!#( + :9'@ %)(
un (*9 % =?3)$2*( d0F:&2( 9'@ %)(A% 2$(V"3!#E?9):G$%)(un
i
("9 %=?3)$2*( :3$29
R63._02:9&2(
i @-+F(*C @-9 %)( $>$2(,9'93 r9'2@ %)( $2G%)"( unii &2@"$;<(*"(3(;(*)9 $2(+ @-!#@ %)(
ui6o :Gq2$E)J" % (*C%'d:_%)@ 3$ @ :($2:3+-((#9'@-%)(*9 3$L= %)@ (*$%q:G+-3)9q2$2(9'293 r9'@-%)(
"!#!v2$2(n′ %)("+-+-(V&2( E?3/%) % i un′
i
*$;<(*"(3(;(*)9ui6#-m(@~E);<($&2(
un′ "3$ ;(*"((T:3@-=+ (*!#("$&%,;<(*)9
u / $D;<*)@ 8F(R&(u ∈ V 6 6
7 @ $N:3EE?("+-+ ( u(*!#@I r("9'E2:"()(*)!# / (V
%) % ("$9'("!=+-(/(/+ :3)!#( v ∈ V , L(v) ≤ α 9&L
("9 %2$2(3)!#(A+-@-$2":3@ )( "$&%)@-$2( $$ @ ("$&%)@ &("!#("$&%#$2+-+-(A9'V
(*%α ∈ R 3$ E?(" %
:Gd:*%)*)@-9'(* ()T:5"$V"!#!# (B+-("9,("$29'("!v=2+-("9 "3$;<(*C ("9(_)!#"9"6
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K("9 %/@-$"+-29q:3$29+' @-$&%)(*)9'(*%)@-$ ("9q("!#@ r("9'E2:"("9)(*)!#"9
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K6H $ E?("%:3+-3)9f:3E2E+-@ &(*+-(D.102"G)4"!#( [ 7&64[36I[ (D9'*EF:Gd: %)@-$
2$#E?@-$&% (*% $D"$;<(_C(,(*%."$9 %')2@ )(:3@-$29'@?$ ("!#@ r("9'EF:"( (*)!#&2@"$&%)@-("$&%K
! :3@-9JEF:39u0
6Z-m(*@ ("9 %>$2()"3$&%'d:@I*%)@-3$A:;<(+ :f 8F$2@ %)@-$ (u0 3$ u0 ∈ K
6
HJIKLKI ORQPOTS V?X3YZK [\N]_\2^*] w#i"h ]qc X \ n ]:M&] ]_j w k \ X \ nY e<]qe&] V x h-X jd^ K ]_^ Z ]_j w k1` X [jh b c X \ n ]_j&]_\?cd] "b YTi"h ]x
]/` h [ ^)|^ Y J ]_^ Z c X \ n ]:M&] ] Z ^*] wYTs c X \ ZTY \[2] Y \ k_j Y ]_[ j'] w ]_\ Z ^)[j KlanGX3Y j h L OM&]_j c Y cd] <x<x
` X [ jcd] ZTZ ] \ X3ZTYTX \ p | b h-X j_^ J ]_^ Z b3[^d^ Y ^"] wvYTs c X \ ZTY \[F] Y \ k_j Y ]_[j)] w ]_\ Z ^)[j K ` X [j h bqc X \ n ]_j &]_\?cd]*b Yai"h ]x@K= JO : 1 8F$@-%)@-3$ 9'@ un "3$;<(*"((:3@-=2+-("!#(*$% ;<(*)9
u :3+-3)9 L(un)"$;<(_"((,;<(*)9
L(u)6 : R"3$29'&2(*$% 2$N(*!#@I r("9'E2:"(>(_)!# (
V("9 %X(*)!#qE? + :D"$;<(*"(3("$*(?T:G@-=2+-(6FG (
G ("!#!#(6 76I[ 7fE?(*)!#(*%% = %)("$2@ >+ :! "!#(*$*+ 9'@-$ E?3K6
:3E)4*9D+-("90J2E?G%)024"9'(*9#9'J +' ("$9'("!=+ (
Epi(J) / 82$2@; + : (*! :GS&2( 6 76 7 6 ("9 %D2$*$;<(*C (0(_)!# (
R × V 3$N@-+ ("9 % :329'9'@X(*)!#E? + : "3$ ;(*"(("$"( T:3@-=+ (36X $ ("$ 2@ %A:G+-3)90T:"@-+-("!#("$&% +-()*9'2+ %d:G% +,9'@+ : 9'2@ %)(
(vn)%)(*$F T:G@-=2+-("!#("$&%A;<(_)9
v:3$9
K :G+ G)9lim infn→+∞ J(vn) ≥ J(v)
6
9>: ;$29,! :G@ $&%)("$2:3$&%1%)329,+-("9,@ $(3) @-("$&%)9>E?3 82$2@ 6@K= W KI ORQPOMPOW :3E)4*9 / 6 6 %) %)(9'2@ %)(,!#@-$2@-!#@-9):3$&%)(
(un)("9 % =?3:
$"(6 $q"2@ %o:3+-3)9~ G ("!#!#( 6 7&64[[1&F @-+<(*C @-9 %)(m$2(J9'293 r9'@-%)((un′
)"3$;<(*"((":3$&%:3@-=2+-("!#("$&%
;(*)92$(B+-@ !#@ %)(u ∈ V
6BT :3@-9 % :3E )4"9,+ (>G ("!#!#(6 76I[ u ∈ K(*%
J(u) ≤ lim infk
J(unk ) = infv∈K
J(v) .
G (E?@-$&%u("9 %3$B=@ (*$ 2$D!#@ $@-!2! (
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ACBEDGFIHKJCLNMOBEPOQSRUT/JVWDOXZYOF.DOXZYOFIF/XZ[\[\J]FIQ^D2R/_]T/J`F/F/_BEY8acb2Y8J]F-R/QKXZDGFIdfe J$a6QKF-R/J]DOg$J;d6J[\Q^DOQSh["Y8[iBZY6ajP8TUXZkOH^l][cJ`Fdfe XmP6R/QK[\QKFUBnRUQ^XmD+o8ACBEDOFgJpg&MGBEP8Q^R/TUJZqrD8XmYOFBEHKH^XmDOFg&M8JT&g&M8J]TstXZk8R/JDOQ^Td6J]FugXZDOd6Q^R/QKXZDGF\D8_]gJ]FUF/BZQ^TUJ]FcJ$RPGBET/vwXZQxFcF/Y6yFUBED2R/J`Fud6Jz[cQKD8QK[uBEHKQSRU_Zo44e Xmk6|@J]g$R/Q^vCJ`F@Rjdfe Y8D8Jg$JT/RUBZQ^DOJ[uBED8QKlTUJ"k>J]BZYOg$XmY8P~P8HKYOF;P8TUBER/Qxb2Y8JZq>P8Y8QxF/b2Y8Jcg$J`F;gXZDOd6Q^R/QKXZDGF;dfe XZP8R/QK[\BZH^Q^R/_cF-J]T/XmD2RH^JtP8H^YGFCF/XZYOZJD2R=Y8R/QKH^QxF-_]J]F;PGXmY8T;R/JD2RUJTpd6Jtm8m'm 'wPOBZT-vwXmQKF=[c][\JcFUBEDOFBZXZQKTF-Yzd6_[\XmDmRUT/J]TF-XmDJarQxF-R/JDGg$JIo84e Qxd6_JWm_D8_]TUBZH^Jpd6J`Fg$XZDGd6QSRUQ^XmDOFdfe XZP6RUQ^[uBEHKQ^R/_WJ]F-RHKB[c][\Jtb2Y8J"gJHKH^J"b2Y8Qq>H^XmTUFUb2Y8JWHe XZDgBEHxg$YOH^JHe J$arR/TUJ[tY8[d+e Y8DOJvwXZDGgR/QKXZD~F/Y8TRq>gXZDOF/QKF-R/Js_]g$TUQKT/Jpb2Y8JF/B\d6_]T/QKZ_]JCd8XZQ^R=Fe BED8DrY8HKJT`oLNJ`FtgXZDOd8QSRUQ^XmDOF"mXZD2Rcd8XZDOgF]e J$a6P8TUQ^[\JT\sHe BZQKd8Jd6JHKBd6_TUQKZ_JjP8T/J][\Q^l]T/Jg$XmDOd6Q^R/QKXZDOFdfe XmTUd8T/Jp`.XZYuF-J`g$XmDOd6JgXZDOd8QSRUQ^XmDOFIdfe XZT&d6TUJmo2XZYGF.XZk6RUQ^J]DOd6TUXZDOF4F-Y8T/R/XmY6R4d6J`FIg$XmDOd6Q^R/QKXZDOF' rZr¡`¡]Ownr¡jdfe XZP6RUQ^[uBZH^Q^R/_mq [uBZQKFtHe Y6RUQ^HKQKFUBnRUQ^XmD¢d6JHKBd6_]T/QKZ_]JjF-J`g$XmDOd6JjXmYHe QKD2R/TUXrd8YOgRUQ^XmDdfe Mr£rP>XER/MOl]F/J]Fd6Jg$XmDrZJ$a6Q^R/_;PGJ]T/[\J$R/R/TUXZD2RBEYOFUF/Q¤d+e Xmk6R/J]D8Q^Td8J]Fg$XmDOd6Q^R/QKXZDOF¡`¦¥¡`8¤§`2¡mq6J$Rd6Jd6QxF@RUQ^D8mY8JTJ]D2R/TUJC[\QKD8QK[\BcJR[\BEa6Q^[uB8oLNJ`Fpg$XZDGd6QSRUQ^XmDOFCdfe XmP6R/QK[uBEHKQSRU_"m_D8_]TUBZH^QxF-J]D2R=HKBjT/J][uBET&bmYOJ_HK_[\JD2R&BEQKT/JcF-Y8QKnBED2R/J¨fF-Q
x0J]F-R;Y8DPGXmQ^D2RWd6J"[\QKD8Q^[tY8[©HKX6gBEHªd6JJF-Y8TCHe QKD2R/JTUnBEHKH^J
[a, b] ⊂ RJ_RUBZDmRCY8D8JtvwXZDOg$R/QKXZDd6_TUQ^nBZk8H^JWF/Y8T
[a, b]$qOBEHKXZT&FNXZDB
J ′(x0) ≥ 0F-Qx0 = a , J ′(x0) = 0
F/Qx0 ∈]a, b[ , J ′(x0) ≤ 0
F/Qx0 = b .«[\JF/QCJHKH^JJ]F-Rk8QKJD¬gXZD8DrY8Jd6YHKJ]gRUJY8T`q®TUBZP8PGJ]H^XmDOFHxBd8_[\XZDOF-R/T&BnRUQ^XmD¯d6JgJ$R-RUJTUJ$h[\BZTUb2Y8J°¨=F/Q
x0 ∈ [a, b[qXmDP>JY8Rg&M8XZQxF-QKT
x = x0 + hBZJ]g
h > 0P>J$RUQSR~JR~_]g$TUQKT/J
±m²
Ecol
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repr
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±m² ! "#$%&'#)(* +,
J(x) ≥ J(x0)q8d+e X.-
J(x0) + hJ ′(x0) + o(h) > J(x0)qOg$Jpb2Y8Q+d6XZDOD8J
J ′(x0) ≥ 0JDzd6Q^rQ^hFUBED2RPOBZT
hJ$R;JDvBEQxF/BZD2RR/JDGd6T/J
hZJT&F
0oOA;J[\[\JXmk6R/QKJD2R/hXmD
J ′(x0) ≤ 0F/Qx0 ∈]a, b]J]D¬g$XmDOF/QKd6_]TUBZD2R
x = x0 − ho/J][\BZTUb2Y8XmDOF_]mBZH^J][cJ]D2RgZe J`F@RHKB¢g$XZDGd6QSRUQ^XmDdfe XZT&d6T/JZb2Y8JF-Q
x0 ∈]a, b[J$R\F-Q
J ′ J`F@R\d6_TUQ^nBZk8H^JjJD x0q XZDB~BZH^XmTUF
J ′′(x0) ≥ 0JDJ0>JR]q XZDB
J(x0) +h2
2J ′′(x0) + o(h2) ≥ J(x0)
P>XZY8ThBmF/F/J1pP>J$RUQSRo'BF-R/T&BnR/_]ZQKJ~dfe XZk6RUJD2R/QKXZDJRd6Jd6_[\XmDOF@RUTUBER/QKXZDd6J`FjgXZDOd6Q^R/QKXZDGFd6J[\Q^D8QK[uBEHKQSRU_J`F@Rd8XZDOggHKBZQ^TUJ ¨IXZD°R/QKJD2RjgXZ[\P6R/J~d8J]FgXZD2R/T&BEQKD2R/J]F
x ∈ [a, b]d8BEDOF\He J$a6J[\P8HKJg$Q^h d6J]FUF-YGFUP>XZYOT"RUJ]F-R/JTcHxB~[\Q^DOQ^[uBEHKQ^R/_zd6J
x0dOBEDOFcd6J]Fcd6QKT/J`gRUQ^XmDOF"PGBET/R/Qxg$Y8HKQ^l]T/J`F"b2Y8QTUJ]F/PGJ`gR/J]D2RtHKJ]FgXZD2R/T&BEQKD2R/J`Fu
x0 + hBmJ]g
h > 0F-Qx0 ∈ [a, b[, x0 − h
BmJ]gh > 0
F-Qx0 ∈]a, b]
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A;_]F/XZTU[uBEQxFwJR=D8XZYGFD8JWHKJT&BEP8P>JHKH^J]T/XmDOFP8HKYOFF/£6F@RU_[uBnRUQKb2Y8J][\JD2R&$qOD8XmYOFF/Y8P8P>XmF/JTUXZDOFb2Y8JVJ]F-RWY8D J]F/POBZgJcd8JCd=Q^HKk>JT/RT/_]JHq>JR"b2Y8J\H^Jug$TUQ^R/lTUJ
JJ]F-RWY8D8J\vwXZDGgR/QKXZD g$XmD2R/QKD2YOJcsnBZH^J]Y8TUF d8BZDOF
RoE¦J4P8TUX6d6Y8Q^RªFUgBZHKBZQ^TUJNdOBEDOF
VJ`F@R R/XmYn|@XZYOTUF'D8XERU_ 〈u, v〉 J$RªHKB;D8XmT/[\JNBmF/F/X6g$QK_J
‖u‖ oLNXm[c[\J]DeP$XmDOFCPOBETpQKDmRUT/X6d6YOQ^TUJ"HxBD8XERUQ^XmDd6J\d6_TUQKZ_JcP8TUJ[\QKlTUJtd6JJP8Y8QxFUbmYOJcDOXZYOFCJDBZY8TUXZDOF k>J]F/XZQKDtP>XZY8T4_]g$TUQKT/Jd6J]F.gXZDOd8QSRUQ^XmDOF.dfe XZP6RUQ^[uBEHKQ^R/_Zom+XZT&FUbmY¦e QKH6£"BCP8HKYOF-QKJYOTUFªnBETUQxBEk8HKJ]FgEe J]F-R-h snh d6QKT/JF/Q4He J`F-PGBZg$J
VD+e J`F@RtPOBZF
R$q'HxBf-kGXmD8D8Jhg\D8XZR/QKXZD RUM8_XmT/Qxb2Y8Jd6Jjd6_]T/QKnBEk8QKHKQSRU_ZqBZP8P>JHK_JWd6Q@0>_]T/J]DmRUQKBZk8QKH^Q^R/_BZYF/JDOFd8J*T8TU_]g&M8JR]q6J]F-R=d6XmD8D8_JpPOBZTHKB\d6_:OD8Q^R/QKXZDF/Y8Q^nBZDmRUJZo
i kj w§`54>mln$o5lWo5lqpr&stvu3wxeyz@| h r~hut rJ s.5)ret5yxxrI ttvrW|ky"s.y
u ∈ V |zyhxs.|reR yhu*s.htv|z@y xst cyreut5|z@y!|xhyr6s.y QQ~6yu*yhr
u6 tvzyQ.tuy
xry h CyztvrQ| tvcy¡~ reutvrexey*xV L ∈ V ′ uyzvz@ywxey
J(u+ w) = J(u) + L(w) + o(w) ,| ¢y~
limw→0
|o(w)|‖w‖ = 0 .
@ ² oK`
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k (%) %)# % ±Z²m±pr | yhzvz@y L z@|stv¢y x'z@|s t cyhrYuZt5yzvz@y xz@yc|s t5yhrYuSsy
Jyr
uyhu r&r u y
L = J ′(u) jr6=f ln$o5lWo¦BA;_h:OD8Q^R/QKXZD ² oKZoKzJ]F-RjJDvBZQSRBZHKBZk8HKJ~F-Q
VJ]F-RF/JY8HKJ[\JD2RjYODJ]F/POBZgJpd8JBEDOBmg&MXZDzD+e Y6R/QKH^QxF/JPOBZFd8JP8TUXrd8Y8QSR=FUgBZHKBZQ^TUJWd8BEDGF@ ² oK`-$oGLNJP>JDOdOBED2R]q>F-Q
VJ]F-RY8DJ`F-PGBZg$Jud6Jd=Q^HKk>JT/R]q+XmDP>JY6RtP8T/_`g$QxF-J]TpHKBTUJHxBnR/QKXZD@ ² oK`;JDQxd6JD2R/Q@:GBZDmRVJRF-XmDd6YOBEH
V ′ ZTZgJBEY";M8_]XZTUl[\J"d6JTUJPOT/_`F-J]DmR&BnRUQ^XmDzd8J/QKJ]Fc1`8o^mo^8o IDJh0¤J$R]q¤QKH¦JarQxF-R/J"YODY8D8QxbmYOJp ∈ V R/J]H¦b2Y8J 〈p, w〉 = L(w)
qGd6XmDOg\@ ² o^®d6JrQKJD2RJ(u+ w) = J(u) + 〈p, w〉+ o(w) ,
BmJ]glimw→0
|o(w)|‖w‖ = 0 .
- ² oZ9;DcDOXER/JBZYOF/F/Q8POBZT-vwXmQKF
p = J ′(u)qmg$Jb2Y8Q8P>JY6RNP8T/R/J]TIsWg$XZD8vwYOF-QKXZD¦o^oKoE¦B;vwXZTU["YOH^J- ² oZ'J`F@RF-XmY8ZJ]DmRNP8H^YGFKf-DOBnRUY8T/J]H^HKJg=b2Y8J"- ² o^qZDOXERUBZ[\[cJ]D2R®F/Q
V = Rn XZYV = L2(Ω)
oªBZTNgXZD2R/TUJZqJHKH^JtPGJ]Y6RCR/TUJtdfe QKDmRUJTUP8TU_$RUBER/QKXZDY8DPGJ]YP8HKYOFCd6_HKQKg]BnRUJ"P>XZY8Tpd6J]F;J]F/POBmg$J]F;d6Jd=Q^HKk>JT/RCP8HKYOFf-gXZ[\P8HKQKb2Y8_`FgCgXZ[\[\J
V = H1(Ω)wmXZQKTHe .a6JT&g$Qxg$J\ ² oKZoVUZo •ACBEDGFHxBuP8HKY8POBET/R=d6J`F=BEPOP8H^QxgBER/QKXZDOF]q8QKH'F/Y6y\RCF/XZY8mJD2Rd6J"d6_R/J]T/[\QKD8JT;HKBcvwXmT/[\JHKQ^DO_]BEQKTUJg$XZD2RUQ^DrY8J
L = J ′(u) ∈ V ′ g]BET.XmDtD+e BpPOBmFªkGJ`F-XmQ^Dud6JHe Ja6P8T/J`F/F/QKXZD"Ja6P8H^Qxg$Q^R/J;d6J p = J ′(u) ∈VHKXZT&F/b2Y8J
V ′ J]F-R;Qxd6JD2RUQS:O_\s V o IDP8T&BnRUQKb2Y8JmqGQKHªJ]F-RCPOH^YOF=vBZgQ^HKJ\d6JRUT/XmY8ZJ]T=He JarPOT/J`F/F/Q^XmDJ$a6P8HKQKgQSRUJd6JLb2Y8JgJHKH^JWd6J
pqOg$Xm[c[\JpHKJC[\XmDmRUT/J]D2RHKJ]FJ$a6JT&g$Qxg$J`F®F/Y8QKnBED2RUF]o
?+2nZZln oZlYoZl "!$# %"&')(+*-,. ( !/ . 021 ( 3! ( 5476 J 8u & "98 0 !:;: (!/3<=: ( L1, L2 > 4 ( ? 8
J(u+ w) ≥ J(u) + L1(w) + o(w) ,J(u+ w) ≤ J(u) + L2(w) + o(w) ,
- ² o ± 0. =: J 9:5@64 ( > 0A3. BC L1 = L2 = J ′(u) &?+2nZZln oZlYoDFE-r¡]¡`r¤§]rGIHKJ= ( a !MLI= * A(+. ( 4 0( @:ON * 4P ( !/ 1 "9 ( !Q:P! V ×V & J= ( L !BLI= * . ( 4 0( 1 ( 3!/B:P! V &"R , :O J(u) = 1
2a(u, u)− L(u) & !/JC:OQ6348 ( > 0A. ):P! V P@"! 〈J ′(u), w〉 = a(u,w)− L(w) , "!3S5"!9 u,w ∈ V &
?+2nZZln oZlYoTUJ= ( A !3/ *V0 ( 1 :5N * 4P ( !/ N ×N Pb ∈ RN &XW ! x ∈ RN <"
, :O J(x) = 12Ax · x − b · x &
Y!/J9:56348 ( > 0A. C!/ J ′(x) = Ax − b , !5!
x ∈ RN &?+2nZZln oZlYoZ R [ , 8\6 .+]^3_ 1/( 1 %"&3& `a0 > 1 V = L2(Ω)
#Ω4C 0 9B!"! > 8b6
RN ' < a(u, v) =∫
Ω uv dx<cC
L(u) =∫
Ω fu dx0 > 1 f ∈ L2(Ω) &d^ ( 69 ( ?0 V C
V ′ <* e"! J ′(u) = u− f &?+2nZZln oZlYoDf R g , /6 .+]h^3_ 8 1( 1 9%"&3& `i0 > 1 V = H1
0 (Ω)#Ω4C 0 9V!3"! > 8@6
RN ' "! .+] * ! ( Q6"! , =6"! ( : 190. 0(
〈u, v〉 =
∫
Ω
(∇u · ∇v + uv) dx.
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±m² R ! "#$%&'#)(* +,
R , :; a(u, v) =∫
Ω∇u · ∇v dx <eP L(u) =
∫
Ωfu dx 0 > 1 f ∈ L2(Ω) & 8V# 0 !* ( /:-LI= * . . * 8 ' "!/ J ′(u) = −∆u − f
6 0 : V ′ = H−1(Ω) & @"!/3< : ( "( 638 ( ? V PV ′ < 0. =: J ′(u) = u0
u09:5 .+] ! ( "!/ :; . ! ( -6 0 : H1
0 (Ω)6
−∆u0 + u0 = −∆u− f 6 0 /: Ωu0 = 0
:P!∂Ω
?V+2nZmln oZlYoUJ= ( Ω!3$! > h A = /4g6 RN # , ! 0 :O 9:5 ( \6 0 ! 1C0 :
N = 1 0 > 1 Ω =]0, 1[ 'P& J= ( L = L(p, t, x)!/ LI 1 ( " 1 "9 ( ! :C!3 RN×R×Ω
<C6348 ( > 0A. ,\0 0,, =h p P t :P! 1 C :; *-A. 3<63)64 ( > 4C9: ,\0 h ( . . C: ∂L∂p
P ∂L∂t
(+, : 18( ( 3/C:Q:P!1 PQ8/:O *A3. &"R , :; V = H1
0 (Ω)PJ(v) =
∫
Ω
L(∇v(x), v(x), x)dx &3& "98 !/ J C:OQ6348 ( > 0A. B:P! H1
0 (Ω)P@!/
〈J ′(u), w〉 =
∫
Ω
(
∂L
∂p(∇u(x), u(x), x) · ∇w(x) +
∂L
∂t(∇u(x), u(x), x)w(x)
)
dx .
`"& J ( N = 1C
Ω =]0, 1[< * "98 "!/3<: ( u ∈ H1
0 (0, 1): 0 ( :OL 0( J ′(u) = 0
< 0. =: u> 4 ( ?
d
dx
(
∂L
∂p
(
u′(x), u(x), x)
)
− ∂L
∂t
(
u′(x), u(x), x)
= 0 ,@ ² o R2
, C:;"!/ ,\0 h5"!9M6 0 /: .+] ( 58 > 0. . ]0, 1[ & & J ( L 64 , 8\6 ,/0 :b63 x # (& & L = L(p, t) ' P@: ( u ∈ C2(]0, 1[)
C:OV!/):O . ! ( 631\. 0 :;:O)6 .+] 49"! 0 ( -6 ( 488 ( . . 7# 9%"& ' < * 8X"! . 0 ! 0 9 ( 54
L(
u′(x), u(x))
− u′(x)∂L∂p
(
u′(x), u(x))
C:O 1 /:5 0 5V:C!3 .+] ( 58 > 0. . [0, 1] &r6=f'Mln oZlYoT H¤JarQxF-R/J;d+e BZY6R/TUJ]FND8XERUQ^XmDOF4d8JCd6Q@0¤_TUJD2R/QxBEk8QKHKQSRU_ZqrP8HKYOFIvBZQ^kOH^Jpb2Y8J;gJHKH^JBZY~F-J]DOFd6J¡T8TU_]g&M8JR]o.BZT=J$a6J][cPOH^JmqOXZDTUJDGg$XZD2RUT/JWF/XZY8mJD2RHKBjd6_h:ODOQSRUQ^XmDF-YOQ^nBED2RUJZoW9;D~d6Q^Rb2Y8JHxBvwXmDOgRUQ^XmD
JqEd6_:OD8QKJF-Y8TªY8DcZXmQKF/QKDOBEmJNd6J
u ∈ V s=nBEHKJYOTUFªd8BZDOFRqnJ`F@RId6Q@0¤_TUJD2R/QxBEk8HKJBZYF/JDGFd6JBnRUJ]BEY8ajJD
uF]e QKHfJ$a6QxF@RUJ
L ∈ V ′ R/JH'bmYOJ∀w ∈ V , lim
δ0+
J(u+ δw)− J(u)
δ= L(w) .
@ ² oVUZ4e QKD2R/_]T/RWd8Jg$J$R/R/JuD8XZR/QKXZD J]F-Rb2Y8JuHKBm_TUQS:Gg]BnRUQ^XmDd6J~- ² oVUZ;J`F@RP8HKYOFBZQKF/_Jub2Y8Jug$J]H^HKJd6J- ² o^oILNJP>JDOd8BZD2R]qIF/Q®YOD8JjvwXmDOgRUQ^XmD¢d6_]T/QKnBEk8HKJzBEY¢F/JDGFcd8JTOT/_`g&M8J$R\He J]F-RuBEYOFUF-QBEY°F-J]DOFd8JBnRUJ]BEY8a¤q4HKBTU_]gQ^POT/X6b2Y8JJ]F-RvBZYOF/F/JZq4[c][\JJDd8Q^[\JDGF-QKXZD :ODOQ^JmqNg$Xm[\[cJH^J~[\XmDmRUT/JHe J$a6J[\P8HKJF-YOQ^nBED2Rd8BZDOF
R2
J(x, y) =x6
(y − x2)2 + x8
P>XZYOT(x, y) 6= (0, 0) , J(0, 0) = 0 .LNXmDrZJDOXZDOFNbmYOJZqrd8BEDOF®g$Jpb2Y8Q¤F-Y8Q^R]qrD8XmYOF®d6Q^TUXZDGFNb2Y+e Y8D8J=vwXZDOg$R/QKXZDjJ`F@Rd6_TUQKBZk8HKJ=HKXZT&F/b2Y+e JHKH^JHe J]F-R=BEYzF-J]DOFd6J*T8TU_]g&MOJ$R]q8FUBEY8v'[\JD2R/QKXZDzJarPOH^Qxg$Q^R/Jd6Yzg$XmD2R/T&BEQKT/Jmo •
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
k (%) %)# % ±Z² U Ia6BZ[\Q^D8XmDOF[uBEQKD2R/J]DOBED2RH^J`FP8TUXZP8TUQ^_R/_`F®d8JWkOBZF/JWd6J]F®vwXmDOgRUQ^XmDOFg$XmDrZJ$a6J`FNd6_]T/QKnBEk8HKJ]F]o
p¢4YB)4>¡]§]Z4Gln$o5lWohZ tvuJxry| zSt5~Q| uZt r st cyreut5|z@y+sy
Vs| r6
R yh|¢y uZt r6 xtv¢| reuy reuwhxtv|zyhrYu yh
Jyhu~ reyQky*x
V ,- ² o 2
J(v) ≥ J(u) + 〈J ′(u), v − u〉 ∀u, v ∈ V ,@ ² oE
〈J ′(u)− J ′(v), u− v〉 ≥ 0 ∀u, v ∈ V .@ ² o 2
p¢4YB)4>¡]§]Z4Gln$o5lWof tvuJxrWy | zSt5~Q| uZt r st hcyreut5|z@ysy V s| r6
Ryuα > 0 yh|¢yhut rextv|rYu yh reuQwxtv|z@yreuy
Jyu
α ~ rY¢yQy x
V ,- ² o V2
J(v) ≥ J(u) + 〈J ′(u), v − u〉+ α
2‖v − u‖2 ∀u, v ∈ V ,
@ ² oK ² 〈J ′(u)− J ′(v), u− v〉 ≥ α‖u− v‖2 ∀u, v ∈ V .
@ ² oKZ`jr6=f ln$o5lWo ¬+J]FugXZDOd8QSRUQ^XmDOF@ ² oK ² "J$R~- ² oEtXmDmRuY8DOJQKD2R/J]T/P8TU_$R&BnRUQ^XmDZ_]XZ[\_$hR/TUQKb2Y8JIF/Q^[\P8HKJZo9 IHKH^J`F+F-QKZDOQS:OJ]D2R¦bmYOJIHKB®vwXmDOgRUQ^XmDWg$XZDrmJ$a6J
J(v)J`F@RfRUXZYn|@XmY8T&F+BEYWd6J`F/F/YOF+d6JIF/XZDP8HKBZDjR&BED8mJD2RJD
ug$XmDOF-Qxd6_]T/_pg$Xm[c[\JCY8D8JCvwXZDGgR/QKXZDzBnyuD8Jpd6J
v$oO+J`Fg$XZDGd6QSRUQ^XmDOFp@ ² oKZ`J$R"@ ² o m®F/XZD2Rd6J]FMr£rP>XER/MOl]F/J]Fd6JgX2J]TUgQ^rQ^R/_mo IDPOBET/R/Qxg$YOH^QKJT`q8F-QJ(u) = 1
2a(u, u)− L(u)BZJ]gaY8D8JNvwXZTU[\JNk8QKHKQ^D8_`BEQKT/JF-£r[\_$RUT/Qxb2Y8JNgXZD2R/QKDrY8JF-Y8T
VJ$RLYOD8J4vwXmT/[\J®H^QKD8_`BEQKT/J®g$XZD2RUQ^DrY8JF-Y8T
Vq¦BZH^XmTUFpHe IarJ]TUgQKgJ ² o^mo j[\XZD2R/TUJb2Y8J~@ ² oKZ`;J`F@RJa6BmgRUJ[\JD2RHxBd6_h:ODOQSRUQ^XmDd6JuHKBg$XrJT&g$QKrQSRU_pd8J
ao •
i r4>'¡]§n6§`54>)o HfF/Y6y\Rd6JCd6_[\XmDmRUT/J]T4HxB4T/XmPGX2F-Q^R/QKXZD ² oKZoVUpJDXZkOF/JTUnBED2RNb2Y8J=HKJpgBZFα = 0
d8XZD8D8J"HKB7ITUXZP>XmF/Q^R/QKXZD ² o^mo RGo8«XZD2RUT/XmDOFb2Y8J@ ² o VmQK[\P8HKQKb2Y8Jj- ² o^ ² o¤LNXZ[\[\JJJ]F-R
αh g$XmD2mJ$a6JZqrXmDm_TUQS:OJWBZQKF/_[\JD2RwPGBETTU_]g$YOT/TUJDOgJ`Nb2Y8JZq6P>XZY8TRUXZY6R
k ≥ 1q
J
(
(
1− 1
2k
)
u+1
2kv
)
≤(
1− 1
2k
)
J(u) +1
2kJ(v)− α
2k+1
(
1− 1
2k
)
‖u− v‖2 ,
dfe X.-2k
[
J
(
u+1
2k(v − u)
)
− J(u)
]
≤ J(v) − J(u)− α
2
(
1− 1
2k
)
‖u− v‖2 .
IDjvBZQKFUBED2R®R/J]DOd6TUJkZJ]TUF
+∞ q6XmDRUT/XmY8ZJ\@ ² oK ² $o3ªXmY8TXZk6RUJD8QKT@ ² oKZ`IQ^H+F/Y6y\Rdfe BZd8d6Q^hR/QKXZD8D8J]T@ ² oK ² NBZJ`gCH^Y8Q^h [c][\JCJ]Dz_]g&MOBZD8ZJ`BED2RuJRvo«~XZD2R/TUXZDOF¦bmYOJ=@ ² oKZ`>QK[\P8HKQKb2Y8JC@ ² o V2oPªXmY8T
u, v ∈ V J$Rt ∈ R
qXmDWPGX2F-Jϕ(t) = J(u+
t(v−u)) o=H^XmTUF ϕ J`F@R d6_TUQ^nBZk8H^J.JR'd6XmDOgIg$XmD2R/QKD2YOJIF-Y8T RqJ$R
ϕ′(t) = 〈J ′(u+t(v−u)), v−u〉 qd6JF-XmT-RUJCb2Y8JmqOdfe BEP8TUl]Fp- ² oKZϕ′(t)− ϕ′(s) ≥ α(t− s)‖v − u‖2 F/Q
t ≥ s . @ ² oK`Z
Ecol
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lyte
chni
que.
Ne
pas
repr
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±m² ! "#$%&'#)(* +,
< XZQ^Rθ ∈]0, 1[
o= IDzQKD2R/_mTUBZD2R®He QKD8_]mBEHKQ^R/_u@ ² oK`ZNd6Jt = θ
st = 1
JRd8Js = 0
ss = θ
q6XZDXmk6R/QKJD2Rθϕ(1) + (1− θ)ϕ(0) − ϕ(θ) ≥ αθ(1− θ)
2‖v − u‖2 ,gZe J`F@R/h sEh d8Q^TUJc- ² o V2o
?V+2nZmln oZlYo X! ] !/LI" 1 ( J 6348 ( > 0A. V:C!3 V 9:5 :O ( 1 5 * 9 1 > _ V: (C :;8! . * 9Q: (J(v) > J(u) + 〈J ′(u), v − u〉 ∀u, v ∈ V 0 > 1 u 6= v ,
"!-8 1 =〈J ′(u)− J ′(v), u− v〉 > 0 ∀u, v ∈ V 0 > 1 u 6= v .
;'JTU[\Q^D8XmDOF¦gJ$R/R/JNF/XZYGF@h F-J`gRUQ^XmDCJ]D"d6_:OD8QxF/FUBED2R¦HxB 2 mnG r"¡`r4>K2 ;d8JJoG/J][uBET&bmYOXZDOFRUXZY6Rdfe BEk>XZT&djb2Y+e QKHfJ]F-R®RUT/l`F®vBZgQ^HKJCd8JCm_D8_]TUBZH^QxF/JT®HKBcA=_:OD8Q^R/QKXZD ² o^mo^Cd6JWd6Q@0>_]T/J]DmRUQKBZk8QKH^Q^R/_BZYg]BZFpdfe Y8D8JcvwXZDGgR/QKXZD
fd8_h:OD8QKJuF-YOT
VsnBEHKJY8T&FWd8BEDGF;YOD BZY6R/TUJcJ]F/POBZgJcd6J d=Q^HKkGJ]T-R
WJ$R=DOXZDPOBZFF/JYOH^J][cJ]D2R=d8BEDOFRoW9;Dd6QKTUB\b2Y8J
fJ]F-R=d6QS0¤_TUJD2RUQKBZk8H^JBEYF/JDOFd8J*T8TU_]g&M8JR&J]D
uFe Q^HfJa6QKF-R/JWY8DOJpBZP8P8HKQKg]BnRUQ^XmDHKQ^DO_]BEQKTUJpgXZD2R/QKDrY8J
Ld6J
Vd8BEDOF
WR/J]H^HKJb2Y8J
f(u+ w) = f(u) + L(w) + o(w) ,BZJ`g
limw→0
‖o(w)‖‖w‖ = 0 .
@ ² o^ ± 9;DBEP8P>JHKH^J
L = f ′(u)HxBd8QS0¤_TUJD2R/QKJHKHKJtd6J
fJD
uof¦Bd6_h:ODOQSRUQ^XmD¢@ ² oK ± J]F-R;Y6RUQ^HKJcPGXmY8Td8_h:OD8QKTHxBud6_TUQ^m_Jpd6J
f(u) = J ′(u)bmYOQ+J]F-RY8D8JBEPOP8H^QxgBER/QKXZDzd6J
Vd8BEDGFF-XmDd6YGBEH
V ′ oi kj w§`54>mln$o5lWo tvu
JxrWy rW~uZt r s.y
Vs.|re
R prstvuCwhxeyJyu¡s.yhx t
s.htv|z@yyhru ∈ V t
Jyhu)stv¢|.hz@ys| r6xr ttvrW|G.ysy
uyhutK|¡s.htvy
J ′(u)yhu
s.htv|z@y yru pr r uy J ′′(u)
z@|stv¢y3yQ~ rWsy¡syJyhruwxt ¢t ,y
J ′(u+ w) = J ′(u) + J ′′(u)w + o(w) ,|y~
limw→0
‖o(w)‖‖w‖ = 0 .
;'JHKH^Jb2Y+e J]H^HKJjJ`F@R\d6_h:GD8Q^JHKBd6_TUQ^m_JjF/J]gXZDOd8JJ]F-Rcd6Q^yg$QKH^Js~_]BZH^YOJT"J]DP8T&BnRUQKb2Y8Jjg]BETJ ′′(u)w
J]F-RY8D_]H^_][cJ]D2R;d6JV ′ oWdJYOT/J]YOF-J][\JD2R]q8J]DHKBcvBZQKFUBED2R=BZZQKTF-Y8T v ∈ V XZDzXmk6R/QKJD2RYOD8JWk8T&BZJCvwXZTU[\JWk8Q^HKQKD8_]BZQ^TUJWg$XZD2RUQ^DrY8JF/Y8T
V × V b2Y8JWHe XZDzD8XZR/J]TUBJ ′′(u)(w, v)
JDzH^QKJYJRPOHKBmg$Jd6J(J ′′(u)w) v
oEXmYOFªHKBZQKFUF/XZDOFªBEYtH^J`gRUJY8T.HKJF-XmQ^Dcd6JP8T/XmY8ZJ]T'HKJT/_`F-YOHSR&BnR._HK_[\JD2RUBZQ^TUJF/Y8QKnBED2R]oDNr ln$o5lWo Yt
Jyu,xrWy rW~uZt rs.yhx ts.htv|z@y s.y
Vs|re
R yzvz@yht y
J(u+w) = J(u)+J ′(u)w+1
2J ′′(u)(w,w)+o(‖w‖2), |y~ lim
w→0
o(‖w‖2)‖w‖2 = 0 ,
@ ² o^GR2
J ′′(u)yhu)t5syreut ,y xry Cy¡htvzStvrW|tvcy ~ reutvrexey x V × V
IDPOTUBER/QxbmYOJpgZe J`F@RJ ′′(u)(w,w)
b2Y8JWHe XmDg]BEHxg$Y8HKJZo
Ecol
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lyte
chni
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k 6 ! "# % ±Z² ?+2nZZln oZlYoUJ= ( a !@LI= * A(+. ( 4 0( :5N * 4P ( !/ 1 "9 ( !/@:P! V ×V & J= ( L !LI= * . ( /4 0( 1 "9 ( !/:P! V &R , :; J(u) = 1
2a(u, u)−L(u) & !/ J 9:5 638! _LI ( :)64 ( > 0A. i:C!3 V C "!J ′′(u)(v, w) = a(v, w) , ! 5! u, v, w ∈ V & M,,3. ( "!/8 1 49:P! . 0 0 ! _ _ *-,. C:@63C: ^3_ 8 1( 1 9: 9%"&3& < %"&3& < 9% && &
¦XZT&F/b2Y8JJJ`F@Rd6JY8acvwXmQKF®d6_]T/QKnBEk8HKJ=XmDuT/JR/TUXZY8mJHKB"g$XZDGd6QSRUQ^XmDYOF/Y8JHKHKJ;d6JCg$XZDrmJ$a6QSRU_¨rF-QHKB\d6_]T/QKZ_]JpF/J]gXZDOd8J;J`F@RP>XmF/QSRUQ^mJZq8BZH^XmTUF®HxB"vwXmDOgRUQ^XmDzJ]F-RgXZDrZJa6JZo
?+2nZZln oZlYo -!/ : ( J C:Og6! _ LI ( :[64 ( > 0A. :C!3 V . 9: 1 \6 ( ( ":g63C:W , 3: ( ( /: 9%"&3& P %"&3& :;C: , 1 ( > * 8 49"! ( > 0. 959:
J ′′(u)(w,w) ≥ 0C
J ′′(u)(w,w) ≥ α‖w‖2 ∀u,w ∈ V .@ ² oKOUZ
#&%'#( %'# !$#%')+*,Z[,/) 021 ^: ` 38?@7N19:b _+5'_f3°<¤741N3r5e`ª?/1N3_<¤741)__
=XZYOF\gXZ[\[\JDePXZDOF"POBETtvwXZTU["Y8HKJTcHKJ]Fcg$XZDGd6QSRUQ^XmDOFcd6J[\QKD8QK[\BZH^Q^R/_HKXZT&F/b2Y8JHe JDOF/J[tk8H^Jd6J]FgXZD2R/T&BEQKD2R/J]FKJ`F@Rg$XmDrZJ$a6Jmqrg]BZFX.-HKJ]Fg&MOXmF/J]FF/XZD2RP8H^YGFF-QK[\P8H^J`FwDOXZYOFF/Y8P8P>XmF/JTUXZDOFR/XZYE|@XZY8T&Ftb2Y8J
KJ`F@RcvwJTU[\_jD8XmDrQxd6JjJR\b2Y8J
JJ]F-R\gXZD2R/QKDrY8JF/Y8TcY8DXmY8ZJ]T-R\g$XmDmRUJDOBZD2R
Ko+4e Qxd6_]JuJ]FUF-J]DmRUQ^J]H^HKJud6Y T/_`F-YOHSR&BnRb2Y8Q4F/Y8QSRJ`F@Rb2Y8Jmq+PGXmY8TpR/XmY6R
v ∈ K q¦XZDPGJ]Y6RWR/J`F@RUJTHe XZP6RUQ^[uBZH^Q^R/_d6Jud8BZDOF'HKBf-d6QKTUJ]gRUQ^XmD"BZd8[cQxFUF-QKk8HKJg
(v−u) gBZT u+h(v−u) ∈ K F/Qh ∈ [0, 1]
oL*F' 4GH2 ln$o o5l E! $ =f'6§`54>&2)>?; 2#"'m8¡u4> G+H tvu
u ∈ K ~ reycy prx 3 ywxey J yus t cyhrYuZt5|.hzyyru Wt u yhuxr tvreu sy ¡tvretv¡x z ~|zsy
Jx
K |z 〈J ′(u), v − u〉 ≥ 0 ∀ v ∈ K .
- ² o^ mYtu ∈ K ht y!$&% $('yhu)t J yhu~ rY¢yQy | z u yhu)xr"¡tvretv¡x z Q| zs.y
Jx
K jr6=f ln$o o9;DtBEPOPGJ]H^HKJC- ² o^ mqf-Q^D8_`b2YOBnRUQ^XmDtdfe IY8HKJT gZo HrF]e BEZQ^R.dfe YOD8J®gXZDOd8QSRUQ^XmD' rZr¡`¡]Own;d+e XmP6R/QK[uBEHKQSRU_®b2Y8Qrd6JrQKJD2R rZr¡`¡]8n2§¡]'¥¡]O¤§]pF-Q
JJ]F-R gXZDrZJarJmo H2vBEY6RBEYOFUF-Q4T/J][\BZTUb2Y8J]Tb2Y8JZq d8BEDOFtd6JY8a g]BZFWQK[\PGXmT-R&BED2RUF]q4@ ² oK2p¡]¢n k2 w§~¡`Bªrr¤§)!>^ =f 6§]Z4GM2,>@?C'm
J ′(u) = 0o\ ID\P8T/J][\Q^J]T HKQKJY+qZF/Q
K = Vqv−u d6_`g$TUQSR.RUXZY6R V H^XmTUFUb2Y8J
vd8_]g$TUQ^R
VqrJR®d6XmDOgt- ² o^ mªJ]DmRUTUB+*xDOJ
J ′(u) = 0orA"e BEY8R/TUJ=POBZT-R`q2F/Q
uJ]F-RNQKD2R/_]T/QKJY8Ts
KqmHxB[c][\Jg$XZDGg$HKYOF-QKXZDFe QK[cP>XmF/JZo •
i r4>'¡]§n6§`54>)oX XmY8Tv ∈ K J$R
h ∈]0, 1], u+ h(v − u) ∈ K q8JRd6XZDOgJ(u+ h(v − u))− J(u)
h≥ 0 .
@ ² oK E9;DJD~d6_`d6Y8Q^Rt@ ² oKm®J]DzvBEQxF/BZD2RR/JDGd6T/J
hZJ]TUF
0oO¦Bjd6JY6a6QKl[\J"BmF/F/JT/R/QKXZDzd6YRUM8_XmT/l][cJd6_]gXZY8HKJpQ^[\[\_]d6QxBnRUJ[\JD2R=d6J- ² o Zo
Ecol
e Po
lyte
chni
que.
Ne
pas
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±m² ! "#$%&'#)(* +,
?V+2nZmln oD oZl J= ( K !3 1 " > _ )LI * 4g" > ( 6i6 V &eW ! x ∈ V <" 1 1 . 0, 1 ( xK ∈ K63x:P!
K# > ( . 4C= * `"&3&9%'
‖x− xK‖ = miny∈K‖x− y‖.
8 !/ . 0)1 \6 ( ( i4 1 C:;: 0( C :P!i: 0 95# %"&3' :O 0* 8/V _\0\1 5 * 9 # 9` &/'P&?V+2nZmln oD oD J= ( A !3/ *V0 ( 1 4C . . g6 ] =D6 p × n P
b ∈ Rp & R 1 ": ( 6 . , A3. * 0 ! _g* ( \6"9: 190 4C:
infx∈Rn
‖Ax− b‖2.8X"! 1 , A. * 0 6 * P 5! !:!B:; . ! ( BP 4 1 ( .+] 49"! 0 ( "-6 ]^ ! . 1 = 9:, "/6 0 5 &?V+2nZmln oD oT R , /6 .+]^3_ *-,. &&
infx∈KerB
J(x) =1
2Ax · x− b · x
0 > 1 A * 0 ( 1 V:ON * 4C ( "!/ 1C0 49 6 ] =D6" n <C B 63 0(+. . m×n # m ≤ n 'P& "! ] (+. _( :O5)!b:; . ! ( b: ( A C:O , 3: ( ( > bPY"! ] . . @9:5M! ( "!: ( A C:O 634 ? ( , 3: ( ( > & "!/V5"!9 , ( Q63 *B( (+* ! * x ∈ Rn > 48 ( ?
Ax− b = B∗p 0 > 1 p ∈ Rm.
?V+2nZmln oD oZ R $ , 8\6 .+]h^3_ *-,. &&%"& 8 !/ .+] 49"! 0 ( i6 ]^ ! . > 48 ( ? 4C,\0 . , ( Q63 *)( (+* ! * u ∈ H10 (Ω)
63
infv∈H1
0 (Ω)
J(v) =1
2
∫
Ω
|∇v|2dx−∫
Ω
fv dx
9:5 , 4 1( :;4 * 9 . 0 LI= * ! . 0 ( " > 0 ( 0 ( . . ∫
Ω
∇u · ∇v dx =
∫
Ω
fv dx ∀ v ∈ H10 (Ω).
# R P! > 0( /: ( !3a4C:C! . 0 M6 . 0gW , 3: ( ( " & ` & & '?V+2nZmln oD oDf J= ( K ! 1 " > _ LI8 * 4[" > ( 63i63 V <:O ( a !3/LI= * A(+. ( 4 0( :ON * 4P ( !/ 1 ( 3!/ 1 = 1/( > :P! V <C :O ( L !LI= * . ( 4 0( 1 "9 ( !:P! V & "!/
J(v) = 12a(v, v)−L(v) 0 6 * P !3!3 ( !/ , ( 63 *B( (+* ! * 6 0 /: K </854 u & "98"!/
u9:5 0 !:;: (.+] ! ( "!/ :; . ! ( )6! , A3. * i# 0,, . 4 ( /49"! 0 ( " > 0 ( 0 ( "3/ . . '
u ∈ K Ca(u, v − u) ≥ L(v − u) ∀ v ∈ K .
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k 6 ! "# % ±Z² V?+2nZZln oD oUJ= ( J1
PJ26! _ LI" 1 ( /: 1 " > _ C: 1 "9 ( !/C:X:C!3! ,\0 ( 1 > _ LI8 * 4C" > ( 63 K ⊂ V &=R V:P! ,, :Ob"!/ J1
:;8! . * 9 9:5Y64 ( > 0A. & 8 "! u ∈ KC:5V!3 *B( (+* ! * 6J1 + J2
: ( CQ:;8! . * 9 : (
〈J ′1(u), v − u〉+ J2(v)− J2(u) ≥ 0 ∀ v ∈ K .
¦J]F"T/J][\BZTUb2Y8J`FWF-YOQ^nBED2RUJ]F]q¦b2Y8Q4F/XZD2R"d6J]FtBEP8POH^QxgBER/QKXZDOFF/QK[cPOH^J`Fd6Y;M8_]XZTUl[\J ² o 8o^mqZXZD2RcD8XZYOF\d6XmD8D8J]TtHe QKD2R/Y8Q^R/QKXZD°d6JHKBD8XERUQ^XmD¢d6J f-["Y8H^R/QKP8HKQKg]BnR/J]Y8T\d6J'BEmTUBZD8ZJhgub2Y8QF/JT&Bd6_mJHKXZP8P>_JWscHKB < XmYOF-h F/J]g$R/QKXZDzF-Y8QKnBED2R/Jmojr6=f ln$o oT .a8BE[\Q^DOXZDOF HKJg]BZF'POBET/R/Qxg$YOH^QKJTªX-
KJ`[email protected]`F-POBmg$J®BnyuD8JNvwJTU[c_d6J
Vo9;D"F/Y8P8P>XmF/JNd8XZDOgIb2Y8J
K = u0+P q]X- u0 ∈ VJR X.- P J`F@R Y8D"F/XZYOF-h J]F/POBZgJ.ZJ]g$R/XmT/QKJHvwJTU[c_d8J
Vo =H^XmTUF]q8HKXZT&F/b2Y8J
vd6_`g$TUQSR
Kqv − u J]F-RY8D_HK_[\J]DmR=b2Y8J]HKgXZDOb2Y8Jd8J P F/Q+k8QKJDbmYOJ\- ² o^ mN_]b2Y8QKnBEY6Rs
〈J ′(u), w〉 = 0 ∀w ∈ P ,gEe J]F-R-h snh d6Q^TUJJ ′(u) ∈ P⊥ oS 4DPOBZT-RUQKgY8HKQ^J]T]qfF/Q P J]F-RCYOD8J\Q^D2R/J]TUF/J]g$R/QKXZD :OD8QKJ\d+e Mr£rPGJ]T/POHKBZDOFqgEe J]F-R-h snh d6Q^TUJpF-Q
P = v ∈ V , 〈ai, v〉 = 0PGXmY8T
1 ≤ i ≤M ,X-a1, . . . , am
F/XZD2Rd6XZD8DO_]Fd8BZDOFVq®BEHKXZT&FzwHKJH^J`gR/J]Y8Tm_TUQS:OJ]TUBb2Y8J` P⊥ J`F@RHe J]F/POBZgJZJ]g$R/XmT/QKJH®JD8mJDOd8T/_POBET\HxB~vBZ[\Q^HKH^J
(ai)1≤i≤MoI'BgXZDOd6Q^R/QKXZD°dfe XZP6RUQ^[uBEHKQ^R/_Fe _]gT/Q^Rud6XZDOgF-XmYOFHKBtvwXmT/[\J\¨
u ∈ K J$R ∃λ1, . . . , λM ∈ R , J ′(u) +
M∑
i=1
λiai = 0 ,@ ² oK9m
J$R¦HKJ]F¦T/_]JHxFλiF-XmD2R¦BEP8P>JHK_]F' w§`8Bªm6§]r'n¡)2 *D46EGn8EG>oXZYGF+ZJ]T/TUXZDGF>BZY*;M8_XmT/l][\J ² o6o tH^J]Y8TTmH^JpP8HKYOFm_D8_]TUBZH>JRvwXZDOdOBE[\JD2RUBZHo •
jr6=f ln$o ohZ < Y8P8P>XmF/XZDOF[\BZQ^D2RUJDOBZDmRb2Y8JKJ`F@R=YODzgZD8JgXZDrZJa6J;vwJ]T/[\_mqOg$Jb2Y8QF-QKZD8Q@:OJ=b2Y8J
KJ]F-R4Y8DuJ]DOF-J]["k8HKJ=gXZDrZJarJvwJ]T/[\_RUJH>bmYOJ
λv ∈ K P>XZYOTIR/XmY6Rv ∈ K JR4R/XmY6R
λ ≥ 0o IDP8TUJDOBZDmR
v = 0P8Y8QxF
v = 2udOBEDOFW- ² oK2qrXZDzXmk6R/QKJD2Rb2Y8J
〈J ′(u), u〉 = 0 .@ ² o^`Vm
ªBZTg$XmDOF-_`b2Y8JD2R]q+- ² o^ m4Q^[\P8HKQKb2Y8Jb2Y8J〈J ′(u), w〉 ≥ 0 ∀w ∈ K .
- ² o ² IDvBEQ^R]q¦- ² oK2IJ`F@R_`b2Y8Q^nBZH^J]DmRsz@ ² oK]V24JR"@ ² o ² $o6A;BZDOFH^Jg]BZFX.-
K = v ∈ V , 〈ai, v〉 ≤ 0PGXmY8T
1 ≤ i ≤M ,
Ecol
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± ² ! "#$%&'#)(* +,
X.-a1, . . . , aM
F-XmD2Rd6XmD8D8_]Fzd8BZDOFVq®H^J +J][c[\J d6JTOBETnBZF ² o6oK °ZXmQ^Tzg$Q^h d8J]FUF-XmYOFU[\XmDmRUT/JWBZH^XmTUFb2Y8J
u ∈ K J$R ∃λ1, . . . , λM ≥ 0 , J ′(u) +
M∑
i=1
λiai = 0 ,@ ² o 8`
JRHe _2BEHKQSRU_@ ² oK]VmN[\XZD2RUT/Jb2Y8Jλi = 0
F-Q 〈ai, u〉 < 0oO+J`FT/_]JHxFP>XmF/QSRUQSvFXmYzD2YOHKF
λiF/XZD2RJ]DOg$XmT/J\BEPOPGJ]H^_`FW w§`8Bªm6§]r'n¡2 D46EGn8EG>o¦=XZYOFpmJTUT/XmDOF;BZYM;MO_XZTUl[\J ² o 8o^¢UHKJYOTTmH^JpP8HKYOFZ_]D8_T&BEHfJRvwXZDOd8BZ[\JD2RUBZHo •
;'JTU[\Q^D8XmDOF¬gJ$R/R/J F/XZYGF@h F-J`gRUQ^XmDJ]D©d6XZDODOBED2RYOD8J 4>2'§]54> 2,>@4YB'§`8w§` 2 2 rX+ZHr4G2¦n>op4YB)4>¡]§]54>ln$o of prMx ywhxey K = V
yuwxeyJyhusyx t stv¢|.hzyyr
u Wtuyhu)xr tvrYus.y3¡tvretv¡x z ~|zsy
J |z
J ′(u) = 0yu
J ′′(u)(w,w) ≥ 0 ∀w ∈ V .- ² oZm
~ht wxeyCyreu t x u xu v s|re xr" ttvrW|kyCsyu
J ′(u) = 0yhu
J ′′(v)(w,w) ≥ 0 ∀w ∈ V ,@ ² o ±
| z uyu,xr"tvrYtvx z ~|zKs.y
J i r4>'¡]§n6§`54>)o < Q
uJ`F@RY8DzP>XZQKD2R;d6JW[\Q^DOQ^[tY8[iHKX6gBEHq8XmDzFUBEQ^Rd6_@|-sub2Y8J
J ′(u) = 0J$RHxBuvwXmT/[tY8HKJj@ ² oKR2D8XZYOFCd6XmD8D8Jj- ² oZmoW/_]gQ^P8TUX6b2Y8J[\JD2R`q>F/Q
uZ_]T/Q@:OJ- ² o ± qOXZD~_`g$TUQSRYODd6_mJHKXZP8P>J[\JD2Rtd8J;'B£rHKXZT"szHe XZT&d6T/Jd8JY6a¯BZYZXmQKF/Q^DGBEZJud6J1_TUX2pBmJ]g\TUJ]F-R/JuJa6BmgRP>XZYOT=HxB\vwXZDOg$R/QKXZD
φ(t) = J(u+ tw)BmJ]g
t ∈ RJ$R;XmDJD~d6_`d6Y8Q^RpBZQKF/_[\J]DmRCbmYOJ
uJ]F-R;YOD[\QKD8QK["Y8[ H^X6gBZH+d6J
JZXZQKTHxBcA;_h:GD8QSRUQ^XmDV8oKZoK`o
)¦*,Z[,Z[ :b 38?b-?-<` 3_¦:=5 9 _*`I56`.1)_LNM8J]TUg&MOXZDOFN[uBEQKDmRUJDOBZD2R®st_]g$TUQKT/J=d8J]Fg$XmDOd6Q^R/QKXZDOF®d6JC[cQKD8QK[uBEHKQSRU_;HKXZT&F/b2Y8J;He JDGF-J]["k8HKJ
KD¦e J`F@RPOBZF®g$XmDrZJ$a6JmoIHKYOF®P8TU_]g$QxF/_[\JD2R]q2D8XmYOF4_R/YOd8Q^J]T/XmDOFNd8J]FNJDOF/J[tk8H^J`FKd6_:OD8QxF4POBZT®d8J]F.4G¤§]n8¤§`2¡¡2,>^ EGOw§` "XmYjd6J`F®.4G¤§]n8¤§`2¡¡2,>^ EGOw§` XZYHKJ]Fd6J]Y6asHxBvwXmQKF&orXmYOFgXZ[\[\JDePXZDOFPGBETY8D8JWTUJ[uBET&b2Y8JCZ_DO_T&BEHKJCF/Y8THKJ]F 2 wnrE§`54>'¡c62 ¡`¡]57 r¡Zo
i kj w§`54>mln$o o r u xu tvreu v ∈ K z yhreyh hzy
K(v) =
w ∈ V , ∃ (vn) ∈ KN , ∃ (εn) ∈ (R∗+)N ,
limn→+∞ vn = v , limn→+∞ εn = 0 , limn→+∞ vn−vεn = w
yhu| yhz@¡zy~ rys.y3s tvcy~hut re*|.s¡tt5z@yh¡|x tvrYu v
Ecol
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k 6 ! "# % ± m 4DRUJTU[cJ`FcP8H^YGF\Q^[uBEm_]F]q'XmDP>JY6Rd6QKTUJBZYOF/F/QbmYOJ
K(v)J`F@R\He JDGF-J]["k8HKJd8JjRUXZYOFcHKJ]FZJ]g$R/J]Y8TUFNb2Y8QfF-XmDmRNRUBZD8ZJ]DmR&F4J]D
vsYOD8JCgXZY8TUk>J;g$XmD2R/JDrY8J;d8BEDOF
KJR®PGBZFUF/BZDmRNPOBET
vF/QKJ]F-RNY8D8J;nBETUQ^_R/_TU_mY8H^QKlTUJZq
K(v)J`F@RF-QK[\P8H^J][\JD2RNHe J`F-PGBZg$JRUBEDOZJD2RNs
KJDvo =Y6R/TUJ[\JD2Rd6QSR`q
K(v)J]F-R;He J]DOF/J["kOH^J"d8JWR/XmY6R/J`FHKJ]F=d8Q^TUJ]g$R/QKXZDOFP>XmFUF/Q^k8HKJ]F;d6JnBETUQxBnR/QKXZDGFsuPOBZT-RUQ^T=d8J
vbmYOQ+T/J`F@RUJD2RQ^D6:OD8Q^R/_]F/QK[\BZH^J][\JD2Rd8BEDOFKo
4DcP>XmFUBED2Rwn = (vn−v)/εn qnXZDcPGJ]Y6R4BEYOFUF/Qrd8Q^TUJd6J®vBP$XmDt_`b2Y8Q^nBZH^J]DmRUJ®b2Y8J w ∈ K(v)F-QfJR=F-J]Y8H^J][\JD2R=F-QfQKH+JarQxF-R/JpY8D8JF/Y8Q^R/J
wn d8BEDGFVJ$RY8D8JF/Y8Q^R/J
εn dOBEDOFRR/J]H^HKJ]Fb2Y8J
limn→+∞
wn = w , limn→+∞
εn = 0J$Rv + εnwn ∈ K ∀n .
HfJ]F-RNvBmg$QKH^Jpd6JpZ_]T/Q@:OJTb2Y8JpHe JDGF-J]["k8HKJK(v)
b2Y8QfPGJ]Y6RR/TUl]F®k8QKJD$RUT/JCT/_`d6Y8Q^Rs 0 .J]F-RY8Dg ZDOJc¨λw ∈ K(v)
PGXmY8TR/XmY6Rw ∈ K(v)
J$RRUXZY6Rλ ≥ 0
o?+2nZZln oD o "!
K(v)C:Od!3 1 /dLI8 * 4 P !/ K(v) = V
: ( v 9:5 ( 548 ( 8!K & !3- _ *-,3. ) K(v)
C:OV496"! ( 0 &4e QKD2R/_]T/R\d6Yg mD8Jzd6J]F\d6QKT/J`gRUQ^XmDOFtBmd6[\QKFUF/Q^k8HKJ]FcTU_]F/QKd8Jd8BZDOFcH^JT/_`F-YOHSR&BnRuF-Y8QKnBED2R]q.b2Y8Qd6XZD8DOJY8D8J=gXZDOd6Q^R/QKXZDu rZr¡]¡`8tdfe XZP6RUQ^[uBEHKQ^R/_Zo2¦BWd6_][\XZDOF-R/T&BnRUQ^XmD+qR/TUl]FNF-QK[\P8H^JmqEJ]F-R4HxBEQxF@hF-_]JpBZYHKJ]g$R/J]Y8T]op¢4YB)4>¡]§]Z4Gln$o o E! ' =f 6§]Z4G&2,>@?C'2 " Z8¡ E> r' 2n8 H tvu
uxrtvrYtvx z
~Q| zKsyJx
K Yt J yhust cyreut5|z@yCyhru rM|
〈J ′(u), w〉 ≥ 0 ∀w ∈ K(u) .=XZYOF"BEHKH^XmDOFp[uBEQKDmRUJDOBZD2RP8T/_`g$QxF-J]TWHKBzgXZDOd6Q^R/QKXZD D8_`g$J`F/FUBEQKT/Jud6JuHxB[ITUXZP>XmF/QSRUQ^XmD° ² o6od8BEDOFHKJg]BZFX-K
J]F-R~d6XZDOD8_POBZTd6J]Fz4>¤§n8¤§]r¡ 2,>^ E>8§] ¢XZY2,>^' E>8§] >oN¦J]FT/_`F-Y8H^RUBERUFb2Y8JpD8XmYOFXZk6RUQ^J]DOd6TUXZDOFmXZD2RZ_DO_T&BEHKQKF/JTg$J]Y6ad6J`F/J[uBZTUb2Y8J`F; ² o6o ± JR ² o 8o RGo 4>¤§n8¤§]r¡ 2,>^ EGOw§`
ACBEDGFg$JpP8TUJ[\QKJTgBmFXZDzF-YOP8PGX2F-JWb2Y8JKJ]F-R=d6XmD8D8_pPOBET
K = v ∈ V , F (v) = 0 , - ² oRrX-
F (v) = (F1(v), ..., FM (v))J]F-R\Y8D8JBZP8P8HKQKg]BnRUQ^XmDd6J
Vd8BEDGF
RM qªBZJ`gM ≥ 1
oª¦Bg$XZDGd6QSRUQ^XmD rZr¡`¡]8ndfe XZP6RUQ^[uBEHKQ^R/_WP8TUJDOdBEHKXZT&F®HKBtvwXmT/[\JF/Y8Q^nBZDmRUJZoL*F' 4GH2 ln$o o tvu
u ∈ K K
yhus rerW |+!$&% prx 3 hywxey J yustv¢|.hz@yyr
u ∈ K yuwhxey¡z@yh rW~uZt r6Fi(1 ≤ i ≤ M)
rYu~ reutvr yreu stv¢|.hzys|re xr ttvr|ky+sy
u prx ysy zxwxey z@yh ¢y~u yx (F ′i (u)
)
1≤i≤M
reuzSt rW|tvcy yhrYutvrs yrWs.|reuZ z t u yhu¡xr ¡tvretv¡x z ~|z sy
Jx
K tvzyQ.tu yλ1, . . . , λM ∈ R | yhz@h='§]8BªZ6§`r¦n¡¡2'+DN6EOn8E> u yzV3whxey
J ′(u) +
M∑
i=1
λiF′i (u) = 0 .
@ ² oUZ
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± ` ! "#$%&'#)(* +,
i r4>'¡]§n6§`54>)o8LNXZ[\[\JNHKJ]F'ZJ`gRUJY8T&F (F ′
i (u))
1≤i≤M
F-XmDmR H^QKD8_`BEQKT/J][cJ]D2R'QKDOd6_P>JDGd8BED2RUF]qHKJpR/MO_XZTUl[\Jpd6J]FvwXZDOg$R/QKXZDOFQK[\P8H^Qxg$Q^R/J`FP>JTU[cJR=d6JW[\XZD2R/TUJTb2Y8JK(u) = w ∈ V , 〈F ′
i (u), w〉 = 0P>XZYOT
i = 1, . . . ,M , @ ² o 2XmYd8JpvBP$XmD_`b2Y8Q^nBZH^J]DmRUJ
K(u) =
M⋂
i=1
[F ′i (u)]
⊥.
@ ² o E=XZYOFcD8Jzd6_RUBEQKHKH^XmDOF"PGBZFcHKBP8TUJY8mJd6J@ ² o 2Wb2Y8QJ]F-RcY8D°TU_]F/Y8HSR&BnR\g$HxBZFUF-Qxb2Y8Jd6JzgBZHKgY8Hd8QS0¤_TUJD2R/QKJHfJDvBZQSR`q
K(u)J`F@RªHe J`F-POBmg$J4RUBEDOZJD2R.sHxB=nBZT/QK_$RU_
KBEYP>XZQKD2R
uoZLNXm[c[\J
K(u)J`F@RY8DJ]F/POBZgJ=mJ]gRUXZTUQ^J]Hq2XZDP>JY6RP8TUJDOd8T/JCF-YOg]g$J`F/F/Q^mJ[\JD2RwJR −w d8BEDGFNHxBIT/XmPGX2F-Q^R/QKXZD ² o6orq6g$JWb2Y8Q¦gXZDOd8Y8QSR=s
〈J ′(u), w〉 = 0 ∀w ∈M⋂
i=1
[F ′i (u)]
⊥,
gZe J`F@R/h sEh d8Q^TUJb2Y8JJ ′(u)
J]F-R=JDOZJDGd6T/_"POBET;H^J`F (F ′
i (u))
1≤i≤M
D8XERUXZDOF;b2Y8J"HKJ]F=["Y8H^R/QKP8HKQKg]BnhRUJY8T&Fcd6J'BEZT&BEDOZJF/XZD2Rcd6_:OD8QxFtd8J[uBEDOQ^l]T/JYOD8QKb2Y8JoDOJBZY6R/TUJjd6_][cXmDOF-R/T&BnR/QKXZDP8HKYOFm_Xm[c_R/TUQKb2Y8JIJ`F@RPOT/XmPGX2F-_pd8BEDGFHKBcP8TUJYOZJpd6JWHKBiITUXZP>XmF/Q^R/QKXZD ² o6oKZmo
r6=f'Mln oD o¬¦XZT&F/b2Y8JzHKJ]FjZJ]g$R/J]Y8TUF (F ′
i (u))
1≤i≤M
F-XmDmRHKQKD8_]BZQ^TUJ[\JD2RjQKDOd6_]PGJ]D6hdOBED2RUFWXZYH^QKk8TUJ]F&qrXZDzd6Q^R=bmYOJCHe XZDJ]F-RdOBEDOF®Y8DZO¡\ E>'2ro6A;BZDOFH^JWg]BZFg$XmDmRUTUBZQ^TUJZq2XZDPGBETUH^JWd6J"m8¡4Gn E>'2J$R;HKBugXZDOgH^YOF/QKXZDzd6Y+;M8_XmT/l][cJu ² o6o "J`F@RvBZYOFUF-JgXZ[\[\JWH^J[\XmDmRUT/JpHe Ja6J[\P8HKJpF/Y8QKnBED2R]oITUJDOXZDOF
V = RqM = 1
qF (v) = v2 q J(v) = v
q8dfe X-K = 0 q u = 0
qF ′(u) = 0
¨6QKHF]e BEZQ^Rd6XZDGgCd+e Y8Dg]BZFD8XmDTU_mY8H^QKJT`oGLNXZ[\[\JJ ′(u) = 1
q+@ ² oUmID+e B\POBmFH^QKJY+o •ªXZYOT¢k8QKJD g$XZ[\P8TUJDGd6T/J¬HKB P>XZT/R/_]Jd6Y ;M8_XmT/l][cJ ² o6o 8qtD8XZYGFHe BZP8P8HKQKb2Y8XmDOFF/Y8THe .a6J[\P8HKJV8oKZo
minx∈KerB
J(x) =1
2Ax · x− b · x
,X.-AJ]F-RcF/£r[c_R/TUQKb2Y8Jzd6_:OD8QKJP>XmF/QSRUQ^mJdfe XZT&d6TUJ
nq J$R
Bd8JjR&BEQKH^HKJ
m × nBmJ]g
m ≤ no
9;DDOXER/J(bi)1≤i≤m
H^J`Fm
HKQKZD8J`F"d6JBo ACBEDOFtg$JPOT/Xmk8H^l][\JuQ^H®£Bd6XZDOg
mg$XmD2R/T&BEQKDmRUJ]F
bi · x = 0o < QªH^JcTUBZD8d8J
BJ]F-R
mq¤H^J`F
(bi)F/XZD2RpQ^DGd6_P>JDOdOBED2RUFCJ$RpXZDPGJ]Y6RBEP8P8HKQxbmYOJTCHKBgXZDOgH^YGF-QKXZD~- ² oUmF/Q>D8XZD+q2HKJ]F4gXZD2R/T&BEQKD2R/J`F4F/XZD2RNF/XZQ^RNTUJ]d6XmDOd8BED2RUJ]F]qZF-XmQSRg$XmDmRUTUBmd6QxgR/XmQ^TUJ]F&o
HfJa6QKF-R/JWd6XmDOgCY8Dz["Y8H^R/QKP8HKQKg]BnR/J]Y8Td6JW¦BZZT&BED8mJp ∈ Rm R/J]H+b2Y8JpY8DzPGXmQ^D2Rd6Jp[\Q^D8QK["YO[
xm_TUQS:GJAx− b =
m∑
i=1
pibi = B∗p.
< QAJ`F@RcQ^DrmJT&F-QKk8HKJuJ$RcF/Q
BJ`F@R\d6JT&BED8
mq XZDJD¢d6_]d6YOQSR\BEHKXZT&FpHKBnBZH^J]Y8Ttd6J
x¨'XZDB
x = A−1(b+B∗p)q8J$R=gXZ[\[\J
Bx = 0XmDXmk6R/QKJD2R
p = −(
BA−1B∗)−1BA−1b
J$Rx = A−1
(
Id−B∗ (BA−1B∗)−1BA−1
)
b.
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
odui
re.
k 6 ! "# % ± ± IDPOBZT-RUQKgY8H^QKJT`q$XmDBBEQKDOF/QEP8TUXZY8m_IbmYOJIH^J4["Y8H^R/QKP8HKQKg]BnR/J]Y8T'd8JN¦BZZT&BED8mJ
pJ]F-R¦Y8D8Qxb2Y8JNF/Q
BJ`F@Rd6J4T&BED8
mo < Q2gJ4D+e J]F-R¦POBmF+HKJNgBmFq]XZDFUBEQ^R'b2Y+e Q^H2J$a6QxF@RUJ
pF-XmH^Y8R/QKXZDd6J
BA−1B∗p = −BA−1bbmYOQfD+e J`F@RY8DOQKb2Y8JWb2Y+e s"He Bmd8d6Q^R/QKXZDzdfe Y8DmJ]g$R/JYOTd8YD8Xn£2BEYjd8JB∗ P8TUl]F]oe9;DjTUJ$R/TUXZYOZJ;BZQ^DGF-QH^JWTU_]F/Y8H^RUBnRd6JpHe IarJ]TUgQKgJ\ ² o6o ± o
?+2nZZln oD oe44 0. ( :;8 . C: 49:P! . 0 5: 1/( 63C:;:P!/: , ! 1 C 5 > 0 ( 0 95 63 .+]^3_ *-,. &3& minBx=c
J(x) =1
2Ax · x− b · x
,
c ∈ Rm C:5V!3 > 1 58!e634 &
?+2nZZln oD o M,,3. ( "!/8 . 4C= * %"& `"& .+]^3_ *-,3. && PM863496"! ( -"! . C:, ( 5:M63 *B( (+* ! * 6
J:P! . 0 : , 87!3 ( 54V:;Q63C: > 1 5!3: , , C:@63 A 0 :;:O 1( 49: . 0
,3. !/: , P ( 5 > 0. !3 , , &?+2nZZln oD oZln ^ V! (+. ( : 0 . C: 4C:C! . 0 5: , 4 1 4695:XP 1 ! _ 6 .+]h^3_ 8 1( 1 %"&3& < * "S"!/ . 0 :O . !9 ( " 6"! , A3. * 63 ( 637# ^3_ *-,. &&' 9:5@/4 1 C:O: 0( * 8M!3 0 1 61 8 1\. &?+2nZZln oD oZlYl R 4P!\6 ( . 0 , *B( > 0. 8! , , 76"! 0,. 0\1( 8 6 0 :i!g6 *V0( /A = /4 Ω
# > ( . 0 J= 1 ( " & 'P&QW ! 1 . 0 ( 96! ( . , A. * [6 *B( (+*B( : 0 ( "[:C!3K = v ∈ H1
0 (Ω),∫
Ω v2dx = 1
minv∈K
J(v) =
∫
Ω
|∇v|2dx
.
"98X"! 1 , A3. * 0 6 * P! *B( (+* ! * # * "98 0 "!/ K 9:5 1 *,/0\1 , ! . C::P! ( 5C: *B( (+*B( : 0 5C: .+] 0( 63-6! 49= * -63 . . ( 1 c& & ` 'P&1 ( .+] 4! 0 ( 6 ]^ ! . e61 , A3. * bP 8)646! ( !/ . 0 > 0. 8!S6! *B( (+* ! * C:5 A( 8 . 0i, *B( > 0. !3 , , C!/ . C: , ( 5:@6 *B( (+* ! * :;Q63C: > 1 5!3: , , C: 0 :O:; 1( 4C: &?+2nZZln oD oZlUJ= ( A !3/ *V0 ( 1 n× n :ON * 4P ( !/-64 ? ( , : ( ( > BC b ∈ Rn /3! .+&
3& X!/ . C: , A. * 9:
supAx·x≤1
b · x Csup
Ax·x=1b · x
:;e49"! ( > 0. 95: Cd"! ] (+. : Q!3/@:O . !9 ( " & (+. ( :O8 . 49= * 9% & ` & , "!3 190. 1 ! . 81 P 5B:O . !9 ( "BC * "98 ! ] . . BC:5V!3 ( "! &
`"&R ( 6"! ( !K=D6" ,/0 h ( . 6 0 /: .+] 8/:O *-A. 63C: * 0 ( 1 C:g:ON * 4P ( !/C:[64 ? ( C:, 3: ( ( > C:V6 ] =6" n 8 6 ( : 0 V"! A ≥ B
: ( P:;8! . * 9b: ( Ax · x ≥ Bx · x , !5"!9x ∈ Rn & 46! ( -6 . 0 "!/C:5 ( " , 4 1 4695)"!<=: ( A ≥ B < 0. =: B−1 ≥ A−1 &
Ecol
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repr
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re.
± R ! "#$%&'#)(* +,
?V+2nZmln oD oZl=T ^ 4C= ( 1/( 4C ( !/63C: 0 . 9: * . 4 1 ! . 9:6 0 :O) , 4C:;854CC:g5"!9 , ( 9-63 .+] 9: ,\0\1 ,\0 V! LI 1 ( $6 4 ,/0 h ( ( " f(v)634 , /6 0 )63 . 0 > ( 5C:;:O*)( 1 : 1 ,( !/ v ∈ RN & C:i! 0 ( 54C: *V0\1 3: 1 ,( !/C:< 1 *-* . 0 638/: ( 54g6"! 0 ρ <e: 0
> ( 5C:;:O u <=PM: 0 5 *-, 48 0 ! T <=:;iC"! > 8" 1 0 ! _g* * 95:M6 . 0 LI 1 ( " f(v)
ρ =
∫
RN
f(v) dv , ρu =
∫
RN
v f(v) dv ,1
2ρu2 +
N
2ρT =
1
2
∫
RN
|v|2f(v) dv .@ ² o8m . *V0 0i( 6"! ( .+] 8 ,8( 1( /4P ( "! H(f)
64 ? ( ,/0
H(f) =
∫
RN
f(v) log(
f(v))
dv .
8 !/HC:5V:5 ( 1 5 * 8 1 > _ :C!3 .+] C: ,\0\1 i69:bLI" 1 ( /: f(v) > 0 * C:C!3 0A. C:5 . . a!/ H(f) < +∞ &XR *)( (+*B( :; H :C!3 1 P C: ,\0\1 7:;!/: . 9: 1 0( 5C:B63 * * 9# 9% & ` 3' <=CQ" 0 6 * P 0 "! ] (+. _3( :55i!3g!3 ( !/ , ( M63 *B( (+* ! * M(v) & "98X"!/ 1 , ( 9M6 *B( (+* ! * 9:5 !3/a 09_ . . ( 3/B64 ? ( ,/0
M(v) =ρ
(2πT )N/2exp
(
−|v − u|2
2T
)
.
r6=f'Mln oD oZln HfJ]F-RY6RUQ^HKJdfe Q^D2R/TUX6d6Y8QKT/JpHxBtvwXZDOg$R/QKXZD L d6_h:GD8Q^JWF/Y8TV × RM POBZT
L(v, µ) = J(v) +M∑
i=1
µiFi(v) = J(v) + µ · F (v)
b2Y8JjHe XZDBZP8P>JHKH^JDN6EGOKE>r d6YP8TUXZk8HKl[\Jjd8J[\Q^D8QK[\QKFUBnRUQ^XmDd6JJF/Y8T
Ko < Q
u ∈ KJ`F@RY8D[\Q^DOQ^[tY8[ HKXrg]BEH+d6JJF/Y8T
Kq6HKJ ;M8_XmT/l][cJc ² o 8o "D8XZYGFd6QSRBZH^XmTUF®b2Y8JZq8d8BZDOFHKJpg]BZFTU_mY8HKQ^J]T]qrQKH+J$a6QxF@RUJ
λ ∈ RM R/J]H+bmYOJ∂L∂v
(u, λ) = 0 ,∂L∂µ
(u, λ) = 0 ,
POY8QKFUb2Y8J ∂L∂µ
(u, λ) = F (u) = 0F/Qu ∈ K JR ∂L
∂v(u, λ) = J ′(u) + λF ′(u) = 0
dfe BZP8TUl]F- ² o .UZo69;DP>JY6RBZQ^DOF/Q¤_]gT/QKT/J=HKBcg$XmDmRUTUBZQ^D2RUJJ$RHxB"g$XmDOd6Q^R/QKXZDdfe XmP6R/QK[uBEHKQSRU_CgXZ[\[\J=He BZD8DrY6hHxBnRUQ^XmDd8YmTUBmd6QKJD2RpHKBuF-RUBER/QKXZD8DGBETUQSRU_`®d6Yz¦BEmTUBZD8ZQKJD¦o •=XZYOFd8XZD8D8XmDOF\[uBEQKD2R/JDGBED2RY8D8J~gXZDOd6Q^R/QKXZD 2mr¡]¡`8wn¢dfe XmP6R/QK[uBEHKQSRU_~d6Y¯d6JY8arQKl[\JXmTUd8T/Jmop4YB)4>¡]§]54>ln$o o5lWl pr y z@|.~Qy xz@yh) uv yhsx6 h y $&% = yhu r x y wxeyz@yh r~hut re J yu
F1, ..., FM reu)s.yhx t ~ rYuZtvr Cyreu stv|z@y*yu wxeyz@yh
y~huyx (F ′
i (u))
1≤i≤M
reu$zStvrW|tvcy yhrYuKtvrs yrWs.|reuZ tvu λ ∈ RM z@y¡xzuZt zt5~|uyhx
Ecol
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lyte
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k 6 ! "# % ± ¢Usy |c|r.ysZret | zy 6 y $&% 3 z u xu¡tvretv¡x z ~Q| z
usyJx
K¢t ,y(
J ′′(u) +
M∑
i=1
λiF′′i (u)
)
(w,w) ≥ 0 ∀w ∈ K(u) =
M⋂
i=1
[F ′i (u)]
⊥.
@ ² oEV2i r4>'¡]§n6§`54>)o < Y8P8P>XmF/XZDGFbmY¦e QKH+J$a6QxF@RUJpY8Dg&M8J[\QKDzBmd6[\QKFUF/Q^k8HKJWd6Jg$HxBZFUF/J
C2 qOgEe J]F-R-h snhd6Q^TUJY8D8JvwXmDOgRUQ^XmDt→ u(t)
d6J[0, 1]
d8BZDOFVR/J]H^HKJb2Y8J
u(0) = uJRF (u(t)) = 0
PGXmY8T.RUXZY6Rt ∈ [0, 1]
o8.BET®d6_h:GD8QSRUQ^XmD+qZHxBWd6_]T/QKZ_]Ju′(0)
BEP8PGBET/R/QKJD2R4BEYgZD8Jd6J`F4d6QKTUJ]gRUQ^XmDOFIBmd6[\QKFUF-QKk8HKJ]FK(u)
oY9;DzP>XmF/Jj(t) = J
(
u(t)) J$R
fi(t) = Fi
(
u(t)) P>XZY8T
1 ≤ i ≤M. IDd6_TUQ^nBZDmRXmDXmk6R/QKJD2R
j′(t) = 〈J ′(u(t))
, u′(t)〉 JRf ′
i(t) = 〈F ′i
(
u(t))
, u′(t)〉 P>XZY8T1 ≤ i ≤M,J$R
j′′(t) = J ′′(u(t))(
u′(t), u′(t))
+ 〈J ′(u(t))
, u′′(t)〉
f ′′i (t) = F ′′
i
(
u(t))(
u′(t), u′(t))
+ 〈F ′i
(
u(t))
, u′′(t)〉 P>XZYOT1 ≤ i ≤M.LNXZ[\[\J
fi(t) = 0P>XZYOT"RUXZY6R
tJ$R\P8Y8QxFUbmYOJ
0J]F-RtY8D¢[cQKD8QK["Y8[ d6J
j(t)qªXZDJ]D¢d6_`d6Y8Q^R
j′(0) = 0qj′′(0) ≥ 0
qGJRf ′
i(0) = f ′′i (0) = 0
oG¦J]F=gXZDOd8QSRUQ^XmDOFf ′
i(0) = 0D8XZYGF=d6QxF-J]D2RCb2Y8J
u′(0)J`F@RXmT-RUM8XZmXZDOBZH>BZYF/XZYOF-h J]F/POBZgJ;J]D8ZJ]DOd6TU_;POBZT (
F ′i (u)
)
1≤i≤M
b2Y8Q+J`F@R_]mBEHfsK(u)bmYGBEDOdg$J$R/R/JtvBE[\QKH^HKJ\J]F-RCHKQ^kOT/Jq>RUBZDOd6QxFCb2Y8J
j′(0) = 0F/QKZD8Q@:OJtb2Y8J
J ′(u)J`F@RpXZT/R/MOXZZXmDOBEHs
u′(0)o < Q
u′(0)d8_]g$TUQ^RWR/XZY8R
K(u)H^XmTUFUb2Y8JcXZDvBEQ^R"nBETUQ^J]TpH^J`Fg&M8J[\QKDOFBmd6[\QKFUF-QKk8HKJ]F]qfXZDJDd8_]d6Y8Q^Rb2Y8J
J ′(u)JR®HKJ]F
F ′i (u)
BZP8POBZT-RUQ^J]D8D8JD2RBEYj[\][cJpF/XZYOF-hJ`F-PGBZg$JtwHe XmT-RUM8XZmXZDOBZHGd6JK(u)
$oW9;DTUJ$RUT/XmY8ZJpBEQKDOF/QfHKB\gXZDOd6Q^R/QKXZDzd6YzP8TUJ[\QKJTXZT&d6TUJt¨OQ^H+Ja6QKF-R/Jλ ∈ RM RUJH¦b2Y8J
J ′(u) +
M∑
i=1
λiF′i (u) = 0.
@ ² o ±Z² +J]FgXZDOd8QSRUQ^XmDOF
f ′′i (0) = 0
QK[cPOH^Qxb2Y8JD2R=b2Y8J0 =
M∑
i=1
λi
(
F ′′i (u)
(
u′(0), u′(0))
+ 〈F ′i (u), u
′′(0)〉)
,
RUBEDGd6QKFb2Y8Jj′′(0) ≥ 0
d8XZD8D8JJ ′′(u)
(
u′(0), u′(0))
+ 〈J ′(u), u′′(0)〉 ≥ 0.CT mg$Jzs°- ² o ±m² "XZDPGJ]Y6R_HKQK[cQKD8J]TuH^J`Fud6_TUQ^m_J`FtPOT/J][cQKlTUJ]FcJR
u′′(0)P>XZY8TuXmk6R/J]D8QKT~wJDF-Xm[c[uBZDmRHKJ]Fd6J]Y6ad8JTUD8Q^l]T/J`F®_]b2YOBnRUQ^XmDOF&
(
M∑
i=1
λiF′′i (u) + J ′′(u)
)
(
u′(0), u′(0))
≥ 0,
Ecol
e Po
lyte
chni
que.
Ne
pas
repr
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re.
± ! "#$%&'#)(* +,
b2Y8Q¦D+e J`F@RTUQ^J]Ddfe BZY6R/TUJWb2Y8Ju@ ² o ZVm4H^XmTUFUb2Y8Ju′(0)
PGBET&g$XZYOT-RK(u)
o4e JarQxF-R/JDGg$J4d6JIR/JHxF g&M8J[\QKDOF+BZd8[cQxFUF-QKk8HKJ]Fu(t)
JR'HKJ.vBEQ^R'b2Y8JIHe JDOF/J[tk8H^JNd6J]Fu′(0)
d6_]gT/Q^RHxBtR/XZRUBEHKQ^R/_Wd6Yg mD8JWd6J]Fd8Q^TUJ]g$R/QKXZDOFBZd8[cQxFUF-QKk8HKJ]FK(u)
J]F-RY8D8JWgXZDOF/_]b2Y8J]DOg$JWd6YR/MO_XZTUl[\Jd8J]FcvwXZDGgR/QKXZDGF\Q^[\P8HKQKgQSRUJ]Fub2Y8JzHe XmD¢P>JY6RjBZP8P8HKQKb2Y8J]TuZTZgJsHe Mr£rPGXZR/M8l`F-Jb2Y8JzHKBvBZ[cQKHKH^J(
F ′i (u)
)
1≤i≤M
J]F-RH^QKk8TUJcD8XZYGFHKBZQKFUF-XmDOF®H^J`Fd6_$R&BEQKHKF®RUJ]g&M8DOQKb2Y8J`F®BZYHKJ]g$R/J]Y8Tg$XmY8TUBZZJ]Y6aOo
?V+2nZmln oD oZlZ 0. 1 ! . . 0V1 \6 ( ( " /4 1 C:;: 0( Q6 ] , (+* 0. ( 54Q6"!:O 1 "/6 =D6 , ! . C:^3_ *,3. 9: && P &3& "& 4>¤§n8¤§]r¡ 2)>K EG8w§`
ACBEDOF=g$JWd6JY8arQKl[\Jg]BZFXZDzF/Y8P8P>XmF/Jpb2Y8JKJ`F@Rd6XmD8D8_WPOBZT
K = v ∈ V , Fi(v) ≤ 0P>XZYOT
1 ≤ i ≤M , @ ² o ± X.-
F1, . . . , FMF/XZD2RIRUXZYn|@XmY8T&F4d6J`F.vwXZDGgR/QKXZDGFNd6J
Vd8BZDOF
Ror+XmTUFUb2Y8JHe XmDZJ]Y6RNd6_R/JTU[\Q^DOJTHKJ"gZD8J"d8J]F=d8Q^TUJ]g$R/QKXZDOFBmd6[\QKFUF/Q^k8HKJ]F
K(v)q>HxBuF-Q^R/YOBER/QKXZDJ`F@RCY8DPGJ]Y~P8HKYOF=gXZ[\P8HKQKb2Y8_]J"b2Y8JPOT/_`g$_]d8J[\[\JD2RgBZTuR/XmY6R/J`FHKJ]FgXZD2R/T&BEQKD2R/J`Fd8BEDGF~@ ² o ± `cDOJu|@XZY8J]D2RjPOBZFjH^J[c][\J~TmH^JF/JHKXZDHKJcP>XZQKD2R
vX-He XZDg]BEHxg$Y8HKJ
K(v)oS ID J0>JR]q¦F/Q
Fi(v) < 0qfQKHIJ]F-Rg$HxBEQKTb2Y8JZqfP>XZY8T
εF/Y6yFUBE[\[\JD2R P>J$RUQSR`qnXZDcBEY8T&BBEYGF/F/QFi(v+εw) ≤ 0
wXZDcd6Q^Rªb2Y8J®HKBCg$XmD2R/T&BEQKDmRUJiJ]F-R.Q^DGBZgRUQ^mJJ]D
vo < Q
Fi(v) = 0P>XZY8TWg$J]T-R&BEQKDOFCQ^DGd6QKgJ]F
iqfQKH.D+e J`F@RPGBZFpg$HxBEQKTWbmYOJcHe XZDP8Y8QxF/F/J"RUT/XmY8ZJ]TYODmJ]gRUJY8T
w ∈ V RUJH+b2Y8JZq6P>XZYOTε > 0
F-Y6yFUBE[\[\JD2RPGJR/Q^R]q(v + εw)
FUBnRUQKF-vBEQ^RR/XZY8R/J]F®HKJ]FgXZD2R/T&BEQKD2R/J`F dOBEDOF- ² o ± `o HOBWd8XZDOg®vBZH^HKXZQKT.Q^[\P>XmF/JT4d8J]FIgXZDOd6Q^R/QKXZDGF.F-Y8POP8H^_][\JD2RUBZQ^TUJ]F.F/Y8THKJ]FgXZD2R/T&BEQKD2R/J`Fq6BEPOPGJ]H^_]J]F4>2'§]54> ¡2 "=f 85j m6§]Z4GNo CTUXmFUF-Xt[cX6d6XGq8g$J`Fg$XZDGd6QSRUQ^XmDOFmXZD2RpmBET&BED2RUQ^TWb2Y8J\He XZDP>JY6RWvBZQ^TUJud6J]F f-nBETUQKBER/QKXZDOF!gtBEY6RUXZY8Td+e Y8DP>XZQKDmRvB:OD d6JcR/J`F@RUJTF/XZDjXZP6RUQ^[uBEHKQ^R/_Zo H¤JarQxF-R/Jpd6QS0¤_TUJD2R&F4R@£rP>J]Fd6Jpg$XmDOd6Q^R/QKXZDOF®d6JWb2YOBEHKQS:>gBnRUQ^XmDwPOH^YOF®XmYj[\XZQKDOFF/XZPOM8QKF-R/Qxb2Y8_J`FJ$Rm_D8_]TUBZH^J`FU$or=XZYOFBZH^HKXZDGFd6XZD8DOJTY8D8JWd6_:OD8Q^R/QKXZDd6XmDmRHKJpP8TUQ^DOgQ^P>JWJ]F-R=d6JTUJ2BET&d6JTF/Y8T=HKJP8TUXZkOH^l][cJ r6n¡` jF]e QKH¦J`F@R=P>XmFUF/Q^k8HKJ"d6JvBEQKT/Jd6J`FnBETUQxBnR/QKXZDGFT/J`F-P>J]g$RUBZDmRHKJ]FpgXZD2R/T&BEQKD2R/J]F;HKQ^D8_`BETUQKF/_J`Fo+LNJ]FpgXZDOF/QKd8_T&BnR/QKXZDGF=d6J*f/gBEHxg$YOH d6J`FpnBETUQKBER/QKXZDOF!g[cXZR/QKZJ]DmRpHKJ]Fd8_h:OD8Q^R/QKXZDGFF-YOQ^nBED2RUJ]F]o
i kj w§`54>mln$o o5l tvuu ∈ K yhrehy hz@y I(u) = i ∈ 1, . . . ,M , Fi(u) = 0yhu| yhz@¡zc yhrehy hz@y¡syh~ rYuZc| tvreuy8Z§] >r¡yr u
i kj w§`54>mln$o o5lT prMs tvu whxeyz@yh~ reuc| tvreuy !$&% $ reu =f 85j rr¡yr u ∈ K tyuyxz@yCyreu)t)tvzKyc.tuy xrWystvcy~uZt r
w ∈ V uyhzvzywxey*zc rM| tvu x u xu i ∈ I(u) x ht5yr 〈F ′
i (u), w〉 < 0 , x ht5yr 〈F ′
i (u), w〉 = 0yu
Fiyu|rWy
.
@ ² o ± Zr6=f'Mln oD oZlZ ¦Bd6QKT/J`gR/QKXZD
wJ`F@R\JDb2Y8JHxb2Y8JF/XZT/R/JY8D8Jf-d8Q^TUJ]g$R/QKXZDTUJD2R/T&BED2RUJgPOY8QKFUb2Y8JuXZDd6_`d6Y8Q^Rcd6J~- ² o ± ZCb2Y8J
u + εw ∈ K P>XZYOTWR/XmY6Rε ≥ 0
F/Y6yFUBE[\[\JD2RPGJR/Q^R]o®QKJDF8T`q]F/QZRUXZY6RUJ]F+HKJ]F+vwXZDOg$R/QKXZDOF
FiF/XZD2R'BnyuD8J`Fq]XZDWP>JY6R P8TUJDOd6TUJ
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k 6 ! "# % ± A=_:OD8Q^R/QKXZD¬ ² o6oK ± J`F@RC|@YOF-R/Q@:O_DOXZDF/JY8HKJ[\J]DmR\POBZTUgJjb2Y8JgJHKH^J`F@h g$QF-XmD2Rcb2YOBEHKQ@:O_J`FtF/XZYOFd6J]FgXZDOd6Q^R/QKXZDGFN[\XmQ^DOFF-R/TUQKg$R/J`Fq6[uBEQxFF-YOT-RUXZY6RJDTUJmBZTUdjd6JpHe Q^[\P>XZT/RUBZDOg$Jpd6J]FgXZD2R/T&BEQKD2R/J`FBnyuD8J]FdOBEDOFHKJ]FBZP8P8HKQKg]BnRUQ^XmDOFpg$Xm[\[cJWHKJp[\XZD2R/TUJpH^J`FJ$a6J[\P8HKJ]Fd8Y~LNMGBEP8Q^R/TUJWVmo •=XZYOF'P>XZY8mXZDOF BEHKXZT&Ff_DOXZDOgJT¦HKJ]F g$XmDOd6Q^R/QKXZDOF 2mr¡]¡`8wnr¡4d+e XmP6R/QK[uBEHKQSRU_®F/Y8T¦He JDOF/J[tk8H^J@ ² o ± `$oL*F' 4GH2 ln$o o5lf prx 3 y whxey K yhus rer |3!$(% $ whxeyz@yh rW~uZt r6 J yhuF1, . . . , FM
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tFi(u) < 0 ∀ i ∈ 1, . . . ,M. - ² o ±Z±
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J]F-R¦P>XmF/QSRUQ^mJZqP8Y8QxF/b2Y8Jmq2P>XZYOTNRUXZY6RQKDOd6Qxg$Ji ∈ 1, . . . ,M q2XZDBtF/XZQ^R Fi(u) = 0
q6F-XmQSRλi = 0
or¦J=vBZQSRb2Y8Jλ · F (u) = 0
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u+ε(w+δw) ∈ K PGXmY8T R/XmY6R TU_J]Hε > 0
BZFUF-JG14P>J$RUQSR`o H2vBEY6R.J$a8BZ[cQKD8J]TR/TUXZQxFgBZFd8J:GZY8TUJZoZo < Qi /∈ I(u) q2XZDB Fi(u) < 0
J$RFi(u+ ε(w+ δw)) < 0
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i ∈ I(u) J$R 〈F ′i (u), w〉 < 0
q8BZH^XmTUFFi(u+ ε(w + δw)) = Fi(u) + ε〈F ′
i (u), w + δw〉+ o(ε)
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@ ² o ± Umdfe X-
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ε > 0BmF/F/J1pP>J$RUQSR`o± o@ ID6:OD+qOF/Q
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qOBEHKXZT&FFi
J]F-R=BnyuD8JpJ$RFi(u+ ε(w + δw)) = Fi(u) + ε〈F ′
i (u), w + δw〉 = ε〈F ′i (u), w〉 ≤ 0 .
@ ² o ± m
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± 9 ! "#$%&'#)(* +,
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v ∈ V , 〈ai, v〉 ≤ 0 x 1 ≤ i ≤M
,
yu
K =
v ∈ V , ∃λ1, . . . , λM ≥ 0 , v = −M∑
i=1
λiai
.
z x u xu p ∈ V rM|z tv zSt5~Q| uZt r
〈p, w〉 ≥ 0 ∀w ∈ K =⇒ p ∈ K . | c~t wxeyu | reuhOt5syreuy tvz 6 |tvuyr h| tvus$ xrWy¡whxtv|z@yrW~y i r4>'¡]§n6§`54>)oGLNXm[c[\J]DeP$XmDOFªPOBET.[\XmDmRUT/J]T.bmYOJ K J]F-RªvwJTU[c_mo8IY8QxFUbmYOJgJ$R/R/JP8TUXZP8TUQK_$R/_J`F@Rp_rQxd6JD2RUJtHKXZT&F/b2Y8J
M = 1q>D8XZYGF;POT/X6g$_`d6XZDGFPOBZTCTU_]gY8TUT/J]DOg$JJRpF/Y8P8P>XmF/XZDGF;b2Y+e JHKH^JcJ]F-RrT&BEQKJ;HKXZT&FUbmYOJCHKJpD8XZ[tk8TUJpd8JWZJ]g$R/J]Y8TUF
aiJ]F-RQKD6vw_TUQ^J]Y8Ts
Mo
< YOP8PGX2F-XmDOFdfe BEk>XZT&db2Y8JHKJ]F\mJ]g$R/JYOTUF(ai)1≤i≤M
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(vn) =(
−∑Mi=1 λ
ni ai
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v ∈ V o =H^XmTUF;Q^HªJ`F@Rpg$HxBEQKTCb2Y8J\g&MOBZb2Y8JcF-YOQSRUJ (λni )
gXZDrZJ]T/mJdOBEDOFR+
mJT&FWY8D8JuHKQ^[\Q^R/Jλi ≥ 0
wP>XZY8T1 ≤ i ≤ M
pP8Y8QxFUbmYOJH^J`FWZJ`gRUJY8T&F(ai)1≤i≤MvwXmT/[\J]DmRY8D8JWkGBZF/JCd6JpHe J`F-PGBZg$Jpb2Y+e Q^HxFJD8mJDOd8T/J]DmR`oe9;DBcd6XZDGg
v = −∑Mi=1 λiai ∈ K
q8b2Y8QJ`F@R=d8XZDOg;vwJTU[\_Zo< Q.HKJ]FpmJ]g$R/JYOTUF
(ai)1≤i≤MF/XZD2RpH^QKD8_]BZQ^TUJ[\J]DmRWd6_]PGJ]DOd8BED2R&Fq¤QKH.J$a6QKF-R/JcY8D8J\TUJHxBnRUQ^XmDd6JHxBWvwXmT/[\J ∑M
i=1 µiai = 0q2J$RXZDP>JY8RF/Y8P8P>XmF/JT®b2Y8J;He Y8DBZY[\XZQKDOFNd8J]Fg$XrJ$ygQ^J]DmR&F
µiJ`F@RF-R/TUQxgR/J][\JD2RPGX2F-Q^R/Q^v@o < XmQSR=BZH^XmTUF
v = −∑Mi=1 λiai
Y8Dz_HK_[\JD2Rd6J K oªXZYOTR/XZY8R t ≤ 0q6XZD
P>JY8R=BEYOFUF-Qf_`g$TUQ^TUJv = −
M∑
i=1
(λi + tµi)aiq8JRXZDzPGJ]Y6R=g&M8XmQKF/Q^T
t ≤ 0P>XZY8Tb2Y8J
λi + tµi ≥ 0 ∀ i ∈ 1, . . . ,M JR ∃ i0 ∈ 1, . . . ,M , λi0 + tµi0 = 0 .LNJpT&BEQxF-XmD8D8J][cJ]D2R[\XmDmRUT/Jpb2Y8JK =
M⋃
i0=1
v ∈ V , ∃λ1, . . . , λM ≥ 0 , v = −∑
i6=i0
λiai
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k 6 ! "# % ± `VªBZT¦D8XZR/TUJIMr£rPGXZR/M8l`F-J4d8JIT/_`g$Y8TUTUJDOgJZqg&MOBmg$Y8Dd8J]F+J]DOF/J["kOH^J`F¦BEPOPOBET&BEQxF/FUBED2R+d8BZDOF+HKJI[\J[tk8T/Jd6Jd6TUXZQ^R/JWd6Ju@ ² o ± ENJ`F@RvwJTU[\_Zq8JRQ^HfJ]DzJ]F-R=d6XZDOgpd6JW[\][cJWd6J K o/=BEQxF-XmD8D8XZDGF[uBEQKD2R/JDGBED2R;POBZT=He BZkOF/Y8TUd8Jj¨¤F-Y8POPGX2F-XmDOF=b2Y8J 〈p, w〉 ≥ 0 ∀w ∈ K J$Rpb2Y8Jp /∈ K oK9;DPGJ]Y6R"BEHKXZT&F;Y6R/QKH^QxF/JTWHKJC;MO_XZTUl[\J`8o^mo^`Vjd6JuF/_POBZTUBER/QKXZDdfe Y8D P>XZQKDmRWJRWd+e Y8Dg$XZDrmJ$a6JuPGXmY8TtF/_PGBETUJT
pJR K b2Y8Q4J`F@RtvwJTU[c_jJ$R]qªsHe _]rQKd6J]DOg$Jmq'gXZDrZJarJuJR"D8XmD2Qxd6Jmo HJ$a6QKF-R/JWd6XmDOg
w 6= 0d8BZDOF
VJ$Rα ∈ R
RUJHxFb2Y8J〈p, w〉 < α < 〈w, v〉 ∀ v ∈ K . @ ² o ± 2
«~BEQxFIBEHKXZT&F]qnXZDd6XmQSRNBmXZQKTα < 0
POY8QKFUb2Y8J0 ∈ K Amdfe BEY6RUT/JPGBET/R]qZP>XZYOT.R/XmY6R i ∈ 1, . . . ,MD8XZYOFjPGXmY8ZXmDOFg&M8XmQKF/QKTjd8BEDGF~@ ² o ± m
v = −λaiBZJ]g
λBETUk8Q^R/T&BEQKT/J][\JD2RuZT&BEDOdfqNg$Jb2Y8Q[cXmD2R/TUJpb2Y8J 〈w, ai〉 ≤ 0
oY9;DXZk8R/QKJD2R=d6XZDGgCb2Y8Jw ∈ K J$R=b2Y8J 〈p, w〉 < α < 0
q8g$JWb2Y8Q+J`F@RQ^[\P>XmFUF-QKk8HKJZo
?+2nZZln oD oZlfUJ= ( A ! *V0 ( 1 i:5N * 4P ( "!764 ? ( , : ( ( > i6 ] =6" n <cP B !*V0 ( 1 63 0(+. . m×n 0 > 1 m ≤ n P 63@ 0 m &R 1 /: ( 6 8 . , A. * Y6 *)( (+*B( : 0 (
minx∈Rn, Bx≤c
J(x) =1
2Ax · x− b · x
,
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Ax− b+B∗p = 0 , p ≥ 0 , p · (Bx− c) = 0.
?+2nZZln oD oZl J= ( f ∈ L2(Ω)!aLI" 1 ( $64 ? ( g:C!3 !["! > 8 A = /4 Ω & W !
ε > 0" 1 /: ( 6 8 . , A3. * B674 "! . 0 ( : 0 ( B:P! ( > 0
minu∈H1
0 (Ω) , ‖u−f‖L2(Ω)≤ε
∫
Ω
|∇u|2dx.
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< :O ( uε = 0
<=:O ( (+. _( :O5 λ > 05 . "! uε
C:O :; . ! ( -6
−∆uε + λ(uε − f) = 06 0 /: Ω,
uε = 0:P!
∂Ω.
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KJ]F-R=d6XZD8DO_CPGBETK = v ∈ V , G(v) = 0 , F (v) ≤ 0 , @ ² o ± Vm
X-G(v) = (G1(v), ..., GN (v))
J$RF (v) = (F1(v), ..., FM (v))
F/XZD2R=d6J]Y6azBZP8P8HKQKg]BnRUQ^XmDOFd6JVd8BEDGF
RN JRRM o]ACBEDOF'g$JID8XmY8ZJ`BEYpg$XmDmRUJ$arR/JmqQKHEvBZY6R'd6XmD8D8JT¦Y8D8JNd8_h:OD8Q^R/QKXZDBmd6_]b2YOBER/JId6JHKBCb2YOBEHKQS:>gBnRUQ^XmDtd6J`F.g$XmDmRUTUBZQ^D2RUJ]F]o9;DcD8XERUJ®R/XZYE|@XZY8T&F
I(u) = i ∈ 1, . . . ,M , Fi(u) = 0He JDOF/J[tk8HKJpd8J]FgXZD2R/T&BEQKD2R/J]Fdfe QKD8_2BEHKQSRU_pBmgRUQ^mJ]F®JDu ∈ K o
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± ² ! "#$%&'#)(* +,
i kj w§`54>mln$o o5l prMs tvu whxeyz@yh~ reuc| tvreuy !$&% reu =f 85j rr¡yr u ∈ K tyuyxz@yCyreu)t zy yQ~uyhx (
G′i(u)
)
1≤i≤N
reuztvrW|tvQyh yhrYu)tvrs yrWs.|reuZyu)tvzKyQ.tuyxry3s tvcy~hut r
w ∈ ⋂Ni=1
[
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]⊥ uyhzvzy¡wxey zc r|tvu x u xu i ∈ I(u)〈F ′
i (u), w〉 < 0 .- ² o R ²
=XZYOF'PGXmY8ZXmDOF BEHKXZT&Ff_D8XmDOg$J]T¦HKJ]F g$XmDOd6Q^R/QKXZDOF ' rm2¡`¡`8wnr¡4dfe XZP8R/QK[\BZH^Q^R/_®F-YOT'He J]DOF/J["kOH^J- ² o ± VmoL3F' 4G¢HrMln oD oZl# tvu
u ∈ K
KyhuCs rYrW | !$&% pr x 3 yMwxey Jyu
F reu¡stv¢|.hz@yh yr
u whxey G yhustv¢|.hzyMs.|rexr ttvrW|G.ysyu yuwhxey+z@y~ reuc|tvrYu yh reuwhxe| zt ,yhyhr
u!|xyr63sy3z@|ZrYtvuZt r$&% $ z t u yhu,xr
¡tvretv¡x z ~|zsyJx
K tvz,yQtuysyh xzSuZt zt5~|uyhxsy |G Q|rkky µ1, . . . , µN yuλ1, . . . , λM ≥ 0 uyhz wxey
J ′(u) +
N∑
i=1
µiG′i(u) +
M∑
i=1
λiF′i (u) = 0 , λ ≥ 0 , F (u) ≤ 0 , λ · F (u) = 0 .
@ ² o RG`'Bd6_][cXmDOF-R/T&BnR/QKXZDd6Y;M8_XmT/l][cJ ² o6oK]VzJ`F@RtY8D8JF/QK[cPOH^JBZd8BZP6RUBER/QKXZDd6Jg$JHKHKJ]F"d8J]F
;MO_XZTUl[\J`F; ² o6o "JR ² o 8o^¢U6qrbmYOJWD8XZYOFHxBEQxFUF-XmDOFBEYH^J`gR/J]Y8TJDzmY8QKF/JWdfe J$a6JT&g$Qxg$Jmo ¦§]n2¡ 4Gnr¡2 r¡c.4G2 w§]Z4G ¡C2'+=f 88jªZ6§`54>
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sjPGBET/R/QKTCdfe Y8DP>XZQKD2Ru ∈ K oO'B\g$XZDGd6QSRUQ^XmDd6Jb2YOBEHKQS:>gBnRUQ^XmDd8JWHKBuA;_h:ODOQSRUQ^XmD ² o6oK ± J]F-R=BZFUF-JG1;m_D8_]TUBZH^Jb2Y8XZQxb2Y8JHKXZQKDdfe $R/TUJ"DO_]g$J`F/FUBEQKTUJ`qO[\BZQKF;POBZT-vwXmQKF;d6QSygQ^HKJtsum_TUQS:GJT;dOBEDOF=H^J`F;BZP8P8HKQKg]BnR/QKXZDGFoL;e J]F-RjP>XZY8T&b2Y8XZQ=H^J`FTUJ[uBET&b2Y8J]Fub2Y8QCF-Y8QKZJ]D2Rjd6XZD8DOJD2Rd6J`Fjg$XmDOd6Q^R/QKXZDOFjd6Jb2YOBEHKQ@:GgBER/QKXZDOFPOH^YOF®F/Q^[\P8HKJ]F;d6XmDOgP8HKYOF4vBZgQ^HKJ]FNsWZ_]T/Q@:OJTNJDuP8T&BnRUQKb2Y8J.[uBEQxFI[\XmQ^DOF4Z_]D8_T&BEHKJ]FQo JZo2[\XZQKDOFF/XZYOZJD2RZ_]T/Q@:O_J`FU$o
r6=f'Mln oD oDenACBEDOFHKJ"g]BZFd6J`F=g$XmD2R/T&BEQKDmRUJ]Fdfe QKD8_2BEHKQSRU_Zq8XmDP>JY8R;Fe QKDOF-POQ^TUJT=d8JWHKBDOXER/QKXZDd6J;g]BZFITU_mY8HKQ^J]T=wQKD2R/TUXrd8Y8QSRUJ;s"HKB/J[uBET&b2Y8J ² o6o VpPGXmY8T4HKJ]F®g$XmDmRUTUBZQ^D2RUJ]FIdfe _2BEHKQSRU_`B :ODzd6JWd6XZD8DOJTY8D8JWgXZDOd6Q^R/QKXZDR/TUl]FF/QK[cPOH^Jb2Y8QfJ]DmRUTUB+*xDOJ;HxB\g$XZDGd6QSRUQ^XmDd8Jpb2YOBZH^Q@:Gg]BnR/QKXZDzd6JHxB"A;_h:GD8QSRUQ^XmD ² o6oK ± o IDJ0>JR]qrP>XZY8Tu ∈ K HKJ]F®g$XZD2RUTUBZQ^D2R/J`FIQKDOBZg$R/QKZJ`FID8Jf |@XmY8JD2R g=POBZFNJ$RF/JYOH^J`F.F-XmDmRIsCP8TUJDOd8T/J®JDug$Xm[\P6R/JHKJ]FIgXZD2R/T&BEQKD2R/J]F BZg$R/QKZJ`F
i ∈ I(u) b2Y8Q8F/XZD2R¤|@YOF@RUJ[\JD2R4d6J`FgXZD2R/T&BEQKD2R/J`FIdfe _mBZH^Q^R/_=JDg$J;PGXmQ^D2R¦9;DuZ_]T/Q@:OJ=BZH^XmTUFIFUBEDGFIP>JQKD8J=b2Y8J=HxBgXZDOd6Q^R/QKXZDF/Y8QKnBED2R/Jb2Y8Q=d6Q^RbmYOJuJ`F@RuY8DP>XZQKD2R\TU_mY8HKQ^J]TcP>XZYOTuH^J`Fug$XZD2RUTUBZQ^D2R/J`F\dfe _]mBZH^Q^R/_
Fi(u) = 0PGXmY8T
i ∈ I(u) (F ′
i (u))i∈I(u)
J]F-RY8D8JpvBZ[\Q^HKH^JWHKQKk8T/J @ ² o R2ZJ]D2R/T&B * D8J@ ² o ± Z$q¤gZe J`F@R/h sEh d8Q^TUJcb2Y8JuHKJ]F"g$XZD2RUTUBZQ^D2R/J`FpF-XmDmRb2YOBZH^Q@:O_J`FoS 4DJ0>JR]q'Q^H4F/Y6y\R"d6JPOT/J]DOd6TUJ
w =∑
i∈I(u) αiF′i (u)
RUJH¦b2Y8J 〈F ′j(u), w〉 = −1
P>XZYOTR/XmY6Rj ∈ I(u) wHe J$a6QKF-R/J]DOg$Jd8J]F4gXrJ$yg$QKJD2RUF
αid6_]gXZY8HKJd6JHe QKDrZJ]TUF/Q^kOQ^HKQSRU_d6JHxBp[uBnR/TUQxg$J
(〈F ′i (u), F
′j(u)〉)ij
o HOJ]F-RNgHKBZQ^TgJP>JDOdOBED2RbmYOJ@ ² o ± Z4D+e Q^[\P8HKQKb2Y8JWPGBZFW@ ² o R2mo •
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k!)#K# %, + m ) Y#$% ± 8jr6=f ln$o o l¢ACBEDOF'HKJ®g]BZF¦d8J]Fªg$XmDmRUTUBZQ^D2RUJ]F¦gXZ[tk8Q^DO_J]F'dfe _mBZH^Q^R/_NJ$R.dfe Q^D8_]mBZH^Q^R/_mq]XmDPGJ]Y6RIBEYOFUF-QrF]e QKDOF/P8Q^TUJTªd6J®HKB=D8XERUQ^XmDtd6JgBmF¦TU_mY8HKQ^J]T'P>XZY8T.d6XmD8D8J]T'Y8DOJ®gXZDOd6Q^R/QKXZDtP8HKYOFªF-QK[\P8H^JbmYOQ®QK[\P8HKQKb2Y8JzHxBgXZDOd6Q^R/QKXZD°d6JzbmYGBEHKQS:Gg]BnRUQ^XmDd6JzHKBA;_h:OD8Q^R/QKXZD ² o 8o^8o.LNJ$R-RUJgXZDOd8QSRUQ^XmDf vwXmT-RUJgcgEe J]F-R-h snh d6Q^TUJp[\XZQKDOFF-XmY8ZJ]D2R®m_TUQS:O_]J`®d6Jb2YOBZH^Q@:GgBER/QKXZDJ]F-R
(
G′i(u)
)
1≤i≤N
⋃
(F ′i (u))i∈I(u)
J`F@RYOD8JCvBE[\Q^HKHKJWH^QKk8TUJZo @ ² o R ± 9;Dcm_TUQS:GJvBZg$QKHKJ[\JD2R4b2Y8J"- ² o R ± 'JD2R/T&B * D8J@ ² o R ² qEgEe J]F-R-h snh d6QKT/Jb2Y8JHKJ]FNg$XZD2RUTUBZQ^D2R/J`FªF-XmDmRbmYGBEHKQS:O_]J]F]o •jr6=f ln$o oc&/J]ZJ]DOBED2R'BZYgBZF'd6J]F'g$XmDmRUTUBZQ^D2RUJ]F+dfe QKD8_2BEHKQSRU_ZqF-Y8POPGX2F-_]J]F¦gXZDrZJa6J]F]qY8D8JtBZY6R/TUJ"gXZDOd6Q^R/QKXZDd6J"b2YOBEHKQ@:GgBER/QKXZDP>XmFUF-QKk8HKJ"J]F-R=HxBjF/Y8QKBZD2R/JZo 9;DF/Y8P8P>XmF/J"b2Y+e QKH J$a6QKF-R/Jv ∈ V RUJH¦b2Y8JWHe XmDBZQSR`q8P>XZY8TRUXZY6R
i ∈ 1, . . . ,M qHKJ]F®vwXmDOgRUQ^XmDOFFi
F-XmD2RgXZDrZJa6J]F®J$R`qXZYk8Q^J]D
Fi(v) < 0 ,XZYk8Q^J]DFi(v) = 0
J$RFi
J]F-RBnyuD8JZo @ ² o RRrIe M2£rP>XERUM8l]F/Jj@ ² o RRr®JD2R/T&B * D8J"b2Y8JHKJ]FCg$XmD2R/T&BEQKDmRUJ]FF/XZD2R;b2YOBZH^Q@:O_J`FJ]D
u ∈ K BEYF-J]DOF=d6JHKBA=_:OD8Q^R/QKXZD ² o6oK ± o" IDJ0>JR]q¤F-Qi ∈ I(u) JRCF/Q Fi(v) < 0
qGBZH^XmTUF]qGd+e BZP8T/l`FHxBg$XmDOd6Q^R/QKXZDd6Jg$XmD2mJ$a6Q^R/_\@ ² o Z〈F ′
i (u), v − u〉 = Fi(u) + 〈F ′i (u), v − u〉 ≤ Fi(v) < 0 .A"e BEY6RUT/JpPOBZT-R`q8F-Q
i ∈ I(u) J$R=F/Q Fi(v) = 0q8BZH^XmTUF
FiJ`F@RBnyuD8JpJR
〈F ′i (u), v − u〉 = Fi(v)− Fi(u) = 0 ,J$R=HxBuA=_:OD8Q^R/QKXZD ² o6oK ± d6J"b2YOBEHKQ@:GgBER/QKXZDzd6J]F;g$XZD2RUTUBZQ^D2R/J`F®J]F-R;F/BER/QxF@vBZQSRUJBZJ`g
w = v − u oIe BBZD2RUBEmJud6JjHe Mr£rP>XER/MOl]F/J@ ² o RR2CJ`F@Rcd6JjD8JjPOBZF"D8_]gJ]FUF-Q^R/J]T"d6JgXZD8DGB *xR/TUJuH^JP>XZQKD2Rcd6J[cQKD8QK["Y8[uD8Q¦d6Jg]BEHxg$Y8HKJTHKJ]Fd8_TUQ^m_J]Fd6J`F®vwXZDOg$R/QKXZDOF
F1, . . . , FMo •
#$t%O(!! % W !®% "#$% &!$#&%'
=XZYOFuBmXZDOF"rY°d8BZDOFcHKBM/J][\BZTUb2Y8J ² o 8o^ ² g$Xm[\[cJ]D2RcQKHJ]F-RuPGX2F/F/Q^kOH^Jzdfe Q^D2R/J]T/POT/_R/JTH^J"gXZY8POH^J(u, λ)
wP>XZQKD2R;d6J"[\QKD8QK["Y8[zq>["Y8H^R/QKP8HKQKg]BnR/J]Y8T;d8Jt¦BZZT&BED8mJ`®g$Xm[\[cJ3B)4G¤§j¡]§]X§]54> '8n2 'D4eEOn8EGr L o2=XZYOF®BEHKHKXZDOFNd8BEDOFNg$JR-R/JpF/J]gRUQ^XmDuP8TU_]g$QxF/JT4HxBWDOBER/Y8TUJ;d8J;g$JPGXmQ^D2RF@R&BnR/QKXZDODOBEQKT/J;g$XZ[\[\JB)4G¤§hX&¡]rJR®[\XZD2RUT/J]TNg$Xm[\[cJ]D2RgJ$R-RUJ=vwXmT/[tY8HxBnR/QKXZDjPGJ]T/[\J$Rd6Jg]BET&BZgRU_TUQKF/JTpY8D[cQKD8QK["Y8[ g$Jb2Y8QNZJ]Y6Rcd6QKTUJjb2Y8JZq F-XmYOF"gJT/RUBEQKD8J`FMr£rPGXZR/M8l`F-J`Fq+DOXZYOFZJTUTUXZDOFb2Y8JH^J`FjgXZDOd6Q^R/QKXZDGF rZr¡]¡`8r¡d6JF-RUBER/QKXZD8DOBZT/Q^R/_~d8Y¦BZZT&BED8mQ^J]D°F/XZD2RzBEYOFUF-Q¡]'¥¡]O¤§]r¡`$o6XZYGF4JarPOH^XmT/J]T/XmDOFIk8TUQKlZJ][\JD2RNHxB"§OF' 4Gn2'z 2' 8§] \b2Y8Q¤JDd6_]gXZY8HKJZo9;Y8R/TUJ=He QKD2R/_TU$R4R/M8_]XZTUQKb2Y8JCd6J=g$JR-RUJCg]BET&BZgRU_TUQKFUBnRUQ^XmD+qEF/XZDuQKDmRU_TU$R®P8TUBER/Qxb2Y8J=d6YP>XZQKD2R®d6J2YOJ=d6J]F®BEHKZXmT/Q^R/MO[cJ`F.DrY8[\_TUQKb2Y8J`FIF/JT&BWQ^HKHKYOF@RUT/_;d8BEDOF4HKB < J]g$R/QKXZD~ ² oVU6o < QKZDGBEHKXZDOF.J]D:ODjb2Y8JHKBcD8XZR/QKXZDzd6JWPGXmQ^D2R/h F/JHKH^J®|@XmY8JWY8DzTmH^J;vwXZDGd8BE[\JD2R&BEH+d8BZDOFHKBc§OF' 4G+2'r¡('`rX.o
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)¦*,,-) z74?-1N3 k_$bcb!_A;J[uBED8QKlTUJBEkGF@RUTUBZQSRUJZq
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L J]F-R;Y8DOJtBEPOP8H^QxgBER/QKXZD~d6JV × Q XZY~dfe Y8D8JcPOBET/R/QKJ
U × P d6JV × Q dOBEDOF R
o¤ACBEDOF;H^Jg]BZd6TUJ4d6Y;M8_XmT/l][cJ= ² o6o WgXZD2R/T&BEQKD2R/J`F¦dfe _2BEHKQSRU_`q]D8XZYGF'BrQKXZDOFU = K
qP = Q = RM J$R
L(v, q) = J(v)+q·F (v)o`'BF/QSRUYOBnRUQ^XmDW_$RUBZQSR Y8DWP>JY"d6Q@0¤_TUJD2R/J®d8BEDOF+H^J®gBZd8T/JId6Y;M8_XmT/l][cJ ² o6oKOUgXZD2R/T&BEQKD2R/J`FWdfe QKD8_]mBEHKQ^R/_`$q¦X.-D8XZYOFtBrQ^XmDOF
U = KqQ = RM q
P = (R+)M J$RJ]DOg$XmT/J L(v, q) = J(v) + q · F (v)oA;XZD8DOXZDOFN[uBEQKDmRUJDOBZD2R4HKB"d6_h:ODOQSRUQ^XmDjdfe Y8DjP>XZQKD2R-h F-J]H^HKJZq2F-XmY8ZJ]DmR4BZP8P>JHK_=_2BEHKJ[\JD2R4[\QKD6h[uBEajXZYzg$XmHo
i kj w§`54>mln$oT$o5lqprMstvuwhxey(u, p) ∈ U × P yhu)xr tvrYu yzvz@yCsy L x
U × P t
∀ q ∈ P L(u, q) ≤ L(u, p) ≤ L(v, p) ∀ v ∈ U .- ² o RUZ
¦JT/_`F-YOHSR&BnR4F/Y8Q^nBZDmRN[cXmD2R/TUJ®HKJHKQ^J]DuJD2R/TUJg$JR-RUJD8XERUQ^XmDd6JP>XZQKDmR/h F/JHKHKJJ$R4HKJ]F4P8T/Xmk8HKl[\J]Fd8Ju[cQKD8QK[\QKFUBnRUQ^XmDBmJ]g\g$XmDmRUTUBZQ^D2RUJ]Fpdfe _mBZH^Q^R/_- ² oRrXmYg$XZD2RUTUBZQ^D2R/J`Fpdfe QKD8_]mBEHKQ^R/_@ ² o ± `_R/YOd8Q^_`Fd8BEDGFpHKBF/J]g$R/QKXZDP8T/_`g$_`d6JD2R/Jmo ªXZY8TF/QK[cPOH^Q@:OJT`q+D8XmYOFpY6RUQ^HKQKF/JTUXZDGFWd6JuD8XZYOZJ]BZYd8J]FQKD8_]mBEHKQ^R/_]FJ]D2R/TUJCmJ]g$R/JYOTUF]qrD8XER&BED2RPOBET/vwXZQxFq ≥ 0
BEYH^QKJYd6Jq ∈ (R+)M o
p4YB)4>¡]§]54>ln$oT$o prMx y wxey¡z@yh r~hut re J, F1, . . . , FM rYu ~ reutvrexeyh x
V yhuwhxey z yhreyh hzy K yhusZrYt | $(% x!$&% $ prr uy P = RM s| r6z@y~|s.y~ rYuZc| tvreuyCs .| ztvu+!$&% "yu P = (R+)M s|rez@y~Q|¡sy~ reuc|tvrYu yh¡s$ tvrW .| ztvu!$(% $ tvu U xr xyhu,s.y
V~ rYu yrW| reu
K x (v, q) ∈ U ×P r y L(v, q) =J(v) + q · F (v) tvu
(u, p)xr tvreu hyzvz@y*s.y L x
U ×P z u ∈ K yhuuyhuxrtvrYtvxz |z
s.yJx
K ¡y zx t J yuF1, . . . , FM
reustv¢|.hzy¡yru r|
J ′(u) +M∑
i=1
piF′i (u) = 0 .
@ ² o R mi r4>'¡]§n6§`54>)oNg$TUQ^mXZDOFHxB\g$XZDGd6QSRUQ^XmDd8JWPGXmQ^D2R-h F/JHKH^J∀ q ∈ P J(u) + q · F (u) ≤ J(u) + p · F (u) ≤ J(v) + p · F (v) ∀ v ∈ U .
@ ² o R E Ia6BZ[\Q^D8XmDOFdfe BEk>XZT&d¢H^J~g]BZFud6J~gXZD2R/T&BEQKD2R/J]Fud+e _]mBEHKQ^R/_Zo 4Y8QKFUb2Y8J
P = RM qIHxB P8TUJ[\Q^l]T/JQKD8_]mBEHKQ^R/_"d8BZDOF- ² o R Z®[\XZD2R/TUJ"b2Y8JF (u) = 0
qOQo Jmou ∈ K o H'TUJ]F-R/JBZH^XmTUF
J(u) ≤ J(v) +p · F (v) ∀ v ∈ U q6b2Y8Qf[\XZD2R/TUJpk8QKJD wJDP8TUJDOBZD2R v ∈ K ®b2Y8J u J]F-RY8D[\QKD8QK["Y8[ ZHKXZkOBZHd8J
JF-YOT
KoACBEDOFH^JugBmFpd6Jg$XmDmRUTUBZQ^D2RUJ]Fpdfe Q^D8_]mBZH^Q^R/_mqfXZD B
P = (R+)M J$RWHxBPOT/J][cQKlTUJcQ^D8_]mBZH^Q^R/_d8Jj@ ² o R E®[\XmDmRUT/JW[uBZQ^D2R/J]DOBED2R=b2Y8JF (u) ≤ 0
JRCb2Y8Jp · F (u) = 0
ofLNJ]gQ+P8TUXZY8mJWJDOgXZTUJb2Y8Ju ∈ K q¦J$RP>JTU[\J$Rtd8Jjd6_]d8Y8Q^TUJ\vBZgQ^HKJ[\J]DmRcd6JuHKBd6JY6a6QKl[\JuQKD8_2BEHKQSRU_bmYOJ
uJ]F-R"Y8D[\QKD8QK["Y8[ ZHKXZkOBZH+d6J
JF/Y8T
Ko
ID6:OD+qªF-QJJRF1, . . . , FM
F/XZD2R"d6_TUQKBZk8HKJ]FWJ]Duq'HKBd6J]Y6a6Q^l][\JQ^DO_mBZH^Q^R/_d8J@ ² o R E[\XmDmRUT/Jpb2Y8J
uJ]F-RY8DP>XZQKD2Rd6JW[\Q^D8QK["YO[iF/BZDOFg$XmDmRUTUBZQ^D2RUJCd6J
J + p · F d8BZDOFHe XZYOZJT/RUqgJtb2Y8QªQK[cPOH^Qxb2Y8Jcb2Y8JtHKBd6_]T/QKZ_]J"Fe BEDOD2YOH^JtJD
uqJ ′(u) + p · F ′(u) = 0
g$v@ofHKB/J[uBZTUb2Y8J ² o6oZ$o
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k!)#K# %, + m ) Y#$% ± ±)+*,,[ ^+745 _ 9_:=1 _f3\:;<_+5
=XZYOF;TUJZJ]D8XZDGFBEYP8TUXZkOH^l][cJtd6J[\Q^DOQ^[\QxF/BER/QKXZD~F/XZYOF;g$XZD2RUTUBZQ^D2R/J`Fdfe Q^DO_mBZH^Q^R/_"PGXmY8T=HKJ$hbmYOJHfHe J]DOF-J]["k8HKJKJ]F-R=d6XZDOD8_pPOBET- ² o ± q6gEe J]F-R-h snh d6QKT/JK = v ∈ V , Fi(v) ≤ 0
PGXmY8T1 ≤ i ≤ m . - ² o R32
+J ;M8_XmT/l][cJt ² o 8o^¢UWB"d6XmD8D8_CY8D8Jpg$XmDOd6Q^R/QKXZDjDO_]g$J`F/FUBEQKTUJ;dfe XZP6RUQ^[uBZH^Q^R/_morACBEDGF®gJ$R/R/JpF-XmYOF@hF-J`gR/QKXZDD8XZYOF;BEHKH^XmDOFmXZQKT;b2Y8Jcg$J$R/R/Jcg$XZDGd6QSRUQ^XmD~J]F-RCBZYOFUF-Q ¡]'¥¡]O¤§]zF-Q HKJ]F=gXZD2R/T&BEQKD2R/J`FJ$RHKBvwXZDOg$R/QKXZDgX 6RpF-XmD2RC4> G+r¡moc 4DJ0>JR]q¤HKBgITUXZP>XmF/QSRUQ^XmD ² o ± o \BEy\TU[\Jtb2Y8Jmq¤F/Q(u, p)J]F-RtY8DP>XZQKD2R-h F-J]H^HKJd8Y¦BZZT&BED8mQ^J]D+q¦BEHKXZT&F
uT/_`BEHKQKF/JuHKJj[\Q^DOQ^[tY8[ d6J
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uTU_]BEHKQxF-JWHKJ[\Q^DOQ^[tY8[d6J
JF-YOT
KqmBEHKXZT&F.Q^H>J$a6QxF@RUJ
p ∈ (R+)M RUJHGb2Y8J(u, p)
F-XmQSR4P>XZQKD2R-h F-J]H^HKJd6Yj'BEmTUBZD8ZQKJD+o.9;DF-Y8POPGX2F-JWd6_`F-XmT/[uBEQxFb2Y8JJ, F1, . . . , FM
F/XZD2Rg$XZDrmJ$a6J]Fg$XmD2R/QKD2YOJ]FF-YOTVo
jr6=f ln$oT$oT LNXZ[\[\JJ, F1, . . . , FM
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K!$&% prtvreu s xtvu)z@y |c|r t5yr L |¢ ~t5
L(v, q) = J(v) + q · F (v) ∀ (v, q) ∈ V × (R+)M .
tvuu ∈ K xr tvrYusy K zy~ rYuZQ|tvreuyh reu wxe|zt Qyh¡| x"yhresy¡z@| ¡5)retvut r
$&% = $ z u yhu xrtvrYtvx z |z)syJx
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(u, p) tvu xr tvrYu yzvz@y sx |G Q|rkt5yr L x V ×(R+)M
x s.yC| ret cyQwxtv|z@yreuy u yz$whxey
F (u) ≤ 0 , p ≥ 0 , p · F (u) = 0 , J ′(u) +M∑
i=1
piF′i (u) = 0 .
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± R ! "#$%&'#)(* +,
k>XZDd6J;TUJ[uBZTUb2Y8J]T4b2Y8J;d6J`FNgXZD2R/T&BEQKD2R/J`F)2,>^ EGOw§` j6¥ r¡Av = b
P>JY8mJD2RFe _]gT/QKT/J;F-XmYOFHxBcvwXZTU[\Jd6Jg$XmD2R/T&BEQKDmRUJ]Fdfe QKD8_2BEHKQSRU_\BnyuD8J`Fd6XZDGgWg$XZDrmJ$a6J]F&Av − b ≤ 0
JRb− Av ≤ 0
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Fi(v)≤ui, 1≤i≤mJ(v),
@ ² o U ² 0 > 1 u1, . . . , um ∈ R & R a:; ,. 0\1 i:O"!: . C: N , 8 C:O9: 6"! 4C= * 9%"& & 63 ! aPe! 1 8 & R a/85 m∗(u) . 0 > 0. !3 *)( (+*V0. )6"! , A. * , 8! A 4a# 9%"& %'P&
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m∗(u) ≥ m∗(0)− pu .- ² o U8`
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k!)#K# %, + m ) Y#$% ± .U A v ∈ U q ∈ P
J (v) = supq∈P
L(v, q) G(q) = infv∈UL(v, q) .
^Jvw,6 ¡xb
V<B - B Q- R - NB ) N < :g infv∈UJ (v) ,
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v ; G(q) = −∞ _?E"F.&! M@', M
q6GSM&'
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. !_%u&/"&' vw,6 x,6 M 0. E E]; ' 0XZjoc-s:|RY] J <" G ^:&:6 6`' ) E2)' v ∈ U , J (v) < +∞ " q ∈ P , G(q) > −∞ E,' ;! 9? 9n?<"&! E">&'%<u-&' FbQ0 E" =@,&!, U5&! ;]^Jvw,6 ¡¢b923"E0;J (u) = G(p) = L(u, p)
6`P7 0'123 0&'2(e" ,-.f")?(&' d E 6OP73% E/"F." E&'@.A"Q23"E;? G,C#,!123 E"G%"E&/"23A"Q'&'% G_&'"EI E!'
(u, p)6 •
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J (u) = minv∈UJ (v) = max
q∈PG(q) = G(p) .
^JvDw,6 ¡Bb
3YopciR[RY~A7*7! `.G!)%u&/"&'^Jvw,6 ¡xbG, J " G UR^JvDw,6 ¡BbG J"Q%D;&'@.'"GL
J (u) = minv∈U
(
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= G(p) .^JvDw,6 ¡"b
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6T89" L∗ = L(u, p)6( W,
v ∈ UUT&!Q J"(?*.&/_54 1 +^JvDw,6 ¡.xBb3; J (v) ≥ L(v, p)
UT74 ) J (v) ≥ L∗ 54 .1 +^Jvw,6 ¡¢b<6%V2323 J (u) = L∗ U`??&V23"E];A J (u) = infv∈U J (v) = L∗ 6(*9 23"E(]!23q23r; G(p) = supq∈P G(q) = L∗ 6t=%?&!,A;23"DUa E E T;=^JvDw,6 ¡BbO9'&!)"O E L∗ = J (u)
6PZ=%u&/"&'#^JvDw,6 ¡.xBb, J 23"E;
L(u, q) ≤ J (u) = L∗ ∀ q ∈ P .^Jvw,6 ¡+b
923q23U,_ E E&(,L(v, p) ≥ G(p) = L∗ ∀ v ∈ U ,
^Jvw,6 ¡-b<") ,%D&/"0?&''23" r^JvDwA6 ¡+Bb IF^JvDwA6 ¡-Bb ; L(u, p) = L∗ UW?#;&e23"E(; (u, p)
J"&'A"EIK E''6
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U L(v, q) ≥ infv′∈U L(v′, q)UQ?
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V6 *9_&'"EA&/"Q'PZ.HF.H&'
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U
J (v) = supq∈(R+)M
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J(v) E&
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+∞ E&',
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G(q) ,
J"O'123VV2(aC,&!23&' a"&!N??a@fhWl:[ Ds:odhZlnY;V'f,!123e,&!2(.^JvDwA6 AvabW?DaO! ?"EF&'" f E"'&!%D&/ ^\ %V"E"9.E"&'?!a&'"%9 J"."F2323"TC,'&'"%>. f, .!H&/"$23 ,23%&!; ^:?< 6-:4 'H&/"$23074 _H`D,aBb<6-_9> E&!23'>?2N&'-.&! E+ Hbf$%123 G=c>$_"bZ?dB>vw,6 x,6 ("G,-.!&/"% vw,6 x,6 + G'% E/"F."G E&!@"D6
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- F (u) ≤ 0 <_ E <] VF<_< u <# T :g B, A ' <f u < A N ' J , V : E p ∈ (R+)M
B p < A D ' G A (R+)M , (u, p) < A( g 0]ZE < L A V × (R+)M A (u, p) ∈ V × (R+)M < f ( - :g dZ : >E<<< E <
F (u) ≤ 0 , p ≥ 0 , p · F (u) = 0 , J ′(u) + p · F ′(u) = 0 .^JvDw,6 xBb
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minv∈VL(v, p) .
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J(v) =1
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N × (R+)M .^Jvw,6 B¡b
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minv∈RN
L(v, q)
=V E'"&'3&!;e&' ;v → L(v, q)
J"TV?"&' E"23"O?A@BC,6$%V"E" E'"&'@%&/u- ∂L
∂v(v, q) = Av − b + B∗q = 0
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6 *9#,"&!A"G,?
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,
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−1
2q ·BA−1B∗q + (BA−1b− c) · q − 1
2A−1b · b
)
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4 C,&'23"e E V23G-aE"&'?'&'123"V E&!23'U&'+' = E&'23'0;>=!0'123&'2(."= G@BE =L(! # JIK E?"&'vDw,6 ¡6 x];?"D@"FH="fq<"E=,"&!'&' E%GV23"E"EG(&'"f('H.&/"$23,Q?D'?5G* E',"&!]'123&'2(.g6 •
bZ23&' Q-aQ#C,?&'?%?%."&/T;&723"E9!N!a"&!+"E' G'123 Q9&'"EI E'!_23&'IK2(.C "G*0"$%.&!, J,C56=\7Y,iBsY ~AR*7 M*(' 2H H#&2BC' D, ,DB2F#(&%* #<'
A =
1 0 4 2 3 5−3 2 −1 2 −5 2−4 2 −2 0 −1 2−2 4 −1 6 −2 2−1 2 −6 3 −1 1
.
R=S>B$>35+-O, >=D&(' >' >F>382 /) ' %i W / #82 DB * j W >3#(&>@+-(/ ' > D#$' >F>382 D&' / #82$9 (' > 1E1 1T >QD&(' W %$#<'0T4 ) # $023 W >3 *) > ' %8;- =' .>A2 1B23 @'01=#$Z D(5D&' 82
aij #OQ#82 ' D
A43/ #82 Z,DBE#(82
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@'08P Q#, +- / *S1B>F-#(9X! >A2?P3'1,+-$' 2F#(I H#8E,D=D,B2?23OH#82 ' D,A
= !#$NW%'# (> =?"E"N E?"&'+ G%"-,&' =;!; 9.'&'?Da"&! 9 Q% E/"Fa" = E?<"&! =%?%<I
" 6 # &'H-.' f;A74 =,"E9.!&'?Da"&'RUA!HF232(a"&'('&!%D&/U, EF>"EF&/"%9L0.E"f?F$-&'M?F$-.&/"E#F&' E+,> EM&'23.E"F?#?<$?<$>%Fa"&'''6
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−∆u = f
Ωu = 0
E∂Ω,
^Jvw,6 "b
$)Ω J"f @BE"T%=,
RN " f ∈ L2(Ω)U J"e%D;&'@.'"VL>!23&'&'23&' ."&'(74 %H&'
minv∈H1
0(Ω)
J(v) =1
2
∫
Ω
|∇v|2dx−∫
Ω
fv dx
^Jvw,6 $+b
^:@B&/0! O E&/"&' ¡6 ¢6 "b<6`8= D@B @AL_:4 `C<?&'? -,6 ¢A6 ++;M^Jvw,6 $+b023"d&!;23&!&'2)2 <"DU5L3g4 `C,?&'?#vw,6 ¢A6 ,UW; E%;A."&'54 O'9 J">! .2)!a"&!p@.&!a"&!'!>^JvDwA6 "b<6,PR9 E&'H&'u?D."&']$ E&!;fQ:4 %H&!0^JvDwA6 $+BbO J"T%@A&!"6$ `.eC,23'U E&Z^Jvw,6 "b
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k 6' )#$ ± ZV23A,%'&' Ef!9%<2(a"&'N74 V232NFO%! J"&!;^
u J"`'%!?23A"Z.2( E `:4 .?"&'
, f.? fb<U,!N E',"&!M J"'>,%!.?23"Q;&23&'&!23&' E=*0 E2323=:4 %H&'>%! J"&!;
,%.2(."&'_"G>:4 %<H&'""&''' .? GCA"%&' 6 *9# E> E023A"E<Q;!)"$%&'>! ,-.!&/"%23"74 E E?&'LM^JvDwA6 $"b_XZY[R\Zs*moYrhZi.s:|XBsh`YdXZYods:|Zsodsc snj5| 23"E"F." J)%H&'UR,&'"5j5odhWl*zoY|csniaY+p23%?D&*;#^:d,-'ab<UR,"! E&'H&/u-?Da"&'+$, E&!;9 J"G""G. E E&Z&'23.E"F">;?'!,3^JvDwA6 +Bb<689 '' f&'"EA&/G]PZ.HF.H&'_ E E?&'%>L:4 %H&'9&'2(.!0^JvDwA6 $+BbV&'];A=?'!<I ?&
=,% E"G- VG?"EF.&!" 6 W?G&/ e"&''&! E fG"&/"9. J"?=(&'A"EA,&' "f@.&!.'&'A"<23%D&!&/
e ∈ L2(Ω)N "Tf?"EF&'" e = ∇v6 ='. G^Jvw,6 $+bT J"T%D;&!@'"
Lmin
v∈H10(Ω) , e∈L2(Ω)N
e=∇v
J(v, e) =1
2
∫
Ω
|e|2dx−∫
Ω
fv dx
.
*9#&'A"EA,&/"Q_PZ.HFH&'+&'"23%D,&*.&/>Q?>'123
M(e, v, τ ) = J(v, e) +
∫
Ω
τ · (∇v − e) dx ,
D@B?)r2N/"&''&'?D.",)PZ.HFHτ ∈ L2(Ω)N 6'*9%!&'23&'N2(&'"-." e
,>,"&/>'PRH.FH&'M?F$?F$%
L(v, τ ) = mine∈L2(Ω)N
M(e, v, τ ).
%V2323e→M(e, v, τ )
J"=nE"23"Q?A@BC,U&!7C,&' J">M&!;&'A"=,23&'&'2)2MUA"G?'?5:.?&''N23A"E9;
L(v, τ ) = −1
2
∫
Ω
|τ |2dx−∫
Ω
fv dx +
∫
Ω
τ · ∇v dx.^Jvw,6 $-b
*9#@B%&/u-9 Q&';A'>'123,&!2(.7 E E?&'%N+PRHF.H&! ^JvDw,6 -Bb J"G&'d^Jvw,6 $+b(
maxτ∈L2(Ω)N
L(v, τ )
)
= J(v) ,
<"9;9''123,-7 J"(
minv∈H1
0(Ω)L(v, τ )
)
= G(τ ) =
−1
2
∫
Ω
|τ |2dx E& − divτ = f
Ω
−∞ E&'76^Jvw,6 "wb
*9#"Q2(.&'A"-"%?G'% E/"Fa"=,&'?&'-.:6 zAj-iamopY~A7Z:~< E >A#, V (u, σ) )ZE < L(v, τ ) A H1
0 (Ω)×L2(Ω)N
L(u, σ) = maxτ∈L2(Ω)N
minv∈H1
0(Ω)L(v, τ ) = min
v∈H10(Ω)
maxτ∈L2(Ω)N
L(v, τ ).
G E < u < = A = ) J(v) B H10 (Ω) σ < 9 , =
B G(τ ) B L2(Ω)N J(u) = min
v∈H10(Ω)
J(v) = maxτ∈L2(Ω)N
G(τ ) = G(σ),
e: 9 <f, :: σ = ∇u
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±m±Z² ! "#$%&'#)(* +,
(YopciZ[RY~RZ PR,'123-.V^Jvw,6 "wb= >&'A"<%"Fa"&!+$, E&!;A?!&/6 %V2323max G(τ ) = −min(−G(τ ))
U&'( 4 H&/" 23&'&'23&' E :4 %H&' ^g&/" ?23'%23"F.&'Dbd ?"EF&'" 23%?D.&!; 1
2
∫
Ω|τ |2dx
.23&O:4 E2N'3 0?F$-.23 ,3?"EF&'" cDsR[RYopY|5cX`ods DsnkWlnYUf?4 J"EI LaI ,&/+@B%&/u.A" g4 %D;&!'&', .? −divτ = f
Ω6V>
?"E"9n2)!."&'+,-.!U'>,%!?23"v-.F!"=?23239'2N/"&''&'?D."GPZH.FH0
*0?"EF.&!">54 %D;&''&!, −divτ = f6*_=."E? E%;A?Tbf$%123)vw,6 6'v> J"G;A74 #
" , G(τ ) ≤ J(v)
?N;&R23<"954 "&/, G = E,9' =%H&' 9,&!2(.'>r'6t=2(aF; ;:4 M E E&
J(u) = G(σ) =1
2
∫
Ω
fu dx
;A&5R4 J"Q&'+74 ."E>;9* 23&/"&!%#"EFD@&'R .? GCA"%&' 6 •y#zopj5|'i.cDsnj5|O`PZ@BQ,EF&/"q"E==? E%D;"=&'2323%D,&*a"G %V!!.&' vw,6 x,6'vv^:p2(aF;-";A74 N?"EF.&!"354 %HB.!&/"%]] 4 %?&/"0?2323 Cr&'%H'&/"% N]3 >,I E% Fb E&O?)"$%123)R4 %"F.&/"- J"E&'A"(^:&'?&*bL]2N0u-&O, ?A"EF.&'A" 6 *=DUR !N'1230f^Jvw,6 "wb9&! M]&'u&/"%),N?"EF&'" 9&' ;0!]?"EF&'" −divτ = f !&'r= ;>""=&'"
x ∈ Ω68=%D23&' U!% E'"Fa"9 J"@F&R"&'Z J"9:.?&'' @B&/
F;&:6[O_B5"DU.G? J"E?<"&!M_
G(τ ) ≤ L(v, τ ) ≤ J(v),
") &/" ;A3' ),'123 0,&'2(V<"),-.f,23"E""d&!;3&'A")323&'&!2N2u"
σU- E?"&'@B23"D6 *=DU?2323
u J"9 E'"&'p]^JvDwA6 "b<U7 E&'23'N&'"%H.F."&'-a-aE"&'
23"E;
J(u) =1
2
∫
Ω
|∇u|2dx−∫
Ω
fu dx = −1
2
∫
Ω
|∇u|2dx = −1
2
∫
Ω
fu dx .
# &7M,%u-&/"σ = ∇u
UM%D,&/"G ^Jvw,6 "bG; −divσ = fU"Q_"&'"G,?
G(τ ) ≤ J(u) = G(σ),
?4 J"EI L.I &/;σ J"Q'>&!"G,2(.C,&'2)2 ,
GU?
(u, σ) J"='>&!"Q E'' L(v, τ )
6
)¦*,,Z[ 7 `.1=9_7 =38? ` b!_*9 % E,"M&'?&:4 `C23' -,6'v6 - 54 ,'123,?232(+"&'2('p!% , J"123
!&'%D&/I ;-,F."&!;6 *9M? E&!10'N , J"123,&5%"&'W'&'%D.&'N"9:4 &'?A(^::4 %"F." J"123ab
y(t) J"9LN@.! G.
RN
dy
dt= Ay + Bv + f
,0 ≤ t ≤ T
y(0) = y0
^Jvw,6 "AvDb
$)y0 ∈ RN J">:4 %"Fa"&!&/"&*.Tr , J"123U
f(t) ∈ RN J"+"230 E?Uv(t) ∈ RM J">!
?232(.- ;A&V23<")54 H&/) EN!] , J"123UZ"A<"
B E" ,Cp2(."E&'? 0? J"F" )
&'23 E&' f E?<"&!@ N ×N
"N ×M
6
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k 6' )#$ ±Z± *9_@B,"Q?<$&' E&/9!)?232(
v>2(&'1>L)23&'&'23&' EQ_?&/"1;-,Fa"&*;
J(v) =1
2
∫ T
0
Rv(t) · v(t)dt+1
2
∫ T
0
Q(y− z)(t) · (y− z)(t)dt+1
2D (y(T )− zT ) · (y(T )− zT ) ,
)z(t)
9"EF J?<"&/ ?&!'UzT E&/"&!_u-' ?&''U,"
R, Q, D"E&' Q2(."E&'? ,23%I
"E&!; V E&/"&'@B VA"V E'R J"f E E%=%<u-&'G E&/"&'@B6AtG2(.F; T;Q!>?"&'
y(t),%_9*N@a&*.'vLN"EFD@B ^JvDwA6 "Avab<6
Wf@&/Q'&!;V! V% E/"Fa" Q74 ,"&!23&' a"&!#,%?%D," U f?F$&' E&! E E Q,=?F$EI?F$<
v. f:4 E-.?9&''E"
L2(]0, T [; RM ) fn?<"&! Q,
]0, T [.
RM =?D.E%9&!"%IH.F!6-^gPT4 E? ! `-a", `?"&' O?"&'A OR4 J"T2(.!$ E23"Z- O) E?,=&'!<E"D6 b W9"&/?23"N74 %@"'' 9?"EF&'" E,!#?232(Ur&'A"EA,&/">?A@BC,9n23%_@A&!
K,
RM ;A&% EA">:4 E2)! ?232( 23&' E E&'' 6-PR,!123>23&'&!23&' a"&!_ J"=,?
infv(t)∈L2(]0,T [;K)
J(v).^Jvw,6 "¢b
%V2323 .Q@B%<&'u=;'> , J"123)^Jvw,6 "AvDbG J"Q&'M E%6
TYoopYd~A7Z* f+aB ) f(t) ∈ L2(]0, T [; RN ) v(t) ∈ L2(]0, T [; K) ' FB ,A -> :g y(t) ∈ H1(]0, T [; RN ) < VQ ,e : 0A [0, T ] y]zopj5|'iacDsnj5|T %V% E/"F."e74 C,&' J"?G<"V54 &'?&/"% J"T&! ?A) E&
f"
v E"T?"&'A 6
`R4 J"> ' >,&0] ?&'!). ')?,L2 6'*9r"&''&' E)!]n2N')C,!&'?&/")% EA"Fa"&',! E'"&'
y(t) = exp(tA)y0 +
∫ t
0
exp(
(t− s)A)
(Bv + f)(s) ds
;&-23"f9@%&/u-Vg4 C,&' J"?":4 &'?&/"%>y.
H1(]0, T [; RN )6,P7PR2323,6 x,6 x)
,&/"QuM;y J"G?"&!A E
[0, T ]6
*9_,"G' G23"EQ:4 C,&' J"?"Q:4 &'?&/"%>!)?232(-,9"&'2('6
iajh`j Ds*Dsnj5| ~ARZ < E =,]A u ∈ L2(]0, T [; K) ) ) B e : B : u < JB < F=,.∫ T
0
Q(yu − z) · (yv − yu)dt +
∫ T
0
Ru · (v − u)dt
+D(yu(T )− zT ) · (yv(T )− yu(T )) ≥ 0 ,
^Jvw,6 "xb
, K v ∈ L2(]0, T [;K) yv < :g p B (.F ] v y]zopj5|'iacDsnj5|T(*9]?2323?=-a2(.F;<Q;
v → y J"Q=?"&'+&]36-OM5<"DU
.G'&'%Da&'"%,3^JvDwA6 "vab+yv = yv + y
U)yv J"G E'"&'r,
dyv
dt= Ayv + Bv
0 ≤ t ≤ T
yv(0) = 0
^Jvw,6 "Bb
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±m± ! "#$%&'#)(* +,
"y J"G E'"&'M
dy
dt= Ay + f
0 ≤ t ≤ T
y(0) = y0
` J"?!.&'0;y ,%-r-. >
v"
v → yv J"'&'%D.&/ ?"&'A)
L2(]0, T [; K)
H1(]0, T [; RN )6 `.>? E%D;"DU
v → J(v) J">0n?<"&!;-,Fa"&*;N E&'"&'@B)
v^:'
%?&' E%23A"DUa!9 E2323V54 e23V;-AF."&!;e"T74 en?<"&!3] ab<U?J J"O?A@B<CU
"23q23E"23">?A@B<C)?D.!_2(a"E&!?R J"N,%u-&!N E&/"&'@B6 %V2323
L2(]0, T [; K) J"
]?A@BC,Q23%=#@A&!U,'@bf$%123 -,6 ¢A6 )<23",9??'LN:4 C,&! J"?<"QLN:4 &!?&'"% &'"]_23&'&!2N2
up^JvDwA6 "¢b<6V)4 "E_-.E"DUT!p?-,&/"&! 54 "&'2('&/"%M%? E .&'r"
E]3 .A"Nbf$%123(vDwA6 ¢A6'v3 J" 〈J ′(u), v − u〉 ≥ 06 W,?D.!?'')H.F&'"DU5!(23%"$A,
*N' Q 9"G*N' Q E&'23' J"=,?D'?'
limε→0
J(u + εw)− J(u)
ε= 〈J ′(u), w〉.
%V2323J(v)
J"G;,Fa"&!;A='?'?5 J""E1 f E&'23'9&' ;yu+εw = yu + εyw
6 *9],"&!A"&' E%23"^JvDwA6 "xBb_2(.F;-.A";A
yu − yv = yu − yv6
(YopciZ[RY~RZNO<C'&!?&'"F."!3?&/"&'74 ,"&'2('&/"%),#^Jvw,6 "¢b=3+:.&'"?D'?!%!0HF&'"
J ′(w)G""
w ∈ L2]0, T [^:<"9 9 E'23"G,'N23&'&!2N2
ub<U?N;&7 J"
"&''>Q' G23%"$A fA23%&!;A ,>23&!&'23&! a"&'d^:@&/Q! # ?"&'pvDwA6 ¡b<6 *9_),"A∫ T
0
J ′(w)v dt =
∫ T
0
Rw · v dt +
∫ T
0
Q(yw − z) · yvdt
+D(yw(T )− zT ) · yv(T ) ,
^Jvw,6 "¡b
$)v J"G9n?<"&!r;'?-;9
L2]0, T [6 •
PZr?&/"&' %? E &/M<" E] "_54 "&'2(.!&/"%^JvDwA6 ".xBb) J") :.&'" &!C,'&/"F.' \O5"DU5," J":4 ,"&'2('&/"%3
u&'O J">%? E &/3>?F$-;A0n?"&'p" J"
vN?D'?'
g4 %"Fa"?.E E.A"yv6U_=."E:.,0@&/9?"E"0&]3?'"%> J"9:4 &'23 E E&'&''&/"% 74 "&'
NC, E E&'C,'&'?&/"3J ′(u)
L#.E"&/0M^Jvw,6 "¡b<6& W?A"?"E"3,&]3?/"% d?,E"=L * "&'M74 zAc3cXj-s:|50;&7 J"G Q&!% =' G' G,n-, =>!)"$%.&'?"E'0,"&!2(.:67S_"E ?2323"Q?%D,> E,9:4 C,23'0%"-,&'%N 9?"E"0 E JIK E?"&'^: Z, 7g4 &!%eH%%F.'f. 7!QtG2(aF;AfvDwA6 ,6 +G?&/I E E Fb<6 WZ'e,'123^Jvw,6 "¢b+,%u-&'"fg4 %"Fa" &'"
p?2323=* E'"&'M&!;>,
dp
dt= −A∗p−Q(y − z)
0 ≤ t ≤ T
p(T ) = D(y(T )− zT )
^Jvw,6 "b
$)y J"!3 E'"&',]^Jvw,6 "AvDb=,!3?232(.-,
u65P7N2 74 %"Fa"> J&'A"9@A&!A"9?0;
?4 J")!_2(a"E&'?(J&'"A∗ ;&T-.F!" (^Jvw,6 "b<6OPT4 &'"%q"N3:4 %"F.")J&!"0 J");R4 &'23"Q54 "&/=<C, E E&'M<C'&!?&'",
J ′(u)6
z,ji.mopY~RZ* Z]< ) J < u < 5G,J ′(u) = B∗p + Ru .
^JvDw,6 ""b
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k 6' )#$ ±Z±m± (, : << ] n:g pZ : n >F<< E3 < N - B B ]<
∫ T
0
(B∗p + Ru) · (v − u) dt ≥ 0 ∀ v ∈ L2(]0, T [; K).^Jvw,6 "+b
3YopciR[RY~A7Z PZpn2N'^JvDwA6 "$"b E+H%%F.'&! Er3","w ∈ L2]0, T [
J ′(w) =
B∗p + RwU,;&/"E"L?D'?'
p.>^Jvw,6 "bf#,"&'!&' .A"fg4 %"Fa"
y?.E E.A"GL!N?232(
w67PR bf$%123(vDwA6 ,6 M0C, E E&'C,'&'?&/" ,+HF&'"r,&'C+0*3% E!,"&' E!%23"F&/f,0 , J"123VJ&'"Q^JvDwA6 "Bb<6%=4 J"Of&5%?Vn23A"F.'fD@B?f!Q2N'^Jvw,6 "¡b;A&:UQ?F$-;A=n?<"&!#" J"
vU,%? E E&/"F&/"=!0% E!,"&'+,# , J"123N^JvDw,6 "vabD@B?!
?232(v6 •
y]zopj5|'iacDsnj5|T # &/"p*> E!,"&'#0^JvDw,6 "Bbe"
yv?!'90^JvDw,6 " b<6PT4 &!%G J"Q,Q2)/"&''&'
^Jvw,6 "b(.yv"r^Jvw,6 "Bb(.
pUe54 &'"%H.]-.(.E"&' ("],M?23.(' 3% E/"F." 6 T'
,%?&' E%23"DU+?D.!?'N!(;-.A"&/"%> E&!@"00,C#2(&!1< 9,&5%" 6UbR"974 .F7U-&'"%H.F""Q(""?23"= ?-,&'"&' &'&/"&!'
yv(0) = 0"
p(T ) = D(y(T )− zT )U
∫ T
0
(
dp
dt· yv + p · dyv
dt
)
dt = D(y(T )− zT ) · yv(T ).^Jvw,6 "-b
N4 ."E>.E"DU,M,"&!'&' "G' Q%D;-a"&! Q_"&'"∫ T
0
(
dp
dt· yv + p · dyv
dt
)
dt = −∫ T
0
Q(y − z) · yv dt +
∫ T
0
Bv · p dt.^Jvw,6 +wb
*9,%D&'"Ng4 %H'&/"% "E_^Jvw,6 "-b9"3^Jvw,6 +wb90 E&'23'&/u-?Da"&'p):4 C, E E&! ^Jvw,6 "¡b! ,%&'@B%
∫ T
0
J ′(u)v dt =
∫ T
0
Ru · vdt +
∫ T
0
Bv · pdt,
?;A&7' % E/"F." 0^JvDw,6 ""b"0^JvDw,6 "+Bb<6
3YopciR[RY~A7Z %V2323"f.Ig"EIK]&'__XRYRs:|RYAi''123N^JvDw,6 "BbQ;A&5%u&/":4 %"F." &'"`.uO E&'23'&/u-Z:4 <C, E E&'0
J ′(v) T?T`&' U.:4 &*,%e&/?"E&'?T J"W:4 &'"EA?<I"&'r74 _PZH.FH&' E E?&'%)M,'123,>23&'&'23&' ."&'^Jvw,6 "¢b<6 *9M? E&!1<0:4 %D;-a"&'54 %"F."^Jvw,6 "AvDb?23239=?"EF&'""E9,C(@a&*.' &'-,%"
v"
y"Q_%u&/"
'PRHF.H&!+?2323=*N E2323,J(v)
<"9,:4 %D;-."&'+74 %"Fa"=2N/"&''&'%.pU,?4 J"EI L.I ,&'
L(v, y, p) =
∫ T
0
Rv(t) · v(t)dt +
∫ T
0
Q(y − z)(t) · (y − z)(t)dt
+D (y(T )− zT ) · (y(T )− zT ) +
∫ T
0
p ·(
−dy
dt+ Ay + Bv + f
)
dt
−p(0) · (y(0)− y0) ,
)p J"3'_2N/"&''&'?D."(,MPZ.HF.H_ *?"EF.&!"p^JvDw,6 "vab "E
v"
y6.23'!<I
23A"DUW! 3?&/"&' (54 "&'2(.!&/"%_d^Jvw,6 "¢b 4 "&'"3 &' .A"3;A#'+PRH.FH&' J" J"Fa"&'&/U?4 J"EI L.I &/;
∂L∂v
=∂L∂y
=∂L∂p
= 0.
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±m± R ! "#$%&'#)(* +,
PZp,23&'1M,%&'@B%_#*?&/"&' 74 ,"&!2(.'&'"%d^JvDw,6 ""b<Ue* E?-M#g4 %D;."&'@B%&/u-%Q.eg4 %"Fa"f &'"
pU"V!9"E&! E&'123G:4 %D;-."&'3@B%&/u-%Q.eg4 %"Fa"
y6 E&! J" e ET'G&/"
;A9?9?D'?R J"Q23A"e.23:U2(.&' Q;A74 #H%%<F7&!7,9! G%D;."&'#:4 %"Fa" &'"D6 •
-aE"&/, bf$%.123 vDwA6 6 3+,"= E&/"9"&'=, =,,&!%<"% 9;'&/"F."&'@B G>!3 E',I"&!
y"0,)!_?232(0"&'2(.!
u^:@B&/g4 `C,?&'?MvDwA6 6'vab<UZ E&/"0? J"E&/ )23%"$A
,23%&!;>,023&'&!23&' a"&!+(^Jvw,6 "¢b=-a9+'H.&/"$230>" ,>H.F&'"D6LO+:4 E?N?"EF&'" E0!M?232(-,U7?4 J"EI L.I &/3 E&
K = RM UR"N''N?.3' '&' g4 .- , E"V"E@ '&7,=?232(G;&-G:4 %<"F."GJ&!"p^:"Q,?=!0?232(
"&'2(.!u = −R−1B∗p
_@B<E"M3^Jvw,6 "+bEb<6i.j-hOj DsnDsnj5| ~ARR f aB p K = RM f = 0 z = 0 N zT = 0 - P (t)
:: p [0, T ] . <,F < Z J N 3 < :g rA NB
dP
dt= −A∗P − PA + PBR−1B∗P −Q A 0 ≤ t ≤ T
P (T ) = D
^Jvw,6 +,vDb
' < P (t) < -Q : Q , K t ∈ [0, T ] p p(t) = P (t)y(t) y#zopj5|'i.cDsnj5|O W#","
t ∈ [0, T ]U:4 '&'?D."&' ;&:UQL
y0 ∈ RN U:&/"+?E E-,(y, p, u)(t)
^:$)u = −R−1B∗p
J" !+?232(.-)"&'2(.!Db J")?!&/23"N'&!%D&/6VT''_ J"&! J?<"&!@>_@B<E"_,:4 `C,?&'? vw,6 6'vU?& ?"&'@B
RN RN 6 `.G? E%D;A"DU,:4 ,I'&'?D."&'
y(t) → p(t) J"'&'%D.&/#,
RN RN UR"&'eC,&' J" 2(a"E&!? P (t)
54 F,N"'! ;A
p(t) = P (t)y(t)6LOd,%&'@."?"E"NC, E E&'p"p,"&'!&' .A"! >%D;-."&' ^Jvw,6 "AvDb
"0^JvDw,6 "Bb<U-M,"&'"dP
dty = −A∗Py − PAy + PBR−1B∗Py −Qy
""y(t)
^g;& J"=;!?-;9&' ;Ay0g4 J"<b<6 *9]M,%D&'"fg4 %D;."&'+, ^JvDwA6 +,vab<6 *9
"&'"!#?-,&/"&!du-'P (T ) = D
?Dap(T ) = Dy(T )
"y(T )
J"N;'?-;). RN 6
*9M,23"Q:4 &'?&/"%,! E'"&'M ^Jvw,6 +,vDb<6
=\7Y,iBsY ~ARZ:~ R@>$>3=+-K = RM W f = 0 W z = 0 W B2 zT = 0
9XY *82X+- W $*.2A82t ∈ [0, T ] W
p(t) · y(t) = Dy(t) · y(t) +
∫ T
t
Qy(s) · y(s) ds +
∫ T
t
R−1B∗p(s) ·B∗p(s) ds .
Q1,-$'0+->,/ '0<' >?23t0 ∈ [0, T ]
23+-y(t0) = 0 W # > y(t) = p(t) = 0
$*-2A82t ∈ [0, T ]
9 82F $1B2FZD 1> 2F#&29
=\7Y,iBsY ~ARZ R@I823$' / 1,+-$' E#( 82K5 #X $>' 2 ' 687*9 -9 )B2G- 1 H6,79 -9 : $* > &>A2 Q5#$#I$* ' +-
∂y
∂t−∆y = v + f
#&>]0, T [×Ω
y = 0>
]0, T [×∂Ωy(0) = y0
#&>Ω
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k 6' )#$ ±Z± U
y0 ∈ L2(Ω) W f ∈ L2(]0, T [×Ω) W v ∈ L2(]0, T [×Ω)>A2@ # D*Q# W B2K*='$'0@' >3
infv∈L2(]0,T [×Ω)
J(v) =
∫ T
0
∫
Ω
v2dt dx +
∫ T
0
∫
Ω
|y − z|2dt dx +
∫
Ω
|y(T )− zT |2dx,
z ∈ L2(]0, T [×Ω)
B2zT ∈ L2(Ω)
9
G\RYAiBs Y ~RZ*Z11# ' >3/ D8' D,O$1,D1,82 / 1,+-#82' *=>*>,9
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23A,%'&' E%>-.
−∆u = f + v
Ωu = 0
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u J"'3%!?23A"9@BE"&'?D.e,)!#232)F.0"
v J"N.?(N?"E! ;A&` EF#!
@.&!.']74 ,"&!23&' a"&!R6&*9 E]3d@BE"ω ⊂ Ω
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vmin ≤ vmax
L2(ω)6 ` J"A"-r ""9?3;&Z E&/"0;0!
n?"&' O,L2(ω)
EA"O%<"-, R-.`%9 Ω\ω 6$*90%u&/"O.!. `:4 E2N'f, `?"E! 23&' E E&''
K =
v ∈ L2(ω)"Z;
vmin(x) ≤ v(x) ≤ vmax(x)
ω"
v = 0
Ω \ ω
.^Jvw,6 +xb
# &f ∈ L2(Ω)
U7'bf$%.123(¡6 ¢6 ¢_ &/"0;R4 &'TC,&! J"))&!;N E'"&'u ∈ H1
0 (Ω)6'*9
?F$<?<$=L>?"E'f!232),Ff,f;R4 ''="G(,%!?23"u0 ∈ L2(Ω)
6*9(%u&/"9n?"&'M? "
J(v) =1
2
∫
Ω
(
|u − u0|2 + c|v|2)
dx ,^Jvw,6 + Bb
)u J"!3 E'"&',]^Jvw,6 +B¢bN^:"?0%M
vb"
c > 065PR0'12354 "&'23&' ."&'
4 %?<&'"inf
v∈KJ(v) .
^Jvw,6 +B¡b
iajh`j Ds*Dsnj5| ~ARZ:~ < E >AMA : v ∈ K A V- B By]zopj5|'iacDsnj5|T(*9]2(.F;>;!Nn?<"&!
v → u J"=&]3>,
L2(Ω).
H10 (Ω)
6 `.? E%;A"DU
J(v) J"9n?"&';-AF."&!; E&/"&'@B0
vU5,?0!'N J"9?A@BC,6[T'') J"
23q23nE"23A"9?A@B<CN&! ;J(v) ≥ c‖v‖2L2(Ω)
65)4 "EN-aE"DUK J"r?A@BC,0n23%
#@,&!,>L2(Ω)
6 `.Q? E%D;A"DU,'bf$%123 -,6 ¢A6 23"G,??'L):4 <C&' J"?>"GL:4 &'?&/"%_&'A"=,23&'&'2)2 ,3^JvDw,6 +¡b<6
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p%u&O?2323)g4 &!; E!,"&' .
H10 (Ω)
,
−∆p = u− u0
Ωp = 0
E,∂Ω.
^Jvw,6 +$b
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i.j-hOj DsnDsnj5| ~ARR:~-~ Z :: p , J(v) < < . A K f pJ ′(v) = p + cv ,
p < B v d< 5), B _ E< - :g 5EF<
F . )Z : f A : v < −∆u = f + v B. Ω , u ∈ H1
0 (Ω),^JvDw,6 +$"b
−∆p = u− u0 B Ω , p ∈ H10 (Ω),
^JvDw,6 ++Bbv = 1IωP[vmin(x),vmax(x)]
(
−p
c
)
,^Jvw,6 +$-b
1Iω < > - :g - F# ω - < g + .. B. ω rB Ω \ ω M
P[vmin(x),vmax(x)] < 9 <E <A]9- E :g - FA Ng < [vmin(x), vmax(x)]
T , P[vmin(x),vmax(x)]w = min(
vmax(x), max(
vmin(x), w(x)))
y#zopj5|'i.cDsnj5|O %V2323M ! ` E&/"&' vw,6 6 ,Uf!p23%"$A]!p' 3 _<"(*' E&!23'>>?D.'?'G'>HF&'"G J"
limε→0
J(v + εw) − J(v)
ε=
∫
Ω
J ′(v)w dx .
%V2323J(v)
J"=;AAF."&!;'>?D.!?7 J"Q"E1 Q E&!23'>"QM"&'"∫
Ω
J ′(v)w dx =
∫
Ω
((u− u0)uw + cvw) dx,
$)uw J"9,%.
−∆uw = w.
Ωuw = 0
E∂Ω.
^Jvw,6 -wb W# E&'23'&/u-#:4 C, E E&' , HF&'"_ "&''&' Ep:4 %"F."MJ&'"##?! ,f 2)'"&'!&'^JvDw,6 -wBb-.
p<"N^Jvw,6 +$bQ.
uw"G_&'"1H>.G-aE"&!
∫
Ω
∇p · ∇uw dx =
∫
Ω
(u− u0)uw dx∫
Ω
∇uw · ∇p dx =
∫
Ω
wp dx
`.G?23.F.&! E+>? G,C(%HB.!&/"% =#+%&/"G;∫
Ω
J ′(v)w dx =
∫
Ω
(p + cv)w dx,
74 $)r:4 C, E E&',+HF,&'"D67PZ3?&/"&'r%? E &/N" E*]3 "074 ,"&'2('&/"%),%0-.!bf$%123 vDw,6 ¢6!v J"
∫
Ω
(p + cv) (w − v) dx ≥ 0 ∀w ∈ K .^Jvw,6 -,vDb
Tp"w%HB.VL
v-aE"" .V E)"&/" '3
ωU5&' >r:&' .A"",N!
F BM,>?"E"'>@B `%<UM," !?'&' E (^JvDwA6 -Avabd^:, ;abe""Q&!"x,
ω(
p(x) + cv(x)) (
w(x)− v(x))
≥ 0 ∀w(x) ∈ [vmin(x), vmax(x)] .
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v"
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L(v, u, p) =1
2
∫
Ω
(
|u − u0|2 + c|v|2)
dx +
∫
Ω
p(∆u + f + v) dx,
)p J"f'92N/"&''&'?D.",f,9PRH.FH9e*?A"EF.&'A"0^JvDwA6 +B¢bT"E
v"
u6 23!'23A"DU
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∂L∂v
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±m± ! "#$%&'#)(* +,
r6=f'Mln oDf oZl°=XZYOF;D8XmYOFCH^QK[\QSRUXZDOFCBEY6a~F/JY8HxFpBEHKZXmT/Q^R/M8[\J`F=d6_R/JTU[\Q^DOQKF-R/J`F=JRCD8XmYOFDOJd6QxF-XmDOFTUQKJDd6J`F"BZH^mXZTUQSRUM8[\J]F"d8JR@£rPGJF@RUX6g&MOBZF-R/Qxb2Y8JwTUJ]gY8Q^RcF/Q^[tY8H^_mq'BZH^mXZTUQSRUM8[\J]Fm_$hDO_$R/Qxb2Y8J]F]q>J$RUgZo $o$9;Y6R/TUJ"HKJvBZQSRWb2Y8J"HKJY8TpBEDGBEHK£rF/JWvBZQSRWBEP8P>JHªsHxBuRUM8_XmT/QKJtd8J]F;P8T/XmkOBEkOQ^HKQSRU_]Fb2Y8JD8XZYOFªD¦e BZkGXmTUd6XmDOF POBZFªd8BZDOF.g$JgXZY8T&FU$qH^J]Y8T.Y6R/QKHKQKFUBnRUQ^XmDtJ`F@RªR/TUl]Fªd6Q@0¤_TUJD2R/Jmo\ XmY8TIFUg&M8_$h[uBER/QxF-J]T"F-QK[\P8HKJ[\JD2R]qªd6QxF-XmDOFb2Y8JHKJ]FtBEHKZXmT/Q^R/M8[\J`FWd6_$RUJTU[cQKD8QxF@RUJ]FtF-XmDmRtH^J`FP8HKYOFJygBZgJ]FP>XZYOTHKBc[\QKD8Q^[\QxF/BER/QKXZDd6JpvwXmDOgRUQ^XmDOFg$XmDrZJ$a6J`Fq2RUBZDOd6QxFb2Y8JWHKJ]FBZH^mXZTUQSRUM8[\J]FF@RUX6g&MOBZF-R/Qxb2Y8J]FP>JTU[\J$R/R/JD2R=dfe BEPOP8T/X6g&M8J]T®d8J]F[\Q^DOQ^[uBCEGZ4e7ª8e¢J$RPOBmF®F/JYOH^J][cJ]D2RHKX6gBZY6aONd8J;vwXmDOgRUQ^XmDOFDOXZDzg$XZDrmJ$a6J]FpscY8DzP8TUQ^aR/XmY6R/JvwXZQxFBZFUF-JG1;_]H^J]Z_pJDzP8T&BnRUQKb2Y8Jo •
)¦*,,Z[ &b47I5O? 3q_9_ 3 _ I5e`ª9;?!_¦1N3`<` k`.1°<¤741N3656`.?-1N3_LNXm[c[\J]DeP$XmDOF"PGBET"_R/YOd8Q^J]TtHxB~TU_]F/XZHKY6R/QKXZDPOTUBER/QxbmYOJd6JP8T/Xmk8HKl[\J]Ftdfe XmP6R/QK[\QKFUBnRUQ^XmDJDHe BEkOF/JDOgJWd6JWg$XZD2RUTUBZQ^D2R/J`Fo < XmQSR
JY8D8JpvwXZDGgR/QKXZD
αh gXZDrZJarJpd6Q@0>_]T/J]D2R/QxBEk8HKJWd6_h:ODOQ^JF/Y8THe J]F-hPGBZg$JWd6J*d=Q^HKk>JT/RT/_]JH
Vq6XmDg$XZDGF-Qxd6lTUJCH^JWP8TUXZkOH^l][cJWFUBEDOFgXZD2R/T&BEQKD2R/J
infv∈V
J(v) .@ ² o V2Z
Ate BEP8TUl]FjH^JM;M8_XmT/l][cJV8o6o Q^HCJ$a6QxF@RUJ~Y8D8JY8D8Qxb2Y8JF/XZHKY6RUQ^XmDuqgBZTUBmgR/_]T/QxF/_Jdfe BEP8TUl]FjHKB
/J[uBET&b2Y8J\ ² o6oWPGBETHe _`bmYGBnR/QKXZDzdfe IY8HKJTJ ′(u) = 0 .
5EW4On§GF 2 "EGn62'r¤§ ) Bª8¡ 4eB §]84e BZH^mXZTUQSRUM8[\Jzd8JmTUBmd6QKJD2Rg$XZDGF-QxF@RUJs"f/F/Jzd8_P8HxBZgJT gdfe Y8D8JQ^R/_TU_J
un J]DF/Y8QKnBED2RuHKBHKQKZD8Jd6J"P8H^YGFZT&BEDGd6JWPGJ]D2R/J"BmF/F/X6g$QK_JWs\HKBuvwXZDOg$R/QKXZD~g$X 6RJ(v)
o¤¦Bud6QKT/J`gRUQ^XmDzd8J"d6J]FUg$J]D2R/JgXZTUT/J`F-P>XZDGd8BED2R's;g$J$R/R/JHKQ^mD8JNd6J®P8HKYOFªZT&BEDGd6JIP>JD2R/JQxF/F/Y8JNd8Jun J]F-R.d6XZD8DO_J®POBET H^J®ZT&BZd8Q^J]DmR
J ′(un)o 4DJh0¤J$R]qGF-QfHe XmDg&M8JT&g&M8J
un+1 F/XZYGFHKBtvwXZTU[\Jun+1 = un − µnwn ,
@ ² o V ± BmJ]g
µn > 0P>J$RUQSR\J$R
wn Y8DOQSR&BEQKT/JjdOBEDOFVqªgEe J]F-RcBZJ`g\H^Jg&M8XZQ^ad6JjHxB~d6QKT/J`gRUQ^XmD
wn =J ′(un)
‖J ′(un)‖b2Y8J=He XZDP>JY6RJ]F/P>_TUJTNR/TUXZY8mJTIHxBP8HKYOF®P>J$R/Q^R/JCnBEHKJY8Td6J
J(un+1)wJ]DjHe BZkOF-J]DOg$J
d+e BZY6R/TUJ]FQKD6vwXZTU[uBnRUQ^XmDOFg$XZ[\[\JWHKJ]Fd6_]T/QKZ_]J]FF-YOPG_]T/QKJY8TUJ]FXmYHKJ]FQ^R/_]T/_]J]FBZDmRU_TUQ^J]Y8TUJ]F&oLNJR-R/JCTUJ[uBET&b2Y8JF-QK[\P8HKJ=D8XmYOF4gXZDOd6YOQSR`qmPOBZT/[\QGH^J`F4[\_$RUM8X6d6J]F®d6YuR@£rP>Jt- ² o V ± .b2Y8QGF/XZD2RBZP8P>JHK_J]F f-[\_$RUM8X6d6J]Fpd6Jd6J`F/gJD2R/Jhg&qfsHe BZH^mXZTUQSRUM8[\Jtd6JEGn62'r¤§ )Bª8¡+4eB §]8-q>d8BZDOFHKJ]b2Y8J]HXZDTU_]F/XZY6RuY8DOJ~F-YOg]g$J`F/F/Q^XmD°d8JP8T/Xmk8HKl[\Jd6J[\Q^D8QK[\QKFUBnRUQ^XmD¯s Y8D8J~F/JY8HKJnBETUQKBZk8H^JTU_J]H^HKJzw[\[\JuF-QVD+e J]F-RpPGBZFpd6J\d6Q^[\J]DOF-QKXZD :OD8QKJ`$o PGBET/R/QKTpd8J
u0 b2Y8J]HKgXZDOb2Y8J\d8BZDOF V qXmDgXZDOF-R/TUY8Q^RHKBuF/Y8Q^R/J(un)
d6_h:GD8Q^JWPGBETun+1 = un − µn J ′(un) ,
@ ² o V R2X.-
µn ∈ RJ`F@Rg&M8XmQKF/Q+s\g&MOBZb2Y8Jp_$R&BEP>JCR/JH'bmYOJ
J(un+1) = infµ∈R
J(
un − µJ ′(un))
.@ ² o V.UZ
LNJR=BEHKZXmT/Q^R/M8[\Jpg$XmD2mJTUZJCg$XZ[\[\JpHe QKDOd6QxbmYOJWH^JWTU_]F/Y8H^RUBnRF/Y8QKBZD2R]o
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kk'#)(* +,K +%, $, ±Z± VL*F' 4GH2 ln$of o pr x yCwhxey J yhu
α ~ reycy s t hQyhrYuZt5|.hz@y yu whxey
J ′ yu t h~tvut5yhr x u xu, rW¡syV ~e yhu stvcy¡whxey
∀M > 0, ∃CM > 0, ‖v‖+ ‖w‖ ≤M ⇒ ‖J ′(v)− J ′(w)‖ ≤ CM‖v − w‖ .@ ² o V m
z z |z tvuvCyCsy Q|st5yreu |¢ utv |zK~ rY¢yZkywxeyhz wxey tvu u0 z@|Cxtvuy (un)sZret5y |!$&% yhu!$&% ~ rY¢yZky¢y z@|C zSxuZt r u s.y!$(% i r4>'¡]§n6§`54>)o'B vwXZDGgR/QKXZD
f(µ) = J(
un − µJ ′(un)) J]F-RvwXZT/R/J][cJ]D2Rg$XmD2mJ$a6JJ$Rd6_TUQ^nBZk8H^JF/Y8T
RF-Q
J ′(un) 6= 0AOF/Q^DOXZD+qGXZD~Bud6_-|-sg$XmD2mJTUZ_mq
un = u $oO+J"P8T/Xmk8HKl[\J"d6J[cQKD8QK[\QKFUBnRUQ^XmD°- ² o V.UmCBd8XZDOgck8QKJD Y8D8JF/XZHKY6RUQ^XmDY8DOQKb2Y8Jmq¦gBZTUBmgR/_]T/QxF/_J"PGBETWHxBgXZDOd8QSRUQ^XmD
f ′(µ) = 0q8gJb2Y8Q+Fe _]gT/Q^R=BEYGF/F/Q
〈J ′(un+1), J ′(un)〉 = 0 .@ ² o V E
LNJ]g$Qf[\XmDmRUT/JWb2Y8Jd6J]Y6a¡f/d6QKT/J`gR/QKXZDGFd6JWd6J]FUg$J]DmRUJgWg$XZDGF-_`g$Y6RUQ^mJ]FF-XmD2RXmT-RUM8XZmXZDOBZH^J`Fo4Y8QKFUb2Y8Jp@ ² o V E+Q^[\P8HKQKb2Y8Jb2Y8J 〈J ′(un+1), un+1−un〉 = 0
qXmD"d6_]d8Y8QSR.d6J®Heαh gXZDrZJarQ^R/_d6J
Jb2Y8J
J(un)− J(un+1) ≥ α
2‖un − un+1‖2 . @ ² o V2
IY8QxF/b2Y8JHxBF/Y8Q^R/JJ(un)
J`F@RNd8_]g$TUXZQxFUF/BZDmRUJ[cQKD8XmT/_]J"wPOBZTJ(u)
q2JHKHKJ=g$XmD2mJTUZJ®J$R4He Q^D8_]mBZH^Q^R/_@ ² o V3m4[\XZD2R/TUJWbmYOJun+1 − un RUJDOdmJT&F
0o8Ate BZY6R/TUJWPOBET/R]q6He
αh g$XZDrmJ$a6QSRU_Wd6J
JJRH^JpvBZQSRbmYOJHKBF-Y8Q^R/J
J(un)J]F-R"k>XZTUD8_Jj[cXmD2R/TUJD2Rtb2Y8JHxB~F/Y8QSRUJ
(un)J]F-R"kGXmT/DO_J~¨ Q^H®J$a6QxF@RUJY8D8Jg$XZDGF@R&BED2R/J
MRUJHKH^Jb2Y8J
‖un‖ ≤M .4g$TUQ^nBZDmRW@ ² o V 2ªPGXmY8T
v = un JRw = un+1 JR®Y8R/QKH^QxF/BZD2Rp- ² o V Zq2XZDZXZQ^R4vBmg$QKH^J][\JD2Rb2Y8J
‖J ′(un)‖ ≤ CM‖un+1 − un‖ q8gJbmYOQ+[\XZD2R/TUJb2Y8J J ′(un)RUJDOdzmJT&F
0oOIe
αh g$XmDrZJ$a6Q^R/_Wd6J
Jd6XmD8D8JWBEHKXZT&Fα‖un − u‖2 ≤ 〈J ′(un)− J ′(u), un − u〉 = 〈J ′(un), un − u〉 ≤ ‖J ′(un)‖ ‖un − u‖ ,bmYOQ+Q^[\P8HKQxbmYOJ
α‖un − u‖ ≤ ‖J ′(un)‖ qOdfe X-zHe XmD~d6_]d6YOQSR=HxBug$XmDrZJTUZJ]DOg$Jpd6JWHe BZH^mXZTUQSRUM8[\JZo
jr6=f ln$of oT H'J`F@R=Y8R/QKH^Jcd6J"D8XZR/J]THe QKDmRU_TU$RCP8TUBER/Qxb2Y8J"d6JHxBd6J]T/DOQ^l]T/JQKD8_]mBEHKQ^R/_"d6Jg$J$R/R/Jd6_][\XZDOF-R/T&BnRUQ^XmD¨+XmY6R/TUJtHxBP8TUJY8mJ\d8JcHxBg$XmDrZJTUZJ]DOg$JmqOJ]H^HKJd6XZDOD8J\Y8D8Ju[uBn|@XmTUBER/QKXZDBEQxF-_][cJ]D2Rg]BEHxg$Y8HxBEkOH^JWd6JpHe J]T/TUJY8Tun − u o •
5EW4G§GF 2'"EGn62 r¤§ ) Bª8¡Cj +4e BZH^mXZTUQSRUM8[\J"d6JmTUBmd6QKJD2R=suPOBmF:8a6JtgXZDOF/QxF@RUJ"F-QK[\P8HKJ[\JD2R=J]D~HKBjg$XZDGF@RUT/YOg$R/QKXZD~dfe Y8D8JF-Y8Q^R/Jun d8_h:OD8QKJWPOBZT
un+1 = un − µJ ′(un) ,@ ² o VZV2X-
µJ]F-R'Y8DWPGBET&BE[\l$RUT/J.P>XmF/QSRUQSv:Oar_moLNJR-R/JN[\_$R/MOXrd8J4J]F-R'd6XmDOg.P8HKYOF'F-QK[\P8H^J®b2Y8JIHe BZH^mXZTUQSRUM8[\Jd6J"mTUBmd6Q^J]D2R;suPOBmFXZP8R/QK[\BZHq>P8Y8QxFUbmY¦e XmDvBEQ^RCsg&MGBZb2Y8J_RUBEP>J"He _]g$XmD8XZ[\QKJtd6JtHKBT/_`F-XmH^Y6RUQ^XmDd6J°- ² o VkUZo+JTU_]F/Y8HSR&BnRF-YOQ^nBED2Rz[\XZD2R/TUJF/XZYOFb2Y8JHKHKJ]FMr£2P>XERUM8l]F/J]FXZDPGJ]Y6R~g&M8XZQxF/Q^TH^JPOBET&BE[\l$RUT/J
µPGXmY8TBZFUF-YOT/J]T®HxB\g$XmD2mJTUZJ]DOg$Jmo
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± R ² ! "#$%&'#)(* +,
L3F' 4G¢HrMln oDf oZ pr x 3 y¡wxey J yuα ~ reycyCst cyreut5|z@yyu whxey
J ′ yu t ~tvut5yhr xV ~e yhu stvcy¡wx tvzyQtuy xry3~ reu|rYu y C > 0
uyhzvzywxey
‖J ′(v) − J ′(w)‖ ≤ C‖v − w‖ ∀ v, w ∈ V .@ ² o^ ²Z²
z t 0 < µ < 2α/C2 z |z tvuvCys.y Q|st5yreu |¢)Ky¡~ reyZ.y )whxeyzwxey* tvuu0 z|Cxtvu y (un)
s.5)ret5y | !$&% ~ reyhZky3¢y z@|C zSxuZt r u sy $(% i r4>'¡]§n6§`54>)o3 X2F-XmDOF
vn = un−u oLNXm[\[cJ J ′(u) = 0q]XZDB
vn+1 = vn−µ(
J ′(un)−J ′(u)
) qOdfe X-zQKHf2QKJD2R‖vn+1‖2 = ‖vn‖2 − 2µ〈J ′(un)− J ′(u), un − u〉+ µ2
∥
∥J ′(un)− J ′(u)∥
∥
2
≤(
1− 2αµ+ C2µ2)
‖vn‖2 ,
@ ² o^ ² d+e BZP8T/l`F@ ² oK ²m² 'J$R4He
αh g$XmD2mJ$a6Q^R/_Zo < Q
0 < µ < 2α/C2 qmQ^HGJ]F-RIvBZgQ^HKJd6JmXZQKTIb2Y8J 1−2αµ+C2µ2 ∈]0, 1[
q6JRHKB\g$XmDrZJTUZJ]DOg$JCF-JWd6_`d6Y8Q^R;d6Ju@ ² oK ² `o
r6=f'Mln oDf oDf =D8JBZd8BZP6RUBER/QKXZD"F/Q^[\P8HKJNd8JNHxB=d6_][\XZDOF-R/T&BnRUQ^XmDP8TU_]g$_`d6JD2RUJZq`HKBZQKFUF-_]J4BEYHKJ]g$R/J]Y8TNJ]DjmY8QKF/J;d+e JarJ]TUgQKgJZqmPGJ]T/[\J$Rd6J;[cXmD2R/TUJT®HKBtg$XZDrmJTUZJDGg$JJDjT/J][cPOHKB.PBED2Rp@ ² o^ ²Z² PGBETHe Mr£rP>XER/MOl]F/JP8HKYOFvBZQ^k8HKJj- ² o V 2o H¦vBEY6RCD8XERUJT=BZYOFUF-Q bmYOJZqOP>XZYOT=He BZH^mXZTUQSRUM8[\Jpd8J"ZT&Bnhd8Q^J]DmRcszPOBZF :OarJmq¦szHxBd6QS0¤_TUJDGg$Jjd6YmTUBmd6QKJD2RszPOBZFWXmP6R/QK[uBEHq+HxBF-Y8Q^R/JJ(un)
D+e J`F@RPGBZFDO_]g$J`F/FUBEQKTUJ[\JD2R[cXmD8XERUXZD8Jmo •r6=f'Mln oDf o H=J$a6QKF-R/J~d8J~D8XZ[tk8TUJY6a¢BZY6R/TUJ]FuBZH^mXZTUQSRUM8[\J]Fud6J~d8J]FUg$JD2RUJ~d6YR@£rPGJ- ² o V ± \b2Y8JDOXZYOFjD8Jd6_`g$TUQ^TUXZDGFuPOBZFjQKgQo 9;DTUJDOgXZD2R/TUJD8XZRUBE[\[\JD2Rzd8BEDOFg$JR-R/Jg$HxBZFUF-Jd+e BZH^mXZTUQSRUM8[\J]FIHxBt[c_R/M8X6d6Jpd6YZT&BZd6QKJD2R®gXZDn|@YOZY8_pd8BEDGF4HxBZb2Y8J]H^HKJ=HxBtd8Q^TUJ]g$R/QKXZDjd6Jpd6J`F/gJD2R/Jwn d6_]PGJ]DOd D8XmDF/JY8HKJ[\J]DmR"d8YmTUBmd6QKJD2R
J ′(un)[uBEQxFWBEYOFUF-QId8J]Fd6QKTUJ]gRUQ^XmDOFWd6Jud6J`F/gJD2R/JY8R/QKH^QxF-_]J]FBEY6aQSRU_T&BnRUQ^XmDOFuP8TU_]g_]d6J]DmRUJ]F]oN=XZYOFjP8T/_`F-J]D2R/XZDGFg$JR-R/J[c_R/M8X6d6Jd8BEDGFHxB < XZYGF@hF/J]g$R/QKXZD ± oKZoVU6q8P>XZY8T;H^JtgBZF=POBET/R/Qxg$YOH^QKJT=d+e Y8DOJvwXZDGgR/QKXZDOD8JHKH^Jcb2YOBZd6T&BnRUQKb2Y8Jd8YR@£rPGJ 1
2Ax ·x− b · x o •r6=f'Mln oDf oLNXm[\[cJ]D2R.g&MOXZQxF-QKT'J]D2R/TUJNHKJ]F.d6J]Y6atBZH^mXZTUQSRUM8[\J]F d6JZT&BZd6QKJD2R.bmYOJND8XmYOFmJD8XmDOFCd6JtmXZQKT]q>J$RWP8HKYOFCZ_]D8_T&BEHKJ[\J]DmRCJD2R/TUJcH^J`FCd8QS0¤_TUJD2R/J`F=[\_R/M8X6d6J`FWd6Jt[\QKD8QK[cQxFUBnR/QKXZDDrY8[\_]T/Qxb2Y8Jb2Y8QWJarQxF-R/JD2R =DP8TUJ[\QKJTzg$TUQSRUlTUJd6Jg&M8XmQSagXZDOgJTUD8JHKJgX 6Rd6J g&MOBZb2Y8JQ^R/_]TUBER/QKXZD+o.BZT®Ja6J[\P8HKJZq6D8XmYOFHe BZXmDOFd6Q^R]q8g&MOBmbmYOJCQ^R/_]TUBER/QKXZDd6JWHe BEHKZXmT/Q^R/MO[cJpd6JpZT&BZd8Q^J]DmRsPOBmF :8a6JcJ]F-RW[cXmQ^DGF;g&M8l]T/J\b2Y+e Y8D8JcQSRU_T&BnR/QKXZDd6J\ZT&BZd6QKJD2RCsjPOBZFCXmP6R/QK[uBEHo42Qxd6J][c[\J]DmR`qF/QHe XZDPOBET/R\d8JHxB~[\[\JQSRU_TU_J
un qªY8D8JQ^R/_T&BnRUQ^XmDd6Y°ZT&BZd6QKJD2RtsPOBmF"XmP6R/QK[uBEHd6_]gT/X *KRPOH^YOFHKBvwXmDOgRUQ^XmD gX 6Rb2Y+e Y8D8JuQ^R/_]TUBER/QKXZDd6Y ZT&BZd6QKJD2RsPOBZF :8a6JZo9;DJDBETUTUQ^mJ\d8XZDOg\BEYd8JY6a6QKl[\Jug$TUQSRUlTUJtd6Jug&M8XmQSafqfF-XmY8ZJ]D2R;P8HKYOFWd6_R/JTU[\Q^DGBED2R]q+b2Y8QªJ]F-Rg$J]H^Y8QId8JtHxB +w§]r¡`¡]2 .4G >2¢E>r'md6JCHe BEHKZXmT/Q^R/MO[cJmq2b2Y8QW:8a6J;H^JCD8XZ[tk8TUJ;dfe QSRU_T&BnRUQ^XmDOF4DO_]g$J`F/FUBEQKTUJ]FIP>XZYOT®TUJDGd6T/JHe JTUT/J]Y8T ‖un − u‖ QKD6vw_TUQ^J]Y8TUJpscY8DOJpR/XZHK_T&BEDGg$J ε :8a6_JBcP8TUQ^XmT/Qo.BETuJa6J[\P8HKJZqIHe QKD8_]mBEHKQ^R/_@ ² oK ² [\XmDmRUT/Jb2Y8JHxB g$XZDrmJTUZJDGg$Jd8JHe BEHKZXmT/Q^R/M8[\Jd6JmTUBmd6QKJD2RscPOBmF:8a6JWJ]F-RBEYz[\XZQKDOFZ_]XZ[\_$RUT/Qxb2Y8JZqrP8Y8QxFUbmYOJ‖un − u‖ ≤ γn‖u0 − u‖ BZJ`g
γ =√
1− 2αµ+ µ2C2.
Ecol
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lyte
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pas
repr
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kk'#)(* +,K +%, $, ± RGLNJ$R-RUJ=TUJ[uBET&b2Y8J=gXZDOd6YOQSRdfe BZQ^HKHKJY8T&Fq2F-XmYOFITU_]F/JTUZJd+e Y8DOJ;BEDGBEHK£rF/JP8HKYOFNPGXmYOFUF-_]JZqrsWP8TU_$vw_TUJTPGXmY8T=HKJ"POBZTUBZ[\l$R/TUJ
µHKBBZH^J]Y8T=[\_`d6QKBZD8J
α/C2 d8BEDGFHe QKD2R/JTUnBEHKH^J ]0, 2α/C2[ qGnBZH^J]Y8T;b2Y8Q[cQKD8QK[\QKF/Jg$J]H^HKJd6J
γo\ 4D\vBEQ^R]qEXmDtP>JY6RI[\XmDmRUT/J]TIb2Y8JHKBpg$XmDrZJTUZJ]DOg$J®d6J]FId8JY6a\BEHKZXZTUQ^R/M8[\J]F_$R/YGd6Q^_`Fg$Q^h d6J]FUF-YGFpJ]F-RJh0¤J]gRUQ^mJ[\JD2R"m_Xm[c_R/TUQKb2Y8J\d8BZDOFg$J]T-R&BEQKDOFWgBmFpPOBET/R/Qxg$YOH^QKJTjg$Jb2Y8QF-QKZD8Q@:OJCbmYOJ=HxB"b2YOBED2RUQSRU_ ‖un−u‖1/n B"Y8D8JCH^QK[\QSRUJ :ODOQ^JmqrgXZ[\P8TUQKF/J=F-R/TUQKg$R/J][cJ]D2RNJD2RUT/J 0
J$R1q6HKXZT&F/b2Y8J
nRUJDOdZJ]TUF
+∞ o •?+2nZZln oDf oZl W "!3 V = R2 C J(x, y) = ax2 + by2 0 > 1 a, b > 0
< * 8b!/.+] 0. 3= ( * 263 0 6 ( 9 ,/0 :7 , (+*V0. 1 > 8[8 !3/$:O! . ( 548 0 ( : ( a = b
! : (x0y0 = 0
< C!/ . 0$1 " > 8 1 C:O 34C * 4P ( !/g6 0 : . C: 0 !9: 190 : & !\6 ( 8 0 !/:O: (Y. 01 > 88 1 )63 .+] 0. 3= ( * B6 0 6 ( 9 ,\0 : ?_ , !e!/ . . 9: > 0. !3:b6"! ,/0 0* P µ. 01 " > 8 1 B:O , =6"! ( . . 3< , !e"! . . > 0. 8!X9:5 . . . 0a,. !:B 0,8( 63
)+*,, &b I7I5O?@3 _ 9 _ 3 _ 456`.9=?!_'143 ]<` ` )_¦< <¤741N3r5e`ª?/1N3_ 9;Dz_R/YOd6QKJW[uBEQKD2R/JDGBED2RHKBcTU_]F/XZHKY6R/QKXZDzd6JpP8TUXZk8HKl[\J]Fd+e XmP6R/QK[\QKFUBnRUQ^XmDBmJ]gCg$XZD2RUTUBZQ^D2R/J`F
infv∈K
J(v) ,@ ² o^ ² m
X-JJ`F@RpY8D8JtvwXZDOg$R/QKXZD
αh g$XZDrmJ$a6Jtd6Q@0¤_TUJD2R/QxBEk8HKJtd8_h:OD8QKJ\F-Y8T
Kq¤F/XZYOF-hJ]DOF/J["kOH^Jcg$XZDrmJ$a6JvwJTU[c_ND8XZD"2Qxd6J®d6J4He J`F-PGBZg$JNd6J,d=Q^HKkGJ]T-R T/_]JH
Vo+J;M8_XmT/l][cJNV8o6o BZFUF-Y8TUJ4BZH^XmTUFfHe J$a6QKF-R/J]DOg$JJ$RpHe Y8D8Qxg$Q^R/_\d6J"HxBjF/XZHKY6RUQ^XmD
ud8J- ² oK ² Z$qGgBZTUBmgRU_TUQKF/_Jdfe BEPOT/l`FHKJ¡;M8_]XZTUl[\J ² o6oKPOBETHKB\g$XmDOd6Q^R/QKXZD〈J ′(u), v − u〉 ≥ 0 ∀ v ∈ K .
- ² oK ²m± < JHKXZD¢H^J`FcBZH^mXZTUQSRUM8[\J]Ft_$RUYOd6QK_]F\g$Q^h d8J]FUF-XmYOFq D8XmYOF\F-J]T/XmDOF"PGBET/vwXZQxFtBZ[cJ]D8_]F\sP8TU_]g$QxF/JT\d6J]FM2£rP>XERUM8l]F/J]FF/Y8P8P8HK_[\J]DmR&BEQKT/J`FF-Y8THe JDOF/J[tk8H^J
Ko
5EW4G§GF 2'"EGn62 r¤§ ) Bª8¡Cj +~ >rB'4 ']rZ§]54>4e BZH^mXZTUQSRUM8[\Jd8JmTUBmd6QKJD2RcsPOBZF:8a6JF]e Bmd8BEP6RUJBZY¢g]BZF\d6Y°P8T/Xmk8HKl[\J- ² o^ ² Z"BZJ`gg$XZD2RUTUBZQ^D2R/J`FstPGBET/R/QKTd6JpHKB\TUJ[uBET&b2Y8JpF-Y8QKnBED2R/JmoªXZY8TRUXZY6RTU_J]H
µ > 0q¦@ ² oK ²m± IF]e _]g$TUQ^R
〈u−(
u− µJ ′(u))
, v − u〉 ≥ 0 ∀ v ∈ K .- ² oK ² R2
XERUXZDOFPK
He XZP>_T&BnRUJY8Tpd6JcP8TUXE|@J]g$R/QKXZDF/Y8TpHe J]DOF-J]["k8HKJtgXZDrZJa6JKqfd6_:OD8Q.BEYM;M8_XmT/l][\J`6oKZoK ² d8JjP8TUXE|@J`gR/QKXZDF/Y8TcY8DgXZDrZJarJ~ZXmQ^TtHKB /J[uBET&b2Y8J~6oKZoKZ`$o =H^XmTUF]q'dfe BEPOT/l`F"g$JR/M8_]XZTUl[\JZq¤@ ² oK ² Rr.D+e J]F-RT/QKJDdfe BZY6R/TUJ;b2Y8JCHxBtg]BET&BZgRU_TUQKFUBnRUQ^XmDd6J
ugXZ[\[\J;HxB"POT/XZ|@J]gRUQ^XmDXZT/R/M8XmZXZDGBEHKJ;d6J
u− µJ ′(u)F/Y8T
Ko =Y6R/TUJ[\J]DmR=d8QSR`q
u = PK
(
u− µJ ′(u))
∀µ > 0 .@ ² o^ ² Um
H¦J]F-RvBZgQ^HKJd6JmXZQKTb2Y8Jj@ ² oK ² UZ4J]F-R=JDvBEQ^R=_`bmYOQ^nBEHKJD2R=s- ² oK ²m± $q6J$R;g]BET&BZgRU_TUQKF/Jpd6XZDOgHKBWF/XZHKY6R/QKXZDud6J"@ ² oK ² ZoE4e BZH^mXZTUQSRUM8[\Jd6JEGe2'2¤§ )Bª8¡ j +j Gr B ¢4 '`rE§]Z4G¢wXmYP8H^YGFF-QK[\P8HKJ[\JD2Rd6JWZT&BZd8Q^J]DmRP8TUXE|@JR/_`®J`F@RBZH^XmTUFd6_:OD8Q+POBZTHe Q^R/_T&BnRUQ^XmD
un+1 = PK
(
un − µJ ′(un))
,@ ² oK ² 2
X-µJ`F@RY8DPOBZTUBZ[clR/TUJ;P>XmF/Q^R/Q^vK:8a6_mo
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± R2 ! "#$%&'#)(* +,
L3F' 4G¢HrMln oDf o pr x 3 y¡wxey J yuα ~ reycyCst cyreut5|z@yyu whxey
J ′ yu t ~tvut5yhrCxV!sy~ r6u|reuy
C tv!$(% $&%(% z t 0 < µ < 2α/C2 z | z tvuvCys.y3c|s t5yhrYu |¢Ky|y~ y~hut r ~ reyZ.y wxeyz wxey¡ tvuu0 ∈ K z@|xtvuy (un)s.5)ret5y | !$&% $(%('~ rY¢yZky*yh z@| zxut r u sy!$&% $&%
i r4>'¡]§n6§`54>)oI'Bjd6_][cXmDOF-R/T&BnR/QKXZDTUJPOT/J]DOd~g$J]H^HKJtd8Y ;M8_]XZTUl[\J ² o U8o RjJD~XmkOF-J]T/nBZDmRb2Y8JCHxBP8TUJY8mJ;d6J\@ ² oK ² `ª[\XZD2RUT/JCP8HKYOF®Z_DO_T&BEHKJ[\JD2RNb2Y8JCHe BEPOP8H^QxgBER/QKXZDv 7→ v− µJ ′(v)J`F@R=F-R/TUQxgR/J][\JD2Rg$XZD2RUTUBmgRUBZD2R/JCH^XmTUFUb2Y8J
0 < µ < 2α/C2 qOgEe J]F-R-h snh d6QKT/Jpb2Y8J∃ γ ∈]0, 1[ ,
∥
∥
(
v − µJ ′(v))
−(
w − µJ ′(w))∥
∥ ≤ γ‖v − w‖ .4Y8QKFUb2Y8JHKBPOT/XZ|@J]gRUQ^XmD
PKJ`F@RvBEQKk8HKJ[\JD2Rzg$XZD2RUTUBmgRUBZD2R/Jdfe BEPOT/l`F@6oZ$qNHe BZP8P8HKQKg]BnRUQ^XmD
v 7→ PK
(
v − µJ ′(v)) J`F@RF-R/TUQKg$R/J[\J]DmRg$XmD2R/T&BZgR&BED2R/Jmq2gJWbmYOQ>POT/XmY8ZJCHKBcg$XmDrZJTUZJ]DOg$J;d6JpHKBF/Y8Q^R/J
(un)d6_:OD8QKJpPOBET- ² oK ² m4ZJ]TUF®HxBuF-XmH^Y6RUQ^XmD
ud8J\- ² o^ ² Z$o
?V+2nZmln oDf oD J= ( V = RN CK = x ∈ RN 5 . "! ∑N
i=1 xi = 1 &Q^3_,. ( 1( 5.+] , 4 0 5!3Y63 , 1 ( 7=h 3 0. PKP ( 58 , 4P5 6 0 /: 1 190 : . 0 LI= * ! . [# % & %')58 * -63 * ! . (+,. ( 1C0 5!3 63 0 0 3 &
5EW4On§GF 2,>Z"¦J®TU_]F/Y8H^RUBER P8TU_]g_]d6J]D2R [\XZD2RUT/Jb2Y8JHKB;[\_$R/MOXrd8Jd8J®mTUBmd6QKJD2R s;POBZFK:8a6JBZJ`gIP8TUXE|@J`gRUQ^XmDJ`[email protected]^QxgBZk8HKJNsYOD8JNHxBETUZJNg$HxBZFUF-J®d6JNPOT/Xmk8H^l][\J]F dfe XmP6R/QK[\QKFUBnRUQ^XmD"g$XmDrZJ$a6J4BmJ]gIgXZD2R/T&BEQKD2R/J`Fo«BEQxF'gJ$R-RUJ®gXZDOgH^YGF-QKXZDWJ`F@R HKBZT/mJ[\JD2R'YODH^J]Y8TUT/J®d6YP>XZQKDmR.d6J®rY8J4P8T&BnRUQKb2Y8Jmq`g]BET'He XZP>_T&BnRUJY8Td8JNP8TUXE|@J`gRUQ^XmD
PKD+e J`F@RªPOBmF gXZD8DrYtJ$a6P8HKQKgQSRUJ[\JD2RªJDcZ_DO_T&BEH>¨`HxB;P8T/XZ|@J]g$R/QKXZD"dfe Y8Dc_HK_[\JD2R
v ∈ V F/Y8TY8Dg$XmD2mJ$a6J;vwJTU[c_b2Y8J]HKgXZDOb2Y8Jpd6JVP>JY6RR/TUJCR/TUl]Fd6Q^yg$QKH^Js\d6_R/JTU[\Q^DOJT¦
=D8JJ$a8gJP6RUQ^XmDQK[\PGXmT-R&BED2R/JWg$XmDOg$J]T/DOJZq6JDd6QK[\JDOF/QKXZD:OD8QKJwP>XZY8TV = RM qOHKJ]FF/XZYGF@hJ]DOF/J["kOH^J`F
Kd8JWHKBtvwXZTU[\J
K =
M∏
i=1
[ai, bi]@ ² o^ ² Z
BZJ`g_mJD2R/YOJHKH^J][cJ]D2Rai = −∞ XmY
bi = +∞ P>XZY8TWgJT/RUBEQKDOF;Q^DGd6QKgJ]Fio IDJh0¤J$R`qfQ^HªJ]F-RBZH^XmTUF vBmg$QKH^J;d6JZXmQ^TNbmYOJZqmF/Q
x = (x1, x2, . . . , xM ) ∈ RM qy = PK(x)
BWP>XZYOT4g$Xm[cP>XmFUBED2RUJ]Fyi = min (max (ai, xi), bi)
P>XZY8T1 ≤ i ≤M ,
- ² oK ² mBZY6R/TUJ[\J]DmRzd6Q^R]qQ^HWF/Y6y\Rt|@YGF@RUJd6J+f@R/TUXZDOb2Y8J]T!gH^J`Fzg$XZ[\P>XmFUBED2R/J`Fjd6J
xo=LNJR-R/JP8TUXZP8TUQK_$R/_F/QK[cPOH^JmqO|@XmQ^D2R/J`FtBZY6aT/J][uBET&bmYOJ]F"F/Y8TtHKBd6YOBEHKQ^R/__D8XmDOg$_]Jjd8BEDGF"HxB < J`gRUQ^XmD ² o ± q¦nB~D8XmYOFgXZDOd8Y8Q^TUJ;s"Y8DjDOXZY8mJH¤BEHKZXZTUQ^R/M8[\JZo IDJh0¤J$R]q6[\][cJCF-Q¤HKJ;POT/Xmk8H^l][\J=P8TUQ^[uBZHOvBZQSRQ^D2RUJTUZJDOQ^TYOD¢J]DOF-J]["k8HKJ
Kd8J]F\F-XmH^Y8R/QKXZDOF\BZd8[cQxFUF-QKk8HKJ]F\F-YOTuH^J`bmYOJHHKBP8TUXE|@J`gRUQ^XmD
PKD8JzPGJ]Y6Ru$RUT/Jd8_$R/J]T/[\QKD8_JcJ$a6P8HKQxg$Q^R/J[\J]DmR`q¤H^J\P8TUXZk8HKl[\Jcd6YOBEH.F/JT&BuvwTU_]b2Y8J][c[\J]DmRpP>XmF/_cF/Y8TpY8DJ]DOF/J["kOH^Jd8JHKB;vwXZTU[\JW@ ² o^ ² Zq`R@£rP8QKb2Y8J][\JD2RIF-Y8T
(R+)M oEACBEDOF4g$JgBmFqnHKJP8TUXZk8HKl[\Jd6YOBEHOPGJ]Y6RI$RUT/JTU_]F/XZHKYpPOBET'HKB[c_R/M8X6d6J®d6YmTUBmd6Q^J]D2R¦sPOBmF :8a6J4BmJ]gªP8TUXE|@J`gR/QKXZD¦qJ$R'HKB;F-XmH^Y6RUQ^XmDd6YP8TUXZkOH^l][cJPOT/QK[uBEH.P>XZY8TUTUBJDGF-Y8Q^R/J\R/TUJcXmk6R/J]DrY8J\JD T/_`F-XmH^nBZDmRpY8D P8TUXZk8HKl[\J\d6Ju[\Q^DOQ^[\QxF/BER/QKXZD ¡]8 ¡.4G¤§]n8¤§`GoILNJ]F"T/J][uBET&bmYOJ]FWF/XZD2RtsHxBkOBmF-Jd6JjHe BEHKZXmT/Q^R/M8[\Jdfe 1`B®BOq¦b2Y8Q4J`F@RtJDvBZQSRYOD8JW[\_$R/MOXrd8Jd6JpT/J`g&M8JT&g&M8Jpd6JWPGXmQ^D2R/h F/JHKH^Jmo
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kk'#)(* +,K +%, $, ± R ±LNXmDOF/QKd6_]T/XmDOF®H^JWPOT/Xmk8H^l][\Jpd8JW[cQKD8QK[\QKFUBnRUQ^XmDgXZDrZJarJ
infF (v)≤0
J(v) ,@ ² o^ ² V2
X-JJ]F-RY8D8JpvwXmDOgRUQ^XmD8D8J]H^HKJg$XmD2mJ$a6Jpd6_h:ODOQ^JF/Y8T
VJ$RFY8D8JpvwXmDOgRUQ^XmDg$XZDrmJ$a6JCd8J
VF/Y8T
RM o < XmYOFtH^J`FtMr£rP>XER/MOl]F/J]Fcd6YI;MO_XZTUl[\Jd6J pYOM8DJRC;'YOgZJ]T ² o ± o ROq'HKBT/_`F-XmH^Y8R/QKXZDd6J@ ² o^ ² V2ITUJrQKJD2Rs"RUT/XmY8ZJ]TY8DzPGXmQ^D2R-h F/JHKH^J(u, p)
d8Yz'BEmTUBZD8ZQKJDL(v, q) = J(v) + q · F (v) ,
@ ² oKZ ² F-Y8T
V × (R+)M o POBET/R/QKTd6JWHxB\A=_:OD8Q^R/QKXZD ² o ± o^pd6YzPGXmQ^D2R/h F/JHKH^J∀ q ∈ (R+)M L(u, q) ≤ L(u, p) ≤ L(v, p) ∀ v ∈ V ,
@ ² o^mZXZDd8_]d6Y8Q^Rb2Y8J
(p − q) · F (u) ≥ 0PGXmY8TCR/XZY8R
q ∈ (R+)M qfdfe X-XZD~RUQ^TUJZq¤P>XZY8TCRUXZY6RpT/_]JHµ > 0
q(p− q) ·
(
p−(
p+ µF (u)))
≤ 0 ∀ q ∈ (R+)M ,g$JWb2Y8QqGdfe BZP8TUl]FW@6oK`$q6[cXmD2R/TUJCb2Y8Jp = PRM
+(p+ µF (u)) ∀µ > 0 ,
- ² oKZZPRM
+
d8_]F/Q^mDOBED2RHKBcP8TUXE|@J`gRUQ^XmDd6JRM F/Y8T
(R+)M o=Y 2Y d6J\g$JR-R/JuPOT/XmP8T/QK_$RU_"JRWd8JcHxBzF-J`g$XmDOd6JcQ^D8_]mBZH^Q^R/_\d8BEDGFc- ² o^mZ`$q¤D8XZYOFCP>XZY8mXZDOFQ^D2R/TUX6d6Y8QKT/Jt!>^O5EW4Onw§OF'"2)> m t¨fsPGBET/R/QKT;dfe Y8D_]H^_][\JD2RCb2Y8J]HKgXZDOb2Y8J
p0 ∈ (R+)M qXZDzg$XZDGF@RUT/Y8Q^RH^J`FF/Y8QSRUJ]F(un)
JR(pn)
d8_$R/J]T/[\QKD8_J`FPOBETHKJ]FQSRU_T&BnR/QKXZDGFL(un, pn) = inf
v∈VL(v, pn) ,
pn+1 = PRM+
(pn + µF (un)) ,
@ ² oKZ ±
µ_RUBED2RY8DzPOBZTUBZ[clR/TUJ;P>XmF/Q^R/Q^vK:8a6_mo
L*F' 4GH2 ln$of o prx y*wxey J yuα ~ reycy st hcyreut5|z@y wxey F yhu)~ reycy
yhu t ~tvut5yhrYrWy sy V s.|reRM ~Y yu s tvcywx tvzyQ.tu y*xrWy3~ r6u | reuy C uyhzvzy¡wxey
‖F (v)− F (w)‖ ≤ C‖v − w‖ ∀ v, w ∈ V ,- ² o^mR2
yhu wx tvzyQtuyxr tvreu yhzvzy (u, p)sx |c| rkt5yhrI!$&% $($&%x V × (R+)M z t
0 < µ < 2α/C2 zc |z tvu8 y s G|)|~ reyh.y wxeyhzKwxey* tvu,zc hz@ yhrYutvrYtvuZt5| z p0 z@|xtvuy(un)
sZrYt5y | !$(% $&$ ~ reyhZky3¢y z@| zxuZt r u sx hz h y $(% $(% i r4>'¡]§n6§`54>)o*/=BEP8P>JHKXZDOFjdfe BZkGXmTUdb2Y8J~He J$a6QKF-R/J]DOg$Jdfe YOD8JF/XZHKY6R/QKXZD
ud8J@ ² oK ² V2d6_]gXZY8HKJ;d6JCg$J]H^HKJ;d6YjPGXmQ^D2R-h F/JHKH^J
(u, p)ZXmQ^TNHKBIT/XmPGX2F-Q^R/QKXZD ² o ± oZ$qmBEHKXZT&FIb2Y8JCF-XmDuY8D8Qxg$Q^R/_J]F-RY8D8Jug$XmDOF-_`b2Y8JDOgJud6JuHe
αh gXZDrZJarQ^R/_\d6J
Jo¦A;Ju[\[\JZq
pn _RUBED2R :8a6_mqfH^JuP8TUXZkOH^l][cJud6J
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± RR ! "#$%&'#)(* +,
[\QKD8QK[cQxFUBnR/QKXZDd8BZDOF- ² oKZ ± NB\k8QKJDzY8D8JF/XZHKY6R/QKXZDzY8DOQKb2Y8Jun oOAte BZP8T/l`FHe .a6J]TUgQKgJu ² o6o 8qHKJ]FQKD8_`bmYGBnR/QKXZDGFdfe 4Y8H^J]T=FUBnR/QxF-vBEQ^R/J]FPGBET
uJ$Run F]e _]g$TUQKZJD2R
〈J ′(u), v − u〉+ p ·(
F (v) − F (u))
≥ 0 ∀ v ∈ V ,- ² oKZ¢UZ
〈J ′(un), v − un〉+ pn ·(
F (v)− F (un))
≥ 0 ∀ v ∈ V .- ² oKZ m
4T/J]DOBED2RNF/YOggJ]FUF-QKZJ][\JD2Rv = un d8BZDOF=- ² o^mOUZ JR
v = ud8BZDOF=- ² o^mm'JR®BZdOd6QSRUQ^XmD8DOBZDmR`qXmDXmk6R/QKJD2R
〈J ′(u)− J ′(un), un − u〉+ (p− pn) ·(
F (un)− F (u))
≥ 0 ,d+e X.-J]DY8R/QKH^QxF/BZD2RHeαh g$XmDrZJ$a6Q^R/_pd6J
JJRJDzPGX2F/BZD2R
rn = pn − p
rn ·(
F (un)− F (u))
≤ −α‖un − u‖2 . @ ² o^m ZAte BEY6RUT/JPOBZT-R`qnHKBCP8TUXE|@J`gRUQ^XmD
PRM+
_$R&BED2R.vBEQKk8HKJ[\JD2R4gXZD2R/T&BZg$RUBED2RUJ®dfe BEPOT/l`F@`8o mqnJ]D\F/XZYGF@hRUTUB£2BED2RW@ ² o^m`mIs@ ² oKZ ± .XmDzXZk6RUQ^J]DmR‖rn+1‖ ≤ ‖rn + µ(F (un)− F (u))‖ ,F/XZQ^R
‖rn+1‖2 ≤ ‖rn‖2 + 2µrn ·(
F (un)− F (u))
+ µ2‖F (un)− F (u)‖2 .R/QKH^QxFUBED2R"@ ² oKZRr.JR"@ ² o^m ZqrQ^H¦2QKJD2R
‖rn+1‖2 ≤ ‖rn‖2 +(
C2µ2 − 2µα)
‖un − u‖2 .< Q
0 < µ < 2α/C2 q6XmDP>JY8RR/TUXZY8mJT β > 0R/J]H¦b2Y8J
C2µ2 − 2µα < −β qOdfe X-β‖un − u‖2 ≤ ‖rn‖2 − ‖rn+1‖2 . @ ² o^m92
LNJ`g$Q6[\XZD2RUT/JBEHKXZT&F bmYOJ®HxBpF-YOQSRUJ ‖rn‖2 J]F-R4d6_`g$TUXZQxF/FUBED2R/J;¨nH^J[\J]["k8TUJ®d8Jd6T/XmQSRUJ®d8J@ ² o^m92RUJDOdzd6XmDOgCZJ]TUF0q8g$JWb2Y8Q+JD2RUTUB+*xD8JWb2Y8J
un R/J]DOdzZJ]TUFuo
=Q^DOF/Qq¦He BZH^mXZTUQSRUM8[\J\d+e 1]BBP>JTU[\J$R"dfe BEPOP8T/X6g&M8J]TWHKBzF/XZHKY6R/QKXZDd6J~- ² o^ ² Vm=JDTUJ[chPOHKB.PBED2RjgJP8T/Xmk8HKl[\J~BZJ`ggXZD2R/T&BEQKD2R/J`F\POBETYOD8JF/Y8QSRUJd6J~POT/Xmk8H^l][\J]Fud6J~[\QKD8QK[cQxFUBnR/QKXZDFUBEDGF4g$XmD2R/T&BEQKDmRUJ]F;@ ² o^m ± o g&MOBZb2Y8J;QSRU_T&BnRUQ^XmD+qZHxB"d6_R/J]T/[\QKDOBnRUQ^XmDjd6Jpn J]F-RN_]H^_][\JD2RUBZQ^TUJZqPOY8QKFUb2Y+e XZDzBcF/QK[cPOH^J][cJ]D2R
pn+1i = max (pn
i + µFi(un), 0)
PGXmY8T1 ≤ i ≤M ,d+e BZP8T/l`F@ ² oK ² mo H+vBEY8RCBEYGF/F/Q+D8XZR/JTCb2Y8JHKJ3;M8_XmT/l][\Ju ² oVU6o V\D8J"d8QSRCT/QKJD~d6J"HKBgXZDrZJ]T-hmJDOgJd6JHxBWF-YOQSRUJ
(pn)o 4DuvBEQ^R]q2g$J$R/R/J=gXZDrZJ]T/mJDOgJND¦e J`F@R4PGBZFIBmF/F/Y8TU_JF-XmYOF.HKJ]F4M2£rP>XERUM8l]F/J]Fd8YRUM8_XmT/l][\JZq+b2Y8QID+e BZFUF/Y8T/J]D2RWd+e BZQ^HKH^J]Y8T&FpPOBZFpHe YOD8QKgQSRU_jd6JuHe _]H^_][\JD2R
p ∈ (R+)M RUJH4b2Y8J(u, p)
F-XmQSRP>XZQKD2R-h F-J]H^HKJuwmXZQKT®HxBC/J[uBZTUb2Y8J\ ² o ± o^WJRHe .a6JT&g$Qxg$J\ ² o U8o ± g$Q^h d6J]FUF-XmYOF&o
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kk'#)(* +,K +%, $, ± RU HGT/J`F@RUJsCvBEQKT/JHKJHKQ^J]DuJD2R/TUJHe BEHKZXmT/Q^R/MO[cJdfe 1`B®BCJR4HKBCRUM8_XmT/QKJd6JHxBd6YOBZH^Q^R/_mqZg$Xm[\[cJD8XZYOF He BmXZDOF d6_-|-sCBZD8D8XmDOg$_mo¢/=BEP8P>JHKXZDOFªdfe BEk>XZT&d"b2Y8J®HKJ®POT/Xmk8H^l][\J®d6YGBEH8d6JW@ ² oK ² VmfF]e _`g$TUQSR
supq≥0G(q) , @ ² o^m]V2
X-+q6POBZTd6_h:GD8QSRUQ^XmDG(q) = inf
v∈VL(v, q) ,
@ ² o^ ² J$R"b2Y8JuHKJu["Y8H^R/QKP8HKQKg]BnR/J]Y8T"d6J'BEmTUBZD8ZJ
pJ]F-RY8D8JjF/XZHKY6R/QKXZD d6Y P8TUXZk8HKl[\Jd6YGBEH=- ² oKZ`Vm$o
IDvBZQSR`qrF/XZYGFNd6J`F4Mr£rP>XER/MOl]F/J]F®BZFUF-JG1=Z_]D8_T&BEHKJ]F]qEXmDjP>JY6R[\XZD2R/TUJT®bmYOJ G J`F@Rd6QS0¤_TUJD2RUQKBZk8H^JJ$RCb2Y8J"H^JtZT&BZd6QKJD2R G′(q) J`F@R=POT/_`g$QxF-_][cJ]D2R=_2BEH's F (uq)q>X-
uqJ`F@RCHe Y8D8Qxb2Y8JtF/XZHKY6R/QKXZD~d8YP8T/Xmk8HKl[\JWd6JW[\Q^D8QK[\QKFUBnRUQ^XmD- ² o^ ² $o= IDzJh0¤J$R]q8vwXZTU[cJ]H^HKJ[\J]DmRXmDB
G(q) = J(uq) + q · F (uq),J$RJDzd6_]T/QKnBED2RPOBETT&BEP8P>XZT/Rsq
G′(q) = F (uq) + (J ′(uq) + q · F ′(uq))u′q = F (uq),s gBZYOF-Jd8JHKB g$XZDGd6QSRUQ^XmD dfe XmP6R/QK[uBEHKQSRU_P>XZY8T
uqo9;D ZXmQSR¢BZH^XmTUFb2Y8J!>^O5EW4Onw§OF'
2)> m ")>K2¡]§W8'§n=f jz 2§GF4 2' 2 EGn62'r¤§ )B 8¡*je+ >2B'4 '`2Z§]Z4GBKBªZ=f' rO+B'4Y7 ZHr 2' 86P8YOQKFUb2Y8J4HxBd6J]Y6a6Q^l][cJ4_]b2YOBnRUQ^XmDd6J=- ² oKZ ± >P>JY6RªFe _]gT/QKTUJpn+1 = PRM
+
(
pn + µG′(pn)) wHKJ"g&MOBZD8ZJ][\JD2R=d6J"F/Q^mD8JWPOBZTTUBZP8P>XZT/R=s~@ ² oK ² 2N2QKJD2R;d8YvBEQ^R;b2Y8JHKJP8TUXZk8HKl[\Jd6YOBZHN- ² oKZ`VmNJ]F-R=Y8DPOT/Xmk8H^l][\J"d6J[uBna6QK[\QKFUBnRUQ^XmDJR=D8XmD~d6J[\Q^DOQSh[cQxFUBnR/QKXZD>o'¦JuH^J`gRUJY8TtZ_TUQ@:OJT&BRUT/l`FWvBmg$QKH^J][cJ]D2Rtg$JR-R/JBZFUF-J]T-RUQ^XmDd8BZDOFWHKJjgBmFWPOBZT-RUQKgY8H^QKJT_$R/YGd6Q^_scHe Ja6JT&g$Qxg$JWF-YOQ^nBED2R`o
?+2nZZln oDf oT ,,3. ( "!/8 .+] 0. 3= ( * -6 ] 0 070 ! , A3. * 63 . 0 *V0 D!/ % & & `#ILI 1 ( "3/ . . -"! 0 6 0 ( !/)P 1 0( 5C: 0 [/C:@876 (+* 8/: ( ? ( 'C& J (S. 02*V0 ( 1 B C:O6 0 M < 1 "! (0 :;:P! .+] ! ( 1( 54b63 p 6 ] 0, 9: . 0 *V0 "!/ % & & ` < * "98S!/ . 0 :C! ( 5pn 1 > 8 > : p &
= r'O¡`6§`54>I2 r¡\4>¤§n8¤§`r¡=XZYOFCg$XmDOg$HKY8XmDOFg$JR-RUJ"F-XmYOF-h F/J]g$R/QKXZDzJDd8_]g$TUQKBZD2Rk8T/QKlmJ[\JD2R=YOD~BEY6RUT/J[\Xn£mJDdfe BEP6hP8T/X6g&M8J]TNYODP8TUXZkOH^l][cJCd6Jp[\Q^D8QK[\QKFUBnRUQ^XmDBmJ]g=gXZD2R/T&BEQKD2R/J`FIPOBZT®YOD8JWF-Y8Q^R/Jpd6JCP8TUXZk8HKl[\J]Fd6J[cQKD8QK[\QKFUBnRUQ^XmDF/BZDOF"gXZD2R/T&BEQKD2R/J]F$A+gEe J]F-R"HxBP8TUXrg_]d6YOT/Jd6JB4 r'O¡`6§`54>¯d6J]Fcg$XmDmRUTUBZQ^D2RUJ]F]oXZYGFu_rQ^R/XZDGFd6J~POBZT/HKJTuQxg$Q;d6Jf-[c_R/M8X6d6JhgzXmYf-dfe BEHKZXmT/Q^R/MO[cJhgzg]BETuHxB PG_]DOBEHKQKFUBnRUQ^XmD¯d6J]Fg$XZD2RUTUBZQ^D2R/J`FD+e J]F-RtPGBZFtY8D8Jj[\_R/M8X6d6Jzs~P8TUXZPOT/J][cJ]D2R"POBETUHKJT`o'¦BT/_`F-XmH^Y8R/QKXZDJh0¤J]g$R/QKZJd6J]FP8T/Xmk8HKl[\J]FcF/BZDOFcg$XZD2RUTUBZQ^D2R/J`F"b2Y8JD8XZYGFtBZH^HKXZDOFcg$XmDOF-R/TUY8Q^TUJjd6XmQSR\$RUT/JT/_`BEHKQKF/_Js~He BZQKd8Jd6JHe Y8D~d6J`F=BEHKZXmT/Q^R/MO[cJ`Fd6JWHxB < XZYGF@h F-J`gRUQ^XmD ² oVU6o6oGLNJ$R/R/JTU_]F/XZHKY6RUQ^XmDP>JY8R;dfe BEQKH^HKJY8T&FF/XZY8HKJ$hZJTCd6J]FCd6Q^yg$Y8H^R/_`FqfgBZT=HKJ"P8TUXZkOH^l][cJ f@P>_DGBEHKQKF/_gj@ ² oK`mZJ`F@RpF-XmY8ZJ]DmRf@[uBEHªg$XmDOd6Q^R/QKXZD8DO_gwZXmQ^THxB < XZYGF@h F-J`gRUQ^XmD ± o^mo mo
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± R ! "#$%&'#)(* +,
=XZYOFD8XmYOF®P8HxBP$XmDOFNPGXmY8TF-QK[\P8H^Q@:OJ]TdOBEDOF®HKJpg]BZFNX-V = RN qrJ$RD8XZYGF®gXZDOF/Qxd6_TUXZDOF®d6JDOXZY8mJ]BEYH^JWPOT/Xmk8H^l][\Jpd8JW[cQKD8QK[\QKFUBnRUQ^XmDgXZDrZJarJ
infF (v)≤0
J(v) ,- ² o^6`
X.-JJ`F@R\Y8D8JvwXmDOgRUQ^XmD°gXZDrZJarJg$XmDmRUQ^DrY8Jzd6J
RN d8BEDGFRJ$RFY8D8JvwXmDOgRUQ^XmD°gXZDrZJarJgXZD2R/QKDrY8JWd6J
RN d8BZDOFRM o
ªXZYOTε > 0
q6D8XZYGFQ^D2R/TUX6d6Y8QxF-XmDOFBEHKXZT&F®H^JpP8TUXZkOH^l][cJWFUBEDOFgXZD2R/T&BEQKD2R/J`Finf
v∈RN
(
J(v) +1
ε
M∑
i=1
[max (Fi(v), 0)]2
)
,@ ² o^Zm
dOBEDOF HKJ]b2Y8JH6XmD2R.d8QSRIb2Y8JH^J`F.g$XmDmRUTUBZQ^D2RUJ]FFi(v) ≤ 0
F-XmD2R f-P>_DOBZH^QxF-_]J]F!g&o9;DtPGJ]Y6R4BEHKXZT&F _D8XmD6hgJT;H^JTU_]F/Y8H^RUBER=F-Y8QKnBED2R]q>b2Y8Q'[\XmDmRUT/Jb2Y8JmqOP>XZY8TεP>J$RUQSR`qGHKJ"POT/Xmk8H^l][\J@ ² o^Zm f-BZP8P8TUXrg&MOJkOQ^J]DgCHKJWP8TUXZk8HKl[\Ju@ ² oK`6o
p4YB)4>¡]§]54>ln$of o5ln prx 3 hy wxey J yhu~ reutvrexey uZt5~u y yhrYu~ rY¢yQy yutvrG)ret5y z tvrOrYt wxeyz@y rW~uZt r6 Fi
rYu ~ reycyhyhu3~ rYuZtvrYxey x 1 ≤ i ≤ M yhu*wxeyzc yhrehy hz@yK =
v ∈ RN , Fi(v) ≤ 0 ∀ i ∈ 1, . . . ,M
yhur rOt5sy rr u|reuuz xrYt5whxey zxut rIs.y+!$&% $ $Myu x ε > 0 uε
z xret5wxey zxuZt rMs.y!$&% $ rM||z
limε→0
uε = u .
i r4>'¡]§n6§`54>)oWIe JDGF-J]["k8HKJK
_$R&BED2Rjg$XmD2mJ$a6JvwJTU[\_Zq4He J$a6QKF-R/J]DOg$JJRjHe Y8DOQKgQSRU_d6Jud8_]g$XmY8HKJD2Rjd6Y&;M8_XmT/l][\JzVOo^mo ± JRd6JHxBF@RUT/QxgRUJ~g$XZDrmJ$a6QSRU_d6J
Jo4A=JP8H^YGFqIHxBvwXZDGgR/QKXZD
G(v) =∑M
i=1 [max (Fi(v), 0)]2J]F-RªgXZD2R/QKDrY8JNJRªgXZDrZJa6J4P8Y8QxFUbmYOJNHxBvwXmDOgRUQ^XmDtd6J
Rd8BZDOF
Rb2Y8QGsxBZFUF-X6g$QK_
max (x, 0)2J`F@R4gXZDrZJa6JNJRNg$TUXZQxF/FUBED2RUJZo9;DtJ]Dud6_]d6YOQSRNb2Y8JHxBCvwXZDOg$R/QKXZD8DOJHKH^J
Jε(v) = J(v) + ε−1G(v)J]F-RF-R/TUQxgR/J][\JD2RgXZDrZJa6JZqmgXZD2R/QKDrY8JZqrJR®QKD:OD8QKJpsHe QKD:OD8QfPOY8QKFUb2Y8J
G(v) ≥ 0q+gJubmYOQIQ^[\P8HKQxbmYOJuHe J$a6QxF@RUJDOgJcJRHe Y8D8Qxg$Q^R/_d8J
uεoªLNXZ[\[\J
G(u) = 0qfXmDP>JY8R_`g$TUQ^TUJ
Jε(uε) = J(uε) +G(uε)
ε≤ Jε(u) = J(u) .
- ² oK` ± LNJ`g$Qf[\XZD2RUT/JWb2Y8J
J(uε) ≤ Jε(uε) ≤ J(u) ,- ² o^R2
JRNd6XmDOgb2Y8Juε
J`F@RNkGXmT/D8_d+e BZP8T/l`F.HxBWg$XZDGd6QSRUQ^XmD3f@QKD:ODOQ^JsWHe Q^D:GD8Qgo9;D\P>JY6Rd6XZDGgJ$arR/T&BEQKT/Jd8JHxBvBE[\QKH^HKJ(uε)
Y8D8JF/Y8QSRUJ(uεk
)b2Y8Q=gXZDrZJ]T/mJjZJ]TUFcY8DOJHKQK[cQ^R/J
u∗HKXZT&F/b2Y8J
εkRUJDOdmJT&F
0o 9;DBBEHKXZT&F
0 ≤ G(uεk) ≤ εk(J(u) − J(uεk
))dfe BEP8TUl]F@ ² oK` ± o .BmF/FUBED2RtsHKBHKQK[cQ^R/JmqOXmDXmk6R/QKJD2R
G(u∗) = 0q8b2Y8Q+[\XZD2RUT/Jb2Y8J
u∗ ∈ KoGLNXZ[\[\J@ ² o^Rr4QK[\P8HKQKb2Y8Jb2Y8J
J(u∗) ≤ J(u)q+XZD BzBEHKXZT&F
u∗ = uq+gJubmYOQ4g$XmDOg$HKY6RWHKBzd6_][\XZDOF-R/T&BnRUQ^XmD+q¤R/XmY6R/J`FpH^J`FWF-Y8Q^R/J`FJarR/T&BEQ^R/J]F
(uεk)g$XZDrmJTUZJ]BZD2RNZJ]TUF®HxBc[c][\JWH^QK[\QSRUJ
uo
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kk'#)(* +,K +%, $, ± R ?+2nZZln oDf oZ ^ ,. !:@63C: N , 9:OC:@63 . 0gW , : ( ( % & & % <3-:P! ,, :OB"! . C:LI 1 ( /: J P
F1, . . . , FM:; 1 ( * 9)6 ( 49 ( 0A. C: & R K5g63 "! > 0 ! I(u).+] :; *-A. )69: 1 0( 5C: 0\1 ( > C:@ u <=CQ"-:P! ,3, 3:OB!/ . C: 1 0( 5C:M:O"9Q! 0. ( ? 4CC:8
u 0 !a:;8/: 63 . 0 4 ? ( ( " %"& `"& &X^ ? <c7:P! ,3, 3:Oi!/ . 9: > 1 58!: (F ′i (u)
)
i∈I(u):O . ( 4 0( * 8 ( /634 , /6 0 5:9< 1 g"! ( 0 :;:P! .+] ! ( 1/( 54g69: * ! . (+,. ( 1C0 58!:-6 0 0 3λ1, . . . , λM
5 . :V!/ J ′(u) +∑M
i=1 λiF′i (u) = 0
< 0 > 1 λi = 0: ( i /∈ I(u) & 8 0. =:!/3< , "!3 5"!9 ( /6 ( 1 i ∈ 1, . . . ,M
limε→0
[
2
εmax (Fi(uε), 0)
]
= λi .
jr6=f ln$of o5lWl¢=XZYOFjZJTUTUXZDOFusHKB < XZYOF-h F/J]g$R/QKXZD Zmo 8o ± YOD8JBZY6R/TUJ~[\_$RUM8X6d6Jd6JPG_]DOBEHKQKFUBnRUQ^XmDPGBETQKDmRUT/X6d6YGgR/QKXZDzd6JCvwXZDGgR/QKXZDGF)f@kGBETUT/QKlTUJ]F!g&o •
)+*,, ^+3 =79 _ 9 __367419;DF/JpP8HKBmg$JpJDd6QK[\JDOF/QKXZD:OD8QKJ
V = RN o .a6P8HKQKb2Y8XmDOFHKJWP8T/QKDOgQ^P>Jd6JWHxBt[\_R/M8X6d6Jd6JJ RUXZD+o < XmQSRFY8DOJ;vwXmDOgRUQ^XmDd6Jpg$HxBZFUF/J
C2 d6J RN dOBEDOFRN o < XZQ^R u Y8D1_TUX"TU_mY8HKQ^J]T®d6J
FgZe J`F@R/h sEh d6QKTUJ;b2Y8J
F (u) = 0JR
F ′(u)[uBER/TUQKgJCQKDrZJ]TUF/Q^kOH^Jmo
D8JpvwXmT/[tY8H^JWd6J*; B£rH^XmT®BZYzZXZQxF/Q^DOBZZJCd6JvD8XmYOFd6XZDOD8J
F (u) = F (v) + F ′(v)(u− v) +O(
‖u− v‖2)
,gEe J]F-R-h snh d6Q^TUJu = v − (F ′(v))
−1F (v) +O
(
‖v − u‖2)
.¦B[\_R/M8X6d6Jpd6J;=J RUXZDgXZDOF/QxF@RUJ;sTU_]F/XZYGd6T/J;d6J=vB.P$XmDQSRU_T&BnRUQ^mJ=g$JR-R/JC_]b2YOBER/QKXZDJ]DjD8_]ZHKQShZJ]BZD2RHKJpT/J`F@RUJZoªXZYOTY8Dg&M8XZQ^ajQKD8QSRUQKBZHu0 ∈ RN q8XZDzgBZHKgY8HKJ
un+1 = un − (F ′(un))−1F (un)
P>XZY8Tn ≥ 0.
@ ² o^Um/BEP8P>JHKXZDGF;b2Y8JtHe XZDD8Jtg]BEHxg$Y8HKJ"POBmF=He Q^DrZJ]TUF/Jtd8J"HxB[uBnRUT/Qxg$J
F ′(un)d8BZDOFt- ² o^UZ®[uBZQKFbmYOJHe XZDjT/_`F-XmY6RIY8DjF/£6F@RUl[\JHKQ^D8_`BEQKT/J=POBETIHe Y8D8J;d8J]FN[c_R/M8X6d6J`FIJarP>XmF/_J`F4sWHxB < J`gRUQ^XmD ± oKZoA=Y~P>XZQKD2R;d6JrY8Jd8J"He XmP6R/QK[\QKFUBnRUQ^XmD+q8HxBu[\_$R/MOXrd8J"d6J"=J R/XmD~Fe QKDmRUJTUP8TUl$R/Jd8JHKB[\BZD8QKlTUJF-Y8QKnBED2R/Jmo < XZQ^R
JY8DOJ=vwXmDOgRUQ^XmDjd6Jpg$HxBZFUF-J
C3 d6J RN d8BEDGFRqrJ$RF/XZQ^R
uY8Dj[\QKD8Q^[tY8[ HKX6gBEHd6J
Jo < QXZDPGX2F-J
F = J ′ qªXZDP>JY6RBZP8P8HKQKb2Y8J]TtHxB~[\_$RUM8X6d6JP8T/_`g$_`d6JD2R/JPGXmY8T\T/_`F-XmYOd6TUJHKB\g$XmDOd6Q^R/QKXZDD8_]gJ]FUF/BZQ^TUJCd+e XmP6R/QK[uBEHKQSRU_J ′(u) = 0
oGLNJP>JDOd8BZD2R]q6XZDPGJ]Y6R=BEYGF/F/Q¤JDrrQKFUBEmJT®HKB[c_R/M8X6d6Jd6J=J R/XmDtgXZ[\[\J®YOD8JN[\_R/M8X6d6Jd6J[\Q^DOQ^[\QxF/BER/QKXZD+o gBZYOF/JNd6Y\d6_mJHKXZP8P>J[\J]DmRd6J*;'B£rHKXZTJ(w) = J(v) + J ′(v) · (w − v) +
1
2J ′′(v)(w − v) · (w − v) +O
(
‖w − v‖3)
,@ ² oK` 2
XZDP>JY6RBZP8P8TUXrg&MOJTJ(w)
BZYjZXmQKF/QKDOBEmJd6JvPOBZTNYOD8J=vwXmDOgRUQ^XmDjb2YOBZd8TUBER/QxbmYOJZom'B"[\_R/M8X6d6Jd6J=J RUXZD¢gXZDOF/QxF@RUJBEHKXZT&F"s[\Q^DOQ^[\QxF-J]TcgJ$R-RUJBEP8P8TUXa6Q^[uBER/QKXZDb2YOBZd6T&BnRUQKb2Y8JJ$RusQ^R/_]T/J]T]o
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± R3 ! "#$%&'#)(* +,
¦Jt[\QKD8QK["Y8[©d6JcHxBPOBZT-RUQ^J\b2YOBZd6T&BnRUQKb2Y8Jtd6Y~RUJTU[cJ\d6J\d6TUXZQ^R/Jtd8J- ² oK` mJ`F@RWd6XmD8D8_cPOBETw = v − (J ′′(v))−1
J ′(v)F/Q+HxB\[\BER/TUQKgJ
J ′′(v)J`F@R=d6_:OD8QKJP>XmF/QSRUQ^mJZoY9;DzT/JR/TUXZY8mJWBEHKXZT&FNHxBvwXmT/[tY8HKJCQ^R/_]TUBER/QKZJu@ ² oK`.UZo4e BnBED2R&BEZJP8TUQKDOg$QKPOBEHGd6JHKBp[\_R/M8X6d6J=d8J=J R/XZD\J`F@RNFUBpg$XmD2mJTUZJ]DOg$J®k8QKJDuP8HKYOFIT&BEPOQKd6Jb2Y8JpHKJ]F[\_R/M8X6d6J`FP8T/_`g$_`d6JD2R/J`Fo
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xry rW~ut r sy"~z@|QyC2 s.y
RN s|reRN yu u xr
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t y F (u) = 0yhuF ′(u)
tvreyht5z@y z yc.tuy3xr cyhz ε > 0uyhzwhxey t
u0 yu|hy ~Qeys.y u | xMhyr6 ‖u − u0‖ ≤ ε z@| hu8 sysy3y u r'sZrYt5y |!$&% $ ~ rY¢yZky ~e yhu stvcy wxeyz@|xtvuy (un)
~ reyh.yyhu yutvzyQtuyxrWy~ r6u | reuy
C > 0uyzvz@ywxey
‖un+1 − u‖ ≤ C‖un − u‖2 . @ ² o^ Zi r4>'¡]§n6§`54>)o ªBZTNgXZD2R/QKDrY8QSRU_d6J
F ′ Q^HGJ$a6QxF@RUJ ε > 0R/JH>b2Y8J
F ′ J]F-R4QKD2mJT&F-QKk8HKJJDuRUXZY6RP>XZQKD2Rd6JpHKBtkGXmY8HKJpd8JpgJD2R/TUJuJRd6JpTUB£mXZD
εo < Y8P8P>XmF/XZDGFbmYOJ
un F/XZQ^R®TUJ]F-R/_CP8TUX6g&M8JCd8JuqBZYF-J]DOFX- ‖u− un‖ ≤ ε
qGd6XZDOgF ′(un)
J`F@R=QKDrZJ]TUF/Q^kOH^JmoGLNXZ[\[\JF (u) = 0
qOXZDd6_]d8Y8QSRCd6J- ² o^UZun+1 − u = un − u− (F ′(un))
−1(F (un)− F (u))b2Y8Qq8POBZTd6_mJHKXZP8P>J[\JD2R=d6J ; B£rH^XmTBZY6R/XmY8Td6J
un qOd6JrQKJD2Run+1 − u = (F ′(un))
−1O(
‖un − u‖2)
.LNXm[\[cJ ‖u − un‖ ≤ εq¤XZDJDd6_`d6Y8Q^Rpb2Y+e QKH Ja6QKF-R/JtY8D8J\g$XZDGF@R&BED2R/J
C > 0wQKDOd6_]PGJ]DOd8BED2RUJd8J
nJ$RWHKQ^_]JBEY[\Xrd8Y8H^Jud6JgXZD2R/QKDrY8Q^R/_ud6J
F ′ J$R"d8J F ′′ F/Y8TWHxBkGXmY8H^Jud6JugJD2R/TUJ u J$R"d6JT&B£ZXmDεNR/J]H^HKJb2Y8J
‖un+1 − u‖ ≤ C‖un − u‖2. @ ² o^82< QεJ`F@RªF-Y8yuFUBE[\[\JD2RªPGJR/Q^RId6JN[uBZD8Q^l]T/J®s=gJ®b2Y8J
Cε ≤ 1q]XmDtd6_`d6Y8Q^Rªd6Jp@ ² oK`mfb2Y8J
un+1TUJ]F-R/Jtd8BEDOFHKBuk>XZYOH^J"d8Jtg$J]D2R/TUJuJ$Rpd6JT&B£ZXmD
εo¤LNJ]HKBPGJ]T/[\J$RCd6JZ_]T/Q@:OJ]TPOBET;T/_`g$Y8TUTUJDOgJHe M2£rP>XERUM8l]F/Jb2Y8J ‖u − un‖ ≤ ε
PGXmY8T"RUXZY6Rn ≥ 0
qªJ$R@ ² oK`82pJ]F-R\k8Q^J]DHxBgXZDOgH^YGF-QKXZDd8_]F/Q^TU_Jmo
r6=f'Mln oDf oZl=T®Q^J]D~F OT]qGQ^H'vBEY6Rpg$XmDOF-J]T/mJTsuHe J]F/P8T/Q^RCb2Y8Jtg&MOBZb2Y8JQ^R/_]TUBER/QKXZD~d6J"HKB[\_R/M8X6d6J"d8J"J R/XZD- ² o^UZND8_`g$J`F/F/QSRUJWHKBT/_`F-XmH^Y6RUQ^XmDdfe YOD~F-£6F-R/l[\JWHKQ^DO_]BEQKTUJZqOgJ"b2Y8Q¦J`F@RgX 6RUJY6afo8A;JpP8H^YGFq8HxBcg$XZDrmJTUZJDGg$J=T&BEPOQKd6J\d8QSRUJf-b2YOBmd6TUBER/Qxb2Y8Jg-4d8XZD8D8_]J;PGBET@ ² oK` E.D¦e BHKQKJYb2Y8JF-QFJ`F@R"d8Jg$HxBZFUF-J
C2 q+J$R"F/Q u0 J]F-RtBZFUF-JG1\P8T/X6g&M8Jud6J u q+Mr£rPGXZR/M8l`F-J`Fpk8QKJDP8HKYOFTUJ]F-R/TUQxgR/QKZJ`FbmYOJg$JHKHKJ]Fb2Y8JWD8XmYOFBrQKXZDOF®Y6RUQ^HKQKF/_J`Fª|@YOFUb2Y+e s\POT/_`F-J]DmR`o 0¤J]g$R/QKZJ[\J]DmR`qO[\[\JdOBEDOFNd6J]F®gBmF.R/TUl]F®F/Q^[\P8HKJ]FNd8BEDOFRq2HKBW[\_R/M8X6d6Jpd6J;=J RUXZDP>JY6Rd6QKZJTUZJ]TIP>XZY8Tg$J]T-R&BEQKD8J]Fd8XZD8D8_]J]FpQKD8Q^R/QxBEHKJ]F
u0 AfQ^HIvBEY6RWD8XZR/JTBZYOFUF-QIb2Y8J\HxBgXZDrZJ]T/mJDOgJtb2YOBmd6TUBER/Qxb2Y8J- ² o^ E=D8JF/JP8T/X6d6YOQSRub2Y+e BZYmXZQxF-QKDOBEmJdfe Y8D1]_TUXTU_mY8H^QKJT`qªg$XZ[\[\JjHKJ[cXmD2R/TUJHe BZP8P8HKQKg]BnRUQ^XmDd6JHKB[\_R/M8X6d6JCd6J=J R/XmDsWHKBWvwXZDOg$R/QKXZDF (x) = ‖x‖2 d8BZDOF RN qZP>XZY8TNHKBmb2Y8JHKH^J=HKBWgXZDrZJ]T/mJDOgJD¦e J`F@Rb2Y8JZ_Xm[\_$R/TUQxbmYOJZod.BETuBEQKH^HKJYOTUF]q.F-QXZDBEPOP8H^Qxb2Y8JHKB[\_$R/MOXrd8Jzd8Jz=J RUXZD°PGXmY8T\HKB[\QKD8QK[cQxFUBnR/QKXZD dfe Y8D8JcvwXZDGgR/QKXZD
Jg$Xm[\[cJcJ$a6P8HKQxbmYO_\gQSh d6J`F/F/YOF]q>QKHIF/JtP>JY6R"b2Y8JcHxB[\_R/M8X6d6J
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J ′′(v)^:_!N2(a"E&!?>%&'@B%
F ′(v)b<6P7 ;',!123 J"G9HF
"F.&'!'Nr&'+ E&JR4 J"=-. =:.?&''23">,C#&' >,%&'@.'Ur"=23A&/u-<9!323%"$A,0
8=a"r=%@A&'"<>?D.'?'>?"E"N2(a"E&'?J ′′(v) = F ′(v)
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n ≥ 0.
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G′(u) 0
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6 `.G? E%D;A"DU, E&R_ E E;A(
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R J"G&!"% E "54 &!",%"G?"=.!H&/"$23?232323%"$A>,>23&!&'23&! a"&'R6 *9&!"EA,&/"G!PRH.FH&' L(v, µ) = J(v) + µ · G(v)
U E 9,%&'@B% Q.QFE"=LvU L′ <" L′′ U-"M@B%<&'u>;:4 %D;-a"&'
(un+1, λn+1) = (un, λn)−(
F ′(un, λn))−1
F (un, λn)
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w∈RN
G(un)+G′(un)·(w−un)=0
Qn(w) ,^Jvw,6'vDx¢b
a@?
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L(un, λn) + L′(un, λn) · (w − un) +1
2L′′(un, λn)(w − un) · (w − un)
)
,
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,",9,_&!"un 6
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(
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x∈Rn"R;
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m ≤ nJ$R;b2Y8JWHKJT&BED8ud6J
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mo IDJh0¤J$R`qGF/Q
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infx∈Rn
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c · x.
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x∈Rn"7;
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BZJ`gλ ≥ 0
o =Q^DGF-Q4-Zmo mIJ`F@R_`bmYOQ^nBEHKJD2Rsinf
(x,λ)∈R(n+m)"R;
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x = x+ − x− BZJ`gx+ = max(0, x)
J$Rx− = −min(0, x)
q XZDXZk6RUQ^J]DmR=b2Y8Ju-ZZoZNJ`F@R_]b2Y8QKnBEHKJD2R=sinf
(x+,x−,λ)∈R(2n+m)"Z;
Ax+−Ax−=b+λ, x+≥0,x−≥0,λ≥0c · (x+ − x−).
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minx1≥0,x2≥0,x3≥02x1+x2+3x3=6
x1 + 4x2 + 2x3 .-Zmo Um
< Y8THxB T'QKZYOT/J Zmo^°D8XmYOFBZXmDOFR/T&BZg$_He J]DOF/J["kOH^Jd6J`F(x1, x2, x3)
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o>LNXZ[\[\JxJ]F-RY8DP>XZQKD2R=d6JW[\Q^DOQ^[tY8[zq8XZDzB\D8_]gJ]FUF/BZQ^TUJ[\J]DmR
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o$9;Dd8_h:OD8Q^R"BEHKXZT&F;Y8D8JcvBE[\QKH^HKJud6JtP>XZQKD2RUFzε = x + εy
POBZTUBZ[c_R/TU_JPGBETεoc IDPOBET/RUBZDmRpd6JcHKBjBZH^J]Y8T
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εXZD~TUJ]F-R/Jcd8BEDGF
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zε ≥ 0J`[email protected]^XmH^_]JZo
Y6RUT/J][cJ]D2R.d8QSR`qzε0 ∈ Xad
PGX2F/F/l]d8J®BZYtP8HKYOF(k−1)
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JRCYOD8J"vBZ[\Q^HKH^Jd6J(k − 1)
gXZHKXZD8DOJ]F(ai)
o vwXmTUgJtd6J\d6QK[cQKDrY8JTCHxBuRUBZQ^HKHKJtd6Jcg$J$R/R/JtvBE[\QKH^HKJZq¤XZDXZk6RUQ^J]DOd6T&B:ODOBEHKJ[\J]DmRY8DOJpvBE[\Q^HKHKJWH^QKk8TUJCJRY8D8JF/XZHKY6R/QKXZDXZP8R/QK[\BZH^JpkOBmF-Qxb2Y8JZo
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A$o POT/l`FPGJ]T/[tY6RUBER/QKXZDzd6JF-J`Fg$XmH^XmD8D8J]F]q6XZDP>JY6R_`g$TUQ^TUJ
A = (B,N)JRx = (xB , xN ),d6JjF/XZT/R/Jjb2Y+e XZDB
Ax = BxB + NxNo;'XZY6RUJjF-XmH^Y8R/QKXZDBZd8[cQxFUF-QKk8HKJuPGJ]Y6R\F]e _]g$TUQKT/J
xB =B−1(b − NxN ) ≥ 0
J$RxN ≥ 0
o.+JzF-Xm[\[cJR\BmF/F/X6g$QK_jsBJ`F@Rud6_h:GD8QWF]e QKHJ$a6QxF@RUJ`"POBET
xN = 0J$RxB = B−1b ≥ 0
o < QXZD¢d8_]g$Xm[\PGX2F-JBEYOFUF-Qc = (cB , cN )
d8BZDOFtgJ$R/R/JzkOBZF/JZq
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BZH^XmTUFXZDP>JY6Rpg$XZ[\POBZT/J]THKJ"g$X 6R;dfe Y8D8JcF-XmH^Y6RUQ^XmD~BZd6[\QxF/F/QKk8H^JtbmYOJHxg$XZDGbmYOJxBZJ`gWg$J]H^Y8Qªd6JHxB\F-XmH^Y8R/QKXZDkOBZF/QKb2Y8J
xoY9;DzZXZQ^RvBZgQ^HKJ[\JD2Rb2Y8J
c · x ≤ c · x F-QfJR=F-J]Y8HKJ[\JD2RF-QcB ·B−1b ≤ cB · B−1(b−NxN ) + cN · xN .
-ZZo m9;DJDd6_`d6Y8Q^RHKB\gXZDOd6Q^R/QKXZDdfe XZP8R/QK[\BZH^Q^R/_WF/Y8QKBZD2R/JZop4YB)4>¡]§]54>lYlWo o Yx r6 wxey z@| zxuZt r |¢t5whxey3|¢ ~t5y B yhu r r s.krW QQy ~Y yhu s tvcywxey B−1b > 0 KrWy ~ rWstvut rr~Qyh|tvcy yux |rYu y xCwhxey ~yuZuy zxuZt rM|¢t5wxey| ~t5Qy B tvu uZtv |z@yCyhuwxey
cN = cN −N∗(B−1)∗cB ≥ 0.@mZoE
y3¢y~u yx cN yhu| yhz@>2Z§]r'2 2¡\.4'§]¡\n 2' w§`¡ i r4>'¡]§n6§`54>)o < XZQ^R
xY8D8JCF-XmH^Y6RUQ^XmDkOBmF-Qxb2Y8JD8XmDd6_m_D8_]T/_]J=BZFUF-X6gQ^_]Js
Bo < Q
cN ≥ 0qBZH^XmTUF®P>XZY8TRUXZY6RUJF-XmH^Y8R/QKXZDzBZd6[\QxF/F/Q^kOH^J
xXZDzB
c · x− c · x = cN · xN ≥ 0 ,
POY8QKFUb2Y8JxN ≥ 0
o4A;XZDOgHxBg$XmDOd6Q^R/QKXZD @mZoEcJ`F@RjF/Y6yF/BZD2R/JP>XZY8Tjb2Y8JxF-XmQSRXmP6R/QK[uBEHo
/_]g$QKP8TUX6bmYOJ[\JD2R]q8F/Y8P8P>XmF/XZDGFb2Y+e QKH+JarQxF-R/JWY8D8JgXZ[\P>XmFUBED2R/Jid8J
cNb2Y8Q¦F/XZQ^R=F@RUT/QxgRUJ[\JD2RDO_mBER/QKZJmq
(cN · ei) < 0oªXZY8T
ε > 0XmDd6_:OD8Q^R"BEHKXZT&F=Y8DZJ`gR/J]Y8T
x(ε)POBZT
xN (ε) = εeiJRxB(ε) = B−1(b−NxN (ε))
o.BETg$XmDOF@RUT/YGgR/QKXZDAx(ε) = b
J$R]q8gXZ[\[\JB−1b > 0
qrPGXmY8Td8J]F;BZH^J]Y8T&F;F/Y6yF/BZ[\[cJ]D2R;P>J$R/Q^R/J`Fpd6JεXZDB
x(ε) ≥ 0q¤d6XZDGg
x(ε) ∈ Xado>Ate BZY6R/TUJ"PGBET/R]q
x(0) = xJ$R`qOg$Xm[\[cJ
ε > 0q6XmDBc · x(ε) = c · x(0) + ε(cN · ei) < c · x,
gJWbmYOQ+[\XZD2R/TUJWbmYOJxD¦e J`F@RPOBmFXZP6RUQ^[uBZHoOA;XZDGg;HxB\g$XmDOd6Q^R/QKXZD@mZoE4J]F-RD8_]gJ]FUF/BZQ^TUJZo
r6=f'MlWlYoD o¬ACBEDGFHKJ gBZd8T/Jd8JHxBF4T/XmPGX2F-Q^R/QKXZD mZo6o 8q®F-QpHxBF/XZHKY6R/QKXZDkOBZF/QKb2Y8JgXZDOF/Qxd6_TU_JNJ]F-R.d8_Z_]D8_TU_JmqHKBCg$XmDOd6Q^R/QKXZDz@Zmo Z+TUJ]F-R/JF/Y6yF/BZDmRUJN[uBZQKFªD+e J]F-R.P8HKYOFªD8_]gJ]FUF/BZQ^TUJZo•
9;Dd6_]d6YOQSRd6JCHKBITUXZP>XmF/QSRUQ^XmDZmo 8o Y8D8JC[c_R/M8X6d6JCP8T&BnRUQKb2Y8JCPGXmY8Td6_]gT/X *KRUT/J;HKBtnBEHKJY8Td8J~HKBvwXmDOgRUQ^XmDg$X 6Rc · x s PGBET/R/QKTjdfe Y8D8JF/XZHKY6R/QKXZDkOBmF-Qxb2Y8J
xD8XZD¯d6_]Z_]D8_TU_JJRjD8XZDXmP6R/QK[uBEHKJ`$o8LNXm[\[cJ
xJ]F-R®D8XmD6h XZP6RUQ^[uBEHKJZq2QKH>Ja6QKF-R/JCY8D8Jpg$Xm[cP>XmFUBED2RUJ=d6YjmJ]g$R/JYOTd8J]F®g$X 8RUFTU_]d8Y8QSR&F
cNR/JHKHKJ=b2Y8J
cN ·ei < 0ok9;Dud6_:OD8Q^RNBEHKXZT&F
x(ε)g$XZ[\[\Jg$Q^h d6J]FUF-YGFo84Y8QKFUb2Y8JHKJ=g$X 6Rd8_]g$TUX+*xRH^QKD8_]BZQ^TUJ[\J]DmRBZJ`g
εq6XZDBcQ^D2R/_]T/RstP8TUJDOd8T/JCHKBcP8HKYOF®ZT&BEDGd6J;nBZH^J]Y8TPGX2F/F/Q^kOH^Jpd6J
εRUJHKH^JtbmYOJWHe XZDTUJ]F-R/Jd8BZDOFXad
o¤L;e J]F-R=HKJP8TUQ^DOgQ^P>Jd6JHe BEHKZXZTUQ^R/M8[\JWd6Y~F/Q^[\P8HKJ$a6J"b2Y8JWD8XmYOFPOT/_`F-J]DmRUXZDOF®[uBEQKD2R/J]DOBED2R]o
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D8Q^R/QxBEHKQKFUBnRUQ^XmD¬wPOMOBZF/J -¨ XmDg&M8J]TUg&M8JuYOD8JkGBZF/JQ^DOQSRUQKBZH^JB0 R/JHKHKJb2Y8JHKB~F/XZHKY6RUQ^XmDkOBmF-Qxb2Y8JWBZFUF-X6g$QK_J
x0 F-XmQSRBZd8[cQxFUF-QKk8HKJx0 =
(
(B0)−1b0
)
≥ 0.
R/_]TUBER/QKXZDOF®wPOMOBZF/J -4¨s;He _$R&BEP>Jk ≥ 0
q]XmDtd6QxF-P>XmF/JNd+e Y8DOJNkOBmF-JBk J$Rªdfe YOD8J®F/XZHKY6RUQ^XmDkOBmF-Qxb2Y8JcBmd6[\QKFUF/Q^k8HKJ
xk oK9;DgBZHKgY8H^JcHKJ\gX 6RWTU_]d6YOQSR ckN = ckN − (Nk)∗(Bk−1)∗ckB
o< QckN ≥ 0
qIBEHKXZT&Fxk J`F@RjXmP6R/QK[uBEHJRjHe BEHKZXmT/Q^R/MO[cJzJ`F@R :ODOQo < Q^D8XmD+q4QKHJarQxF-R/JY8D8JnBETUQKBZk8HKJjM8XmTUF-hkGBZF/Jjdfe Q^DOd8QKgJ
iR/J]H^HKJzbmYOJ
(cN · ei) < 0qªJ$RtXmDD8XZR/J
aiHxBgXZHKXZD8DOJg$XmT/TUJ]F/P>XZDOd8BZD2R/Jpd6J
AoW9;DzP>XmF/J
xk(ε) = (xkB(ε), xk
N (ε))BZJ]g
xkN (ε) = εei , x
kB(ε) = (Bk)−1(b− εai).
< XZQ^R=XmD~PGJ]Y6RCg&MOXZQxF-QKTε > 0
BEYOFUF-Q'ZT&BEDOdzb2Y8JtHe XZDmJY6RpBZJ]gxk(ε) ∈ Xad
o¤ACBEDOFg$JWgBmFq6HKJp[\Q^D8QK["YO[d6YzP8TUXZmTUBZ[\[cJ;H^QKD8_`BEQKT/JpJ]F-R −∞ o < XZQ^RQ^HfJa6QKF-R/JpY8D8JWnBZH^J]Y8T[uBna6Q^[uBZH^J
εkJRY8DQKDOd6Qxg$J
jR/JHxFb2Y8JpHxB
jh l[\JWg$XZ[\P>XEhF/BZD2R/J;d8J
xk(εk)Fe BED8DrY8HKJZoe9;DXmk6R/QKJD2RBEQKDOF/QGY8DOJ;D8XmY8ZJ]H^HKJ=kGBZF/J
Bk+1 d6_]d6YOQSRUJCd6JBk JDTUJ[\P8HxBP]BED2RF/B
jh l[\Jg$XmH^XmD8D8JuPOBZTHxBg$XZHKXZDOD8J
aio''BzD8XZYOZJHKHKJ\F/XZHKY6RUQ^XmDBZd6[\QxF/F/QKk8H^JpkOBmF-Qxb2Y8J
xk+1 BcY8Dg$X 8RQKD6vw_]T/QKJY8TXmY_]mBZH+scgJHKY8Q¦d6J xk o H TUJ]F-R/JtY8DgJT/RUBEQKD~D8Xm["k8TUJ]FCd6JtPGXmQ^D2RUFCP8T&BnRUQKb2Y8J`F=sP8TU_]g$QxF/JT;dOBEDOF;He BEHKZXmT/Q^R/MO[cJtd6YF/Q^[chP8H^Ja6JZo8XmYOFHKJ]FPOBmF/F/XZDGFNT&BEPOQKd6J][\JD2RJDzT/J]rY8JZoi E> 2 2n2¡`Zr Z2§u +m6E>
9;DBpR/XmYn|@XZY8T&Fc ·xk+1 ≤ c ·xk q2[\BZQKF4Q^H>P>JY6R®£\BmXZQKT._2BEHKQSRU_=F-QGHKB"F-XmH^Y6RUQ^XmDBZd6[\QxF/F/QKk8H^JkOBZF/QKb2Y8J
xk J`F@R\d6_m_D8_]T/_]JZqªBEYOb2Y8J]HNg]BZFtXZDP>JY6RcR/TUXZYOZJTtbmYOJεk = 0
F-QJ]H^HKJjD+e J]F-RtPGBZFd6_m_D8_]T/_]JZqfHxBd8_[\XZDOF-R/T&BnRUQ^XmDd6JuHxB$ITUXZP>XmF/QSRUQ^XmD¢mZo6o j2BET&BED2R/Q^RpY8D8JuQKD8_2BEHKQSRU_\F-R/TUQxgR/Jo9;DB\d6XmDOgpg&MOBED8m_Cd8JCkGBZF/JpFUBEDGF®BZ[\_HKQ^XmT/J]T®HKJWg$X 6RW¨OgEe J]F-RH^JpP8M8_]D8XZ[\l]D8JWd6Yg$£6g$HxBEmJCb2Y8QPGJ]Y6R®J][\PG`g&M8JTNHe BEHKZXmT/Q^R/M8[\J=d8J;g$XmDrZJTUZJ]T]o H>J$a6QxF@RUJ;d6J`F4[\Xn£mJDOFId8J;F]e J]DP8TU_["YOD8Q^T`qZ[uBZQKFJDzP8T&BnR/Qxb2Y8JpH^JWg£rgHKBZZJCD+e BEP8POBZTUB+*KRª|-BZ[\BZQKF]o
4D~He BEkGF-J]DOg$J"d8J"g$£6g$HxBEmJZq8He BEHKZXZTUQ^R/M8[\Jd6Y~F/Q^[\P8HKJ$a6JPGBET&g$XZYOT-RY8DF-XmYOF@h JDGF-J]["k8HKJWd6J]FF-Xm[c[\JRUFd6JXad
JDd6QK[\Q^DrYOBZD2Rjd6JuvB.P$XmD F-R/TUQxgR/JCH^Jg$X 8R]o LNXm[c[\JuQKH4£BzY8DD8XZ[tk8TUJ:OD8QGd8J;F/XZ[\[\J$RUF]qEHe BZH^mXZTUQSRUM8[\Jd6XZQ^R4D8_`g$J`F/FUBEQKT/J][\JD2R.R/TUXZY8mJTIYODF-Xm[c[\JR4XZP6RUQ^[uBZHGd6J=gX 6R[cQKD8QK[uBEHoW9;DB\d6XmDOgWd6_][cXmD2R/TU_;HKJWT/_`F-YOHSR&BnRF-YOQ^nBED2R`oD4r lYlWo o5lnWt u xu yh zxuZt r6'|s tt5z@y|t5wxeyh
xk s xtvu yh | z |z tvuvCy+s xtv z@yQky reu r r&s .rWcyh |z zc |z tvu8 y ~ rY¢yZky"yrIxrIr Ccyrets$ hu| y
P8T/QKXZTUQfHKJWD8XZ[tk8T/JWdfe Q^R/_T&BnRUQ^XmDOFd6JWHe BZH^mXZTUQSRUM8[\JWd6YF-QK[\P8H^Ja6JWPGJ]Y6R=$RUT/JBZYOFUF-QfmTUBZDOdbmYOJtHKJtDOXZ["kOT/Jcd6JcF/XZ[\[\J$R&F\b2Y8QªJ]F-RCJa6PGXmD8JD2R/QKJHªPOBZT;T&BEPOPGXmT-RpBEYDOXZ["kOT/Jcd6JtnBZT/QxBEk8HKJ]FnA2ZXmQ^TNHe .a6JT&g$Qxg$J"ZZo6oZ$o3®QKJDjb2Y+e Q^H>J$a6QxF@RUJ;d6J`F4Ja6J[\P8HKJ]FCBZg]BZd6_][\QKb2Y8J`FUªX.-jgEe J]F-RNJ0>J`ghR/QKZJ[\J]DmRWHKJcg]BZF]q¤JDP8TUBER/Qxb2Y8JcgJ$RBEHKZXmT/Q^R/MO[cJcg$XmD2mJTUZJJ]DYODD8Xm["k8TUJ\dfe _RUBZPGJ`FCb2Y8Q.J`F@RY8D8JpvwXZDGgR/QKXZDP>XZHK£2DOXZ[\QKBZH^JWd6J
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ckNF@RUT/QxgRUJ[\JD2R"DO_mBER/QKZJ`FqQKHIvBZY6RvBZQ^TUJ\Y8Dg&M8XZQ^adOBEDOFWHe BEHKZXZTUQ^R/M8[\JZoS4H^YOF/QKJY8T&F"F-R/T&BnRU_ZQKJ]FWF/XZD2RP>XmFUF-QKk8HKJ]F]q+[uBEQxFWJDm_D8_]TUBZH¤XZDzg&M8XZQxF-Q^RHKB\POH^YOFDO_mBER/QKZJmo
'§]O¡`6§`54>LNXm[c[\J]DmRWRUT/XmY8ZJ]TWY8D8JF/XZHKY6R/QKXZD BZd8[cQxFUF-QKk8HKJukOBZF/QKb2Y8J\HKXZT&FWd6JuHe QKD8Q^R/QxBEHKQKFUBnRUQ^XmD 5/BZP6hP>JHKXZDGF4b2Y8JHxBg$XmDOd6Q^R/QKXZDjd+e Bmd6[\QKFUF-QKk8QKH^Q^R/_
xB = B−1b ≥ 0D+e J]F-R4POBmFI_]2Qxd6J]DmRUJ=J]DuZ_DO_T&BEHo
< XZQ^RtXZDJDgXZD8DOB+*KRcY8D8JsgBZYOF/Jd6JHKBF@RUT/YOg$R/Y8TUJjd6Y¢P8TUXZk8HKl[\JZoe.BETcJ$a6J[\P8HKJZq PGXmY8TcH^JPOT/Xmk8H^l][\J@mZo R2Wb2Y8QNPGX2F/F/l]d6Jm
nBETUQxBEk8HKJ]Fd+e _`gBET/R]q − IdmJ`F@RcY8D8JjkGBZF/Jjd6JHxB~[uBnRUT/Qxg$J
f-ZHKXZkGBEHKJg=d6J]FgXZD2R/T&BEQKD2R/J]F®dfe _2BEHKQSRU_;d6J\@mZo R2o < Qb ≤ 0
qmHKJpZJ`gR/J]Y8T(x0, λ0) = (0,−b) J]F-RBZH^XmTUF®Y8DOJF-XmH^Y6RUQ^XmDBmd6[\QKFUF/Q^k8HKJpkOBZF/QxbmYOJCP>XZYOT@Zmo RroACBEDOF HKJNg]BZF'Z_DO_T&BEHq$XZDtQ^D2R/TUX6d6Y8Q^R'Y8DOJ4D8XmY8ZJ]H^HKJ4nBETUQxBEk8HKJ
y ∈ Rm q`YOD"D8XmY8ZJ`BEYWZJ`gRUJY8TgX 6Rk > 0
q6J$RY8DzD8XmY8ZJ`BEYP8T/XmZT&BE[\[\J;HKQKD8_]BZQ^TUJinf
x≥0, y≥0Ax+y=b
c · x+ k · y @mZo mX.-uXZDBWPOT/_`BEHxBEk8HKJ[\JD2RI[tY8HSRUQ^POH^QK_=POBZT −1
R/XmY6R/J`FIHKJ]F®g$XmDmRUTUBZQ^D2RUJ]F.dfe _2BEHKQSRU_=g$XmT/TUJ]F/PGXmDOd8BZDmRsd6J]Ftg$XZ[\P>XmFUBED2R/J`FWD8_]mBnRUQ^mJ]Fd8Jbd6JjR/JHKHKJ]F"F/XZT/R/J`F"b2Y8J
b ≥ 0oª+JmJ]g$R/JYOT
(x0, y0) =(0, b)
J]F-RpYOD8JjF/XZHKY6R/QKXZD BZd6[\QxF/F/QKk8H^J\kOBmF-Qxb2Y8JuP>XZY8TgJuP8T/Xmk8HKl[\JZo < e QKH4J$a6QxF@RUJ\Y8D8JF/XZHKY6R/QKXZDBmd6[\QKFUF/Q^k8HKJd6YP8TUXZZT&BE[\[\JjHKQKD8_]BZQ^TUJzXZTUQ^mQ^DOBZHW@Zmo^tJ$RF-Q=H^Jz[\Q^DOQ^[tY8[ d6J-Zmo^"D¦e J`F@RPGBZF −∞ q'BZH^XmTUFWHKJ]FtF-XmH^Y8R/QKXZDOF"XZP6RUQ^[uBEHKJ]F"d6J-ZZo mCm_TUQS:GJD2R"D8_`g$J]FUFUBEQKT/J][cJ]D2Ry = 0
oe IDBZP8P8HKQKb2YOBZD2R®He BZH^mXZTUQSRUM8[\J;d6YzF/Q^[\P8HKJ$a6JWs-Zmo 2qrXZDR/TUXZY8mJ;BZQ^DOF/Q¤Y8D8JWF/XZHKY6R/QKXZDBZd6[\QxF/F/Q^kOH^JkGBZF/QKb2Y8J\P>XZY8Tj-Zmo^;F]e Q^HIJ]D J$a6QKF-R/JuYOD8JZo < e QKHID+e JD J$a6QxF@RUJuPOBZFugEe J]F-R-h snh d6QKT/JuF/QXad = ∅ $qXmD~HKJcd6_R/J`gR/J\gBZT=HKJt[\QKD8QK["Y8[©d6Jz@mZo mJ]F-RCBnR/R/J]Q^D2RpPOBETCY8DmJ]g$R/JYOT
(x, y)BZJ]g
y 6= 0o
42Qxd6J][c[\J]DmR`q]P>XZYOT+R/TUXZYOZJTfT&BEPOQKd6J][\JD2R¦Y8D8J®F-XmH^Y8R/QKXZDWBZd6[\QxF/F/QKk8H^J4PGXmY8TN-ZZoK`>XmDBQKDmRU_TU$Rs\g&M8XmQKF/QKTkRUT/l`FZT&BEDOdjPOBETT&BEP8P>XZT/Rs
co
G2n¡]Z4G&2 ~ 7 O¡];'JH¤b2Y8JD8XmYOFIHe BmXZDOF.d8_]g$TUQ^R4He BEHKZXmT/Q^R/MO[cJd8YF-QK[\P8H^Ja6J=d6J[uBZDOd6JHe QKDrZJ]TUF/Q^XmDusg&MOBZb2Y8J_RUBZPGJ~d6JHKB kOBmF-J
Bk q4gJb2Y8QP>JY6R$R/TUJzR/TUl]Fjg$X 8R/JY8a¢P>XZY8THKJ]FP8T/Xmk8HKl[\J]Fd8JZT&BEDOd6JR&BEQKH^HKJBmJ]gkGJ`BEYOgXZY8P¯d6J~gXZD2R/T&BEQKD2R/J`FtPOY8QKFUb2Y8JHe XmTUd6TUJd6JBk J]F-R_2BEHBZY°DOXZ["kOT/Jd6JgXZD2R/T&BEQKD2R/J`FU$o¢9;D"P>JY8R'RUQ^TUJT POBET/R/Qrd6Y"vBEQ^Rªb2Y8J
Bk+1 DOJNd6Q@0>l]T/Jd6J Bk b2Y8J®POBZT'Y8DOJ®gXZHKXZD8D8JP>XZYOT[\J$R-RUT/JBEYP>XZQKDmRtY8D8J[\JQKHKH^J]Y8TtF-R/T&BnRU_ZQKJZoX IDJh0¤J$R`q F/QNgZe J`F@R"HxBjhl][\Jjg$XmH^XmD8D8Jb2Y8Q
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Bk+1 = BkEk BZJ]gEk =
1 l1o o o ooo0
1
oooljooo
1
0
ooo o o oln 1
,
J$REk J]F-RvBZgQ^HKJWs\Q^DrZJ]TUF/JT
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lj
1 −l1o o o ooo0
1 −lj−1
1−lj+1 1
0
ooo o o o−ln 1
.
9;DY6RUQ^HKQxF-Jd6XmDOgCHKBcvwXmT/[tY8HKJZq8F/XZYOFNvwXZTU[\JpvBZg$R/XZTUQxF-_]JZq(Bk)−1 = (Ek−1)−1(Ek−2)−1 · · · (E0)−1(B0)−1.
?+2nZZlWlYoD oT 4C:;!\6 ,/0 .+] 0. = ( 3* B6"!: (+*-,. _ . , 0*-* . ( /4 0( min
x1≥0, x2≥0, x3≥0, x4≥0, x5≥0x1 + 2x2
:O!/: . C: 1 "9 0( 5C:
−3x1 + 2x2 + x3 = 2−x1 + 2x2 + x4 = 4x1 + x2 + x5 = 5
?+2nZZlWlYoD oZ 4C:;!\6 ,/0 .+] 0. = ( 3* B6"!: (+*-,. _ . , 0*-* . ( /4 0( min
x1≥0, x2≥02x1 − x2
:O!/: . 9: 1 0( 5C: x1 + x2 ≤ 1Px2 − x1 ≤ 1/2
# , "!3 0 : ] 0( 638b6 ] ! 69:O: ( $P( 96! ( )63C: > 0 ( 0A. C:@6 ] 4 190 'P&?+2nZZlWlYoD oDf 4C:;!\6 ,/0 .+] 0. = ( 3* B6"!: (+*-,. _ . , 0*-* . ( /4 0(
minx1≥0, x2≥0, x3≥0, x4≥0
3x3 − x4
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± m . +%¡, #$ ) %) )#K#
:;!/: . C: 1 "9 0( 959:
x1 − 3x3 + 3x4 = 6x2 − 8x3 + 4x4 = 4
)4).,Z[, &b47I5O? 3q_9_W74?-1N3 ?-1N3^+5O? _¦:5 A;JP8YOQKFH^J`FuRUTUBnBEY8a°d6J pMGBZg&M8QxBEDJ$R WBETU[\BZT nBZT\BZYd6_kOY6Rd6J`FjBZD8D8_J`F]V3 ² qIY8D8JDOXZY8mJHKH^JWg$HxBZFUF/Jdfe BZH^mXZTUQSRUM8[\J]F]q8d6Q^RUFd6JWP>XZQKD2RUF=Q^D2R/_]T/QKJYOTUF]q6J]F-R=BEP8PGBETUYP>XZY8T=T/_`F-XmYOd6TUJpd8J]FPOT/XmZT&BE[\[\J]F+HKQKD8_]BZQ^TUJ]F]o`¦J4D8Xm[ d6Jg$J$R/R/Jg$HxBZFUF/JNdfe BEHKZXmT/Q^R/M8[\J`F+rQ^J]D2Rªd6Jg$Jb2Y+e BEYtgXZD2R/T&BEQKTUJd8J®HxB;[c_R/M8X6d6Jd6Y\F/Q^[\P8HKJ$a6Jpb2Y8QqEPOBET&g$XmY8T&BED2R¦HKJ]FªF-Xm[\[cJRUF]qnT/J`F@RUJ®F/Y8TªHKJ®k>XZT&d"d8YtP>XZHK£Zl`d6T/J
Xadªg$J`F.BEHKZXmT/Q^R/M8[\J`F'd6JP>XZQKD2RUFªQKDmRU_TUQ^J]Y8T&F'_]ZXmH^Y8J]D2RªsCHe QKD2R/_TUQKJY8TId6J
XadJRID8JT/J-|@XZQKZD8J]DmRF/XZDkGXmTUdb2Y+e stgXZDrZJ]T/mJDOgJZom=XZYOFBZH^HKXZDGF®d8_]g$TUQKT/JCQKgQ¤Y8Dd6JWg$J]FBZH^mXZTUQSRUM8[\J]F®bmYOJCHe XZDzBEP6hP>JHKHKJBEYOFUF-QIO5EW4Onw§OF'2'§]n '`2Z§G4>wn°Zr¤§n8Go HI£Bzd6J]Y6aQxd6_]J]FWD8XmY8ZJ]H^HKJ]Fpd8BZDOFgJ$R/R/J[c_R/M8X6d6J ¨ªP8TUJ[\QKlTUJ[\JD2R]qªXmDP>_DOBZH^QxF-Jg$J]T-R&BEQKD8J]F\g$XmD2R/T&BEQKDmRUJ]FtsHe BZQKd8Jd6JP>XER/J]D6hRUQ^J]HKFNXZYvwXZDOg$R/QKXZDOFf@kOBZT/TUQKlTUJ]F!gKA2d8JY6a6QKl[\J[\JD2R`q2XmDuY6R/QKHKQKF/J;YOD8J=[\_$RUM8X6d6J;d8J;J R/XmDPGXmY8TPGBZFUF-J]Td+e Y8DOJWQSRU_TU_JWscHKBuF/Y8QKBZD2R/JZoA;_]gT/QKZXmDOFg$JR-R/JW[\_R/M8X6d6JF/Y8THKJCPOT/XmZT&BE[\[\J;HKQ^DO_]BEQKTUJCF-RUBZDOd8BET&d
infx∈Rn
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c · x. -ZZo Vm9;Dzd6_:OD8Q^RY8DzPGXZR/JD2RUQ^J]H+H^XmmBZT/Q^R/M8[\Qxb2Y8J;P>XZYOT
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i=1
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ªXmY8TY8D POBET&BE[\lR/TUJd6JP>_DGBEHKQKFUBnRUQ^XmDµ > 0
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x∈Rn"Z;
Ax=b, x>0µπ(x) + c · x. -Zmo^m`
/J[uBET&b2Y8XZDOFb2Y+e JDP8T&BnR/Qxb2Y8JCHKB\g$XmD2R/T&BEQKDmRUJx > 0
D+e J]DJ`F@RPOBmFY8D8JWgBZTJHKH^JpD+e J]F-Rª|-BE[uBEQxFBmgRUQ^mJ¨+b2YOBZDOdXZD[cQKD8QK[\QKF/J~@mZoKZ`=XZDD8JuP>JY6RPGBZF f/Fe BEP8POT/X6g&M8J]T!gtd6Yk>XZT&dd6Jx > 0F/XZYGFPGJ]Q^D8Jd8J;vBZQ^TUJ f@Ja6P8H^X2F-J]T!gCH^JpP>XER/J]D2R/QKJH
π(x)ZJ]TUF
+∞ o¦J®POT/QKDOg$QKP>Jd8J®He BEHKZXZTUQ^R/M8[\JNd8JNRUTUBE|@J]g$R/XZQKTUJNg$J]D2R/T&BEHKJ4J]F-RId6J[\Q^D8QK[\QKF/JT-ZZoKZ+PGBETªY8D8J[\_R/M8X6d6Jpd6JpJ RUXZDPGXmY8T®d8J]F®nBEHKJY8T&FNd8J;P8HKYOFNJDP8H^YGFNP>J$RUQSRUJ]Fd6Jµo= IDJ0>JR]qrHKXZT&F/b2Y8J
µRUJDOdmJT&F,1]_TUXOq6HKJpP8T/Xmk8HKl[\JpPG_]DOBEHKQKF/[email protected]]TUF®HKJWP8TUXZZT&BE[\[\J=HKQKD8_]BZQ^TUJu@Zmo V2o?V+2nZmlWlYoD o M"!< : ( Xad
9:5 A = /4 / > ( 633< # & 'B0 6 * Ci!/ ! ( "!:; . ! ( b , (+*V0. xµ & 1 ( . 9: 1 "/6 ( ( /: 6 ] , (+*V0. ( 54@P 63496"! ( "!<8: ( # & 'd0 6 * C!/! ( !/B:O . ! ( ) , (+*V0. x0 < 0. =: xµ 1 > 3 > : x0 . =: !/ µ 5/6 > 8: P48 &)4).,Z[, \:`b/? 3^
'B"RUM8_XmT/QKJpd6JpHKBcd6YOBZH^Q^R/_ud8_@|-sc_mXrb2Y8_]J;HKXZT&F®d6JpHKB < XmYOF@h F/J]gRUQ^XmD~ ² o ± o ± .J`F@RR/TUl]F®Y6RUQ^HKJJ]DP8TUXZZT&BE[\[uBnRUQ^XmDHKQKD8_]BZQ^TUJZo.LNXZDOF/QKd8_TUXZDOFs~D8XZY8mJ]BZYHKJP8TUXZmTUBZ[\[cJ\HKQ^DO_]BEQKTUJjF@R&BEDOdOBET&d
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6) ( " # ! %, ! ± ±bmYOJCDOXZYOFBZP8P>JHKH^J]T/XmDOF®P8T/QK[uBEHIPOBETXmP8PGX2F-Q^R/QKXZDzBEYzd6YOBZHx
infx∈Rn
"R;Ax=b, x≥0
c · x, @mZoK`mX-
AJ]F-R=Y8DOJ[uBnR/TUQxg$Jd6JWR&BEQKH^HKJ
m× n q b ∈ Rm qGJ$R c ∈ Rn o"ªXZY8T p ∈ Rm q8XZDQKD2R/TUXrd8Y8QSRH^J'BEZT&BEDOZQKJDjd6Ju-ZZoK`mL(x, p) = c · x+ p · (b−Ax), @mZoK ±
X-zHe XZDzBuF/JY8HKJ[\JD2R,f/d6YOBZH^QxF-_hgCHKJ]FgXZD2R/T&BEQKD2R/J]Fd+e _]mBEHKQ^R/_ZoY9;DzQKD2R/TUXrd8Y8QSRHxBtvwXZDOg$R/QKXZDd6YOBZH^JBZFUF-X6g$QK_JG(p) = min
x≥0L(x, p),bmYOQqOBZP8TUl]FgBZHKgY8Hq6nBZY6R
G(p) =
p · b F/QA∗p− c ≤ 0
−∞ F/Q^DOXZD+o @mZoKRr+JpP8TUXZk8HKl[\JWd6YOBZH¦d6Ju@mZoK`ZNJ`F@Rd6XmDOg
supp∈Rm
"R;A∗p−c≤0
p · b. @mZoKOUZIe J]F/POBZgJpd8JpF/XZHKY6RUQ^XmDOFBZd6[\QxF/F/QKk8H^J`Fd6YzP8TUXZk8HKl[\JWd6YOBZHI@Zmo^¢UZ4J]F-RD8XERU_
Pad = p ∈ Rm R/JH'bmYOJA∗p− c ≤ 0 .
/BEP8P>JHKXZDGFbmYOJWHe J]F/POBZgJCd8JF-XmH^Y6RUQ^XmDOFBZd8[cQxFUF-QKk8HKJ]Fd6Ju@mZoK`m4J`F@RXad = x ∈ Rn RUJH+b2Y8J
Ax = b, x ≥ 0 .+J]FpPOT/XmZT&BE[\[\J]FHKQKD8_]BZQ^TUJ]F\@mZoK`mJRu-Zmo^¢UZ;F-XmDmRd6Q^RUFpJ]DM2 8§] >o+Ie Q^D2RU_TU$RWd6Jg$JR-R/JD8XERUQ^XmDrQ^J]DmRcd6YTU_]F/Y8H^RUBnRcF-Y8QKnBED2Rcb2Y8Q®J]F-RtYODgBmFPOBZT-RUQKgY8H^QKJTcd6Y;MO_XZTUl[\Jjd8Jd6YOBZH^Q^R/_ ² o ± oKZmoL*F' 4GH2 lWlWo o5lWl Yt,!$&$ $ x !$&$ $ |¡xrWy|z@yx utvC| z@y )ret5y | z tvz yQ.tu yx ∈ Xad
zxut r utvC| z@y s.yM $($ $ yhup ∈ Pad
zxuZt r utv |z@y syM!$&$ $ 'wxt¢t ,yreu(
minx∈Rn
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c · x)
= c · x = p · b =
(
maxp∈Rm
"R;A∗p−c≤0
p · b) @mZoK2
y zx x yup rYu zSxuZt r6 utv |z@yhsy!$&$ $ yu!$&$ $ t)yuyxz@yCyreu t)yzvz@yh¢t ,yreuz@yh*~ rWstvut re3s uZtv |zStvu s.yx.r yhu xe~.yh
Ax = b, x ≥ 0, A∗p− c ≤ 0, x · (c−A∗p) = 0.@Zmo^ E
Yt¡ $($ $ xm!$&$ $ '|MxrWy+¢| z@yhx uZtv |zy"tvrOret5y |z z yr6yCz@y syh zSxuZt r6|s¡tt5hz@yh¡s.y*z | xuZQy z CyCyhu,Gt5s.y
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± R . +%¡, #$ ) %) )#K#
r6=f'MlWlYoD oZl DOJg$XmDOF-_`b2Y8JDOgJ~QK[c[\_`d6QKBER/Jd8Y ;M8_XmT/l][\JZZo6oKZd6Jd6YOBZH^Q^R/_J`F@RCbmYOJZq¤F-Qx ∈ Xad
JRp ∈ Pad
F-XmDmRCd6JY6a~F/XZHKY6R/QKXZDGF=BZd8[cQxFUF-QKk8HKJ]Fd6J@mZoK`Z®JRc-ZZoKOUmqTUJ]F/P>J]gRUQ^mJ[\JD2R`q6JHKH^J`FZ_TUQ@:OJD2Rc · x ≥ b · p.A;JW[\[\JZqGF-Q
x ∈ XadJRp ∈ Pad
m_TUQS:GJD2Rc · x = b · pBZH^XmTUF
xJ]F-R F/XZHKY6RUQ^XmDXZP6RUQ^[uBZH^J4d8J;@mZoK`m>J$R
pd6J;-ZZoKOUmonLNJ]F¦d8JY6aWP8TUXZP8TUQK_$R/_`FfP>JTU[cJR-RUJD2Rd8JR/TUXZYOZJTvBZg$QKHKJ[\JD2RCd8J]F;kGXmT/D8J`FP>XZY8TCHKJ]F;BZH^J]Y8T&FXZP8R/QK[\BZH^J`F=d6J@mZoK`ZJ$Rc-Zmo^¢UZqGJ$Rd8JpR/J]F-R/J]TF-QfYODzgXZY8POH^J
(x, p)J]F-RXmP6R/QK[uBEHo •
i r4>'¡]§n6§`54>)o < Y8P8P>XmF/XZDOF.b2Y8JXad
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F-XmD2RªDOXZDurQKd8J]F]o < XZQ^Rx ∈ Xad
JRp ∈ Pad
oLNXm[\[cJx ≥ 0
J$RA∗p ≤ c q8XZDzB
c · x ≥ A∗p · x = p · Ax = p · b,POY8QKFUb2Y8JAx = b
o IDPOBET/R/Qxg$Y8HKQKJT`qIg$J$R/R/JQKD8_2BEHKQSRU_QK[\P8HKQKb2Y8J~b2Y8JHKJ]F\nBEHKJYOTUFcXZP8R/QK[\BZH^J`Fd8J]FWd6J]Y6a~P8TUXZk8HKl[\J]F]q¤P8TUQ^[uBEH.JRd6YOBEHq+F/XZD2R:GD8Q^J`Fq+d8XZDOgcb2Y+e QKHxFWBZd6[\J$R/R/J]DmRd6J`FWF-XmH^Y8R/QKXZDOFXmP6R/QK[uBEHKJ]FtJDmJT/R/Yd8Y+J][c[\JZZo6o ± o Ie _2BEHKQSRU_@mZoKmWJ$R\HKBg$XZDGd6QSRUQ^XmDdfe XZP6RUQ^[uBZH^Q^R/_-Zmo^ ENF-XmDmRBZH^XmTUF®Y8D8JgXZDOF/_]b2Y8J]DOg$Jpd6Y+;M8_]XZTUl[\JWd6Jd6YOBZH^Q^R/_\ ² o ± o^mZo< YOP8PGX2F-XmDOFc[uBEQKDmRUJDOBZD2R\b2Y8JHe Y8Dd6J]F\d6J]Y6aP8TUXZkOH^l][cJ`F"POT/QK[uBEHXZY°d6YOBEHBmd6[\J$RuY8D8JnBZH^J]Y8T=XmP6R/QK[uBEHKJ :OD8QKJZocªXZY8T:8a6JTHKJ]F;QKd8_J]F]qGBmd6[\J$R/R/XZDGF=b2Y+e Q^HªFe BEmQKFUF/Jpd8Y~P8TUXZk8HKl[\J"d6YGBEHY8DtBZT/mY8[\JD2R'F/£r[c_R/TUQKb2Y8J4vwXZDOg$R/QKXZD8DOJNP>XZY8T HKJ4P8TUXZk8HKl[\JNP8T/QK[uBEH o HKXZT&F]qHKJ®¦J[\[\J;mZo6o ±BEyuT/[\Jub2Y+e QKHIJarQxF-R/J\Y8D8JF/XZHKY6R/QKXZDXZP8R/QK[\BZH^J
pd6J~-Zmo^¢UZo < Q
XadD+e J]F-RPOBZFprQxd6JZq¦XZD F-JTUJ$RUT/XmY8ZJud8BZDOFHxBF/QSRUYOBnRUQ^XmDP8T/_`g$_`d6JD2R/JgJjb2Y8Q:GD8QSRcHKBd6_[\XZDGF@RUTUBER/QKXZD+o¦«~XZD2R/TUXZDOFWd8XZDOgb2Y8J
XadD+e J]F-RPGBZFN2Qxd6JCJDY6R/QKH^QxFUBED2R®J]DOg$XmT/J;H^Jp+J][\[cJpd6J TOBZT nBZF; ² o 8o^ ro3 XmY8T
p ∈ Rm qXmDQKD2R/TUX6d6Y8Q^RH^J`FZJ]g$R/J]Y8TUFd8JRm+1
b =
(
b−b · p
) JRp =
(
p1
)
.
9;DZ_TUQ@:OJtb2Y8Jb · p = b · p − b · p ≤ 0
q¤PGXmY8T=RUXZY6Rp ∈ Pad
ofA"e BEY8R/TUJ"POBZT-R`qGHxBjgXZDOd6Q^R/QKXZDp ∈ Pad
P>JY6R=F/JWTU__]gT/QKTUJp ∈ C =
p ∈ Rm+1 R/JH'bmYOJ pm+1 = 1, A∗p ≤ 0 BZJ`g
A =
(
A−c∗
)
.
LNXm[\[cJb · p ≤ 0
P>XZYOT=RUXZY6Rp ∈ C q¤HKJc¦J[\[\Jtd6JCTOBZT nBZFt ² o 8o^ tD8XZYGF;d6Q^RWbmY¦e QKHªJ$a6QKF-R/J
x ∈ Rn R/J]H¦b2Y8Jx ≥ 0
J$Rb = Ax
qGgEe J]F-R-h snh d6Q^TUJpbmYOJx ∈ Xad
bmYOQ+D+e J]F-R=d6XZDOgCPOBmFrQKd6JmoT QKDOBEHKJ[\JD2R`qEF-Y8POPGX2F-XmDOFªbmYOJNHxB=nBZH^J]Y8TªXZP6RUQ^[uBEHKJd6Y"POT/Xmk8H^l][\JNP8TUQK[\BZHrJ]F-R-Zmo^Z −∞ o
< QPad
D+e J]F-RNPOBmF4rQxd6JZq2PGXmY8TIRUXZY6Rx ∈ Xad
JR4RUXZY6Rp ∈ Pad
q2XZDjBc ·x ≥ b · p o IDjPOT/J]DOBED2RYOD8JF/Y8Q^R/Jj[\QKD8QK[cQxFUBED2R/JjdOBEDOF
XadXmDXZk6RUQ^J]D2R
b · p = −∞ q g$JbmYOQNBEkGF-Y8T&d6Jmo'A;XZDOgPadJ`F@RprQKd8JZo DTUBZQKF/XZD8DOJ[\JD2R;F/Q^[\QKHKBZQ^TUJ"[\XmDmRUT/J\b2Y8JZq¤F/Q HxBBZH^J]Y8TCXZP6RUQ^[uBEHKJtd8J-ZZoK`mJ`F@RQKD:ODOQ^JmqOBEHKXZT&F
XadJ]F-RrQKd6Jmo
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mnBZT/QxBEk8HKJ]FJ$R
ng$XmDmRUTUBZQ^D2RUJ]Fdfe Q^DO_mBZH^Q^R/_`FUbmYOJH^JWPOT/Xmk8H^l][\JWP8T/QK[uBEH@Zmo^Zb2Y8QB
nnBETUQKBZk8H^J`Fq
mg$XmDmRUTUBZQ^D2RUJ]Fdfe _]mBZH^Q^R/_`FJR
ng$XmDmRUTUBZQ^D2RUJ]Fjd+e QKD8_2BEHKQSRU_]F&oA"e BEY6RUT/JPOBZT-R`q XmDP>JY6RjgXZDOF-R/TUY8QKT/Jd6J]F\BZH^mXZTUQSRUM8[\J]FtD2YO[c_]T/Qxb2Y8J]FtRUT/l`FtJygBZgJ]FtPGXmY8T\HKBT/_`F-XmH^Y6RUQ^XmDd6J@Zmo^Z"bmYOQ®Y8R/QKH^QxF-J]D2RcHKJ]Fcd6JY8avwXZTU[cJ`F"POT/QK[uBEHKJjJ$Rud6YOBZH^Jzd6Y¢P8T/XmZT&BE[\[\JH^QKD8_]BZQ^TUJZo
?+2nZZlWlYoD o (+. ( :O8 . 0 6! 0. ( 54 , ! 4C:;!\6 . 02*V0( i# P@: 0 /: 1C0. 1 ! . : 'V. , 0*-* . ( 4 0(
minx1≥0, x2≥0, x3≥0, x4≥0
8x1 + 9x2 + 4x3 + 6x4
:O!/: . C: 1 "9 0( 5C:
4x1 + x2 + x3 + 2x4 ≥ 1x1 + 3x2 + 2x3 + x4 ≥ 1
?+2nZZlWlYoD o e! > . , A. * 6! 0. 63 # & `3' . =:;"! ] 6! 0. ( :O 0 !/:O: ( . 01 0( 95 x ≥ 0
< 1] C:O 6 ( )! ] ( 96! ( . 0 " 0 (
L(x, p, q) = c · x+ p · (b−Ax) − q · x0 > 1 q ∈ Rn 5 . "!/ q ≥ 0 & *-,\0 8 0 > 1 # & 3' P ( 95 , 4P5 . 0 /! > . . > 0 ( 0A3. @6"! 0. q & ^ 646! ( B"! ] (+. ] N 0g,/0 :@6 ] ( 548P 6! 0. ( :O 0 !:;: (c. 0i1 "9 0( 5 x ≥ 0 &?+2nZZlWlYoD oY4 ( ? 8d!/ . , A. * )6"! 0. 67# & ' 9:5 a/! > 0 ! # & `3'C&?+2nZZlWlYoD oZlnJ= ( v ∈ Rn < c ∈ Rn < A !3/ *V0 ( 1 m×n C b ∈ Rm &=R 1 /: ( 6 8. , 0*-* . ( 4 0(
infv≥0
Av≤b
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supx∈P
c · x ,@mZoZZ
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supx∈X
J(x) = supx∈ X
J(x) = supx∈ X
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JJ`F@RgXZDrZJarJmqGXmDB
J(y) ≤ ∑1≤j≤k αjJ(xj) ≤max1≤j≤k J(xj) ≤ supx∈X J(x)
qªJ$R\P8Y8QxF/b2Y8Jg$J]gQNJ`F@RcrTUBZQ4P>XZY8TtR/XmY6Ry ∈ gX
Xq¦XmDB
supx∈ X J(x) ≤ supx∈X J(x)o .BETWBEQKH^HKJYOTUF]q¤PGXmY8TCR/XZY8R
z ∈ g$XXq>XZDP>JY6RW_]gT/QKTUJ
z =limk→∞ yk
qrBZJ`gyk ∈
gXXorLNXm[c[\JYOD8JvwXmDOgRUQ^XmDg$XZDrmJ$a6J
Rn → RJ`F@RND8_]gJ]FUF/BZQ^TUJ[\JD2RgXZD2R/QKDrY8J\gv@o IarJ]TUgQKgJWV8o6oEq2XZDzB
J(z) = limk→∞ J(yk) ≤ supx∈ X J(x)q8JRPOY8QKFUb2Y8JgJ]gQ4J]F-R"rT&BEQIP>XZY8TR/XZY8R
z ∈ g$XXq'XZDB
supx∈ X J(x) ≤ supx∈ X J(x)oª+J`F"BEY8R/TUJ]FQKD8_]mBEHKQ^R/_]Fª_RUBZDmR.RUT/QKrQKBZH^J`FqnXmDuB;[\XmDmRUT/_-Zmo ± on¦XZT&F/b2Y8J
XJ]F-RIkGXmT/D8_mq g$X
Xb2Y8Q8J`F@R4BZYOFUF-Qk>XZTUD8_mqmJ`F@Rg$XZ[\POBmgR`omAte BZP8T/l`F4HKJRUM8_XmT/l][\J=d6J;«~Q^D mX F rQ'ZZo ± oKZq g$X
X =g$X4J$arR/T gX
Xq2J$RJ]DBEP8P8HKQxbmYGBED2R -ZZo ± q
supx∈ X J(x) = supx∈ X J(x) = supx∈ X J(x) =supx∈X J(x)
qGgJWbmYOQ+P8TUXZY8mJc-ZZoRro'BbITUXZP>XmF/QSRUQ^XmDuZmo ± o RDOXZYOF'F-Y8mZl]T/J4d8JNg$XmDOF/QKd6_]T/J]T+He J]D2mJHKXZP8P>J4gXZDrZJarJNd6J4He J]DOF/J["kOH^JBmd6[\QKFUF/Q^k8HKJ
X = P ∩ Zn d6JWD8XZR/TUJpP8T/Xmk8HKl[\JpQ^D8Q^R/QxBEHN@Zmo 8`$oi kj w§`54>mlYlWoT$of pr | yhzvz@yr Gr54YBB4¯r¤§]5H2n s$ xr z scy P ⊂ Rn z yrey z yC~ reyQky¡sy3z yr6yCz@yCsyh tvreuZ*yreut5y*s.y P wxey3z r r uy Pe =
gX(P ∩ Zn) ¦JRUJTU[cJf@J]DrZJHKXZPOPGJJ]D2R/QKlTUJg=J]F-R4RUTUBmd6Q^R/QKXZD8D8J]HG[uBZQKFNH^_]Zl]T/J][cJ]D2RIR/TUXZ[\P>JY8TC¨2d+e XmTUd6QKDOBZQ^TUJZqYOD8JWJDrZJ]H^XmP8P>J;J`F@RY8DzXmk6|@J$RP8HKYOFmT/X2Fq6BEHKXZT&Fb2Y+e Qxg$Q
Pe ⊂ Po
4G4G8MlWlYoT o WtJ : Rn → R
yhu~ reyQky yu)t P ⊂ Rn yhu,xr z Qs cy |z
supx∈P∩Zn
J(x) = supx∈Pe
J(x) .@Zmo .UZ
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JJ`F@RHKQ^DO_]BEQKTUJZq.HKJPOT/Xmk8H^l][\Jzs d6TUXZQ^R/Jd6J"@mZoUm J]F-RNY8DPOT/XmZT&BE[\[\J®HKQKD8_]BZQ^TUJg$HxBZFUF-Qxb2Y8J¨mXZDBBZQ^DOF/QGgXZDOgJD2R/TU_HxBd6Q^yugY8H^R/_;dOBEDOFH^Jg]BEHxg$Y8Hq6XZYHe BEPOP8T/Xa6QK[uBnR/QKXZD¦qrd8YP>XZHK£Zl`d6TUJ
Peo H+£BtYODzg]BZFX.-jRUXZY6R=d6J]rQ^J]DmRvBmg$QKH^Jmo
i kjª'§]54>mlYlWoT$o pr stvuwhx xr z scy P yhu,xrB)4> WHk2'n~r¤§]mtP = Pe =XZYOF\BZH^HKXZDOFc[uBEQKD2R/J]DOBED2Rcd8XZD8D8J]Tud6J]F\g$XmDOd6Q^R/QKXZDOF\F/Y6yF/BZD2R/J]FwPOT/_`g$QxF-J`FU"PGXmY8Tub2Y+e YODPGXmH^£ml]d6TUJCF/XZQ^RJD2RUQ^J]T]o
)I).,,[ ` 365O?/<_ 367.3` b!_ _¦1N3:=1;? 79=:b!`.?@56_=DPGXmH^£ml]d6TUJpbmYOJHxg$XZDGbmYOJCP>JY8R=Fe _]gT/QKT/J
P = x ∈ Rn | Ax ≤ b @Zmo mBZJ]g
A ∈ Rm×n J$Rb ∈ Rm o HIgXZDrrQ^J]DmRWd6J\D8XZR/JTWb2Y+e Y8DR/JH4PGXmH^£ml]d6TUJ"J`F@RWP8HKYOFpZ_]D8_T&BEHbmYOJjHKJjP>XZHK£Zl]d8T/J
Xadd8J]FcF-XmH^Y6RUQ^XmDOF"Bmd6[\QKFUF/Q^k8HKJ]Fcd6YPOT/XmZT&BE[\[\JH^QKD8_`BEQKT/JjF-RUBZDOd8BZTUd¯gv@oA=_:OD8Q^R/QKXZDmZo6oK`C¨>JDJh0¤J$R`q
XadJ`F@R=PGBET;d8_h:OD8Q^R/QKXZDQ^DOgH^YGF=d8BZDOFHKJtgZD8JP>XmF/Q^R/Q^v4d6J
Rn qJ$RD8XmYOFBZXmDOFcJDXZY8R/TUJzd8_@|-s D8XZR/_b2Y8JXad
F]e QKHJ`F@RuD8XmD6hrQxd6JZqNBRUXZYn|@XmY8T&Fud6J]FuP>XZQKD2RUFJ$arR/TU_[uBEY8acg$J;bmYOQGD+e J]F-R4POBmFIHKJ=gBmFIP>XZY8TNY8DuP>XZHK£Zl]d8T/J=b2Y8J]HKgXZDOb2Y8Jgv@o/J[uBZTUb2Y8JWmZo6oEo¦BgBZTUBmgRU_TUQKFUBnRUQ^XmDd6J`FWPGXmQ^D2R&FWJ$arR/TU_[uBZY6ad6JXad
¦J[\[\J~Zmo 8o Um;F]e _$R/J]DOdg$J]PGJ]DOd8BZDmRd6JWHKBc[uBEDOQ^l]T/JWF/Y8Q^nBZDmRUJZoD4r lYlWoT$o Kr tvreu yc.uch |zsx z sQy P sZret | "!$&$ 'QMyhur~Qyh|tvcy Cyreu zxut r"s$ xr ¢u y
A′x = b′ A′ yuxrWy x |ut5~QytvrY¢yt5hzy y synzStS rys.y
A yhu b′ yu)zy*y~huyhx CCsyh ztSrWyh~ cyh rs| reuy3sy b i r4>'¡]§n6§`54>)o < XZQ^R
xY8DcPGXmQ^D2RIJ$arRUT/_][\BZH6d6J
PqnJ$RIF-XmQSR
I(x) = 1 ≤ i ≤ m | Ai ·x =bi
wHe JDOF/J[tk8H^Jd8J]Fg$XmD2R/T&BEQKDmRUJ]FBZg$R/QKZJ`FNJ]Dxq8X.-
Aid6_`F-QKZDOJCHxB
ihl][cJpHKQ^mD8JWd6J
Ao < QfHKBvBE[\Q^HKHKJ Aii∈I(x)
qOD+e J`F@RPGBZFd6JT&BEDOnq8XmDzPGJ]Y6RR/TUXZYOZJTY8DzmJ]gRUJY8TDOXZDzDrY8H
yRUJH¦b2Y8J
Ai · y = 0P>XZYOTR/XZY8R
i ∈ I(x) ofLNXZ[\[\J x J]F-RH^JW[\QKH^QKJYd6J`FPGXmQ^D2R&F x− εy JR x+ εyqOb2Y8QF-XmDmRNk8Q^J]Djd6J`FI_HK_[\J]DmR&F4d6J
PF-QεJ`F@RBZFUF-JG1PGJR/Q^R]qrXZDgXZD2R/TUJ]d8QSR4He J$arR/TU_[uBEHKQ^R/_=d6J
xo QKDOF-QqXZDPGJ]Y6RWR/TUXZY8mJTCY8DF-XmYOFCJDOF/J[tk8HKJ
I ′ ⊂ I(x)d6Jg]BET&d6Q^DGBEH
nRUJHIb2Y8J\HKB[uBnRUT/Qxg$J
n × nd6XZD2RpH^J`F;HKQKZD8J`F;F/XZD2RpH^J`FAiqfBmJ]g
i ∈ I ′ qfJ]F-RpQ^DrZJ]TUF/QKk8H^Jmo>¦JcF/£6F@RUl[\J Ai · x = bi, i ∈ I ′qgBET&BZg$R/_]T/QxF-JCBEHKXZT&F
xo
¦Jc¦J[\[\JmZo ± o u[\XZD2R/TUJcJDPGBET/R/Qxg$Y8HKQ^J]TCb2Y+e Y8DPGXmH^£ml]d6TUJ"D+e Bjb2Y+e Y8DD8XZ[tk8TUJ*:OD8Q.d6JPGXmQ^D2RUFJarR/TU_[uBEY6afo?+2nZZlWlYoT oZl 4 1(+, "! * 8 "!@: ( x 9:5 !3 , ( 9d6 P > 48 ( ?0 A′x = b′
<0 > 1 A′ C b′ 1 *-* B6 0 /: . *-* & & < 0. =: x C:OV! , ( _ 4 *V0.+&¦JC¦J[\[\JtmZo ± o F/Y8ZmlTUJ=dfe _$R/YGd6Q^J]TH^J`F®g]BZFNX-HxBcF-XmH^Y6RUQ^XmDjdfe Y8DF/£6F@RUl[\JCH^QKD8_]BZQ^TUJ=J`F@RJD2R/QKlTUJZo
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p4YB)4>¡]§]54>lYlWoT$o tvuA ∈ Zn×n xrWy |ut5~Qy tvreyht5z@y yh |¢yuZt r6xtv|rYu yh reuwxtv¢| z@yreuy
$ detA = ±1
x u xu b ∈ Zn r| A−1b ∈ Zn i r4>'¡]§n6§`54>)o4e QK[\P8HKQKg]BnR/QKXZD ⇒ TU_]F/Y8H^R/JpBEYOFUF/QSR ER®d8J]FNvwXZTU["Y8HKJ]Fd6JLNT&BE[\JT`ok/_]g$Q^hPOT/X6b2Y8J[\J]DmR`q>F/Y8P8P>XmF/XZDGF=b2Y8J
AZ_]T/Q@:OJHe BZFUF-J]T-RUQ^XmD6o¤«XZD2R/TUXZDGF=dfe BEk>XZT&d~b2Y8J
A−1 J]F-RCsgXrJ$yg$QKJD2RUF;JD2RUQ^J]TUF]o 4DP8TUJDGBED2R;P>XZY8TbHKJih l[\J"mJ]g$R/JYOTCd6JcHxBkOBmF-JcgBEDOXZD8QxbmYOJtd6J
Rn qXmD\ZXZQ^R4b2Y8JHxBihl][cJgXZHKXZD8DOJd6J
A−1 qZb2Y8Q>g$XDOgQKd8JBZJ]g A−1bqZJ]F-R4sWg$XrJyugQ^J]D2RUFIJ]D2R/QKJT&FoLNXm[\[cJ\g$J`g$Q J]F-RCrT&BEQ PGXmY8TCR/XmY6R
1 ≤ i ≤ nqGXmDB
A−1 ∈ Zn×n o¤A;XZDGg detA−1 ∈ ZqfJ$R
1 = detA detA−1 [\XZD2R/TUJWbmYOJ detAd6QKrQKF/J
1qOgZe J`F@R/h sEh d6QKTUJCb2Y8J
detA = ±1o
i kj w§`54>mlYlWoT$o5ln pr stvuwhx xry |ut5~QyA ∈ Zn×n yhu"' '4 2' Ownwhxe| rWs
detA = ±1 yu¡wx xrWy+C| uZt5~y B ∈ Zm×n yhu=§O4O§`8rr¤§' '4 2' Own whxe| rWsu xuy x |ut5~y~| cyyQ.uZc| tvu ysy
ByhusyCs.huyh¡tvrW| reu ±1
x0
IDuPOT/J]DOBED2R4d6J`FIF-XmYOF-h[uBnRUT/Qxg$J`F1×1
qXZD\mXZQ^RIJD\POBET/R/Qxg$YOH^QKJT4b2Y8JHKJ]F4g$XrJ$yg$QKJD2R&FIdfe Y8D8J[uBER/TUQKgJR/XZRUBZH^J][cJ]D2RCY8DOQ^[\X6d6Y8HxBEQKT/JcnBEHKJD2R;DO_]g$J`F/FUBEQKTUJ[\JD2R ±1XZY
0ofIe Q^D2R/TUX6d6YOg$R/QKXZDd6J`F[uBER/TUQKgJ]F®R/XZRUBZH^J][cJ]D2RYOD8Q^[\X6d6Y8HxBEQKTUJ]FJ]F-R[\XER/QKZ_]JpPOBETHKJWT/_`F-YOHSR&BnRF-YOQ^nBED2R`o
4G4G8MlWlYoT oZlYl E B'§`Ow§` 2 r¡\¡G4>'§]Z4G ¡tr¤§`5H2n2¡\H t5yreuD ∈ Zm×n xrWy
|ut5~y u u|z@y yhrYuxretv sxz@| tvcy f ∈ (Z ∪ +∞)m f ′ ∈ (Z ∪ −∞)m g ∈ (Z ∪+∞)n yu g′ ∈ (Z ∪ −∞)n z z@y tvrYu53yQucC| xs x z scy
Q = x ∈ Rn | f ′ ≤ Dx ≤ f, g′ ≤ x ≤ g -Zmo E reu rW~y|tvQyh yhrYu3yhrYuZt5y r |ut5~hxzSt5yh t Q yu* rW r |
Q = Qe yu xu xuy rW~uZt rM~ rY¢yQyJs.y
Rn s.|r6R rM|
supx∈Q
J(x) = supx∈Q∩Zn
J(x) .@Zmo m
i r4>'¡]§n6§`54>)o'9;DPGJ]Y6R=_]gT/QKTUJQ = x ∈ Rn | Ax ≤ b , -Zmo ZVm
X.-bJ]F-RWY8DmJ]g$R/JYOT;J]DmRUQ^J]T :OD8Q.JR
AJ]F-RWY8D8Jc[uBnR/TUQxg$Jud6XZD2RWg&MOBZb2Y8JcHKQ^mD8JtJ`F@RF/XZQ^Rpd8JcHxBvwXmT/[\J ±Di
q'BZJ`gDi
Y8D8JH^QKZDOJjb2Y8JHxg$XmDOb2Y8Jd6JDq'F/XZQ^Rtd6JjHxBzvwXmT/[\J ±ej
q¦X.-ej
J`F@RcH^Jjh l[\JpZJ`gR/J]Y8Td6JWHxBckOBZF/JWgBEDOXZD8Qxb2Y8JCd8J
Rn q8PGXmY8TY8DzQKDOd6Qxg$Jb2Y8JHxg$XmDOb2Y8J 1 ≤ j ≤ n o9;D[\XZD2RUT/Jdfe BEk>XZT&d b2Y8JAJ`F@RRUXER&BEHKJ[\JD2RY8D8QK[\X6d6Y8HxBEQKT/Jmo < XZQ^Rtd6XmDOg
MY8D8JjF/XZYOF-h[uBER/TUQKgJ
k × k Ja2RUTUBZQSRUJd6J A o>«XZD2RUT/XmDOFPOBZTT/_`g$Y8TUTUJDOgJpF/Y8T k b2Y8J detM ∈ ±1, 0 o < Qk = 1
qfg$J]HKBTU_]F/Y8HSRUJtBZYOF/F/Q^RERWd6JcHKBjR/XZRUBZH^JtY8D8QK[cX6d6YOHKBZT/Q^R/_\d6JDo < Y8P8P>XmF/XZDGF=[uBEQKD2R/J]DOBED2RHKJTU_]F/Y8H^RUBnRPOT/XmY8Z_uP>XZY8TR/XZY8R/J]F"H^J`F"F/XZYOF-h [\BER/TUQKgJ]FpgBZT/TU_J`FWd6JAd6Jd6QK[cJ]DOF/Q^XmDBEYP8HKYOF
k − 1q¦J$R[\XZD2R/TUXZDGFpH^J\P>XZY8T
Mo < Q
MgXZD2R/QKJD2RY8D8JuHKQKZD8J\_2BEHKJuszY8D ZJ]g$R/J]Y8T ±ej
q+XZDd8_ZJ]H^XmP8P>JdetM
POBETCTUBZP8P>XZT/R;sjgJ$R-RUJ"HKQ^mD8JZq¤JR;POBZT;TU_]gY8T/TUJDGg$JZqGH^JtT/_`F-YOHSR&BnR;J`F@RCP8TUXZY8m_Zo< QM
g$XmDmRUQ^J]D2RWd8JY6aHKQKZD8J`F;_]mBZH^J`F;BZYF/Q^mD8JcP8T/l`FqdetM = 0
q¤J$RWHKJcTU_]F/Y8H^RUBERCJ`F@RWJDGg$XZTUJ
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Mg$XDGg$Qxd6JZq2BEYg&MOBZD8ZJ][cJ]D2RNd6YjF/Q^mD8Jd6JCg$JT/RUBZQ^DOJ]FIHKQKZD8J`FIP8TUl]F]qmBmJ]gY8D8JF-XmYOF@h [uBnRUT/Qxg$Jd6J
DqZJ$RNgXZ[\[\J
DJ`F@R4R/XER&BEHKJ[\JD2RIY8DOQ^[\X6d6Y8HxBEQKT/Jmq
detM ∈ ±1, 0 qng$Jb2Y8QBZg&M8l]ZJCHKBcP8TUJY8mJWd6JWHKBtRUXERUBZH^JpY8DOQ^[\X6d6Y8HxBETUQSRU_d6JAoLNXm[\[cJ
QJ]F-R"d6XmD8D8_\POBET-Zmo ZVmq+BmJ]g
bJ]DmRUQ^J]TWJ$R
ARUXERUBZH^J][\JD2RWY8D8QK[cX6d6YOHKBZQ^TUJZq+QKHT/_`F-Y8H^R/Jtd6Y+J][\[cJZmo ± o \JRCd6JtHKBgIT/XmPGX2F-Q^R/QKXZDmZo ± o Vub2Y8JHKJ]F;PGXmQ^D2RUF=J$arR/TU_[uBZY6azd8J
QqFe Q^HxFJ$a6QxF@RUJD2R]qGF-XmDmRJ]DmRUQ^J]TUF]o
< Q¦He XmDzF/Y8P8P>XmF/JpJDzXZY8R/TUJpb2Y8JQJ`F@Rk>XZTUD8_Zq
QJ]F-R=g$Xm[\POBZg$R]q8JR=dfe BZP8TUl]FHKJ*;M8_XmT/l][cJd6J«Q^D ZX F 2QpmZo ± oKZq
Q =g$XIJarR/T
QoNLNXm[\[cJJ$arRUT
QJ`F@R\vwXZTU[\_d6JzZJ]g$R/J]Y8TUFcJ]DmRUQ^J]TUF]q
Qe =g$X
(Q ∩ Zn) ⊃ gX.J$arR/TQ = Q
q'J$RcPOBETcBEQKH^HKJY8T&FWHe QKDOgH^YOF/QKXZDQe ⊂ Q
J`F@RtR/TUQ^rQxBEHKJZoIe _mBZH^Q^R/_u-ZZo82IJ`F@R=BZH^XmTUF®Xmk6R/J]D2YOJCJ]DBZP8P8HKQKb2YOBZD2RHKJtLNXZTUXZHKHxBEQKT/JtZZo ± o 8o
?+2nZZlWlYoT oD . : ] 0 ( B6 ] 4P 0A. ( @!3/$4 1(+, !/ 0 ! = . . 0( & & & *-* /:,/0 e _\0*B( /8 . 1C0 :@: , 4 1( 0. 6"! , . N96" Xad = x ∈ Rn | Ax = b, x ≥ 0 69:b:O . !9 ( ":0 6 *B( :;: (+A. C: 6"! , 0*-* . ( /4 0( b:O 0 /6 0 D6c< 0 > 1 A ∈ Zm×n 6 0 m & 8c"!/ . C:6! _g, , ( 4C54C:Q:P! ( > 0 5C:M:; 49"! ( > 0. 959:
& , "!3S5"!9 b ∈ Zm < . C: , ( 95:Q _ 4 *V0 ! _ 6 Xad:O 9 ( :
`"& 5"!959: . 9:Q:;!/: *V0 ( 1 C: m×m 6A:; 63-64C58 *B( 0 ±1
!0 &J= ( *V0( 58 0 9 D ∈ Zm×n <"C 1 /: ( 6348/: Q = x ∈ Rn | Dx ≤ b, x ≥ 0 & 496"! ( 6 .+] 4! ( > 0. 1 "! (, 4 1 963 .+] 49"! ( > 0. 8 1 63C:[6! _ , , ( 4P549: :P! ( > 0 959: #I 4C= * 6 e* 0 -P !: P0. '
& , "!3S5"!9 b ∈ Zm < . C: , ( 95:Q _ 4 *V0 ! _ 6 Q :O"9Q8 ( 8: & D C:O 5 0. * 9b! (+* 6! . 0( &
jr6=f lYlWoT$o5l 9;DD8XZR/J]TUBb2Y8JHKJLNXmT/XmH^HxBEQKT/JzmZo ± oKZcD8JjP>XmF/JBEYOgY8D8JjgXZDOd8QSRUQ^XmDF-Y8TJqnM8XmT/[\QxF¦HxB;gXZDrZJa6QSRU_Zo/ IDcPOBET/R/Qxg$Y8HKQKJT`q`F/Q
J(x) = c·x J]F-R.H^QKD8_`BEQKT/Jmq]HKJgBET&BZg$R/l]T/JIJ]D2R/QKJTd6J]FCF-XmH^Y8R/QKXZDOF=XZP6RUQ^[uBEHKJ]F=D+e J]F-R;POBmF=d6QKT/J`gRUJ[\JD2R;TUJHKQK_tBEY~g]BET&BZg$R/lTUJWJD2RUQ^J]TCd6YZJ]g$R/J]Y8T;d6Jg$X 8Rco •
jr6=f lYlWoT$o5lT¦JpLNXZTUXZHKHxBEQKT/JpZmo ± o^mD8J;d8QSRF-YOT-RUXZY6R4PGBZF®bmYOJR/XmY6R/J`F4HKJ]F®F/XZHKY6R/QKXZDOFXZP6RUQ^[uBEHKJ]FF/XZD2RJ]DmRUQ^l]T/J`FoAte BZQ^HKH^J]Y8T&Fq®H^XmTUFUb2Y8JJJ`F@RH^QKD8_]BZQ^TUJZq®QKHCDOJP>JY6RJD¬$R/TUJBZQ^DGF-Qs [cXmQ^DGFb2Y8JHKBF/XZHKY6RUQ^XmD°XmP6R/QK[uBEHKJD8J~F/XZQ^RY8D8QxbmYOJZqNg]BET\RUXZY6RjkGBETU£rgJD2R/TUJzd6JF/XZHKY6R/QKXZDOFXZP6RUQ^[uBEHKJ]Fd+e Y8DzP8TUXZmTUBZ[\[cJCHKQ^D8_`BEQKT/JCJ]F-R=F/XZHKY6R/QKXZDXZP6RUQ^[uBZH^Jmo •
jr6=f lYlWoT$o5l=Z¬'BgXZDOd8QSRUQ^XmD b2Y8JQF/XZQ^RkGXmT/DO_cD¦e J`F@R"PGBZFpD8_`g$J]FUFUBEQKT/J\P>XZY8T"BEyuT-h[cJ]TbmYOJ
Q = Qed8BZDOF®H^JLNXZTUXZHKHKBZQ^TUJmZo ± oKZ¨6D8XmYOFND8XZYOFF/XZ[\[\J]F®HKQ^[\Q^R/_]FBZY6aPGXmH^£ml]d6TUJ]FkGXmT/D8_`Fq8b2Y8Q+F/XZD2RF/Y6yF/BZDmR&F®JDzP8T&BnR/Qxb2Y8JCPGXmY8T[\Xrd8_HKQKF/JTHxBcP8H^YOPOBET/Rd6J]FP8TUXZkOH^l][cJ`Fg$Xm[thk8Q^DGBnR/XmQ^TUJ]F]qrY8D8Qxb2Y8J[\JD2RP>XZY8T;F-QK[\P8H^Q@:OJ]THe J$a6P>XmF/_Zo.XZQKT ± P>XZYOTP8H^YGFd6JWd6_$R&BEQKHKF]o •
H4J$a6QKF-R/J\d6J\D8Xm["k8TUJY6a~TU_]F/Y8H^RUBERUFpF-YOTWH^J`F;[uBER/TUQKgJ]F;R/XZRUBEHKJ[\J]DmRWY8DOQ^[\X6d6Y8HxBEQKT/J`Fof=XZYOFD8XZYOFk>XZTUD8XmDOFQKgQ+s\d6XZD8DOJTY8D8JgXZDOd8QSRUQ^XmDF/Y6yF/BZD2R/JCR/TUl]FY6RUQ^HKJZop¢4YB)4>¡]§]Z4GlYlWoT$o5lf E 4> Z6n H Yt
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AJ]F-RCg]BETUT/_]JZq
detA ∈ ±1, 0 o < Q A BY8DOJtg$XmH^XmD8D8JDrY8HKHKJZqdetA = 0
o < QABuY8D8J\g$XmH^XmD8D8JBmJ]gF/JY8HKJ[\J]DmRCY8DgX2Jyg$QKJD2R=D8XmD6hDrY8Hq>XZD~d6_]ZJHKXZPOPGJHKJtd8_$R/J]T/[\QKDOBED2RPGBET"T&BEPOPGXmT-R"sg$J$R/R/Jg$XZHKXZDOD8JZq+JRtHe XmDg$XmDOg$HKY6R"PGBET"TU_]gY8TUT/J]DOg$Jb2Y8J
detA ∈ ±1, 0 o HDOJuT/J`F@RUJbmY¦e BgXZDOF/QKd8_TUJTWHKJgBZFX-g&MOBZb2Y8JugXZHKXZD8D8Jd8JABzJ$a8BZg$R/J][cJ]D2RpYODg$XrJ$yg$QKJD2R
1JRpJa8BZgRUJ[\JD2RWY8D g$XrJ$ygQ^J]DmR −1
¨+BEHKXZT&Fqfg&MOBmb2Y8JcgXZHKXZD8DOJcJ`F@Rd6JuF/XZ[\[\J\D2YOH^HKJZq+d6XmDOgdetA = 0
o=XZYOFBEP8P8HKQxbmYOJTUXZDOFcHxB ITUXZP>XmF/QSRUQ^XmDmZo ± oKOUBEY6a°P8T/Xmk8HKl[\J]Fd8J OXER&Fd8BEDGFuHKBF/XZYGF@hF/J]g$R/QKXZDjF/Y8QKnBED2R/Jmo2A;XZD8DOXZDOFNPGXmY8T®He Q^DGF@R&BED2R®J]DJ$a6J]TUgQKgJ=Y8Djg]BZFNX-He XZDP>JY6RgXZDOgH^Y8TUJ=s"HKB[uBZQ^Dzs\HKBtRUXERUBZH^JpY8DOQ^[\X6d6Y8HxBETUQSRU_Zo
?V+2nZmlWlYoT oT E p¢4Y7ª5Hr2'.4G G2§]'nH 1 9V6 ] 0,, . 54 . 4 , ( !/ 0 !1 ! A 63 1 \0 ct C:O . / *-A 6 18. ( 85: 6 > 0 C :;8 > ( : .+] ( /:5 0 [6 ( : 1 Ct ∈ 1, . . . , T & 1 h 0( $/ *-A 76 1 ":; (+. . 8:V63 1\. ( 8 . g4 , \69 0 ! _20,, . : &SR : (+*-,. ( ? . , A. * a2:P! ,, : 0 )!/75!/: . C: 0,3, . :):O-6 * * 7 N , & R 2:C! ,, :;8 0 "! ] (+. N 0 k = 0( C:@63B 0 > 0(+. , 3:O: (+A3. C:< .+] = 0( i 4P 0 1C0 0\1 548 ( :O4 ,/0 Y! ( 95 > 0. . [αi, βi]
< 0 > 1 1 ≤ αi ≤ βi ≤ T< 1 -! ( > ( 8 -L 0( 0A :O 0\1 ( 63C: ,\0 !:;C: &"R 5 Si. ): 0. 0( > :;8 g! 1 ":; (+. . 8 63 1\. ( 9 . V 0 > 0(+. . 0 9M6 .+] ( :O 0 9 αi
.+] ( :O 0 9 βi& R
, 3:O uit = 1: ( αi ≤ t ≤ βi
<Cuit = 0
: ( " & !:O ( ? 8 . , A. *
infx ∈ N
k∑
1≤i≤kxiuit ≥ ct, ∀1 ≤ t ≤ T
∑
1≤i≤k
xiSi .@mZo ±Z²
8b!/ .+] :; *-A. [63C:i:O . !9 ( ": 0 6 *)( :O: (+A. C:i6 1 , A. * , 8!: ] 4 1 ( 1 *-* .+] 8/:O *-A. B63C: , ( 95:M8 ( 8:b6 ] ! , . N96")63 . 0 LI= * 7# & `3' <= . 0g*V0 ( 1 D C:O!* 0 ( 1 Q6 ] ( 58 > 0. . C:9< 1] C:O 6 ( !3/ * 0 ( 1 1 = 1( 85: 0, 1
5 . . "! . C: 1 0,,\0 0( :O:;81 /:O4 1 ! ( > * 8:P!)! 1 . & "98"! ] ! *V0 ( 1 6 ] ( 58 > 0. . 9:7C:558 0. * 8! (+* 6! . 0( & " 18. !3 &)4).,, 5O7ab _9 _;7I3
:ODd6JWd6_h:ODOQ^THKJ]FP8TUXZkOH^l][cJ`Fd6J OXER&Fq8gXZDOF/QKd8_TUXZDOFNY8DzZT&BEP8MOJ=XZTUQKJD2R/_ G = (N ,A)¨
N J`F@RCHe JDGF-J]["k8HKJ"d6J`F;zK2 ¡ZqGJ$R A ⊂ N ×N J]F-R=He J]DOF/J["kOH^Jcd6J]F;6nZ¡mo =DBET&gBZH^HxBED2Rd8Y¢DWYGdiBZYDWYOd
jJ]F-RBEQKDOF/Q®DOXER/_
(i, j)o,9;D¢[tY8D8Q^Rug&MGBZb2Y8JBZTUg
(i, j) ∈ A d+e Y8DOJmBª8Z§] uij ∈ R+ ∪ +∞
J$RWdfe Y8D4 '§cij ∈ R
oª+J fug"d8BZDOF
uijJ]F-R;P>XZYOTf-Y8P8P>JTk>XZYODOdkgo )9;DuF/Jd6XZDOD8J=BEYGF/F/Q8JDug&MOBmb2Y8JDWYOdud6Y\mTUBZP8M8JY8DOXZRIJD2R/T&BED2RIJarXmZl]D8J
bi ∈ RF-Qbi < 0
q¦QKHNF]e BZZQ^Rtdfe Y8D OXZRtF-XmT-R&BED2R]q g$Xm[cP8R/_jBZH^m_k8TUQKb2Y8J][\JD2R&$o9;DBEPOPGJ]H^HKJ4O§Y8D8JvwXmDOgRUQ^XmDx ∈ RA q (i, j) 7→ xij
q6Z_]T/Q@:GBED2RHxBcZ4G,2'r¡uzK2 ¡C2'"JnOF4bi +
∑
j∈N ,(j,i)∈Axji =
∑
j∈N ,(i,j)∈Axij , ∀i ∈ N , @mZo ± `
BZQ^DGF-Q+b2Y8JWHxB\g$XZD2RUTUBZQ^D2R/Jpd6JWP>XmF/QSRUQ^rQ^R/_0 ≤ xij , ∀(i, j) ∈ A .
-Zmo ± Z
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#),+)) ± ± ID¬F/XZ[\[uBED2RH^J`FuHKXZQxFjd6J`FuDWYOd8F@mZo ± `$qIXZDmXZQ^RjbmY¦e Y8DOJg$XmDOd6Q^R/QKXZDD8_`g$J]FUFUBEQKT/JzP>XZY8THe J$a6QxF@RUJDOgJpd+e Y8D OXZRJ]F-RbmYOJWHKB\F/XZ[\[\Jpd8J]F OXZRUFJarXmZl]D8J]F®JD2RUTUBZDmR&FF-XmQSRDrY8HKH^J
∑
i∈Nbi = 0 .
@mZo ±Z± XZYGFF-YOP8PGX2F-J]T/XmDOF;R/XmYn|@XZY8T&FWb2Y8JuHxBg$XZDGd6QSRUQ^XmD°-Zmo ±m± ;J]F-RZ_]T/Q@:O_Jmo D OXZRJ]F-R"d6QSRc62 X¡`¡]57 tF]e Q^H+FUBnRUQKF-vBEQ^RH^J`Fg$XmDmRUTUBZQ^D2RUJ]Fd6JWgBZPOBZgQSRU_
xij ≤ uij , ∀(i, j) ∈ A .@Zmo ± R2
i kjª'§]54>mlYlWoT$o5l pr| yzvz@yB ¢4Y7ª5Hr¡2' 4O§)4 ¦§W' zy c| yztvrW|tvQy
min∑
(i,j)∈Acijxij
F/XZYOFHKJ]FgXZD2R/T&BEQKD2R/J]F(11.31), (11.32), (11.34).
@Zmo ± UZ+J~P8TUXZkOH^l][cJd6J OXZRsg$X 6Rj[\QKD8Q^[tY8[ BZd6[\JRP8HKYOF/Q^J]Y8TUFjF-XmYOF@h P8TUXZk8HKl[\J`FuQ^[\P>XZT/RUBZDmR&Fqg$XZ[\[\JHKJ"P8TUXZkOH^l][cJd8JR/T&BEDGF-P>XZT/R=d6J"He .a6J[\P8HKJtV8oKZoKZqOXZYHKJ"P8TUXZk8HKl[\J"d+e B 0>J`gRUBER/QKXZD~d6JHe .a6J[\P8HKJV8oKZo6o?+2nZZlWlYoT oZ ^3_=,3. ( 1/( 58 . , A. * -63 8 1 9 *B( (+* ! * 1 = 9: , \6 0 9 0 ! , A3. * )6 0 : , =h@6 .+]^3_ *-,. & & & # R 63C:;: ( /8 0a. " 0, & '
=DgBmFfPOBET/R/Qxg$YOH^QKJTfvwXmDOd8BZ[cJ]D2RUBEHmd6YWP8TUXZk8HKl[\JId6J OXZR'sg$X 6R+[\Q^DOQ^[tY8[J`F@R'H^JIPOT/Xmk8H^l][\Jd6Y GXERW[uBna6Q^[uBZHqfXmY B 4e7ª5HrM2' 4G§\P8T/XmP8TUJ[\JD2RWd6QSR`q¦b2Y8QIg$XmDOg$J]T/DOJcF/JY8HKJ[\J]DmRWHKJ]FgBEPGBZg$Q^R/_`FwJRND8XmDHKJ]F4gX 6R&FU$o H¤F-J]TUBWgXZ[\[\X6d6J=d6J=F/Y8P8P>XmF/JT®b2Y8J G Bd6J]Y6auDWYOd8FNd6QKF-R/QKD6hZY8_`FqsJRpqBEP8P>JHK_]F¦T/J`F-P>J]g$R/QKZJ[\J]DmR¦¡O4>¦nZ;J$RB w§`¡ZqR/J]HKF'bmYOJ
sD¦e BPGBZF¦d8J4P8TU_]d6_`g$J`F/F/JY8T i ∈ N | (i, s) ∈ A = ∅ $qfJ$R p D+e BPOBZFWd6JF/YOg]g$J]FUF/JY8Tu i ∈ N | (p, i) ∈ A = ∅ $o< XZQ^R
v ∈ R+o9;D¢BZP8PGJ]H^HKJ 4G§62 ¡`¡]57 2
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vF/Qi = s
q−v F/Q
i = pq
0F/QKD8XZD+o
i kjª'§]54>mlYlWoT$o5l yB'4Y7 ZHr2' 4G§+8 ~ r6tuy u xyh3xr u|s tt5hzy¡s.ys p sy¢| z@yx C|Gtv |zy
?+2nZZlWlYoT oDf "! . , A3. * 6! * 09_(+*V0. C:Oa 1 ( > * 82! 1C0 :,/0 h ( 1 ! . ( 6 , A. * B63 8 1 *B( (+*V0.+& # /6 ( 1C0 ( " =" , ! 0 0 "!95 !3 0 1B0 ! 0, ( 95 > 8 0 9M6 0 /: . 0 634 ? ( ( 6"! , A. * )6"! * 09_(+*V0.+& '
4DP8T&BnRUQKb2Y8JmqrXZDjg&M8J]TUg&MOJ;F/XZY8mJD2R®d6J`FF-XmH^Y6RUQ^XmDOFNJD2R/QKlTUJ]F®dfe Y8DjP8TUXZkOH^l][cJpd6J OXERp¨rPOBETJ$a6J[\P8HKJZq6P>XZY8THKJpP8TUXZk8HKl[\JWd6JCR/T&BEDOF/P>XZT/Rd6JpHe IarJ][\P8H^JVOo^mo^mqrH^J`F[\BZTUg&MGBEDOd6QxF/J]F®stHKQK2TUJTPGJ]Y8ZJ]DmRWR/TUJcd6J`FWg$XmH^QxFq¤JRH^QKrT/J]T;Y8Dd6J][\QSh g$XmH^QxFCPGJ]Y6RD8J\POBZFpBmXZQKTCd6JuF/JDOF]o H.J]F-Rd6XZDOg
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DGBnR/YOT/J]H d6J\F/Jcd6J][uBEDOd6J]TCF/QªY8DPOT/Xmk8H^l][\Jtd6J OXERWsjgX 6Rp[\Q^DOQ^[tY8[©BjBZY6R/Xm[\BER/Qxb2Y8J[\JD2Rd8J]FF/XZHKY6R/QKXZDOFXmP6R/QK[uBEHKJ]FJ]DmRUQ^l]T/J`Fo :ODdfe BEP8P8HKQxbmYOJTHKJtLNXZTUXZHKHKBZQ^TUJ"mZo ± oKZZqrD8XZR/XmDOFb2Y8JpHKBHKXZQ®d6J]FtDWYOdOF"d6J pQ^T&g&M8X 0 @Zmo ± `pP>JY8RcF]e _`g$TUQ^TUJAx = b
q¦X-HKB[uBER/TUQKgJA ∈ RN×A qBZP8P>JHK_JW6§]Z+2,>^'m52 2 ZzK2 ¡XU6nZ¡"d6J G q8J]F-R=d6_h:GD8Q^JpPOBZT
Ai,(j,k) =
−1F-Qi = k
q1
F-Qi = j
q0
F-QKD8XmD+o'B[uBnR/TUQxg$JAJ`F@R'k8Q^J]D"d6_:OD8QKJZq]FUBEY8v6dOBEDOFfHKJNg]BZF+d8_Z_]D8_TU_.X-pHKJ4mTUBZP8M8J.BY8D8J4kGXmYOg$HKJZq`gEe J]F-R-hsEh d8Q^TUJ;YODBZTUg(j, k)
RUJH+b2Y8Jj = k
o D OXZRg$QKTUgY8HKBZD2RF/Y8TY8D8JWk>XZYOgH^JWBtY8D8JgXZD2R/TUQ^kOY6R/QKXZDb2Y8Q>F-J;F-QK[\P8H^Q@:OJ=dOBEDOF4HKBWHKXZQGd6J]F4DWYOd8FNd6J pQKTUg&M8X0@mZo ± qZBZYOF/F/Q8D+e £uBnhR-h Q^HGBEYOgY8D8JP>JT/R/Jd8JCm_D8_]TUBZH^Q^R/_pscF-Y8POPGX2F-J]T®HKJpZT&BEP8MOJCFUBEDOF®k>XZYOgH^Jmq8g$JWb2Y8JpD8XZYGF®vwJTUXZDOFd8BZDOF®HKB\F/Y8QSRUJWd6JpHKBF/J]g$R/QKXZD+o=XZYOFP>XZYOZXZDGFN[uBZQ^D2R/J]DOBED2R_]gT/QKTUJCHe JDOF/J[tk8H^JWd6J`FOXER&FBZd8[cQxFUF-QKk8HKJ]Fx ∈ RA | Ax = b, 0 ≤ x ≤ u . -ZZo ± 2
p4YB)4>¡]§]54>lYlWoT$o5l |C| uZt5~y s tvr~ht5s.yhr~Qy r,xes¢ |Q~s$ xr c| 6y yhuu u | z@yh yreuxrYtv sxz|tvcy i r4>'¡]§n6§`54>)oL;e J]F-RY8D8JWg$XmDOF-_`b2Y8JDOgJpQ^[\[\_]d6QxBnRUJd6JpHKB74T/XmPGX2F-Q^R/QKXZDZZo ± oKOU8o
4G4G8MlWlYoT oZl# E B'§`Ow§` 2 r¡ 4G§]¡ur¤§]2¡\H Ytz@y uZCyreuc|rYu5Cyc rWyhbi rYuyreut5yh yu t z@yh~| |~tvu h uij
rYuyreut hcyh xtvrOret5yh |z z@yh tvrYu5yQ ucC| xs.yz yhreyh hzy!$($ 's.y zSxuZt r63|s tt5hzy s xr z Cy s.y u ~ u¡tvretv |z reuyreut5yh r | uZt5~xzt5y tz [email protected] uZC|s tt5z@yyhu rWyur r Gt5s.y tvzyc.tu y3xr uyhrYuZt5y uZtv |z i r4>'¡]§n6§`54>)o L;e J]F-RY8D8Jg$XZDGF-_`bmYOJDOgJcQK[\[\_]d6QxBnRUJd6Y¢LNXZTUXZHKHKBZQ^TUJZZo ± oKZcJ$R"d8JuHKB4T/XmPGX2F-Q^R/QKXZDZmo ± o^8o
?V+2nZmlWlYoT o X!/ . , A. * V63 1 ! > h!i# ^3_ 1/( 1 & & 'M, 8! :; * 634 . ( :;8 ,\0 X! , A. * b63 8 1 9 *B( (+* ! * <8CQP! > 8 0( /: (. 0 1 " 18. !: ( 63 .+]h^3_ 8 1/( 1 & & &?V+2nZmlWlYoT o , 8//: . , A. * 6 ] 0 1 0 ( < ( 6"! ( -6 0 : .+]^3_ *-,. & & `"&R 1 /: ( 6 8 5"! !: n 0 ":P n ?. . C:9< *V0( : ( 1/( < aij
9:5 ! 49 . ! ( , 4C:;85 . A " 8!X6"! 1 "! ,. (i, j)
<C 1 1 7!/ , * !9 0 ( " σ ∈ Sn<:; . ! ( ) , (+*V0. )63
maxσ∈Sn
∑
1≤i≤n
aiσ(i) .@mZo ± E
& "98X"!/ 1 , A. * BC:O 4! ( > 0. 9 0 ! , A. * . ( /4 0( a *-A C:M8 ( 8:
maxx∈Bn∩Zn×n
∑
1≤i,j≤n
aijxij ,@mZo ± m
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634C: ( " .+] 8/:O *-A3. Y63C: *V0 ( 1 C: A( :55 1 \0 :5 ( "!/C:9< 13] 9:5 6 ( .+] :; *-A. Y69: *V0 ( 1 C:49 . . C: x = (xij)63B 0(+. . n× n 5 . . C:M!/
∀i ∈ 1, ..., n, 1 =∑
1≤k≤n
xik ,
∀j ∈ 1, ..., n, 1 =∑
1≤k≤n
xkj ,
∀i, j ∈ 1, ..., n, 0 ≤ xij
`"& "98Y"! . , A. * 2# & 3' C:5!3 , A. * i6 8 ( 8 1 *B( (+* ! * # 69:O: ( /8 02. 0, 'P& ^ i63496"! ( "! . , . N 6 Bn
C:Ob8 ( 8 & !/ , !9 8 1 1\. !! 0 9 . 0 6 ( 1 ! . 54B6! , A3. * 6 ] 0 1 0 ( & 46! ( @63 1 b"! (, 4 1 963M"!/M5!5 *V0 ( 1 A8( :O5 1 \0 :5 ( "!QC:O A\0 N 1 8b6 ] !3 *-A ? ( 6 *V0 ( 1 9:b6 , 8 * ! 0 ( /:B# 1 4C= * B9:5@6 ( Y'C& & 496"! ( 63 1 i! (X, 4 1 963"!-: ( [!3 A\0. < (+. N 0 n 0 /:bP n ?. . 9:9< 1 \0 !/ 0 0 N 0 Y4P54 , 4C:O954 r ?c. . 9:9<3P 1/0 "!/ ?. . 0 N 0 d4C54 , 4C:;854C r 0 ":< 0 > 1 r ≥ 1
< (+.C:5 , 3:O: (+A3. 6LI= * n 1 "! ,. C: 63B6 0 /:O!3: 63:;=h5 !/ . 9: 6 0 :;8!:Q6 ] ! * * 1 ! ,. 0( 8Q634 4P54 , 4C:;854C: .+] !3 .+] 0 !i# 1 B 4C= * B9:5Q6 ( 'C&?+2nZZlWlYoT o PM _ 8 1( 1 , 4C:O95i! 0. = ( * LI"/6 0* 9 0. 8- 4C= ( )69: 5:9<6 \=D6C=! ./ 8:O" &\R 1 ": ( 6 . , A. * Q6! 8 * 09_(+*V0. 6 ] !3/ :;! 1 s ! , ! ( 5:p6 0 /:B! 0, G = (N ,A)
# > ( . 0 4 ? ( ( " & & 3'C&W !X: (+*,3. ( ? 8 <":C! ,, :;8 0! ] 0 ! 1 !/i63C: 1C0,\0\1( 549: uij/ , 8\6 . 0 > 0. !3 +∞ & W "!3d5"!: I, J ⊂ N <P , ! 5"!9
x = (xij) ∈ RA <" , :; x(I, J) =∑
i∈I, j∈J, (i,j)∈A xij&R i6 ( M"! ] ! ,/0 h ( ( "63 N8638! _ :;!/: 8/:O *A3. 9: S C
S9:5 !3/
s p : ( s ∈ S Pp ∈ S <C .+] 6 ( M!/ u(S, S)
9:5 . 0 !"@63 1 C 5 1 ! , & & "98Y"! , ! 5"!95 1 "! , (S, S)
:O4 ,\0 0 s 6 p <cP , "!3d5"!9 8 0 6 *B( :;: (+A. x6sp6 > 0. 8! v <
x(S, S)− x(S, S) = v .
^ 763496"! ( "!/ . 0 > 0. 8! 63-5"!9 0 6 *B( :;: (+A. 63 s p 9:5 *V0 =49 ,\0 . 0a1C0,\0\1( 54i65!5 1 ! , B:O4 ,\0 0 s 63 p &`"& 0 a6"3/4 ! 8 0 6 *B( :O: (+A3. x 63sp<Y" 634 ? ( . $#% '&()*+ ,- Gr(x) =
(N ,Ar(x)) Ar(x)
634C: ( / .+] 8/:O *-A. 63C: 1 ! ,3. 9: (i, j)5 . :@!/B! A( 8 (i, j) ∈ A P
xij < uij<! A( 8 (j, i) ∈ A P
xji > 0 & !/: ] (+. _3( :55[!3 1 *B( γ 63 s p6 0 /: . 0, a4C: ( 6!/ . Gr(x)< (+. C:5 , 3:O: (+A. -63 1 ":O ! ( 7!g/! > 0 ! 8 0 6 *B( :O: (+A3. x′6
sp6 > 0. 8!:O ( 1 5 * 9 :P! , 48 ( 8! 1 . . B6 x <= * 6 ( ?0 :O! . * 8 . C: > 0. 8!:
xij. =: !/ (i, j)
!(j, i)
9:5B!3 0 1 6! 1 *B( γ &SRVA :O8 > 8d87!"!/: ( x C:5b9 ( P : (. 9: 1C0,\0\1( 54C:M:; 8 ( 8C:< , !9 1 ( : ( x′ 9 ( & &\R @:P! ,3, 3:O * 0( 95 0 X! ] (+. ] N 0B,\0 :e63 1 *B( M63 s p 6 0 : Gr(x)& J= ( S .+] 8/:O *-A3. 69: .7!\6: 0\1C1 C:O: (+A3. 9: 63 , ! ( : s 6 0 /: Gr(x)
<\PS . 1 *-,. 4 * 9 0( M6 S 6 0 /: N & "98!/
(S, S)C:5V!3/ 1 ! , :O4 ,\0 0 s 63 p 63 . 0190,/0\1/( 54V9:5Q4 0. . 0 > 0. !3e6"! 8 x &
& " 18. !3$!/ . 0 > 0. !3 *V09_3(+*V0. $6! 8i6 s p C:54 0. . 0 190,/0\1/( 54 *)( (+*V0. 6 ] !3/ 1 "! , $:O4 ,/0 0 9 s 6 p & # ] 9:5 . [ 4C= * 8 *V09_(+* 0. / 1 "! , *B( (+*V0. < !
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*V09_ / *B( 1 ! <363 \=67C =! ./ 8:O & ' & ^ 646! ( [! 0. = ( 3* , 8 * C 0 9C< . =:;"! . C: 190,/0\1/( 54C: :O9 ( 9:9<6 190. 1 ! . 8! *V09_(+* 0. 63 s p g!5 *, : O(v∗|A|) < v∗ 9:5 . 0 > 0. !3 *V09_3(+*V0. 6 ] ! 63sp<=P |A| C:O . i/ *-A )6 ] 0 1 : &
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'B B'4WEOn86§]Z4G 2 + 8Z=f'"T/J]PGX2F-JF/Y8TIHKJP8TUQKDOg$QKPGJ;dfe XmP6R/QK[uBEHKQSRU_d6J®JHKHK[\BZD+qb2Y8QfP>JY8RF]e _D8XmDOg$J]TNRUT/l`FF-QK[\P8H^J][\JD2Rd8BEDOF®HKJWgBZF®POBZT-RUQKgY8HKQ^J]Td8YP8TUXZkOH^l][cJpd6YP8HKYOFg$XmY8T-Rg&MOJ[\Q^D°¨+F/Q.Y8DPOH^YOFWgXZY8T/RWg&M8J[\QKDdfe Y8D8J\rQKH^HKJAsY8DOJcrQKH^HKJ
BPGBZFUF-JtPOBETWY8DOJcrQKH^HKJ
CqBZH^XmTUF=H^JcF-XmYOF-h g&M8J][\Q^D~BZH^HxBED2RCd8J
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dAB = minC
(dAC + dCB) ,@mZo ± Vm
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6) ( " + K ± X-HKJ
minJ`F@R"POT/QxF"F/Y8THe J]DOF/J["kOH^Jjd8J]FrQKH^HKJ]F
CPOBETtH^J`F/b2Y8J]H^HKJ]FWP>JY8RtPOBmF/F/JTYODg&M8J[\QKDd6J
AsBo4Ie _]b2YOBER/QKXZD @mZo ± Vm"J]F-RY8D¯gBmF\POBET/R/Qxg$YOH^QKJTudfe _]b2YOBER/QKXZD¯d6JzP8TUXZZT&BE[\[uBnRUQ^XmDd6£rDOBE[\QxbmYOJZqZXmYj =f'8§]54>2 cr84qZJHKH^JnB"PGJ]T/[\J$R/R/TUJ=d6JCgBEHxg$YOH^J]T4TU_]g$YOTUF/Q^mJ[\JD2R
dF/Bmg&MOBED2R=b2Y8JdXY
J]F-R;gXZD8DrY8Jb2YOBZDOdX
J$RYF-XmDmRCd6J]F=2QKHKH^J`FZXmQKF/Q^DOJ]F]oO¦BuDOXER/QKXZD~d+e _RUBnRBEP8POBZTUB+*KRCQKgQªDOBnRUY8T/J]H^HKJ[\J]DmRc¨¤D8XZYOFCD8XmYOFpF-Xm[\[cJ`F=P>XmF/_]F;HKJtP8TUXZkOH^l][cJ\d6YgBEHxg$YOH d6J
dABq
AJRB_RUBZDmRd6J]Y6arQKH^HKJ]F:8a6_J`FqfJRWD8XZYOFpmJD8XmDOFpd6J\ZXmQ^Tpb2Y+e QKHIg$XmDr2QKJD2Rd6JcRUBZk8Y8HKJTpHKJ]Fd6QKF-RUBZDOg$J`FdCB
XZYd6J;[uBED8QKlTUJd6YOBZH^JmqdAC
ªP>XZY8T4R/XmY6R/J`F4HKJ]F42QKHKH^J`FIQKDmRUJTU[\_]d6QxBEQKT/J`FCom¦J]Fd6_$R&BEQKHKFd6Yg]BEHxg$Y8H+d6XmQ^mJD2Rk8QKJDF OT®R/TUJ :8a6_]FP>XZY8TvwXmY8TUD8Q^TYODrT&BEQ+BEHKZXmT/Q^R/MO[cJc¨OgZe J`F@Rg$JbmYOJCDOXZYOFvwJ]T/XmDOFd8BZDOFH^J`Fd6JY8aF/XZYOF-h F/J]g$R/QKXZDOF®b2Y8Q¦F/Y8QKZJD2R`o8XZYGFR/T&BEQ^R/J]T/XmDOFdfe BZkGXmTUdY8D8JZJT&F/Q^XmDP8HKYOFWF/Q^[\P8HKJud6Y P8T/Xmk8HKl[\JZqfX.-HKJcR/J[\POFWBZP8POBZTUB+*KRpJ$a6P8HKQKgQSRUJ[\JD2R`q¦d8BEDGFpHKB < XmYOF@hF-J`gR/QKXZD¬ZZo ROozP8YOQKFtT/J]rQ^J]DOd6TUXZDOFtBEYP8TUXZkOH^l][cJd6YPOH^YOF\gXZY8T/Rtg&M8J][cQKD+q.F/XZYOFtY8D8JjvwXZTU[cJP8H^YGFZ_]D8_T&BEHKJZq6d8BZDOFHKB < XmYOF-h F/J]g$R/QKXZDmZo ROo ± o
)I).,,[ 587ab _ _¦1 =7I5O?f741 p1=?LNXmDOF/QKd6_]T/XmDOFY8DPGJR/Q^RCP8TUXZkOH^l][cJd8J"DOBER/Y8TUJ_]gXZD8Xm[\QKb2Y8JmqGb2Y8Q¦[\_]T/Q^R/JtH^JDOXZ[d6J"P8TUXEhk8H^l][\Jpd8JCgXZD2R/T ZHKJ=XmP6R/QK[uBEHfd6QxF/gT/JR®J]DM8XmT/Q1XmD :OD8Qo < XmQSR G = (N ,A)
YODmTUBZP8M8J=XmT/QKJD2RU_Zq["Y8D8Qd+e Y8D°P>XZQxd8Fc : A → R
qIb2Y8JD8XZYGF\Q^D2R/J]T/POT/_R/JTUXZDGF"g$Xm[\[cJzY8D°4 ¦§mo/BEPOPGJ]H^XmDOFbmY¦e Y8DOF rJ]F-RCYOD8JtF/Y8Q^R/J\d6J"DWYGd8F=TUJHKQK_]F=gXZDOF/_]gY6R/QKZJ][\JD2R;POBZT;d6J`F;BZTUg]FqGJ$RWb2Y+e YODg&M8J[\QKD°d6XmD2R\HKJP8TUJ[\QKJT\J$RHKJd6JTUD8QKJTuDWYOd°g$XDGg$Qxd6JD2R\J]F-RjBEP8P>JHK_Zwnm'§Zo4¦JzgX 6Rdfe YODg&M8J[\QKDJ]F-R"POBZTtd6_:OD8Q^R/QKXZDHxB~F/XZ[\[\Jjd6J`FtgX 6R&F"d6JjF/J]FtBET&gF]oKT Q^arXmDOFYODJD2R/QKJTTwHe M8XmT/Q1XmDGq>J$RCY8DD WYOdQKD8QSRUQKBZH
i ∈ N o$9;D~mJY6RpgBEHxg$YOH^J]T=HKJtg$X 6R=RUXER&BEH POBET/RUBZDmRpd6J i JDM8XZTUQ@1]XZDTb2Y8JpHe XmDDOXER/J
vTi = min
(`0, . . . , `T ) `0 = i
c`0,`1 + · · ·+ c`T−1,`T .@mZo R ²
¦BvwXZDOg$R/QKXZDi 7→ vT
i
q'bmYOJuHe XZDPGJ]Y6R"TUJP8TU_]F/JD2RUJTpPOBETWYODZJ]g$R/J]Y8TvT ∈ (R ∪ +∞)N qJ]F-R=R/T&BZd8QSRUQ^XmD8D8JHKHKJ[\JD2RWBEP8P>JHK_J4> E§]Z4G 882'~wJ]DM8XmT/Q1XmD
To¤XmYOFCg$XZDrmJD8XmDOFJ]D_]g$TUQKBZD2R;@mZo R ² IbmYOJ
min ∅ = +∞ qrg$JCbmYOQGTUJrQKJD2R®s"_]g$TUQKT/JvT
i = +∞ b2YOBZDOduQKHGD¦e £BWPGBZFd6Jg&M8J[\QKDOF.d6JH^XmD8ZYOJY8TTPOBET/RUBZDmR.d6J
id8BZDOFªHKJ®mTUBZP8M8Jmo =DcBZY6R/TUJgBmF F/P>_]g$QxBEH6J`F@R
T = 0¨H^J`F4g&M8J][cQKDOFIb2Y8Q>BEP8PGBET&BEQxF/F/JD2RId8BEDGF=@mZo R ² 'F/XZD2RNBZH^XmTUF.d8JH^XmD8ZYOJY8TIDrY8HKHKJZo9;DgXZDrrQ^J]DOd6T&BbmYOJtP>XZY8T;R/XmY6RWF-Xm[c[\JRCd6YZT&BEP8M8JmqGQKHª£~BY8Dg$QKTUgY8Q^RCd6JcH^XmD8ZYOJY8TCDrY8H^HKJcPOBZFUF/BZD2R=POBETpg$JF-Xm[c[\JR]q6J$R=b2Y8JF/XZDzg$X 6RJ]F-R 1_]T/XGo 4DPOBET/R/Qxg$Y8HKQKJT`q
v0i = 0
o=D8J"_]nBEHKYOBnRUQ^XmDDOB ZJ"d8Jj@mZo R ² g$XZDGF-QxF@RUJT&BEQ^R=su_DrY8[\_]T/J]TR/XmYOFHKJ]F;g&M8J[\QKDOFCd6JHKXZD6hZY8J]Y8T
Tq¦d8XZD2R"HKJD8XZ[tk8TUJg$TUX+*xR"J$a6P>XZD8J]DmRUQ^J]H^HKJ[\J]DmRcBZJ]g
To ¦BzP8TUXZmTUBZ[\[\BER/QKXZD d6£rDOBnh[cQxb2Y8JCBtD8XmYOF®PGJ]T/[\J$R/R/TUJCd8J=vBmgR/XmT/QxF/JTg$JpgBEHxg$YOHqrJ$RBEQKDOF/Qfd6J;He Jh0¤J]gRUY8JTJ]DRUJ[\POF®PGXmH^£2hD8XZ[\QxBEHIwHxB\D8XZR/QKXZDzd6JCR/J[\POFPGXmH^£rD8Xm[cQxBEHfJ`F@R=d8_h:OD8QKJd8BZDOFHKBC/J][uBET&bmYOJcmZo 8o m$o4e Qxd6_]JuPGXmY8TgBZHKgY8H^J]T
vTi
J`F@R"d6JcvBZQ^TUJ\BZT/QKJTpHe MOXZTUQ@1]XZDJRHe _$R&BnRQKD8Q^R/QxBEHqfJDgBZHKgY8HKBZD2Rvt
k
P>XZYOTR/XZY8R0 ≤ t ≤ T JR
k ∈ N o" 4DJ0>JR]q8XmDP>JY8R=_]gT/QKT/JCP>XZY8TRUXZY6R i ∈ N q8JR t ≥ 1q
vti = min
k∈N , (i,k)∈A(ci,k + vt−1
k ) ,@mZo RO
v0i = 0 .
@mZo R2Z
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4e _`b2YOBnRUQ^XmD@mZo RO.TU_]F/Y8HSRUJWd6YPOT/QKDOg$QKP>JCdfe XZP8R/QK[\BZH^Q^R/_Wd6J)®JHKHK[\BZD¨6HKJWg$X 8RXmP6R/QK[uBEH¤POBZT-hR&BED2Rd6Jiq2J]D
t_$R&BEP>J]F]q6J]F-R®Xmk6R/J]D2YJDzg&M8XmQKF/QKFUFUBED2RNHKJpP8T/J][\Q^J]T®[\XmY8ZJ][cJ]D2R
i→ kq8d6JC[\BEhDOQ^l]T/Jsp[\QKD8Q^[\QxF-J]T4HKJ=g$X 6R4g$XmDOd6Q^R/QKXZD8DO_POBETIgJP8TUJ[\QKJTI[\XZYOZJ[\J]DmR`qZgEe J]F-R-h snh d6QKT/Jci,k
P8HKYOFHKJg$X 6RIXZP6RUQ^[uBZH6POBET/RUBZDmRId6JkJDcM8XmT/Q1XmD
t−1oZ'Bpg$XZDGd6QSRUQ^XmDtQKD8Q^R/QxBEHKJW@Zmo RrZ'J]F-RªR/TUQ^rQxBEHKJp¨F]e Q^HfD8JWTUJ]F-R/JpBEYGg$Y8Dzg$XmY8Ps;|@XmY8JT`q6XZDD8JpPOB£ZJCTUQ^J]D+oY9;DrQKJD2R=dfe _]gT/QKT/JCHe _`bmYGBnR/QKXZDzd6JpP8TUXEhmTUBZ[\[\BER/QKXZDd6£rDOBE[\QxbmYOJZq8XmYz =f 6§]Z4G&2 cr84qrBZdOBEP6RU_JscD8XERUT/JWPOT/Xmk8H^l][\Jc¨OJHKH^JDOXZYOFªP>JTU[\J$R4d6Jg]BEHxg$Y8HKJT.POBZT.T/_`g$Y8TUTUJDOgJNHxBpF-Y8Q^R/Jd8JZJ]g$R/J]Y8TUF
v0, v1, . . . ∈ (R∪+∞)N o H+J`F@RvBmg$QKH^JWd6JpTU_]F/XZYOd6TUJ;d8J]FP8TUXZk8HKl[\J`FNPOH^YOFm_D8_]TUBZY6aJD[\X6d6QS:>BED2R@Zmo RrZ$o8LNXmDOF/QShd8_TUXZDOF®POBZTJ$a6J[\P8HKJWHKBcD8XmY8ZJ]H^HKJCvwXZDOg$R/QKXZDnBEHKJY8TvT
i = min(`0, . . . , `T )
`0 = i
c`0,`1 + · · ·+ c`T−1,`T + φ`T ,@mZo R ±
X.-φ ∈ (R ∪ +∞)N J]F-R;Y8DZJ]g$R/J]Y8T=TUJP8TU_]F/JD2R&BED2RY8D8JtPG_]DOBEHKQSRU_tBmF/F/X6g$QK_suHe _$R&BnR:ODOBEHo¦XZT&F/b2Y8J
φ = 0qZXZD\T/JR/TUXZY8mJC-Zmo R ² oE¦XZT&F/b2Y8J
φJ]F-RIHKBCvwXmDOgRUQ^XmDuQ^DGd6QKg]BnRUT/Qxg$Jdfe Y8DF/XZ[\[\J$R
j ∈ N q8Qo JZoφk =
0F/Qk = j
+∞ F/Q^DOXZD+q @mZo RR2-Zmo R ± IvwXZT&g$JWstR/JTU[\Q^DOJTd8BEDGFHe _$R&BnR
jq6J$R
vTi
d6XZDOD8JWBEHKXZT&FNHKJWg$X 6R[\Q^D8QK[uBEHfJ]DT_$R&BEP>J]FP>XZYOTNBEHKHKJT4d6J
isjor¦BCvwXmDOgRUQ^XmDnBEHKJY8T=-Zmo R ± Z_]T/Q@:OJRUXZYn|@XmY8TUF;@mZo RO`$qZBZJ`gHKBD8XZY8mJHKH^JgXZDOd8QSRUQ^XmDQKD8Q^R/QxBEHKJWPGXmY8THe _]b2YOBER/QKXZDzd6J-®JHKH^[uBZD
v0 = φ .@Zmo RUZ
L;e J]F-RQxg$QIHKJu[\XZ[\JD2R"d6JuTUJ[uBET&b2Y8JTWb2Y8JuHKJ\R/J][cPGFTb2Y8Q4Q^D2R/J]T/rQKJD2R"d8BZDOFpHe _]b2YOBER/QKXZDd6J
®J]H^HK[uBED¢-Zmo RG`qx@mZo RkUmFe _]gXZY8HKJtr ¡]r ¡j G2n¡]d6YR/J][\POFCP8Mr£rF/QxbmYOJtb2Y8QªQ^D2RUJTU2QKJD2RdOBEDOF\HKJ]FcRUTUBE|@J]gRUXZQKT/J`F-Zmo R ± j¨4d8BEDGFtDOXER/TUJz[\Xrd8_HKQKFUBnRUQ^XmD+qTJ`F@RuHKJzR/J[\POFub2Y8Q=T/J`F@RUJs|@XmY8JT`qªBEYOFUF-QNY8D8JPG_]DOBEHKQSRU_F/Y8TcHe _$R&BnRc§]2n 8d6JHKBR/T&Bn|@J]g$R/XmQ^TUJ
(`0, . . . , `T )g$XZDGd6Y8Q^R-hJ]H^HKJuszg&MOBED8mJTCHKBzgXZDOd6Q^R/QKXZD w§`8d6J\He _]b2YOBER/QKXZD d6Ja®JHKH^[uBED¦oK9;Dd6Q^RP>XZY8TWg$J]HKBzb2Y8JHe _]b2YOBnRUQ^XmD¢d6J$®JHKH^[uBZDJ]F-R\Y8D8J_]b2YOBnRUQ^XmDn 2§4YEGn62 GoNLNJR-RUJQKDrZJ]TUF/Q^XmDd6YRUJ[\POF\J`F@RQKD8M8_]T/J]D2R/JBEY6a¢P8T/Xmk8HKl[\J]Fud6Jd6_`g$QxF-QKXZD ¨4gJD¦e J`F@RjF/XZY8mJD2RubmY¦e sHxB :GDd+e Y8DOJ[\BZY8nBEQxF-JPGBET/R/QKJpb2Y+e XZDzg$Xm[\P8T/J]DOdg$XZ[\[\JD2RQKHfD8JpvBEHKHxBEQ^RP>XZQKD2R.|@XZY8J]TBEYzd6_kOY6R]o4e QKD2R/_TU$Rpd6JHe _`b2YOBnRUQ^XmD~d6J®JHKHK[\BZD@mZo RO`®J`F@Rpb2Y8J
vt P>JY8RCF/JtgBZHKgY8HKJT=suPGBET/R/QKT;d6Jvt−1 JDjR/J][cPGF O(|A|+ |N |) q6X.- |A| J$R |N | d6_]F/QKZD8J]DmRTUJ]F/PGJ`gR/QKZJ][\JD2RNHKJ=D8Xm["k8TUJ;d+e BZTUg]FJR.H^JD8Xm["k8TUJ®d8JNDWYGd8FqnmXZQKT'HxB /J[uBZTUb2Y8JCZmo RGo^NPGXmY8TªP8H^YGF.d6Jd6_$R&BEQKHKF]o QKDOF-Qq
vT PGJ]Y6RIF-Jg]BEHxg$Y8HKJTJ]DRUJ[\POFO(T (|A|+ |N |)) qGscg$XZ[\POBZT/J]TPOBETJarJ][\P8H^JWBmJ]g;H^JCR/J][\POF O(|N |pT )d+e Y8DBEHKZXZTUQ^R/M8[\JDOB vf_DrY8[\_]TUBZDmRNH^J`F4g&M8J][cQKDOF®d6JHKXZD8mY8JYOT
Td8BEDGFIY8DmTUBZP8M8JX.-g&MOBZb2Y8JF/XZ[\[\J$RBJa8BZgRUJ[\JD2R
pF/YOg]g$J]FUF/JY8T&Fo9;D¬Xmk6R/QKJD2Rdfe BZY6R/TUJPGBET/RRUT/l`FF/Q^[\P8HKJ[\J]DmRY8Dg&MOJ[\Q^DXZP8R/QK[\BZHIsjPGBET/R/QKTpd8JuHe _]b2YOBER/QKXZDd8J7®J]H^HK[uBED@Zmo RG`$qª-Zmo RUZW¨fXmDP>XmF/J
`0 = iqPOY8QKFXZDg&M8XZQxF-Q^R
`1T/_`BEHKQKFUBED2RjHKJ[\QKD8Q^[tY8[ d8BZDOF-ZZo ROqNQo JZo
vT`0
= c`0,`1 + vT−1`1
qJ$RPOH^YOF\m_D8_]TUBZH^J][\JD2R`r+1
R/J]HbmYOJvT−r
`r= c`r,`r+1 + vT−r−1
`r+1
o .BETugXZDOF-R/TUYOgRUQ^XmD+qvT
i =
c`0,`1 + · · ·+c`T−1,`T +φ`T
qmg$Jb2Y8QO[\XmDmRUT/Jb2Y8J(`0, . . . , `T )
J]F-R4Y8Dg&M8J][cQKD\XZP6RUQ^[uBZH8PGXmY8THKJWP8TUXZk8HKl[\J\@mZo R ± $o
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6) ( " + K ± EV?+rBª lYlWohZKo5l ªXZY8T=Q^HKH^YGF@RUT/J]T;g$JtbmYOQ¦P8TU_]gl]d6JmqOg$XmDOF-Qxd6_]T/XmDOFHKJ"gBmFdfe Y8D~g&MOBEY60>J]Y8T;d6JRUBna6QN[\BZTUBZYOd8BZDmR"dOBEDOF"HKBzrQKH^HKJjQK[uBEZQKDOBZQ^TUJ\T/J]P8T/_`F-J]D2R/_JF/Y8TtHKB+T QKZY8TUJ~Zmo 8o¦¦B~2QKHKH^JJ`F@RvwXZTU[c_]Jd6JpR/TUXZQxF1XmD8J]F]qed"qOY8Db2YOBET/R/QKJTMrY8POPG_mq Y8D~BZ_TUXZP>XZT/R]q6J$R¬Y8D8JkGBED8HKQ^J]Y8Jd+e X.-H^Jg&MOBZY0¤JY8TTUJrQKJD2RJDzZ_]D8_T&BEH+scrQxd6J`$oG+Jg&MGBEY0¤JY8Td8JpRUBna6Q¦F/XZYOMOBEQ^R/JpvBZQ^TUJ
TgXZY8T&F-J`FqPOBET/RUBZDmRd6J dq6JR[uBEarQK[\QKF/JTHKJ;2BEQKDRUXERUBZHq6b2Y8QfJ`F@RHKB\F/XZ[\[\JCd8J]F®mBZQ^DOFd6J`Fg$XZYOTUF/J]F
cq6J$Rdfe YOD8JP8TUQ^[\Jmq
φqORUTUBmd6Y8QxF/BZDmRFUB\P8TU_$vw_TUJDGg$JWP>XZY8THe _$RUBER :ODOBZH¦d6JHxBud6J]T/D8QKlTUJg$XmY8T&F-JmoY9;DzBT/J]P8T/_`F-J]D2R/_cH^J`FWmBZQ^DOFWd8J]FWg$XmY8T&F-J`FCF/Y8TWHKJ]FBZTUg]Fq¤JRH^J`FWP8TUQ^[\J]FpPGBETd6J`FBET&gFpF/XZT/RUBED2R&Fof¦BP8T/QK[\J −∞ JDBF-QKZDOQS:OJ®b2Y8J4HKJ4R&Bna6Q2D8J4mJY6RªPOBZF :OD8QKT.F/Bª|@XmY8TUD8_JNJD
Bo`=XZYOF F-YOP8PGX2F-J]T/XmDOFPOBET®BEQKH^HKJY8T&F®bmYOJ=HKJRUBEa6Q>B"HKBvBZg$YOHSRU_;d6Jpg&M8XZQxF/Q^T4F/J]F®g$XmY8T&F-J`Fo < e BZZQxF/FUBED2RNd+e Y8DjP8T/Xmk8HKl[\J;d6J[\BEa6Q^[\QxF/BER/QKXZD+q+He _]b2YOBnRUQ^XmDd6JP8TUXZmTUBZ[c[uBER/QKXZD d6£rDOBE[\QxbmYOJFe _]gT/Q^R"BZJ`g\[\BEaBZYH^QKJYd6J[cQKD+q8JR −∞ BEYH^QKJYd6J
+∞ ¨vt
H = max(3 + vt−1H , 10 + vt−1
A )
vtA = max(12 + vt−1
H , 7 + vt−1B )
vtB = max(−5 + vt−1
H ,−3 + vt−1A )
v0H = 0, v0
A = 2, v0B = −∞ .L®BEHxg$Y8HKXZDOF"[uBEQKDmRUJDOBZD2RtHxB~F@RUTUBER/_]ZQKJXZP6RUQ^[uBZH^Jd6YRUBEa6QNJ]DM8XZTUQ1XZD~POBET/RUBZD2Rtd6J
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v1H = max(3 + 0, 10 + 2) = 12
@mZo R mv1
A = max(12 + 0, 7 +−∞) = 12@Zmo R E
v1B = max(−5 + 0,−3 + 2) = −1
@Zmo Rmv2
H = max(3 + 12, 10 + 12) = 22 ,@Zmo R2Vm
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D@B?n = |N | >-, )" m = |A| .? 32)&' 74 ?$ " c
UT? E&' J"+L%u&/ "E&' @B?", ,B)@B?",
h ∈ 1, . . . , nm ^: $D b<U)@?",t ∈ 1, . . . , nm ^:K"F&' b<U
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A 1L
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hk = iL
tk = jU
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isjJ$Rd6JCg$X 8RN[\Q^DOQ^[tY8[zo H¤F]e BZZQ^R®dfe Y8D8J;Z_DO_T&BEHKQKFUBnRUQ^XmDd6YP8TUXZk8HKl[\JCd6YP8HKYOFg$XmY8T-Rg&MOJ[\Q^Dd6_-|-s\_mXrb2Y8_Wd8BZDOFHxB < XZYOF-h F-J`gR/QKXZDZmo RGo^c¨OgXZD2R/T&BEQKT/J][\JD2RBEYzgBZF=d6J]Fd6QxF-RUBEDGg$J]F]qDOXZYOFjD8JF/Y8P8P>XmF/XZDOFuPGBZFQxg$Qpb2Y8J~HKJ]Fg$X 8RUFjF/XZD2RP>XmF/QSRUQSvF]oQ IDJ$a6P8HKXZQ^RUBZD2RjH^J`FD8XZRUBER/QKXZDOFPOT/_`g$_]d8JD2R/J`Fq6QKHfD8XZYGFvBEY6R[uBEQKD2R/J]DOBED2R=gBZHKgY8HKJTvi = inf
T∈N
vTi = inf
(`0, . . . , `T ) T ∈ N, `0 = i
c`0,`1 + · · ·+ c`T−1,`T + φ`T ,@mZoVU ²
X.-HKJ\ZJ]g$R/J]Y8TφPG_]DOBEHKQKFUBED2RpY8D_$RUBER:GDOBEHIBZY6R/TUJtb2Y8J
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φ ∈ (R ∪ +∞)N b2Y8JHxg$XmDOb2Y8JZq¦gJjb2Y8Q4DOJjg&MOBEDOZJ\T/QKJDBZY6aTU_]F/Y8H^RUBnR&FoK9;D D8XZR/J]TUBzb2Y8JviJ]F-R[uBEQKD2R/JDGBED2RszBZH^J]Y8T&FWd8BEDOF
R = R ∪ ±∞ wHKJ]FF/XZ[\[\J]F"sd6T/XmQSRUJd6J-ZZoVU ² ;PGJ]Y8ZJ]DmRtBZH^XmQ^T+∞ q+POBZT"BEQKHKH^J]Y8TUF]q+QKH4J`F@RP>XmFUF-QKk8HKJjb2Y8Jg$J`FF/XZ[\[\J]FD8JtF/XZQKJD2RPOBmFk>XZTUD8_J`FQKD6vw_TUQ^J]Y8TUJ[\JD2R]q>gBZTHe XZD~gXZDOF/Qxd6lTUJd6J]F;g&M8J[\QKDOF=d8JH^XmD6hmY8JYOTjBETUk8Q^R/T&BEQKT/J][cJ]D2RcmTUBZDOd6J`$o4¦BvwXmDOgRUQ^XmDBZH^J]Y8T
vZ_TUQ@:OJzHKB D8XmY8ZJ]H^HKJ_]b2YOBER/QKXZD¯d6J
®J]H^HK[uBEDvi = min
(
φi, mink∈N , (i,k)∈A
(ci,k + vk))
, ∀i ∈ N .@mZoVU6`
.BZT\gXZ[\POBZTUBZQKF/XZDBmJ]g@mZo RO`$q'HxBP8TU_]F/JDOgJdfe Y8D°R/J]T/[\JF/Y8P8P8HK_[\JD2R&BEQKT/Jzd8BEDGFtHKJz[\Q^DRUTUBmd6Y8Q^R4HxBpPGX2F/F/Q^kOQ^HKQSRU_=d6JF]e BZT/TU$RUJTId8BZDOF.D+e QK[cP>XZT/R/J;bmYOJHO_RUBERkBmJ]gHxBpPG_]DOBEHKQ^R/_:ODGBEHKJ
φkqgJWbmYOQ¦d8BEDGFH^Jg]BZFF/PG_`g$QxBEHI@mZo RRrqrQ^D2RUJT&d6QSR=d8JpF]e BETUT/R/JTBZQ^HKH^J]Y8T&FbmY¦e J]D
jo
HCD8XZYGFuvBEY8R[uBEQKD2R/JDGBED2R[\XZD2RUT/J]Tjb2Y8JHKJF/£rF-R/l][\J¢@mZoVU6cPGJ]T/[\J$Rzd6Jg]BEHxg$Y8HKJTjHKBvwXmDOgRUQ^XmDnBEHKJYOT]o /BZP8PGJ]H^XmDOF\b2Y+e XmD¢BZP8PGJ]H^HKJ¡G4>'§]Z4G +8dfe Y8D8Jz_`bmYGBnR/QKXZD¢Y8D8JF/XZHKY6RUQ^XmDb2Y8Q+[uBE|@XZTUJ=RUXZY6RUJ]FHKJ]FBZY6R/TUJ]F]oL3F' 4G¢HrMlWlYoZ$oD | r~hut r ¢| z@yhx v s.5)ret5y |C!$&$ %yhuz@|¡ zxuZt r+C|G.tvC| [email protected]*zc Qwxe|ut rsyyhzvz |rI!$&$ $ i r4>'¡]§n6§`54>)oM9;DBtd6_-|-s"[\XmDmRUT/_Cb2Y8J
vm_TUQS:OJc@mZoVU6`$o < XZQ^R
v′ ∈ RN Y8DOJCF/XZHKY6R/QKXZDb2Y8J]HKgXZDOb2Y8Jd8Jj@mZoVU6`$o8«XZD2RUT/XmDOFb2Y8Jv′ ≤ v
o < XmQSR(`0, . . . , `T )
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iorLNXm[\[cJ
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v′`r≤ c`r,`r+1 + v′`r+1
q2PGXmY8T4RUXZY6R0 ≤ r ≤ T − 1
q6JRv′`T≤ φ`T
o" IDzJDOg&MOB+*xDGBED2Rg$J]FQKD8_]mBEHKQ^R/_]F]qv′i ≤ c`0,`1 + · · ·+ c`T−1,`T + φ`TJ$R=J]DP8T/J]DOBED2RHe QKD:O[tY8[ F-YOTR/XmYOFHKJ]F;g&M8J[\QKDOF
(`0, . . . , `T )PGBET/RUBED2R=d6J
iqv′i ≤ vi
o=Q^DGF-Qv′ ≤ v qGg$Jb2Y8Qf[\XZD2R/TUJWb2Y8J v J]F-R=F-XmH^Y8R/QKXZD[\BEa6Q^[uBEHKJWd6Ju@mZoVU6`$o
?+2nZZlWlYoZ$oZl "! .+] 4! 0 ( 63 . . * 0 2# & 'V, 8! 0 > ( ,3. !/: ( !3:@:O . ! ( /: ? ( C: & # /6 ( 1C0 ( " 1 ": ( 64 !3 " 0, 0 > 1 !-:O! . :O ** P & '?+2nZZlWlYoZ$oD !/:M6 ( /: "! . 0, V9:5 1 0\1C1 C:O: (+A3. , ! φ : ( , "!35!b.7!/6i6"! " 0, < (+. _3( :55 !3 1 *B( 263 i !3 .7!\6 j 5 . !/ φj 6= +∞ &dR > 0K* 8!/ : (b. " 0, ] 0 ,/0 :26 1/( 1 ! ( 63 1 2/4 0 ( LBCaC:5 1 0\1C1 9:O: (+A3. , ! φ < 0. =:.+] 4! 0 ( )6 . . *V0 # & '@, :;: 963!3/i!3 ( !/ :O . ! ( ? ( <=4 0. v &SW "!3 1 . 0 <1 /: ( 6 8 v′ ∈ RN !M:; . ! ( b6 .+] 49"! 0 ( "b6 . . * 0 -# & 'C&R ( 96! ( .+] 8/:O *-A. 6 1 "9 ( ! 0 (
C = i ∈ N | φi ≥ mink∈N , (i,k)∈A
(ci,k + v′k) ,P .+] 1 ( : ( , ! 1 \0 !/ i ∈ C ! .7!\6
π(i)5 . "!
v′i = ci,π(i) + v′π(i) .
R 64 ? ( 0( /: ( ! 0,,3. ( 190 ( π : C → N & 8 !/i"!/ . "!i:O ( i ∈ C < (+. _3( :55!3-9 ( k 5 . "!/ . k * ( 544 πk(i)< ] 0,,\0 ( 8 ,\0 : C & 1\. !)"! v′ ≥ v &
9;DjPGJ]Y6R®TU__`g$TUQ^TUJ"-ZZoVU6.g$Xm[\[cJ;Y8D8J;_]b2YOBnRUQ^XmDjd6J;PGXmQ^D2R):OarJv = f(v)
qrBmJ]gYOD8J;d6_h:OD8Q^R/QKXZD~_]2Qxd6J]DmRUJ"d6J
f : RN → RN o =YOFUF-Q'H^J3;MO_XZTUl[\JmZo ROocF-Y8mZl]T/JhR/hQKH¦d6J"g]BEHxg$Y8HKJTvs"He BEQxd6JWdfe YODzBZH^mXZTUQSRUM8[\JCd8JCP>XZQKD2R:8a6JZoªXZYOTg$XZDGg$HKY8T/JpscHKB\gXZDrZJ]T/mJDOgJ=d6JW[\_$RUM8X6d6J]Fd6JP>XZQKD2RC:8a6JZqªXZD¢T/J`g$XZYOT-R\F-XmY8ZJ]DmR\sd8J]F\BETUZY8[\J]DmR&F"d6JgXZD2R/T&BZg$R/QKXZD+o @g$QqªHe BEDOBZH^£6F/Jd6JHKB g$XmDrZJTUZJ]DOg$JjvwJT&BP8H^Y8RERjBZP8P>JH=sHxBF@RUT/YOg$R/Y8TUJdfe XZT&d6T/Jmo)9;D°[tY8D8Q^RJDJh0¤J$R
RN d6JHe XZT&d6TUJCPGBET/R/QKJH+YOF/Y8J]HqOd8_h:OD8Q g$Xm[cP>XmFUBED2RUJCPGBET=gXZ[\PGX2F/BZD2R/JZq6JRPGXmY8T=gJ$R=XmTUd6TUJZq6He BEP8P8HKQxgBnhR/QKXZDf : x 7→ f(x)
J]F-R®gT/XmQKFUFUBED2R/Jmo8IY8QxF/b2Y8Jf(φ) ≤ φ q2Y8D8J;T/_`g$Y8TUT/J]DOg$JQK[\[\_]d6QxBnRUJ=[\XZD2R/TUJbmYOJ
fr+1(φ) ≤ fr(φ)q4P>XZYOT\R/XZY8R
r ≥ 0oI¦BF/Y8Q^R/J f r(φ)r≥0
b2Y8Q=d6_`g$TUX+*xRugXZDrZJ]T/mJD8_]gJ]FUF/BZQ^TUJ[\JD2RWZJ]TUFpYODmJ]g$R/JYOTv′ ∈ RN o®¦JuH^J`gR/J]Y8TD8XZR/J]TUBzQxg$Q4b2Y8JuHe XZDBEY6RUXZTUQKF/J\HKBBZH^J]Y8T −∞ PGXmY8T.HKJ]FIgX2Jyg$QKJD2RUF.d6Jv′qZFUBEDOF.gJHxB;P>XZQKD2RId6Jg$XmD2mJTUZJ]DOg$Jmo 9;D\B
f(v′) = v′POBET®g$XZD2RUQ^DrY8Q^R/_Cd6JforA"e BEY6RUT/J;POBET/R]q2F-Q
v′′J]F-RNYODPGXmQ^D2R,:8a6J;b2Y8JHxg$XmDOb2Y8J=d6J
fqZXmDjBpR/TUQK2QxBnhH^J][cJ]D2R
v′′ ≤ φ qGd6XmDOgCP8Y8QxF/b2Y8J f J]F-Rg$TUXZQxF/FUBED2R/Jmq v′′ = fr(v′′) ≤ fr(φ)qOP>XZY8TRUXZY6R
r ≥ 0qd6XZDOgCJ]DPOT/J]DOBED2RHe QKD:O["YO[F/Y8THKJ]F
r ≥ 0qv′′ ≤ v′ qGd6XZDOg v′ J`F@RHKJWP8HKYOFZT&BEDGdP>XZQKDmR:OarJd6J
foc4Y8QKFUb2Y8Judfe BEP8TUl]FCHKJC;MO_XZTUl[\JZmo RGo 8qGHxBvwXmDOgRUQ^XmDnBZH^J]Y8TpJ]F-RpBZYOF/F/QªH^J\P8HKYOFpmTUBZDOdPGXmQ^D2R-h:8a6JWd6J
fq6XZDrQ^J]D2R=d6JW[\XZD2R/TUJTHKJWT/_`F-Y8H^RUBERF-Y8QKnBED2R]o
L*F' 4GH2 lWlWohZKoT E! §` mn6§`54>¯¡]'j2¡ 882'n¡/H | xtvuy fr(φ)r≥0s~ uyhz@|
rW~ut r |zyhxvsZrYt5y | !$($ %
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± m . +%¡, #$ ) %) )#K#
T' >H%%F'23"DU5%"F.">%)r%F.",NHF232(a"&',-.23&!;AfA"
@B"e?D.!?'Q3&'A"VuC,U:4 'H.&/"$239? E&' J"F.A"L0? J"E&'== E&'"fr(φ)
UA E&/"QL-aE"&/74 EEIK E'"&'
φ;'?-;>^:3&/"V;
φ J"e E,EIK E'"&'],g4 %D;."&'(,G&'A"OuC
x = f(x) E&
f(φ) ≤ φb<U E&'"9L -aE"&/54 > E JIK E'"&'
φ;'?;A3^g%u&'#A@B "
g4 &'%HB'&/"%abN J" .'%]snzAiacDs*j5| a[RidlnY ,c-l*Y[7i6 W, , N,!123 ),3HF232(a"&',-.23&*;)H%%F.C5UZ"F.2323">?"E'( J"?<$ J"&!;UZ323%<"$A " , 8=a"7U!%N&/"%Fa"&!r EG' 9!&/"&!; N^:@B&/=-a9C,23'>)6 KOt b # c # U ,23&'? OHF.2 I23&!H3 *9"&'2(. %VA"E:U e:6 " FU "$ # ?&:6!U V'23"DU-S UZv -$-¡b<U J"= E@B"G' F&!,6PO4 &'"%q"9:4 &/"%F."&'r E,9' G@.' G J"= E&!23'&!?&'"%6> ='?D. 9,%"23&'&' J"U)23q239?A@BH?9_"23 Vu-&Z,:4 &/"%Fa"&'+ EG' Q@' 6
=\7Y,iBsY ~-~-Z* j5| YAiY|^Y+Y| Yodh 0T|Zs Y82Q+-T> ' G (/ # #>STD&'0D$' 2A>5D82 >A2 ' D23H82 1%$#&2 ' W # > f |N|−1(φ) = v9(MU'<#D*82#<' G #5@D&'0D$' 2 HD82 >A2 ' D23H821%<#82' W 2V>' G >A2 DPA#D,D,B>F>'0I Q φ 4CE&('0^/ D&' D,Q66<9 -9 S$*[D,2A23Q&2' *;V#( > f |N|(φ) <
f |N|−1(φ)9
PT4 `C<?&'?#vv6 6 x( EHH1 54 &!23'%23"Q?2323> E&/"9:4 'H&/"$23)74 &/"%F."&'r E,9' @.I! 6 *9M?'?'0-a=%?E?* E&/"
x0 = φU"
xr = f(xr−1),
r ≥ 0UM@B%&/u."9L
?<$;>%"F.N E&xr = xr−1
U-;AZ? xr = v
"=:4 r 4 .Eq"]^::4 &'"%q"9?" J"= J"9;A>'"23 Q,>?A@BH?> J"9 E@BA"Q?_' Q"&/"=; |N | − 1
b<6 #&/U-M-a@A&!A"GLr = |N | "9r 4 .Eq" ' ,5r &/";R4 &'`C,&' J"N?&'?&/">,N? "> J"E&'?"23"9%HBa"&' 6^TF."&!;UWdH.F2323 Fa23""f;e?<" .!H&/"$23]54 &/"%Fa"&' E0' )@', UW2(.&! '"."N!M@.&!"( E&'@"( " , E JI # &!:UO.'%]'H&/"$23] F,I V!'2(.RUR*aI;A''323"0L JNd' "."0"" ' 0?F,% 0 :4 &/"%F."&' E,0! N@', 6 e J"' fF.&!,>H.F2323f?"E">@.&!">;>! %?&/6
lj-i.sn opY ~-~-R JXZYZjiX YANY,llopc|& G = (N ,A) c : A → R <v ∈ (R ∪ +∞)N b D < r < : << i, k ∈ N
: < :g r← 0 b← @AF.& A K i ∈ N vi ← φi r < |N | b Eb← C , K i ∈ N A K (i, k) ∈ A Q ci,k + vk < vi E vi ← ci,k + vk b← @F& r < |N | v < O > :g R r = |N | : E GAM E = , < 5gB :
u-M74 &'!' J"E9:4 &/"%F."&'p E,=' =@.' U =''123_?<$5="FaC&7%<I E"%G E,T* W&'H,>vv6 ¢UA"V&'"% E E T eL*@',f, T?<$23&' TH&'#opc\Rs:o[Wo "V!H,9a&/"EF&/-aE"F"=
H6 %V2323&' _ _?&/?&/"9,>HB.&!+ J"E&'?"23"Q E&/"&/=^g.'!<
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AU"Q@B&/DU?N;&5F..E"
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)? E UZ"g4 #p.E"&'?'&'vH = +∞ 6 up,N,N!)'123)23&! >%H%%%U E J"EF =-.=C,23!+&!23 .",
11 E9'N,&'CM?F$-;0? E65PT4 %<F.",
f,%u-&
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[f(x)]H = max(0,−8 + xH ,−1 + xA)
[f(x)]A = max(2, 1 + xH ,−4 + xB)
[f(x)]B = max(−16 + xH ,−14 + xA) .
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6) ( " + K ± ±PO4 &'"%<F."&'+ E,Q! =@', ? E&! J"0L)?'?'
x0 = φUx1 = f(x0)
UR666!U E&/"x0 = φ = (0, 2, +∞)
x1H = max(0,−8 + 0,−1 + 2) = 1
x1A = max(2, 1 + 0,−4 +−∞) = 2
x1B = max(−16 + 0,−14 + 2) = −12
x2 = x1 = v, J"
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AU7"
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i")! .?
(i, k) )!+?!#
:4 'H&/"$23 vv6 6 ,,_"E@BF0'9"23H%%&!; !7?.E?"&'H3.'H&/"$23 >. !'&/"E"%Fa"6PR G.!H&/"$23 =,>?"E"923&'''U-"QM-aE"&!?'&'Gg4 .!H&/"$230, FAI V''2(RU E").23&V' N' 0F&*, )N?D'?' ' ?F$23&! 0(?$ ," 23&'&'2N2MU7. 0'#?D. 374 H.F$0&'"%);&`( =?&/?&/" ^g,0?$ "9 E&/"&/fAnb""=' ? " 9@B"Gq"E0%IH."&/ 65 =C#?D G E%?&!,C5UM .&/"9:.&/N?-"9?_23&!C_;A FAI V''2(R6P7 ;G'9H.F$ G 74 0 f,9?&/?&/"DUA]5?"=]isOjh`jl*j sR[RYU?4 J"EI LaI &/;G:4 2N&'"O' T E2323<" O74 NF,"."F ≤ ";f 4 &'N ?F$23&!) i
LjU.'
i ≤ j6 # @B"DU
' Q >- E"G%JL)-a"''23"QF,% >^:.GC,23'=-.Q"23 f?&! E ."<b<6->. '>?D ?"EF&/U7 &/""E@B<+"`.F,N"23 G!&'%D&/UR@B&/>.9C,23' !v O9' ,%"F&'' 6O^ *9r&/";A74 r'H&/"$23N J"rYodh']lns:|RzAc-sniaYr E&W'N"23 =74 C,%?,"&!r J".%.=0? J"F."n&' ! "F&'''),+?A.H02(?F$&'0 9,% U?4 J"?N;R4 +"G E%, 23&'C5U5@B&/N!_tG2(aF;A]vv6 ,6 +,' ,3%<"F&'' N E,*#"&'N"23 , ?D'?:6 b_= n&' N' 0> >.F% UZ&'V E])")54 &!&/"&*.'&! E<
vL
+∞ UZ")74 5?"0 "N E!n&' 9!3 E J"&/""&'vi ← min(φi, mink∈N , (i,k)∈A ci,k + vk)
=?<$;iU-M-a?,F"=!
i-a9.F,0%?<&' E "DUG,"&/=* n?<"&!+@.'DU?0;A&Z,-]_"23 Q!&'%D&/6L_=."E ? -aE"&!?'&'>2(.F;' J"?'&O$)p' ?$ ," > E" E&/"&/ 0p,' ,W >? ?D. U "N23' N .'H&/"$23M3" (H!"^:? 6 # ?<"&! vv6 ¡b<UT!%_'H.&/"$23#S_.IK& J 3d"EF,U@B&/G-aGC,23' 'v g6 •G\RYAiBs Y ~~-ZZ*&>@$&&>@ / 96<9 W >3#8E&(' $ I QHD #&>F>' +-Q->3#DP 8P?<$>,9X&>=>$>F*&> ' D&'L+- B>O$' <>@>F*82=>@82' >9XR@))<D
n*I B2F>T(' <> >$,DB2 ' >
p1, . . . , pn ∈ N W 2[/82'0 ' 231> >B<,D2' E> u1, . . . , un ∈ R W B2K/ &2F P ∈ N Q(' <>H#8<'Q#&+-
/ *5B>A2G*' >$>31 T 2F 9UR@T$>Fxi = 1
>'- HB2= iP H *I B2G#(&>= =>3#DBP 8PA<> W 2 xi = 0> '0&(9R@=E82@H#8<'=' >3/82'0 ' 231G-=>3#DO<>>F&>GD*82#<'823Q $' <>
maxx ∈ 0, 1n
∑
1≤i≤nxipi ≤ P
∑
1≤i≤n
xiui .^Jvv6 ¡¢b
*XD, # W D*&>' W L2F*82 1 ≤ t ≤ nB2
0 ≤ Q ≤ P W $ I H
maxx ∈ 0, 1t
∑
1≤i≤txipi ≤ Q
∑
1≤i≤t
xiui ,^Jvv6 ¡xb
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#E# .*82'0H# $9R=S&23vt = (vt
Q)0≤Q≤P9
6<9 ( '0Hvt *DB2' * vt−1 / #$' Q[/@1,+-#82' $%#Q#82' #=' +-$9
9 1,-$'0K# %( ' 2 H [1>A-H436$6$9 <;9@>A2 2F&>Z[/ 1,D82' */ #( %C' 2H 9 ' +-V/ # %$ ' 2H / Q>$' E#82
maxx ∈ 0, 13
2x1 + 3x2 + 5x3 ≤ 6
8x1 + 2x2 + 9x3 .^Jvv6 ¡Bb
=\7Y,iBsY ~-~-Z* Yods:| XZY jR3ods:|Zs:o[Wo c Y* j5|iac-s:|YXZY+Yodh M*(' 2 G =(N ,A)
%*#( = ' 8231 W $', E#( #&2 ' &> W c ∈ RA W ODB82 W B2 τ ∈ NA W 23&> W B2φ ∈ (R ∪ +∞)N 5<1#( ' 231HJL# $9UR@J 5S>FD,
sB2==#8235 '=' 23
T ∈ N W B2/ DD 2B 8E OD@'0(`0, . . . , `m)
5 *&%^#$ I$' 2#$'0 W #$ 2F#(82K s4A' 9 $9
`0 = s; W23+-@ %<#$'0 2F23#
c`0,`1 + · · ·+ c`m−1,`m +φ`m
>F' 2@'0$'0H#( W >A&>H #@D82B#<'082F@B><DB23V ##&2F= '0@' 23T W ' 9 $9$>F*&>G #QD*82#<'823 τ`0,`1 + · · ·+τ`m−1,`m ≤ T
9<R=H>$>3#=+-(/ '0$(/ @# #&> D&'0D$' 2G<*82G2A&> >Q#$DB>Q82H>G23&> >,9 KS# %( ' 2 H $ $%*#(H#&2 ' 5 #&P=' +-Q$*X1>F*- D,@*I H$9 ' D,#82' *$2 *8EX#$^$ $%*#H#&2 ' H #=' +-Q D@'0HD 82 ='$'0 - 5
1# 5
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59VM* I82
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a⊗ b = a+ b43&&> $ 82F*&> D,B2A2F #<' >F#(82F ' G9 9LY #&> E W "!$#%'&()+*",).- !/ / !$#BW Y 10 W6 <;9!#">A2B D2#( %$1I$ ' +-OD >B$*#(82F W (R ∪ +∞,⊕,⊗)
B>A2 #($ 1, YodsEc|W|7Y,c-[ods:| hZl:['<9 E1C' J-= >32H> #,<' H> +-Q > #,#(, W >3# ^+-Q/ #*' 2 ' W # ' G[/ 2B2B@ ('-% $ W E1C' J- a ⊕ a = a
9 * &$>Q#$02 ' &> W / 1,+-#&2 ' . H# #>F>A(D&' 1 #( $ I H &C' 4JL$'L4366<9 68;=(/ >A2#(82BQ+-(/0O$ (-$' 2 Q#82 ' D @'08P &>
vTi =
⊕
k∈N
Mi,k ⊗ vT−1k ,
#&E,DMi,k = ci,k
>'(i, k) ∈ A W B2 Mi,k = +∞ > '0&(9K!> Q#$02 ' &> W +-$' 82' ' >382O >2QB>&23#82' &>QH#82 ' D&' > +- &&> W H#$' > #(&>H Q>3@' P?#,#5@'08P &> W 1,DC' E82K2F*82K>' H82 W
vt = Mvt−1,"
vT = MT φ .
=#82 / 1,+-#82' *= . H#5436$6$9 *6&; W '0 >Q / 1D ' E82
v = Mv ⊕ φ .^Jvv6 ¡¡b
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k'#)(* +,(# ± kUY82BZ+- @D 82Q@'0$'0 [/D@'0 *&%
t#( #(82
i
k>?2<1 #$
(M t)i,k9
Y82B^+- #=>F* 82' *OH#,<'0H#( H 4366<9 $;G>?2K1%$#( v = M∗φ W M∗ = M0 ⊕M ⊕M2 ⊕
· · · 9 0 B2 *8E #$'0&>' 1 Q 6$6$9 -9 9GY82+-> ' G (/ # #&>OD&'0D$' 2D82S1%$#&2 ' WlimT→∞ MT = +∞ 43 #5H#&2BC' D,Q<*82 2A&>Q B>KD( OD&' 82F>K>F*821%$#(,
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L ZQ/A(1)
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G/9F?>_:KnH9;?YEN<J5IG*?YA?O63 5I6K78.9F:<F=>@?YRN8.L/C:TC<F?OUZRJ=8:TCF:9p6K?jG*8:KL'<4CQ/:P5IL'<4A1?Y6T5DPe:6K6?Oi'Q?O6[3 8.LVP5Pe:TC:<F?9]i.Q/:G*?HQ1<(.N<F9;?
2]3]4]N8Q
5` 0 =8:TC:TC;C8.L/CcG/59d?NW1?H>@G6K?m6K?bG8.:L'<
2]HRN?Xi.Q/:L8.Q/Cd5I>@nL?
5Q!LZQ/A(1, 2)
i.Q/?ML8.Q/C_5(P8LCu9;?HG9;ECF?HL'<FE"U7'5IQ/RJ=?"A1?6[3 5I9;\9F?.]5(P.?R,6K?M<F8Q/9VG/59<;:?6i'Q:X6Q/:jRH89;9F?%CG*8L/Ad` 0 8>@>@?yL8.Q/CuLd3 5(P8.L/CZG/5.ClA1?"C8.6Q<F:K8L!5A>_:TC;C:K\6K?yG8.Q9l6[3 :L/C<;5L'<]m = +∞ ]N?N<rRH8>@>@?m6?m<;?C<
b1(1, 2) < m?C<r5IQ1<;8>,5<;:Ki'Q?>_?L'<dPEH9;:s/E]NL/8Q/CL?XRH5I6TRNQ/68.L/C
G5Cb1(1, 2)
`Yp8Q9JCQ/:P.8L/CM6?wG/59;RH8Q9JCy?LG9;8Io8.L/A1?Q9"Ad3 5\8.9;A 8.L59F9;:P.?~5IQLZQ/A(1, 2, 3)
v)L8Q/CZ5(P.8L/CDU"L8.QP?%5IQw9F?G9FE%C?L'<FE_6?@<;8Q9ZG/59<;:?6 | ]dGQ:TCZ5IQLZQA (1, 2, 3, 4)]
i'Q:O?%Ch<,QL/?So?HQ:K66K?]XRH5I9@:K6^A1EN<;?H9;>@:L?A1?>,5L:n9F?"QL:Ti'Q?S6K5CF86KQ1<F:K8Lz5A>_:TC;C:K\6K?x =
(1, 2, 3, 4, 5)]A1?O<;?H>@G/C
m = 16`2d?Y<;8Q9XRN8>@G6K?N<45:LC:8.\1<F?LeQ,?C<b9F?G9FE%C?L'<FEOUD63 ?HLA19F8.:<
A?6T5Zo?Q:6K6K?`54m<J5IL'<Y59F9;:KPEH?UlQL/?o?Q:6K6K?]e6K5V9;?RJ=?9;RJ=??HLG/9F8o8L/A1?Q949;?H>@8.L.<;?5QLZQ/A
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##() #$!#() /
Gn9F?.].G*8Q9Y9;?A1?%CFRH?HL/A9F?Ul6K5uo?HQ:K6K6?DCQ/:P5IL'<;?]ei.Q/:*9;?HG/9FE%C?L.<;?^6T5lCF86KQ1<F:K8L(1, 2, 3, 5, 4)
]'A1?<F?H>@G/C
18G:K9F?Vi.Q/?
m`*2d?uG/9F81RJ=/5:LSL8.QP?%5IQSLZQ/AS?HWeG/68.9FEZ?%Ch<
(1, 2, 4)]?N<B63 8LR5I6TRNQ6K?
b1(1, 2, 4) = 13 < m 8.Lu?NW1G6K89;?4A18.L/RbQLlG9F?>@:?9rs/6TCA1? (1, 2, 4)]i'Q:.o8Q9;L:<6T5^CF86KQ1<F:K8L
x = (1, 2, 4, 5, 3)]IA1?O<;?H>@G/C
14]'RN?^i'Q:*?C<4>@:?Q1W@i'Q?
m 8.L@G8'C?A8L/R m = 14`'2b3 5IQ1<;9F?
s/6KCjA1?(1, 2, 4)
o8Q/9FL:<YQL?5Q1<F9;?CF86KQ1<;:8.LA1?^<;?H>@G/C14](1, 2, 4, 3, 5)
`.2r?L8.QP?%5IQ,LZQACQ:KP5IL'<VPe:TC:<FE"?C<
(1, 2, 5)]5(P.?R
b1(1, 2, 5) = 17 ≥ m 8LxRN8Q/G?"A18L/R6?SCF8Q/C59F\/9F?G/5I9F<;5L.<@A1?
(1, 2, 5)]p?N<@6?"L8.QP?%5IQ!LZQACFQ:P5L.<@?C<
(1, 3)`8Q/C_6T5I:TCFCF8L/CV6K?M6K?RN<F?HQ/9
s/L:K9^6?VRH5I6TRNQ/6]?H<>@8L'<F9;?H9^5:LC:ri'Q?x = (1, 2, 4, 5, 3)
?C<^QLS<F8.Q9A18.L'<^6?Z<;?H>@G/C14
?C<8G1<;:>,5I6[`
(1)
(1, 2) (1, 3), b1 = 12 (1, 5)(1, 4)
(1, 2, 3) (1, 2, 4), b1 = 13 (1, 2, 5), b1 = 17
16 18 14 14
x = (1, 2, 3, 4, 5), m = 16x = (1, 2, 4, 5, 3), m = 14
Y.` ^L?MG/5I9F<F:K?yA1?y6[3 59F\/9F?yA1?MCEG/5I9J5<;:8.L?H<lEHP5I6KQ/5I<F:K8Ld]#G8.Q9u6K?yG9;8\6KnH>@?A1?P8t.57?Q9XA1?uRH8>@>@?H9JRN?ZA1?D6T52:K7Q9;?@.` `
e !1c0 2d?_G9F8.\6KnH>@?_AQP.8t.5I7.?HQ9BA1?@RN8>@>@?H9JRN?@A5L/C 15IL=5<<J5IL?C<DQ/LRH5C#G/59<;:KRHQ6:K?H9A1QlP.8t.57?HQ/9dA1?XRH8>@>@?H9JRN?b>@EN<;9F:Ti'Q?]A5L/Cd6K?i'Q?6
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L/89;>_? ‖ · ‖∞ ]1?N<;R` | `2d?BG9;8\6KnH>@?DA1QMP.8t.57?HQ/94A1?ZRN8.>_>@?9;RH?Vv>@EH<F9;:Ki'Q?B8Q"7ELEH9J5I6 | ?%Ch<Q/L"?HW1?H>@G6K?DRH6K5.CFCF:Ti.Q/?DA?BG/9F8.\6n>@?DmA1:yRH:6K?vP.8:K9Y6K5O?>@59;i'Q?@.` ` | ` ) !&4/(%#.M c0)) )4(
2r?V6K?RN<F?HQ/9G*8Q9;9J5I:<BRN9;8:K9;?li'Q?l6T5y\*89;L?b1?%Ch<B9;5:KCF8LL5I\6K?]>@5:KCP8t.8L/CRN8>@>@?HL'<
6[3 5I6K789;:<F=>@?CF?RN8.>@G8.9<;?i'Q/5IL/Ay8Ly5IQ/7>@?HL'<F?6?L8>V\9F?A1?CF8>@>@?N<JC4A1QM79J5IG=?.`^8.Q/C5(P.8L/CbG9F8.79J5I>@>@E^6[3 567.89;:<;=>@?i'Q:G/9FE%RNnA?]'?N<X9;ECF86KQyA1?%C4:KL/C<;5ILRN?CjCF:>@:K6K5:9;?CXA1?P.8t.57.?HQ9A1?VRH8>@>@?H9JRN?uA5L/C15L=/5I<<;5Ld]/>,5:KC?HLSo)5I:TC;5IL'<P5I9;:?9^6K?uL8>V\9;?
nA1?lCF8>@>@?N<;C`
m8.:KRH:.Q/LOgh?HQ_A1?49;ECFQ6<;5<JCc<hteG:Ti'Q?](A18L/L/5IL'<#6K?4L/8>l\/9F?4A?XLZQACrA1?j6[3 59F\9;?bA1?YCEG/5I9J5<;:8.L?H<^EHP5I6KQ/5I<F:K8Ld]1?Lo8.L/R<;:8.LSA?
n n
2r?ZRH5Cn = 15
G9;?HL/A"A1EgUVQ/L?D>_:KLeQ1<F?uCFQ9YQL" 0 Q/CQ/?H6[`^8QCXP.?HL8.L/CYA1?D9F?L/RN8.L.<;9F?9jRH?i'Qd3 8LS5IG/G?66K?Z63 'c&$%)*x* !p)1* `
OEH<F9;8.CFG?%R<;:P.?H>@?HL'<]r6?R5I9J5RN<FnH9;?@79;8.C;C:K?H9ZA1?6K5S\*89;L?b1
5GG/59;5#"K< 6K89JC;i.Q/? n − k?%Ch<Z7.9;5L/Ac]:64?C<D>,5QP5I:TCZA1?@>_:KL8.9F?9B6T5"68.L7Q/?HQ9uA1Q<F8.Q9ZG/59<;:?6bRN8>@G6KEH>@?L.<J5I:K9F?@A1?(`1, . . . , `k)
G/5I9(n − k − 2)
o8:TCy6K?>@:KL:>VQ> A1?Cy<F?>_GCMA1?%C59 .H<F?Cy?HL'<F9;?~CF8>@>@?N<;C9;?C<;5L'<;C]RN?H6T5@9;?HPe:?L'<OU@>VQ6<F:KG6:K?H9QL?Z?H9;9;?HQ9OG/59^QL"<F?9F>@?uAc3 89JA19;?
n`*kLSP8:<^:TRN:ri'Q?
A/5IL/C,Q/L5I6K789;:<F=>@?A1?~CFEHG5I9J5<F:K8Lx?N<"Ac3 EP5I6KQ/5<;:8.Ld]b:K6?C<:KL/A1:TCG*?HLCF5\6?Ac3 5(P.8:K9,QL?\*89;L?Zi'Q:rR5IG1<;Q9F?ZCFQ$yCF5>@>_?L'<Y6T5ZhG/='t1CF:Ki'Q?NBA1QSG9;8\6KnH>@?.`% ( 0
1 &&4*0rM)S-('**'0#M . ! ..
)8LL/8L/CQL_G9;?H>@:?9p?NW1?H>@G6K?jAd3 QL/?X<;?H6K6?j\*89;L?XG*8Q/96K?jP8t.57?Q9#A1?YRN8.>@>_?9;RH?4L8.L18.9F:K?HL'<;E`-4m<J5IL'<A8LLEXQLD<F8.Q9rG5I9F<F:K?H6
(`1, `2, . . . , `k)A? G ]%5(P.?R k ≤ n−2
]RH8L/C<F9;Q:TC8.L/Cc6K?7.9;5G=? G′ :L/AQ:<jG/5I946?CF8QCm?HL/CF?H>V\6K?A1?CF8>@>@?N<JC V ′ = V \`1, . . . , `k
]R3 ?%Ch<FUIA1:K9;?O6?7.9;5G=?Ad3 ?L/C?>l\6K?DA1?CF8>@>@?N<JC V ′ ?N<OAc3 ?L/CF?H>l\/6?DAc3 5I9/.N<F?%C E ′ = i, j ∈ E | i, j ∈ V ′ `8Q/C#L8<F8LC#5.R
(G′) 6K?XRH8f1<>@:L:K>lQ/> Ac3 Q/LV5I9;\9;?4RN8.QPe9;5L'< G′ `1v5IGG*?H6K8L/Ci'Q?46T5^L/8I<F:K8LAd3 59F\9;?SRN8Q/P'9J5IL'<@5EH<FEA1EHs/L:K?A5IL/C_6T5qe?RN<F:K8L `1]p8.L8.Q/C,5(P8.L/CV5Q/C;C:OPeQzi.Qr3 QL59F\/9F?_RN8Q/P'9J5IL'<ZA1?,RN8.f1<Z>_:KL:K>lQ> C?,RH56KRHQ6K?V?L~<;?H>@G/C
O(|E| log |E|) G/5I9D6[3 567.89;:<;=>@?A?+*D9FQ/C,5I6[` | kL5_6K5_\*89;L?D:L1oE9F:K?HQ/9F?ZCFQ:P5L.<;?
b2(`1, . . . , `k) =
t`1`2 + · · ·+ t`k−1`k+ min
j∈V\`1,...,`k−1t`kj + min
m∈V\`2,...,`ktm`1 +
5R(G′) .vh.` |
)B5ILCb6K?^RH5CXCG*ERH:K568,6K?Y<F8.Q94G5I9F<F:K?H6?%Ch<XA?^68.L7Q/?HQ9bLeQ6K6K?]CF8:<k = 1
].8L@G*?HQ1<j9;5-@L/?H9<;9Fn%Cm6KEH7n9F?>@?HL'<b6K5u\8.9FL?
b2?HL,L8<;5L.<ji.Q/?A5ILCbRN?R5C]I6K?C459 .H<F?%C `1, `2 ?H< `k, `1 A1Q<;8Q9DA18.:P.?HL'<.N<;9F?@A1:TCh<;:LR<F?%CVvURH8L/A1:<F:K8LwA1?VCFQGG*8.CF?H9Di'Q?l6K?l7.9;5G=?_55IQ~>_8.:LC^<;9F8.:KC
CF8>@>@?N<JC | ]RN?ui'Q:rA8LL?D6T5_L8QP.?H6K6?D\*89;L?
b′2(`1) = minj,m∈V\`1, j 6=m
(t`1j + tm`1) +5.R
(G′) . vh.` |
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15I9;\9F?,Ac3 QL7.9;5G=?,?C<lQL!CF8QCh79J5IG=/?lo8.9F>@EAc3 QL?G/5I9F<VAc3 QL!5I9;\9;?yRN8.QPe9;5L.<D<;8Q/C6?%C,C8.>_>@?H<;C@=8.9F>@:TCVQ/LxCF8>@>@?N<yA1:TCh<;:L/7QEL8<FEy
1]p?N<yAc3 5IQ1<;9F?SG/59<yA1?A1?Q1W59 .H<F?C
A1:KC<F:KL/RN<F?CYA18.L'<1?C<j6[3 ?NWe<F9;EH>@:<FE | `1m5I9Y?NW1?H>@G6K?]eG*8Q9Y6K?7.9;5G=?BA1?6T52#:K7Q/9F?_.` ]6[3 56789;:<F=>@?A? *D9;Q/C,(56o8.Q9FL/:<Y6?
15I9;\9;?^A1?DRN8f<j>@:KL:>,56*9;?HG/9FE%C?L.<;ECFQ9Y6K5 2#:K7Q/9F?V`1`
2d?SRN8f<,A?SRH?159F\/9F?M?C<
b′2(1) = 13`a4L9;?HP.?HL/5L.<@Uw63 5I9;\9;?MA1?S6T5 2:K7Q9;?.` /]8L
P8:<Oi.Q/?9;?H>@G6T5RH?H9b1G5I9
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(1, 3)]eRH59
b2(1, 3) = 14?%Ch<OCFQGE9F:K?HQ/9Bv)?HLyo)5:<%]eEH7.56 | Uu6T5lP56?Q9 m = 14
A1?6T5u>@?:6K6?Q9;?C8.6Q1<;:8.LM9;?HLRN8L'<;9FE?B5(P5IL'<O6T5VPe:TC:<F?ZA1?
(1, 3)`
1
4
5
3
2
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6K5 2:K7Q9;?y.` `/243 59F\9;?ZRN8.QPe9;5L.< G′ ?C<?LS<F9J5I:<;C^7.9;5.CH]16K?CBA1?Q1WS59 .H<F?%C^RN8.LL?RN<;5L.<BRN?N<5I9;\9F?D5IQSCF8>@>@?N<1C8.L'<Y?L"<F9J5I:<;CYG*8:KL.<;:6K6KEC`
0+() hK1#', # r X'c&/1%)*z #e$ #(#e$6Yt5?L!7ELEH9J5I6XG6KQ/C:K?HQ/9;Cl>@5L:KnH9;?C_A1?9;?HG9;ECF?HL'<;?H9u6[3 ?L/CF?H>l\/6?"A1?C_C8.6Q<F:K8L/C_5Ae
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\8.9FL?.` :L/CF:[]L8.Q/C^P.?H9;9F8.L/COA5L/C^6T5Mqe8.Q/CCF?RN<F:K8L.` /` @i'Qd3 8LG*?HQ1<B>@8eAEH6K:KCF?H96?uP.8t.5I7?HQ/9uA1?yRN8.>_>@?9;RH?_G5I9ZQLG9;87.9;5>_>@?_6:KLE%5I:K9F?@?HLP5I9;:T5I\6K?CD?L.<;:n9F?%CH]c?L:L'<F9;81A1Q:TCF5L'<G8.Q9uRJ=/5i'Q? i, j ∈ E QL?,P59F:T5I\6K? xij
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] i, j 7→ tij` RJ=/5i'Q?
<;8Q9%]18LS5C;CF8eRH:?DQLSP.?R<;?HQ9x ∈ 0, 1E <;?H6ri'Q? xij = 1
CF: i, j o)5I:<^G/5I9F<F:K?ZA1Q"<F8Q/9]?H<xij = 0
CF:L/8Ld` OERH:G/9F81i'Q?H>@?L.<%]QL~P.?R<;?HQ9x ∈ 0, 1E 9F?G9;ECF?HL'<F?V6?_C8.Q/Ch79J5IG/=?lA1?
G 5(t.5L'<YG*8Q/95I9/.N<;?CY6K?C i, j <F?H6TCi'Q? xij = 1`6#L8Q/C^o)5IQ1<>@5:L'<;?HL/5L.<^?HW1G9F:K>@?H9^G5I9
A?ClRH8L'<F9J5I:KL'<F?%CD6:KLE5:9;?Cu6?,o)5:<_i.Q/?x ∈ 0, 1E 9;?HG9;ECF?HL'<;?_Q/L<F8.Q9` 0 8>@>@?RJ=/5i'Q?
CF8>@>@?N<^Ac3 QL"<F8.Q9^5@?NW5.R<F?>@?HL'<^A1?HQ1WMP8:TCF:L/C]xPE9F:s/?BLERH?C;CF5:9;?H>@?L.<
∑
k∈V,k,j∈Exkj = 2 ,
G*8Q9O<;8Q1<j ∈ V ` v.` |
2r?CMRN8L'<;9;5:L'<F?%Cvh.` | L?CFQ$yCF?HL'<G/5.CUR5I9J5RN<FEH9;:TC?9,QL+<F8.Q9%]XRH596[3 ?L/CF?H>l\/6?~A?C59 .H<F?%C i, j <;?H6K6?%Cui'Q? xij = 1
G*?HQ1<uL/?,G/5C .H<F9;?@RH8LL/?NW1? ?HLo)5I:<]d8.LG*?HQ<lP8.:9Zi'Q?6K?Cx ∈ 0, 1E CF86KQ1<F:K8L/CMA1?v.` | 9;?HG9;ECF?HL'<;?HL'<?NW5.R<F?>@?HL'<M6K?CMQL:K8L/C"A1:TCgh8.:L'<;?CA?"RH:9JRNQ:<;C` s/LzAc3 EH6K:>@:KL?H9@6K?C@CF86KQ1<F:K8L/CVG/59;5.C:<F?%CH]#8LG*?HQ<@9J5gh8.Q1<F?9V6K?C@RH8L'<F9J5I:KL'<F?C
CFQ:KP5IL'<F?%CH]A1:<F?%C^A1? $%0$ & *0r ]G8.Q9Y<F8.Q1< S ⊂ V , <F?6ri'Q? S 6= ∅ ?N< S 6= V , ∑
k,m∈S,k,m∈Exkm ≤ |S| − 1 .
v.` |kLP8.:<_o)5RH:6K?H>@?HL'<yi'Q?
x ∈ 0, 1E P.EH9;:s/?vh.` | ?N<vh.` I | CF:]p?H<,CF?HQ/6?>_?L'<@CF:]m:K69;?HG/9FE%C?L.<;?,QL<;8Q9 ?HL?,*?H<]#<F8.Q1<VP.?R<;?HQ9_5C;C81RH:EyUQ/L<F8Q/9lPE9F:s/?RH?CuRH8L'<F9J5I:KL'<F?%CH]?H<,9FE%RN:KG9F81i'Q?>_?L'<]mRN8.>_>@?"<F8.Q1<x ∈ 0, 1E PE9F:s5IL'<vh.` | 9F?G9;ECF?HL'<F?"QL?SQL:K8LA:KC)gh8:KL'<F?ZA1?uRN:K9JRNQ:<;C]1:6dCFQ$@<A1?DG9;?HL/A9F?DG*8Q9 S 6[3 ?L/C?>l\6K?ZA1?COCF8>@>@?N<JCOA1?D63 QLi'Q?H6RH8L/i'Q?A1?~RN?%CRH:9JRNQ:<;CG8.Q9"8\1<;?HL:K9 ∑
k,m∈S,k,m∈E xkm = |S| ]j?H<MCF: x PEH9;:s/?6?%C
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(V C) : min∑
i,j∈Etijxij
C8.Q/CY6?%CORN8.L.<;9;5:L'<;?Cx ∈ 0, 1E , (11.63), (11.64),
?N<m8LV8\1<;:?L.<m5IQCFCF:< I<pQL?j\8.9FL/?X:KL1oEH9;:K?HQ9;?X?LVRH8L/CF:KAEH9J5IL'<#6K?XG/9F8.79J5I>@>@?46K:LE%5I:K9F?X9F?6K5IW1E?HLSP5I9;:K5\6?%CYRN8L'<;:LeQ?%C
(V C) : min∑
i,j∈Etijxij
CF8Q/Cj6K?C^RH8L'<F9J5I:KL'<F?%C0 ≤ xij ≤ 1, (11.63), (11.64).
15I6K79;E!6K?L8.>l\9;?!?NW1G*8L?L'<F:K?H6VA?RN8.L'<F9J5I:KL.<;?CA5IL/Cvh.` I | ]:K6V?%Ch<wG8'CFCF:K\6?xA1?9;ENC8.Q/A19;?"? yRH5RH?H>@?HL'<
(V C) ?HLzG9;81RNE%A5IL'<yRN8.>_>@?CFQ:<`YkL+RH8>@>@?HL/RH?U>_:KL:K>@:KCF?H9∑
i,j∈E tijxijC8.Q/Cj6?%CORN8L'<;9;5:L'<F?%C
0 ≤ xij ≤ 1?N<uv` | ]e?HLS8.Q\6K:K5L.<O6K?CORH8L'<F9J5I:KL'<F?CA1?VCF8Q/C[<F8Q/9Vvh.` I | `kL<F9;8QP.?l5I:KL/CF:rQLG9F?>@:?9 x ∈ [0, 1]E
`ckL~P(5?HL/CFQ:<F?@RJ=?H9JRJ=?9CH3 :6/?NW1:KC<F?YQL,CF8Q/Cp?L/CF?H>l\/6? S ⊂ V ] S 6= ∅ ] S 6= V ]IG8.Q9b6K?i'Q?6163 :L/EH7.56:<FE^A?Dv.` | ?C<P':K86KEH?.]IRN?Oi'Q:G*?HQ1<4CF?jo)5I:K9F?Zv)amW1?H9JRN:TRN?Z.` /` | <;9Fn%Cp? yRH5.RN?>_?L'<`q3 :6/Ld3 ?L@?NW1:KC<F?YG/5.CH]8.L_59FE%C8.6Q
(V C) `/q1:d5IQMRH8L'<F9J5I:K9;?^8L"5u<;9F8.QPEQL S <F?6di'Q?_v.` | L/?BCF8:<OG/5CjCF5I<F:TCho)5:<;?]63 8L>@:L/:>@:TC?yU"L8.QP?%5IQ ∑i,j∈E tijxij
C8.Q/CD6?%CuRN8.L'<F9J5I:KL.<;?C0 ≤ tij ≤ 1
?N<v` | ]?HL9J5gh8.Q1<;5L.<63 :L/EH7.56:<FE_A1?VCF8QCh[<F8.Q95.CFCF81RN:KEH?lU S `abL~G*8Q9JCFQ:P5L.<BRN?N<F<F?@CQ/:<;?_A1?V>@:L:>@:TCF5I<F:K8L/CY?H<O9;5Igh8Q1<JCjCFQ/RRN?C;CF:o)CYAc3 :L/EH7.56:<FE%CH]18.L"5\8.Q1<F:<Ys/L/56?>_?L'<OU_QL/?DCF86KQ1<;:8.L"A1?(V C) ` 0 ?N<F<F?y>@EH<F=81A1?yG*?HQ<_C3 :KL'<F?H9;G9;EN<;?H9Z7E8>@EN<;9F:Ti'Q?H>@?L.<u?LG/5I9;6T5IL'<lA1? .0#&4e$ `^8I<;8L/CB?L~?-,?N<
[email protected]'<F8.G?@A1?%CBCF86KQ1<;:8.L/C5.A1>@:KC;CF:\6K?@A1Q~G9;8\/6n>_?
(V C) ]?N<ZC8.:<t : x 7→ ∑
i,j∈E tijxij6T5"o89;>_?6:KLE5:9;?yi'Q?y6[3 8.L>@:L:K>@:KCF?`p2r5>_EH<F=81A1?Mi.Q/?y63 8L!5
?C;i.Q/:KC;CE?^9;?HPe:?L'<XUlA1EHs/L:K94QL/?BCFQ:<F?AERN9;8:TC;CF5L.<;?^A1?G*86Kt'<F8.G?%CP 1 ⊃ P 2 ⊃ · · · ⊃ P kL
G9F?L/AV<F8.Q1<XAc3 5I\*89JAlG*8Q/9P 1 6[3 ?L/C?>l\6K?^A1EHs/L:/G5I9 0 ≤ xij ≤ 1
?N<vh.` | `.kL,>@:L/:>@:TC?Ac3 5\8.9;AtCFQ9
P 1 ]?H<r6?4>_:KL:K>lQ> ?%Ch<#5I<<;?H:KL.<#?HLZQLuG*8:KL'<x1 `%q1: x1 ?C<#A5L/C P ]8Lu5O9;ECF86KQ
(V C) ]CF:L8.LA1EN<;?R<;?H9QL?uRN8.L'<F9J5I:KL.<;?yvh.` I | Pe:K86KEH?ZG/59 x1 9;?HPe:K?HL'<U $%.&p1' x1 A1?P]/RI3 ?C<UA1:K9F?ZU_<F9;8Q/P?H9OQ/LSA?H>@:?CFG/5.RN?
H1 G/59<;:KRHQ6K:?9Y<F?H6i'Q? x1 6∈ H1 ]?H< P ⊂ H1 ]?N<y6[3 EH<;5IG*?CQ/:P5IL'<;?"9;?HPe:K?HL'<yUw>_:KL:K>@:KCF?H9tCQ9,6K?L8.QP?%5IQG*86Kt'<F8.G?
P 2 = P 1 ∩ H1 ]8\1<;?HLeQ?HLFRN8Q/G/5IL'<hP 1 G/59 H `2b3 8Lwo)5I\9;:Ti.Q/?y5I:KL/C:XQL?CFQ:<F?AERN9;8:TC;CF5L.<;?_A?yG8.6t'
<F8G*?C_A18.L.<_8L!G*?HQ<_A1:K9F?M:L'<;Q:<;:P.?H>@?HL'<@i.Qr3 :K6KCDF5IGG9;81RJ=?HL'<P5QP8.:KCF:L5I7?yAQG*8:KL.<
8t?C<O>_:KL:K>,5I6[` )?B<F?66K?Cj<;?RJ=L/:Ki'Q?%CXG*?HQ/P?HL'<O> .H>@?Bo8Q9;L:K9Y6K5_C8.6Q<F:K8LSA1QMG/9F8.\6n>@?
89;:7.:L/56(V C)
]eC:*6[3 8.Ly?C<jR5IG/5\6?BA1?^<;9F8.QP?94Ac3 5IQ1<;9F?%CXRH8QG*?C]eA1?L/5I<FQ9;?RH8>l\/:L/5I<F8.:9;?]G?9F>@?N<F<;5L.<Ad3 5GG9;8eRJ=/?H9OCQ ,C;5I>@>@?HL'<O\:K?HLS6T5_RH6 <FQ9;?D?HL'<F:KnH9;?
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k∈V, m∈S\V,k,m∈Exkm ≥ 2 .
vh.` . |F E/'&
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>@:KL:>@:TC?9J(x)
C8.Q/CY6K?CORN8.L'<F9J5I:KL.<;?C x ∈ X,Fi(x) ≤ 0, i = 1, . . . ,m,Fi(x) = 0, i = m+ 1, . . . ,m+ q,
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]5(P?%RΛ = (R+)m × Rq ?N<
L(x, λ) = J(x) + λ1F1(x) + · · ·+ λm+qFm+q(x) .q18:<
J∗ 6T5ZP5I6K?HQ9b8.G1<F:K>,5I6K?^A1?lv` | `8Q/CX5(P8.L/CmA1EhgUDL/8I<FEA5IL/C46K5 ^?H>,5I9Ji'Q?Z /` ` i'Qd3 8LS5V<F8.Qgh8Q9JCY6[3 :KLEH7'5I6K:<;EZA1?uA1Q/56:<FEDo)5:\6K?
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supλ∈ΛL(x, λ) ≥ sup
λ∈Λinf
x∈XL(x, λ) = sup
λ∈ΛD(λ) ,
vh.` e |8.
D : Λ→ R ∪ −∞; D(λ) = infx∈XL(x, λ)
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X?C<js/L:[] D i'Q:
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R ∪ −∞ ]dRH5I9lG/5I9V?NW1?H>@G6K?y6T5"o8L/RN<F:K8L D A1EHs/L:K?G5I9Yvh.` 6$ | ?%Ch<mRN8.L/RH5(P.?b?N<pG*?HQ1<pG9;?HL/A19;?46T5P(56?Q9 −∞ v6T5P(56?Q9
+∞ Ld3 ?C<G5C#G8'CFCF:\/6?.]R5I9pL8.Q/Cp5(P8LCr?HW1RH6QV6?OR5C
X = ∅ | `(2r89JCFi'Q? J ?C<bRN8.L'P.?NW1?XA? Rn A5ILC R∪+∞ ]%L8QCL/8I<F8.L/COA18.>J = x ∈ Rn | J(x) < +∞ 6? ) A1?
J`qete>@EN<;9F:Ti'Q?H>@?HL'<%]C:
J?C<
Q/L?lo8.L/R<;:8.LRH8L/R5(P?lA1?Rn A5L/C
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##() #$!#() e
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Rn % R ∪ +∞
x ∈A18>J $%*0#$ & 4'e' 6
J x
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J?C<RN8.L'P.?NW1?]
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JA1?
Rn A/5IL/CR ∪ +∞ ]e?HLSA1:KC;5IL'<^i'Q? p ?%Ch<OQLSCF8Q/C[7.9;5.A1:K?HL'<jA1? J ?HL"QLMG*8:KL'<
x ∈ A18> J CF:J?C<
681RH56?>@?HL'<45Q"A1?%CFCFQ/ChBA1?Z6T5Vo8L/RN<F:K8L5#,L?y 7→ J(x) + p · (y − x) ]/R3 ?%Ch<FUIA1:K9;?
J(y) ≥ J(x) + p · (y − x) + o(y − x), 68.9;C;i'Q?y → x .
2d89JC;i.Q/?J?C<ORH8LeP?HW1?]1RN?H<<F?Z:KLE7.5I6K:<FED?C<^Ei'Q:KP5I6K?HL'<F?ZU
p ∈ ∂J(x)` •
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λ 6∈ Λ]eRN?i'Q:cA1EHs/L:<X\:K?HLMQL?o8L/RN<F:K8L
RN8LRH5(P?VA1?Rm+q A5ILC R ∪ −∞ `*2r89JCFi'Q? X ?C<s/L/:]dA18.> D = Λ
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Γ(λ) =5I9;7u>,5W
x∈XL(x, λ) = x ∈ X | L(x, λ) = D(x) "O :% &% 0x ∈ Γ(λ) F (x) = (Fi(x))1≤i≤m+q
P &%$',+-%" /6 D 00#
λ"
e !*#$H1%)* p8.Q9u<;8Q1<x ∈ Γ(λ)
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?C<QLCFQ979J5A:?L.<^A1? D5IQ"G8.:L'<λ`
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JA1?
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x 7→ Ji(x)?%Ch<
RN8LeP.?NW1?]eG*8Q9Y<;8Q1<i ∈ I ]/5I6K89JCH]eG*8Q9Y<;8Q1< x ∈ A18.> J ]
∂J(x) =RN8 ( ⋃
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PΛ6K5 G/9F8gh?R<;:8.L A?
Rm+q A5IL/CΛ]λ 7→ (λ+
1 , . . . , λ+m, λm+1, . . . , λm+q)
`243 /)*/%# $%0r
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λk+1 = PΛ
(
λk +ρk
‖pk‖pk
)
,vh.`e |
8.λ0 ∈ Λ
?C<lRJ=8:TC:b59F\/:<;9;5:9;?H>@?HL'<%]*8.pk
?C<uQLCFQ979J5A:?L.<Zi'Q?H6TRN8.L/i'Q?yA1? D 5IQG*8:KL'<
λk]?N<O8
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ρk → 0,∑
i
ρi = +∞ .vh.` |
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?%Ch<4568.9;Cp6K?^>,5W1:K>lQ> A1?JvG/594A1ENsL:<;:8.L@>.>@?A1?C4CQ9F79J5A1:K?HL'<;C
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]c63 5I9JR(i, j)
EN<J5IL'<B>VQL:pA1QRH8f1<cij
?H<BA1Q~<F?H>@G/Cτij`*8.Q9
s/We?9X6K?CX:TA1EH?%CH]e8LMRJ=?H9JRJ=?H9J5D6K?BRJ=?>@:LA?BRH8f1<j>_:KL:K>lQ> ?H<YA?<;?H>@G/CX<F8<;5I65IQG6QC10]
566T5IL'<^A1QSL Q/A"CF8Q9JRN? s = 15QSL Q/AMGQ:<;C p = 6
]A5ILCY6?Z7.9;5G=?
1, 8
6
2, 2
8, 1
2, 31, 6
5
3, 6
2, 1
2 41, 4
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(1, 2)RN8f<F?
c12 = 1?H<YG/9F?L/A
τ12 = 8QL:<FE%C^A1?D<F?>@G/CH`
6mL8.Q/Co)5IQ1<B<F8.Q1<uAc3 5\8.9;Ao89;>lQ6K?H9DRN?_G9;8\6KnH>@?VCF8QC6T5yo8.9F>@?"vh.` . | `p8Q/9BRH?H6T5]L/8Q/C_9;?HG9;ECF?HL'<F?9F8.L/CVQLxRJ=/?H>@:LxG/59_6K?SP?%R<F?Q9j8'8.6E?HLx ∈ RA <F?H6^i'Q?
xij = 1CF:
(i, j)5IG/G/5I9F<F:K?HL'<X5QyRJ=?H>@:KLd].?H<
xij = 0C:KL8.Ld`'2r?^G9;8\6KnH>@?BA1?G6KQ/CXRN8Q/9<YRJ=?H>@:KLAc3 QL?
CF8Q/9;RH?sUVQ/L"G/Q:<JC
p?HL"<F?>_GC^5IQ"G6KQ/C
TC3 ERN9;:<5I6K89JC
>_:KL:K>@:KCF?H9 ∑
(i,j)∈A cijxijCF8Q/Cj6K?C^RH8L'<F9J5I:KL'<F?C x ∈ 0, 1A,x9F?G9;ECF?HL'<F?BQLRJ=?H>@:KL"A?
sUp,
∑
(i,j)∈A τijxij ≤ T .
vh.`( |
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sUpuC8.Q/C
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D(λ) = infx ∈ 0, 1A
x 4 4 / s p
(
∑
(i,j)∈Acijxij
)
+ λ(
∑
(i,j)∈Aτijxij − T
)
,
= −λT + infx ∈ 0, 1A
x 4 4 / s p
∑
(i,j)∈A(cij + λτij)xij .
λsW1E]I6K?^RH56KRHQ6A1? D(λ)
9;?HPe:K?HL'<bUB9FE%C8.Q/A19;?XQL@G9;8\/6n>_?ORH6K5.CFCF:Ki'Q?OA1?^RJ=/?H>@:L@A1?^RH8f1<>_:KL:K>lQ> $/#$Z.(/)e$Z @e &p$ A5IL/Cp6K?O79J5IG/=?OA1?^RN8.f1<;C
cij +λτij]RH?^i'Q:G*?HQ1<
C?o)5I:K9F?BG/59XG9;87.9;5>@>@5I<F:K8LyA1teL/5>@:Ki'Q?.]'?H<Yi'Q:cG/9F?L/Ay> .H>@?BQLM<F?H>@G/CY6K:KLE5:9;?A5L/Cj6?RH5COAd3 QLS7.9;5G=?DCF5L/CORN:K9;RHQ:<;C]RN8>@>@?uRI3 ?C<O6?uR5CYG*8Q9O6[3 ?NW1?H>@G6K?yvh.` | ` GG6K:Ti.Q/8L/C>@5:L'<;?HL/5L.<X63 5I6K78.9F:<F=>@?A?CQ/979J5A1:K?HL'<v`e | CQ/9XRN?H<X?HWe?>@G6?.`.p8.Q9 λ1 = 0
]:K6L8.Q/Co)5IQ1<<F9;8QP.?H9#6?jRJ=?H>@:KLVA1?jRN8f<>@:KL:>VQ>
1→ 6A/5IL/C#6?X79J5IG=/?4>lQ/L:'A?CpRN8.f1<;C
cij +0τij
6
5
2 4
1
3
1
83
2
1
2
2
1
2b3 ?HL/CF?H>V\6?Γ(0)
A1?%CbRJ=?>_:KL/Cp8.G1<F:K>,5IQ1Wl?%Ch<b9;EAQ:<45Q1WVRJ=?>@:L/C(1, 2, 4, 6)
]?N<(1, 2, 5, 6)9F?G9FE%C?L'<FECY?L"<F9J5I:<;CO7.9;5.CH]i'Q:d8.L'<^G8.Q9RN8.f1<
4` :KL/C:[] D(0) = 4− 0T = 4
` )V3 5G9Fn%CY6K5b9F8.G8'C:<F:K8Lw.` /` ]e8LS5V6K?uCFQ979J5A:?L.<
p1 = τ12 + τ24 + τ46 − T = 4 .
P.?RMQLG5Cρ1 = 1
]Yv.`1 | A18LL/? λ2 = 1]p?N<,6K?SL8QP.?5Q79J5IG/=?M>VQL:^A1?%C@RN8f<;C
cij + 1τij
6
5
2 4
1
3
9
99
3
5
5
4
7
.0 8>@>@?y6?MRJ=?H>@:KL!8G1<;:>,564Ld3 5G/5.CuRJ=/5IL/7E]r8L!56K?y> .H>@?MCFQ9F[7.9;5.A1:K?HL'<
p2 = p1`rabL
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!" ##$ %&'$ ##())#$*+
G/9F?L/5IL'<OG/5I9O?HW1?H>@G6K?ρ2 = 1
]18LS5λ3 = 2
]RH?ui'Q:dA18L/L?Z6?D7.9;5G=?
6
5
2 4
1
3
17
10
4
9
8
6
15
13
.
6'tZ5RH?N<<;?bo8.:KCRN:eA1?HQWZRJ=/?H>@:LCr8.G1<F:K>,5IQ1W Γ(λ3) = (1, 3, 4, 6), (1, 3, 5, 6) ]A1?jRN8.f1< 21`
a4L,G/5I9F<F:TRNQ6K:K?H9%]6?CFQ9F[7.9;5.A1:K?HL'<pRH89;9F?%CG*8LA5IL'<m5IQ,RJ=?H>@:KL(1, 3, 5, 6)
?%Ch<τ13 +τ35 +τ56−
T = 0`%243 567.89;:<;=>@?4A1?YCQ/979J5A1:K?HL'<rC3 59F9/.N<;?XA18.L/R maxλ∈R+ D(λ) = D(2) = 31−20 =
11` 0 8.>_>@?b6K?4RJ=?>@:L
(1, 3, 5, 6)?C<#A1?m<;?H>@G/C
10 ≤ T ?N<A?4RN8.f1<11 = D(2)
]L8.Q/CcP.?HL8.L/CA?Z9FE%C8.Q/A19;?BL/8LC?Q6K?H>@?HL'<6K?ZG9;8\6KnH>@?Z9;?H6RJ=EZ2r579J5IL7.:?Ld]1i'Q:rRN8.L/CF:KC<F?uU@>,5W1:K>@:KCF?H9D(λ)
]#>,5:KClEH7.56?>@?HL'<u6?,G/9F8.\6n>@?y:L/:<;:K56vh.`( | `d2d?yo)5I:<Vi'Q?y6K?y9F?6.RJ=E,2r579J5IL7.:?Lo8.Q9;L:KC;CF?uQL?_C8.6Q1<;:8.LA1QG9;8\/6n>_?l:L:<F:T5I6?C<BRH?HG*?HL/A/5IL'<?NWRH?HG1<;:8.LL?H6[`*abL7ELEH9J5I6[]/:K6t!5QLxCF5Q1<@A?"A1Q5I6K:<;E]p>,5I:TCV6K?C
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λA1? D ]cG?QP?L'<ZCF8Q/P?HL'< .N<F9;?_>_81A1:s/E?CZC;5IL/C<F9;8G5Q7>@?L.<;?H9m6K?^RN8.f1<bG8.Q9459F9;:P.?H9pUDQL?^CF86KQ1<F:K8LvC8.Q/Ch8G<F:K>@56? | A1Q@G9F8.\6KnH>@?O:L/:<;:K56`'kLG5I9;6?Z5I6K89JCYAc3 #e0#($H%c0e$ X/*)e#e$ `
p8Q/9lRN8.L/RN6KQ9;?_RH?N<<;?y:K66KQ/C<F9J5<F:K8LA1?,6T5"9F?6K5IW15I<F:K8Lw2r5I7.9;5L7:K?HL/L?]9;?HPe:KCF:<F8LC6K?,G9;8I\/6n>_?jA1QlP.8t.57?HQ/9dA1?jRH8>@>@?H9JRN?4A/5IL/C#QLl7.9;5G=?bL/8L189;:?L'<FEbRN8.>@G6?H< G = (V , E) ](>lQL/:Ad3 QL/?Yo8L/RN<F:K8LV<F?H>@G/C
t : E → R+, i, j 7→ tij`'kL_G?Q1<bERH9F:K9F?j6K?OG9F8.\6KnH>@?jAQ_P.8t.57?HQ/9
A?YRH8>@>@?H9JRN?OCF8Q/Co8.9F>@?YA1?OG9;879J5I>@>@?46K:KLE5:9;?Y?HL@L8>V\9;?X?L.<;:?9;C]Ei'Q:KP5I6K?HL'<bU(V C)
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∑
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x ∈ 0, 1E ,∑
k∈V,k,j∈E xkj = 2,G*8Q9Y<;8Q1<
j ∈ V ]x9;?HG9;ECF?HL'<;?QLS7.9;5G=?ZRN8.LL?NW1?ZU |V| 5I9/.N<;?CORH8QPe9J5IL'<X<;8Q/CY6K?COCF8>@>@?N<JCH`
):TCh<;:L7.Q8LCDQLC8.>_>@?H<uG/5I9F<F:TRNQ6K:K?H91 ∈ V `rq1:]dG*8Q/9Z<F8.Q1< j ∈ V \ 1 ]*8.L9F?6.RJ=?V6K?CRH8L'<F9J5I:KL'<F?%C ∑
k∈V,k,j∈E xkj = 2]18.LM8.\1<F:K?HL'<^6T5Vo8L/RN<F:K8LA1Q/56? D : RV\1 → R
D(λ) = min∑
i,j∈E tijxij +∑
j∈V\1 λj(∑
k∈V, k,j∈E xkj − 2)CF8Q/Cj6K?C^RH8L'<F9J5I:KL'<F?%Cx ∈ 0, 1E∑
k∈V,k,1∈E xk1 = 2
x9;?HG/9FE%C?L.<;?Q/L"7.9;5G=?ZRN8.LL?HWe?DU |V| 5I9/.N<;?CYRN8.QPe9;5L'<X<;8Q/CY6K?C^CF8>@>@?N<JCH`
Ql<;?H9;>@? −∑j∈V\1 2λjG9;nC]8LV9F?%RN8.LL/5"T<mA5IL/C D(λ)
6?ORN8.f1<m>@:L:K>lQ/> Ac3 QL15I9;\9F?
G*8Q/9r6K?479J5IG/=? D = (V , E) ]A18L'<#RJ=/5.i'Q?45I9/.N<;? i, j ?%Ch<>lQL:K?XAQlRN8.f1< tij +λi+λj]%G8.Q9
i, j ∈ V\1 ]H?H<mA8L'<bRJ=/5i'Q?j5I9/.N<F? i, 1 ]%G8.Q9 i ∈ V\1 ](5G*8Q/9pRH8f1< t1i+λi` 0 8.>_>@?
L/8Q/Cl63 5(P8.L/CZA1EhgU~9F?>,5I9Ji.Q/EyA5IL/Cu6K5~q18Q/CCF?RN<F:K8Lz.` /`.]d6[3 567.89;:<;=>@?yA1? *D9;Q/C,(564A1?6T5qe?%R<;:8.L!.` @G*?H9;>@?N<XghQ/Ch<;?H>@?HL'<ZA1?VR5I6TRNQ6K?H9QL
159F\9;?ZA1?VRH8f1<B>@:L:K>lQ/> ?HL<;?H>@G/C
O(|E| log |E|) ]pRH?"i'Q:jL8.Q/ClG*?H9;>@?N<,A1?">_?H<<;9F?M?HL Q/P'9;?? yRH5RH?H>@?HL'<_63 5I6K78.9F:<F=>@?A?
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9F?>@59;i'Q/5\6K?^G8.Q9Y6?DG9;8\/6n>_?BA1Q"P8t.57?Q94A1?ZRN8.>@>_?9;RH?]'?C<YAQ?ZULE?H6TA?H< *Z59FGd`e2r5\8.9FL?^AQ
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hg jilknm1knoEi & :zrq K ⊂ V 0 )J #0&%
x, y ∈ K %
θ ∈ [0, 1] ( (θx+ (1− θ)y) :% u
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% " K ⊂ V
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‖x− xK‖ = miny∈K‖x− y‖.
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y = xK ± zA 0!2;B
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zTVW8H§3:
dn = ‖x− yn‖ → d = infy∈K‖x− y‖ >A@1K<1S n→ +∞.
?lE<AGFVWE<DZ>A@3:yn :TDRGZ@<3:DF@38HGW:S: ¤K@3BnAAip<0@GW8;L;8;DWK<AGZL~KDR [zTGFVW8;:CS@0I VWEQS@8;G¡DFBKL~K8HVW:X8;L
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2(yn + yp)‖2 + ‖1
2(yn − yp)‖2 =
1
2(d2
n + d2p).
oVX3I3KVxB=E< :Ty 8;GWzS:KX(yn + yp)/2 ∈ K
X:TG ‖x− 12(yn + yp)‖2 ≥ d2 i1pKvVoBTE<3DFz=>Q@:=<AG
‖yn − yp‖2 ≤ 2(d2n + d2
p)− 4d2,
B=:C>A@38[E<AGFVW:>A@3:yn :TDRGZ@<3:JDF@8;GW:S : ¤K@3BnAAi 9E[[:
V:TDRG@<0:=DFI3KB=:S:o8;L~O&:#VFGXY8;L9:=DRG
B=E[IL~:#GXSE<3BCL~KDF@8;GW:yn :=DRGJBTE<Q:TVWm:=<AGW::TVWDZ@<3:CL;8~[8HGW:
xKipKvVK8~L;L;:=@ VWD=X%BTE[[:
K:=DRG
u:TVW[zXQBT:TGFGW:kL;8;[8HGW:xK
KI3I3KVFGW8;:=<AGxKi pKVxB=E<3DFz>A@3:T<QGXQE<lK
d = ‖x− xK‖iA9E[[:]GWE@GW:kL~K
DF@38HGW:_[8;<38;[8;DWK<AGW:k:=DRGxBTE<Q:TVWm:=<AGW:X3L~KZL;8;[8HGW::=DRGxuEVWB=z=[:=<AGx@<38~>A@3:X :TGxK
:=DRG]L;:_DF:=@3L'I&E8;<AG]S:[8~<8;[Z@3[S:
miny∈K ‖x− y‖ i¢ E8;GxK ∈ K
B=:¡I&E8;<AG\S :¡[8;<8~[@3[i3YE@ VkGWE@ Gy ∈ K
:TGθ ∈ [0, 1]
X"I3KV_B=E<Q:Ty 8HGWzS :KX
xK + θ(y − xK)KI3I3KVFGW8;:=<AG]
K:TG]E<K
‖x− xK‖2 ≤ ‖x− (xK + θ(y − xK))‖2.<lSz=:TL~EI3I3K<AGxL;:GW:TVW[:_S:S VWE8HGW:X38;LQ8;:=<AG
‖x− xK‖2 ≤ ‖x− xK‖2 + θ2‖y − xK‖2 − 2θ〈x− xK , y − xK〉,B=:\>A@38'SE<3<3:kI&E@V
θ > 0
0 ≥ −2〈x− xK , y − xK〉+ θ‖z‖2.<NK8;DWK<AGtGW:=<1SQVW:
θ:TVWD X E<CEOGW8;:=<AGtL~K¡B=KVKB#GWzTVW8;DWKGW8;E<0sR£Qi;vw#iI@]z=B=8;IVWEQ>A@:=[:=<AGXDFE8;G
xK>A@8
zTVW8H§1:BT:TGFGW:_BKVKBTGWzTVW8;DWKGW8;E<i"YE@VxGWE@ Gy ∈ K
E<K
‖x− y‖2 = ‖x− xK‖2 + ‖xK − y‖2 + 2〈x− xK , xK − y〉 ≥ ‖x− xK‖2,B=:\>A@38"IVWE@3:_>A@:
xK:TDRGoO8;:=<L~KZIVWEvPR:TBTGW8;E<EVFGWnEmE<1KL;:S :
xDF@V
Ki
g \i\kHm;knoEi & &C V
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x5 8
p 8 % ( 2 > 7",2 M#-%
xn = 〈x, en〉" 78
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n≥1
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v> % : -%
x =∑
n≥1
〈x, en〉en.
h*`"r =g=,v+gfd`"r2 ¢ M 8;L:Ty 8~DRGW:_@<3:_DF@8;GW:(xn)n≥1
S:VWz=:TL~D]GW:=L;L;:>A@:limp→+∞
∑p
n=1xnen = x
XKL;EVWDxI1KvVxIVWEvPR:=BTGW8;E<DF@V
ensN:TGxB=E[[:oB=:TGFGW:DF@8HGW::=DRGxI1KvV]SzT§3<38HGW8;E<U8~<3Sz=I&:T<1S3K<AGW:kS :
pw%E<K
xn = 〈x, en〉XB=:]>A@38 IVWE@3:tLNM @3<8~BT8;GWzoS:tL~K_DF@38HGW:
(xn)n≥1iI?lE<AGFVWE<3D[CK8;<AGW:=<1K<QGDFE<:Ty 8;DRGW:=<3B=:i
pKvVSzT§3<38HGW8;E< S'M @3<:lO3KDF:n8~L;O&:TVFGW8;:=<<3:XI&E@VJGWE@Gx ∈ V
:TGCI&E@ VJGWE@ Gε > 0
X%8;Lx:Ty 8;DRGW:yX
BTE[ZO8;<1K8;DFE<L;8;<3z=K8HVW:Z§1<8;:S :=D(en)n≥1
X&GW:=Lp>A@3: ‖x − y‖ < εi BoVB=:JK@¤n3z=EVW=[:£Qi;i; lE<
I&:T@G9SzT§3<38HV¤@<3:]KII3L;8;BKGW8;E<UL;8~<zK8HVW:Sp
>A@8XA_GWE@G9I&E8;<AGz ∈ V
XuK8HGtB=EVFVW:=DFI&E<3S VW:Spz = zW
XE
zW:=DRGYL~KxIVWEvPR:=B#GW8~E<ZEVFGWn3EmE<3KL;:¤DF@VcL;:%DFE@3DR :=DFI1KB=:¤:TBTGWEVW8;:=L
W:=<m:=<3S VWzI1KvVYL;:=D
pIVW:=[8;:TVWD
:=BTGW:=@ VWD(en)1≤n≤p
i<U:#VFGW@US :sR£Qi;vw#X(z−Spz)
:=DRGtEVFGWn3EmE<3KL\GWE@Gtz=L;z=[:T<QG¤S:WXQSE<3B
:T<lI1KvVFGW8;B=@3L;8;:TVkSpz
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‖z‖2 = ‖z − Spz‖2 + ‖Spz‖2, sR£Qi AwBT:\>Q@8"8~[IL;85>A@:
‖Spz‖ ≤ ‖z‖∀z ∈ V.9E[[:
Spz:=DRG_:T<3m:T<1S VWzZI1KVL~:TD
(en)1≤n≤pX':TG\>A@3:
(z − Spz):=DRG_EVFGWn3EmE<3KL%UBn1KB=@3<S:=D
(en)1≤n≤pX3E<lzTVW8H§3:NKBT8~L;:=[:T<QG]>A@:
Spz =
p∑
n=1
〈z, en〉en.
YE@VpDF@DWK[[:=<AGcmVK<1S"XE<JK
Spy = yBKV
y:TDRG@3<3:%B=E[O38;<1K8;DFE<ZL;8;<3z=K8HVW:¤§1<8~:9S:=D
(en)n≥1i
pKvV]B=E<DFz>A@3:=<AG
‖Spx− x‖ ≤ ‖Sp(x− y)‖+ ‖y − x‖ ≤ 2‖x− y‖ ≤ 2ε.
k<U:=<S zS@8;GxL~KJB=E<Q:TVWm:T<3B=:\S :Spx
:TVWDxiACk:_B=:TGFGW:_B=E< :TVWm:=<B=:_:TGoS:_LNM z>A@1KvGW8;E<sRv£Ai AwtE<
GW8HVW:lim
p→+∞‖Spx‖2 = ‖x‖2,
>A@8'<'M :=DRGxVW8~:T<S'M K@ GFVW:_>A@3:L~KZdEVW[Z@L;:_S:DFE[[CKGW8;E<sRv£Ai w#X&S 8;GW:\S :_pKVWDF:TKLi
d 2?R&,:"`5@21'#BD24$Ld #'(2%)#AC"`2%T(,/9:)E2?+@5@,:2?'('(24$&>?'#!3%)#+-A!)(9/28' d 2;"|5*<#A!"CAJ+IAJ'65-,/28<E! (+5@C #"f9:21"21"@<#A!B?21"8$&2Ef,:9:)M21+`5;[ A <(+-!<EC"@,/5@,:!'"` (,:0QAJ'65@2$&C'('(2 ('#2 BD!'M$&,_5-,/C'S'#>1BD2;"@"-AJ,:+-2 2?5%"` $@"@A!'C5-28$Ld 2?R&,:"`5@21'#BD28$Ld ('(28)#AC"`2=T(,:9:)M21+`5-,/21'('(2]$&>1'(!3 )(+-A!)(9:2
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Uh*`"r =gT,+gfd`"r2 ¢ E8HG(vn)n≥1
L~KluK[8;L;L~:S E<AG¡L~:CDFE@3DR :=DFI1KB=:C:=BTGWEVW8;:=L9:=<m:=<3S VWzJ:TDRGJS :=<3DF:S3K<D
Vs>A@8;GFGW:¡JVW:=<Q@3[z#VWEGW:TV]L;:=D
vn:TG_:=<DF@I3I VW8~[:#VkB=:#VFGK8;<3D=X&E<I&:=@G]GWE@PRE@ VWDDF@3II&EDF:TV
>Q@'M 8~L;DoDFE<AGkL;8;OVW:TDw#i'pKV_KI3IL~8;BKvGW8;E<S@I VWEAB=zSzS: BoVK[J ¢ B#n[8~S G]JB=:TGFGW:\NK[8;L~L;:X"E<EO GW8~:T<QG@3<3:uK[8;L;L~:EVFGWnE<EVW[CKL;:
(en)n≥1i9E[[:
[v1, · · · , vn] = [e1, · · · , en]XE<:=<|SzS @38HG>Q@:CL~:
DFE@3DR :=DFI3KB=:_:=BTGWEVW8;:=L':=<m:=<3S VWzI3KV(en)n≥1
B=E<3B=8~S:Kv:=BB=:=L;@8':T<3m:T<1S VWzkI1KV(vn)n≥1
>A@38&:TDRGS:=<DF:_S3K<3D
Vi5CE<3BX
(en)n≥1:=DRG]@3<:O1KDF:_n38;L;O&:TVFGW8;:=<3<:i
g \i\kHm;knoEi & V
W
% $ % $ 6 s % 3% " q !"8 ' %
A6V
% W
7 1 ! 5 0 C
5
‖Ax‖W ≤ C‖x‖V ∀x ∈ V. P8 01 0
C5 :% 2 7 ++: 9 0 5:% 6 ( !"8 0 0
8 %A % 7
‖A‖ = supx∈V,x6=0
‖Ax‖W‖x‖V
.
C (0!21'65X!' &5-,/9:,:"@2?+IA*9:A%$&>?'(C3,/'#AJ5@,:!'>176 (,:0QAJ9:2?'65@2f$ad C<M>1+-AJ5@2? #+XAJ 9:,/21 $ad A!<(<(9:,:B1AQ5@,:!'21'65@+-2 2;"`<#ACBD2;"\$(2E,/9:)E2?+@5cFn!'4<#AJ+-9:2?+IAfAJ,:'#"@,($Ld C<M>1+-AJ5@21 (+l9:,:'(>1A!,/+-2eB?!'65@,:'O <#9/ &5 J5X76 (2$Ld A!<1<#9/,B?AJ5@,:!'t9:,/'(>;AJ,:+@2 BD!'65-,/'O (2 Y~[ ,
V2;"|5*$&2 $&,:32?'#"@,:!'b^#'(,:2!KAJ9:!+I"e5@C &5@2;"9:21"NAJ<(<#9/,B?AJ5@,:!'#"
9:,:'(>1A!,/+-21"N$&2V$(AJ'M"
W"@!'65*B?!'65@,:'O (21"1K#3A!,:"*BD2 'ad 2;"|5*<(9: #"*06+IAJ,\"`,
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,:'&^#'#,/2C[
g \i\kHm;knoEi & V
$ % *6s % % " 0P0 :V ′ $ ( 1 L%
-% % !oEim1k iLlO &%V '`u-'( % ( 1 66$ 1 %
0 6V
% R"a :%L",2 5:% *
L ∈ V ′
‖L‖V ′ = supx∈V,x6=0
|L(x)|‖x‖ .
)*AJ'#"4 ('y21"@<#ACBD2h$(2 E,/9:)M21+`59AS$( #AJ9:,_5-> AS ('(2 ,:'C5-2?+-<(+->D5-AJ5@,:!'5-+@.;" "@,:34<#9/2 !+ !B?2GAJ 5-T(>[email protected]$&2 c,:21"4Fn0!!,:+ 9:2N5@T(>1!+-.?32M[ [/N$&2 U QVHY 7C #,E<E2?+-32D5c$Ld ,:$&21'65@,/^#2?+ ('h21"@<#A!B?2N$&2E,/9:)E2?+@54"@!'b$& #A!9a<#A!+e,"@!3!+-<(T(,"`32C[
% g oM O 1 Opx|\tQ6l g 1Oim;&m1k oMi\ k 66 V
% %s D % %! 0
V ′ - "N 0&% *-% % L ∈ V ′
,5 y ∈ V 5
L(x) = 〈y, x〉 ∀x ∈ V. 7 ‖L‖V ′ = ‖y‖ "
Uh*`"r =gT,+gfd`"r2 ¢ E8HGM = KerL
iQ¥ L1D=M Km8HG]S'M @3<CDFE@DR :=DFI1KB=:od:TVW[zkS :VBKvV
L:=DRGtBTE<AGW8;< @:i
¢ 8M = V
X&KL;EVWDL:=DRG]8~S:=<AGW8~>A@3:=[:=<AGt<Q@L~L;:_:TGoDF:=@L
y = 0BTE<QQ8~:T<QGi ¢ 8
M 6= VX1KL;EVWDo8;Lc:Ty 8;DRGW:
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)) e
z ∈ V \Mi ¢ E8HGxKL;EVWD
zM ∈MDWK¡I VWEP :=BTGW8;E<DF@ V
MiA9E[[:
z<'M KI3I3KVFGW8;:=<AGtI3KDt
MXz− zM:TDRG¤<3E<<Q@3L :TGXAI1KvV9L;: ¤n3z=EVW=[:\v£Qi;i~ X :=DRG%EVFGWnEmE<1KL"_GWE@G%z=L;z=[:=<AG9S:
Mi ¢ E8HG§1<3KL;:=[:=<AG
z0 =z − zM
‖z − zM‖.
E@Gx:=BTGW:=@ Vx ∈ V
I&:=@GxD=M z=B#VW8;VW:
x = w + λz0K:TB
λ =L(x)
L(z0).
k<zTVW8H§3:JK8;DFz=[:T<QG_>Q@:L(w) = 0
XSE<3Bw ∈ M
i 9:TB=8%I VWE@:Z>A@:V = Vect(z0) ⊕M
ipKvVS zT§1<8HGW8~E<S :
zM:TGoS:
z0XE<lK 〈w, z0〉 = 0
X3BT:\>Q@8"8~[IL;85>A@:
L(x) = 〈x, z0〉L(z0),
S"M E L;:VWz=DF@L;GKvGoSz=DF8HVWz\K:=By = L(z0)z0
sNLNM @3<38;B=8HGWz_:=DRG]z=Q8~S:T<QGW:w#i5CM K@GFVW:I3KVFGXE<K
‖y‖ = |L(z0)|,:#G
‖L‖V ′ = supx∈V,x6=0
|L(x)|‖x‖ = L(z0) sup
x∈V,x6=0
〈x, z0〉‖x‖ .
':[CKvy 8~[@3[ S3K<3DoL~:ZS:TVW<8;:TVkGW:TVW[:S:¡B=:TGFGW:z=mKL;8;GWzZ:=DRG_KvGFGW:=8;<QGI1KVx = z0
X&B=:J>A@8Y8;[I3L;8~>Q@:>A@: ‖L‖V ′ = ‖y‖ i
'+@>;"` (9/5-AJ5821"-"`21'C5-,/219 <MC (+8<E! (0C!,:+]$&>?3!'65-+@21+*9:2 213432 $&2 2MAJ+ ,QA!"4 #[ ([/tFH (5@,:9/22?'b!<&5-,/3,"@AJ5@,:!'MYc21"`5f9:A<(+-!<#+@,:>D5-> C>?C34>?5@+-,:76 (2]"@ (,:0QAJ'65@2]76 (,a2;"|55@! (54nAJ,/5NBDC'&H!+-32=9 d ,/'65- (,_5-,/C'a[
] g oMQ6 1 &zx g j&Q&m1k oMi/\iy oMk im46m/\i!oEiM'a( K
#-%$' 0 ) 6 #% . #6 %4s D %
V x0 /∈ K " O :% ! 4! % % *6
V5 9 :% %
x0K '`u-'( % 5
:% * 8 %L ∈ V ′ α ∈ R
5
L(x0) < α < L(x) ∀x ∈ K .F|&[ !Y
h*`"r =g=,v+gfd`"r2c¦kEGWE<3DxK
L5K\IVWEvPR:TBTGW8;E<S :x0DF@ V
Ki @8;DW>Q@:
x0 /∈ KXQE<UK
xK −x0 6= 0i
¢ E8HGLL~KuEVW[:L~8;<zK8HVW:SzT§3<38;:I&E@ VCGWE@G
y ∈ VI1KvV
L(y) = 〈xK − x0, y〉X9:TGDFE8;G
α =
(L(xk) +L(x0))/2i'CM KIVW=DksRv£Qi;w#XE<lK
L(x) ≥ L(xK) > α > L(x0)I&E@ V9GWE@ G
x ∈ KXQB=:>A@8
KB#n=:L~KS z=[E<DRGFVKGW8;E<'i
! #"A! (+@C'#"X21'&^#' )E21"@!,:' <E! (+$&>134C'65@+-2?+e9/2*T(>[email protected]$&2 1 ,/' ,Cf",O,l!![ [:$Ld ('(20 A!+@,AJ'65-2=$( b5@T(>1!+-.?32]$&2%"@>?<#A!+-AJ5@,:!'aKOnA!,:"-AJ'65f,:'65@2?+-0!21'(,:+f9:A'#J5@,:!'b,:3<MC+`5IAJ'65@2%$ad TOPO<M21+<(9:A!' $Ld AJ<(<( #, [ ,
K21"`5N (' B?!'O0!2?RO2]$ad (' 21"@<#A!B?2]$&2LE,/9:)M21+`5
VKE!' AJ<#<M219/9:2 'L 6 (i
:&lj\k $&2K21' ('<E!,:'65
x #'T6PO<E2?+-<(9AJ'A#'(2
H = y ∈ V | L(y) = α KlA 0!2;BL ∈ V ′ K L 6= 0
K&2D5α ∈ R
K&5-2?9 76 (2α = L(x) ≤ L(y)
K(<E! #+5@! (5y ∈ C [
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%$ ( #
oMQo n(kn 1 #y 'L C n(i/(ljlk >! &4!#% 8 0 . 8% %L !) %
K #6 0%=s % M%0 2 "
Uh*`"r =gT,+gfd`"r2 ¢ E8;Gx@3<I&E8;<AGtdVWE<AGW8~#VW:_S:
K 8~L":Ty 8;DRGW:_KL~EVWDt@3<:kDF@38HGW:
xn ∈ V \KXK:=B
xn → xi&': ¤nz=EVWT[:S :\DFz=I1KvVKGW8;E<v£Ai;i;GCuE@ VW<38HGI&E@V]GWE@ G
n@3<3:_dEVW[:\L;8~<zK8HVW:
Ln<E<
< @L;L~:]GW:=L;L;:_>A@3:Ln(xn) ≤ Ln(y)
I&E@V¤GWE@Gy ∈ K
ik<I&:=@ GtB#nE8;DF8HVLn
S:k<EVW[:1iA9E[[:
V:=DRGxS:kS8;[:=<DF8~E<§3<38;:XQL~K\DFI3n3#VW:k@3<8HGWzS:V ′ :=DRGxBTE[I1KBTGW:XQ:TGt>Q@8HGFGW:k\VW:=[I3L~KBT:TV Ln
I3KVt@<3:DFE@3DR DF@8;GW:XE<I&:=@GDF@I3I&EDF:#V9>A@:
LnB=E<Q:#VWm:t:TVWD@<3:9dEVW[:tL;8;<3zK8;VW:
LX>A@8:=DRG<E<Z<Q@L~L;:XBKvV
S:9<EVW[:1i¥ LADF@ JGp[CK8;<AGW:=<3K<AGpS :¤I1KDFDF:TVoL~KoL;8;[8HGW:¤SK<3D
Ln(xn) ≤ Ln(y)XvB=:t>A@:%LNM E<]PR@3DRGW8H§3:
:=<lz=B#VW8~K<AGLn(xn) = Ln(xn − x) + Ln(x)
:TGx:T<l<3EGK<AGx>A@3: |Ln(xn − x)| ≤ ‖Ln‖‖xn − x‖ =‖xn − x‖ XQI&E@VxEOGW:=<8HV L(x) ≤ L(y)
>A@:=L">A@3:kDFE8HGy ∈ K
i5Oo8;<3DF8NXH = y ∈ V | L(y) = L(x):=DRG]@3<lnA I&:TVWIL5K<lS"M KI3I@38S:
K:=<
xi
(C)*+,-c.)0/'2N/12'/$KI VW:=@3:S@ 9EVWEL;L~K8HVW:v£Ai~i £ l<:D=M zTGW:=<3SI1KDk:=<S8;[:=<3DF8;E<8;< §1<38;: S3K<DB=:tBKD=XL~KDF@38HGW:
LnKO38;:=<J@<3:¤KL;:=@ V9S"M KSnzTVW:=<B=:
LI&E@VL~KkGWEI&EL;Em8;:]uK8;O3L;:X[CK8;DpVW8~:T<J<3:
S8HG>A@:L 6= 0
iN9E[[:\B=E<AGFVW:¡:#y:T[I3L;:X1BTE<3DF8~S zTVWE<DLNM :T<3DF:=[O3L;:KS :=DoDF@8HGW:=DS :
`2JGW:TVW[:=D
I&EDF8HGW8HuD_E@<Q@3L;D=X">A@8Y:=DRG@<B=E<Q:Ty :_u:#VW[zZS :`2S"M 8;<AGWzTVW8;:=@ VQ8~S:i E@GkI&E8;<AG_S :
K:TDRG_SE<3B
I&E8;<AG5VWE<AGW8;TVW:X[CK8;DDF8x:=DRGJ@<3:CDF@38HGW:S:
`2GW:TVW[:=D¡DRGFVW8;BTGW:=[:=<AGI&EDF8HGW8HuD=X8;L¤<M :Ty 8;DRGW:CI1KD
S'M nAI&:#VWI3L~K<lS'M KII3@38":=<xi •
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0 2?5`5-2]AJ'('(2?R&2*2;"|5B?!'#"-A!B?+@>12N49Wd A!'#AJ9:P&"`2*'O (3>?+-,:76 (2=3AQ5-+@,BD,:2?9:9/2CK&2D5f<(9: #"c<#+@>;BD,"`>13421'65AJ &RbAJ9 !!+-,/5@T(321"f &5@,:9/,"@>1"f<E! #+f+@>;"`C #$&+-2]$&21""`P&"`5@.?32;"f9:,/'(>;AJ,:+@2;"]FH'#J5-A!33421'65NBD21 &Rh,"@"@ #"$&2t9:A3>D5-T(&$&2 $&2;">?9:>?32?'65-"4^M'(,:"IY~K 2D5 <E! (+ B1AJ9BD (9:2?+9:21"0QAJ9:2? #+-"42D5 0C21BD5@2? #+-"4<(+-!<(+-21"$Ld #'(2 3AQ5-+@,BD2XA! &5@ AC$ |!,:'65@2fFn,/'65@21+@0C2?'#A!'65l$(AJ'M"a9:2 B?A!9:B? (9C$&2;"a3&$&21" <(+-!<(+-21"a$ad ('%34&$&.19/234>;B?A!'(,:76 (2 Y~[1j! #+c<(9/ M"f$&2]$&>?5-AJ,:9"'(! #"+-2?'O0CQP!!'M" A! &RG! #06+IA C21"NU V 2D5]U QV [
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x ∈ Rn Fn"@,L2?9:9:282DR&,:"`5@2 Y$&2=9 d >176 #AJ5@,:!'bAJ9 !>1)(+@,76 (2="` (,:0QAJ'65@2
Ax = b,F` [:;Y
AAJ<(<#A!+`5-,/21'65 ]9Wd 21'#"@2?3%)#9/2 Mn(R)
$&2;" 3AJ5@+-,:B?21"X+->?219/9:21" B?A!+@+->?2;"X$Ld !+I$&+@2nKC2?5
b ∈ Rn21"`5h (' 0C21BD5@2? #+hAJ<#<M219/> "@21B?!'#$ 3213%)(+-2![e,/21' ".-(+;Ke!' $&,"`<EC"@2 $&2;"hB?>?9:.?)(+-21" H!+-3% (9:21"$&2 0 +IAJ32?+ 7C #, Kj<E! (+ ('(2G3AJ5@+-,:B?2 ,/'O0C2?+I"`,:)(9:2
A$&2 BDC9/C'('(21"
(a1, · · · , an)K\$&C'('(21'C549:A
"`C9/ &5-,/C' $(2 F| [/ YX<MAJ+f"@21"cBDC3<M6"@A!'C5-21"
xi =det(a1, · · · , ai−1, b, ai+1, · · · , an)
detA.
N'<MC (+-+-A!,_5 B?+@C,/+-2 76 (2hB?2D5@5@2hH!+-3% (9:2 2DR&<(9:,BD,/5@2h"@ $5<MC (+ '#C"])E21"@!,:'#"1[(1 AJ,"],:9e' d 21'21"`5 +-,/21'aKcB?A!+G9:21"hH!+-3% (9:21" $&2 0 +-A!32?+b"`C'C5 m1oMm1 OOimyk il(\&\m g O <E! #+bB1AJ9BD (9:2?+2 B?ACBD2?321'C5j9AN"@!9: &5@,:!'4$Ld ('4"@P&"|5-.?32e9/,:'(>1A!,/+-2![ X'42-,2D5;KQ9/21 (+ BD/-&5 2?']5-2?3<#" $Ld 2DR&>;BD &5-,/C'"` (+ (' !+I$&,:'#AQ5-2? (+e21"`5c<(+-!T(,:)(,/5@,/10&,/9nA! &5fB?A!9:B? (9/21+
n+ 1$(>D5@21+@3,:'#AJ'65I" 2?5f"`,!'h &5@,:9:,:"@2=9:A
34>?5@T(&$&2*$& h$(>?0!219/C<(<E2?32?'65 <MAJ+ 9:, !'(24FH! hBD!9:!'#'(2;Y BIT#A!76 (2$(>D5@21+@3,:'#AJ'65$&2?3AJ'M$&2<(9: #"$&2n!3% #9_5-,/<(9:,B?AQ5-,/C'#"1[ 5@!5-A!9 KO9A]3>D5-T(&$&2*$(2 0 +IAJ32?+ '(>1B?21"-"`,/5@2*$&!'#BN<(9: #"e$&2
(n+ 1)!3% (9/5@,:<(9:,:B1AQ5@,:!'M"?KjBD2G76 (,e21"`5 ,:'(21'60O,"@A !2;AJ)(9:220l<MAJ+42DR&2?3<(9:2 <E! (+n = 50
K\"`,e9/2;" B1AJ9BD (9""`C'C5c2,E2;B~5@ #>1"c"@ (+c ('b!+I$&,:'#AQ5-2? (+eH!'MB~5@,:!'#'#AJ'65f 43*, CA #!<M"*Fn ('b3,/9:9/,AJ+I$ $Ld C<M>1+-AJ5@,:!'#"<#AJ+f"@21B?!'#$&2 Y~K&9:2=5@2134<M"f$&2]B?A!9:B? (9L21"`5$&289Wd C+-$(+@2=$&2
4.8 1049 A!'('(>?2;"65 1.132]"`,LC' (5@,:9/,"`2 ('(2G321,/9:9/21 (+-2G3>D5-T(&$&2h<E! #+4B1AJ9BD (9:2?+ 9/2;" $&>?5@21+@3,:'#AJ'65-" 9A 3>D5-T(&$&2h$&2 0 +IAJ32?+ 'ad 2;"|5<#A!" BD!3<E>D5-,_5-,/0C2G<#A!+]+-A!<(<E!+@5 AJ &RAJ9 !C+@,/5@T#342;"]76 (2G'#! #" AJ9:9/C'#"80!C,/+ Fn$&C'65%9/2h'(!3 )(+-2$Ld C<M>1+-AJ5@,:!'#""@2?+IA 5|P6<#,:76 (213421'65c$(289 d !+I$&+-2*$&2
n3 Y~[! #"N0!21+@+-!'M"f$&2? (R 5|PO<E21"*$&2%3>?5@T(&$&2;"N$&2 +@>;"`C9/ (5@,:!' $&24"`P&"`5@.?32;"f9:,/'(>;AJ,:+@2;"0BD219/9:21"$&,_5-21"e$&,:+@2;B~5@2;"?KOBJd 21"`5$&,:+@2N7C #,M<E2?+-32D5`5-2?'65c$&2NB1AJ9BD (9:2?+ 9:A "`C9/ &5-,/C'2DR(A!BD5@2N2?' ('h'(!3 )(+-2^#'(,=$Ld C<M>1+-AJ5@,:!'#"1KX2?5 BD219/9:21"h$&,/5@21" kHm g Q&m;kHE6 KBJd 21"`5$&,/+-2t76 (,*B1AJ9BD (9:2?'65G #'(2 "@ (,_5-2S$&2"`C9/ &5-,/C'#"cAJ<#<(+@&BIT(>121"76 (,aBDC'O0!2?+ !2N0!21+-"9A"`C9/ &5-,/C'h2?R(A!B~5-2
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Cn " v2 5 :% % :%6 u 7 :% * -% ",2 M#-%
‖A‖ = supx∈Cn,x6=0
‖Ax‖‖x‖ .
A!+]AJ)( #"%$&29:A!' CA !24!''(J5-2 $&29Ab3.132nA DC'S9:21"]'(!+-321"=0!2;B~5-!+-,/219/9:2 2?5]3AQ5@+-,BD,:2?9:9/2"@ ()E!+I$&!'#'(>?2C[JN' 0!>1+@,/^#2AJ,"`>13421'65j76 ad ('(2'(C+@32c"` #)MC+-$&C'('(>12 AJ,:'#"@,($&>D^M'(,/22;"|5 )(,:2?'4 ('(2'#!+-342=3AQ5-+@,BD,:2?9:9:2="@ (+ Mn(C)
C "@ (+ Mn(R)[
O C ‖ · ‖ 5 5:% % * :%6 &% Mn(C)
"=%"* &% %
A :% ‖A‖ 20 ! ",2 M#-%
‖A‖ = supx∈Cn,‖x‖=1
‖Ax‖ = supx∈Cn,‖x‖≤1
‖Ax‖.
" ! xA ∈ Cn, xA 6= 0
5 ‖A‖ =‖AxA‖‖xA‖
.
"3 0 % % ; :% 2> ‖ Id‖ = 1. "
AB
% 0 % "ev2 ‖AB‖ ≤ ‖A‖ ‖B‖. g oEil1m?Q&m;knoEi & 2e<(+-2?3,:2?+j<MC,/'65j21"`5 >?0O,:$(2?'651[ 2"@21BDC'#$%"@2$&>?3!'65-+@2 2?'4+@213A!+-76 #A!'6576 (2 9:AHC'#B~5-,/C']BDC'C5-,/'O (2 ‖Ax‖ AQ5@5@2?,:'65l"@!'83AQR&,:3% (3"` #+a9/2 BD!3<#ACB~5 x ∈ Cn, ‖x‖ = 1 [ 245@+-!,"`,:.?3221"`58>?0O,:$(2?'651KL5IAJ'#$(,:"876 (29:2 7C MAQ5@+-,:.?32 2;"|5] #'(2BD!'M"`>;7C #2?'#B?2$(249Wd ,/'(>CA!9/,/5@>‖ABx‖ ≤ ‖A‖ ‖Bx‖ [
O&Ll &C C 9l2DR&,"|5-2%$&21"'(!+-321"c3AJ5@+-,:B?,/219/9:21"f76 (,l'(2%"@!'65"@ ()MC+-$(!'('(>121"fGAJ 1B? ('(2'(C+@320!2;B~5@C+@,:2?9:9:2![ 232?,:9/9:2? (+82?R&2?3<(9:2421'21"`5=9Ah'(!+-32421 #BD9:,$&,/21'('(2$&>?^#'(,:2<#AJ+
‖A‖ =√
∑ni,j=1 |aij |2
[ X' 2,E2?51K#C' A ‖ Id‖ =√nKMB?2%76 (,l' d 2;"|5<MA!"f<EC"-"@,/)(9:2]<E! (+N ('(2
'#!+-3428"@ ()E!+I$&!'('#>?2![ •N' '(!5@2 ‖A‖p
9A'(C+@32b3AQ5@+-,BD,:2?9:9/2 "@ ()E!+I$&!'('#>?2t9:A'(C+@32b0!2;B~5-!+-,/219/9:2t"` (+Cn
$(>D^#'(,:2<E! (+p ≥ 1
<MAJ+ ‖x‖p = (∑n
i=1 |xi|p)1/p Kf2D5 <E! #+p = +∞ <#AJ+ ‖x‖∞ =
max1≤i≤n |xi|[MN'h<M21 &5fB?A!9:B? (9/21+2DR&<(9:,:B?,_5-2?32?'65cBD2?+@5-A!,/'#21"$&2=BD21"'#!+-342;" "@ ()E!+I$&!'#'(>?2;"?[
F)NA!'#"5-! &5fBD2]76 (,l"` (,/5c!'b'(!5@2A∗ 9A43AQ5@+-,BD28A!$ |C,/'65-2]$&2 A [ Y
a6Q!knC &C a
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5 + '()"!$! )#$ #C
<>‖A‖2 = ‖A∗‖2 =
*+/X8*1 * 4! ]3 " & !D!8jE" 8 )/! A ,E>
‖A‖1 = max1≤j≤n (∑n
i=1 |aij |) ,YE>‖A‖∞ = max1≤i≤n
(
∑nj=1 |aij |
) >
O&L\ CG f'(2]3AJ5@+-,:B?2]+@>12?9:9/28<E2? (5.D5-+@2 BD!'M"`,$&>?+->?2]"@!,/5BDC3342% ('(2%3AJ5@+-,:B?2$&2 Mn(R)
K\"`C,_5]BDC3342 ('#243AJ5@+-,:B?2$(2 Mn(C)B?A!+
R ⊂ C[ , ‖ · ‖C
2;"|58 ('#2'(!+-320!21BD5@C+@,:2?9:9/2=$(A!'#"
Cn K&!' <M21 &5$&>D^M'(,/+f"-A +@2;"|5-+@,B~5-,/C' ‖ · ‖R
Rn 76 (,L21"`5fAJ #"-"`, ('#28'(!+-320!21BD5@C+@,:2?9:9/2]$#AJ'#"
Rn [jC (+ ('#2%3AQ5-+@,BD2]+->?219/9:2 A ∈ Mn(R)KE!'t<E2? &58$&!'MB]$&>D^M'(,/+=$&21 &R
'(!+-321"3AQ5-+@,BD,:2?9:9/2;""` ()E!+I$&C'('(>?2;" ‖A‖C2D5 ‖A‖R
<#AJ+
‖A‖C = supx∈Cn,x6=0
‖Ax‖C‖x‖C
2?5 ‖A‖R = supx∈Rn,x6=0
‖Ax‖R‖x‖R
.
<(+-,/C+@,EBD2;" $&21 &R $&>D^#'#,_5-,/C'#" <E2? #0!2?'65'.D5@+-2N$(,:"`5@,:'#B~5-21"1[ 3*+ !B?2AJ (R HC+@3 (9/2;"X2?RO<#9/,BD,/5@2;" $&29 d R&2?+IBD,BD2 [/C[/CKMC'S"-AJ,/5876 ad 2?9:9:21"=BD '#B?,:$(2?'65="@, ‖x‖C
21"`5= #'(24$(21"='(C+@321" ‖x‖1K ‖x‖2
K! ‖x‖∞
[ 0 21<M21'#$(AJ'65;K(<MC (+f$Ld AJ &5-+@2;"c'(C+@321"0C21BD5@!+-,:2?9:9/2;" C'b<M21 &5NA 0!C,/+ ‖A‖C > ‖A‖R[
jA!+NA!,/9:9/21 (+I"?K$(AJ'M"f9A <(+-2? (0C2%$&2 9:AX+@C<M6"`,/5@,:!' [:![4!'SA)M2;"`C,/'S$&2%9AG$&>?^#'(,/5@,:!' "@ (+C$&2*9:A '(!+-32N"@ ()E!+I$&!'('#>?23 .1342="@,E9A]3AJ5@+-,:B?2N2;"|5+@>12?9:9/2C[ 0 d 21"`5<MC (+-76 (C,MC'G &5-,/9:,"`2
C$(AJ'#"9A)>?^#'(,/5@,:!' [/C[/=$&289A4'(!+-32="@ ()E!+I$&!'#'(>?2C[ •hg jilknm1knoEi CQ
A % 61 Mn(C)
"jv2 %:4%0 %: %A 25
ρ(A) % I& *% - &%/ %! % 6A"
2+-A PC!' "@<M2;B~5@+IAJ9ρ(A)
' d 2;"|5=<#A!"* ('#24'#!+-342"@ (+ Mn(C)[ 'S2,E2?51KL!'<E2? &5]A 0!C,/+
ρ(A) = 0A 0!2;B
A 6= 0Fn<(+-2?'#$&+-2!KM<#AJ+N2DR&2?3<(9:2!KM ('(2 3AJ5@+-,:B?285@+-,AJ' C (9:A!,/+-2]A 0!2;B8$&21" ?>1+@6"
"` (+ 9:AS$&,A !C'#AJ9:2;YD[ 0 21<M21'#$(AJ'65;Kl9:2G9:2?332hB?,$&21"-"@! #"83C'C5-+@2h7C #2hBJd 21"`5 #'(2G'(C+@32G"@ (+9 d 2?'#"@2?3 )(9:2=$(21"c3AQ5-+@,BD2;"e'(!+-3AJ9:21"1[
O C U
5' 0 % ' %0<U∗ = U−1 B ‖UA‖2 = ‖AU‖2 =
‖A‖2" :% 5
A 0 % -% - <
A∗A = AA∗ B - :% ‖A‖2 =ρ(A)
"
hg oEil1m?Q&m;knoEi 0 !332U∗U = Id
K#!'bA
‖UA‖22 = supx∈Cn,x6=0
‖UAx‖22‖x‖22
= supx∈Cn,x6=0
〈U∗UAx,Ax〉〈x, x〉 = ‖A‖22.
)%d AJ &5-+@2=<#A!+`5;K&9/2]BITMAJ' C2?32?'65f$&2=0QAJ+-,:A!)(9/2y = Ux
0!>1+@,/^#2 ‖x‖2 = ‖y‖2K(2D5$&C'#B
‖AU‖22 = supx∈Cn,x6=0
‖AUx‖22‖x‖22
= supy∈Cn,y 6=0
‖Ay‖22‖U−1y‖22
= supy∈Cn,y 6=0
‖Ay‖22‖y‖22
= ‖A‖22.
,A21"`5t'(!+-3AJ9:2!K2?9:9/22;"|5 $&,A C!'#A!9/,"@A!)(9:2S$(A!'#" ('(2)#A!"@2!+@5@T(C'(!+-3>?2$&20!21BD5@21 (+-"
<(+@C<(+-21"e2D5fC' $&>;$& (,/5$&21"f+@>;"` (9/5-AJ5-"<(+->1B?>1$&21'65-"76 (2 ‖A‖2 = ‖ diag(λi)‖2 = ρ(A)[
N'tB?!3<#AJ+-2*3A!,/'65-2?'#A!'C5c9A4'(!+-32=$ad ('#283AJ5@+-,:B?2AA 0!21B*"@!' +-A PC!'h"@<E21B~5-+-A!9
ρ(A)[
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M )$ )($ * # ( $*+
F QoM oE1knm1knoEi C ‖ · ‖ 5 5:% :%68 &% Mn(C)"ev2
ρ(A) ≤ ‖A‖. % 5 &% 0 %
A/ &% %
ε > 0 2! 5:%
:%!%8 ‖ · ‖ < 5 "9 6A
εB 5
‖A‖ ≤ ρ(A) + ε.F` [ CY
g oEil1m?Q&m;knoEi !,/5λ ∈ C
('#280 A!9/21 (+f<(+-!<#+@28$&2A5@2?9:9:2%76 (2
ρ(A) = |λ| K#2?5 x0 6= 0 #' 0C21BD5@2? #+c<(+@C<(+-2*AC"@"@&BD,:>4FAx0 = λx0 Y~[EN'tA
‖λx0‖ = ρ(A)‖x0‖ = ‖Ax0‖ ≤ ‖A‖‖x0‖,$ad .x9Wd C'x$&>;$& (,/5
ρ(A) ≤ ‖A‖ [ 0 !332 9:2b0!2;B~5@21 (+<(+-!<(+-2 x0 <E2? &5 .?5@+-2 B?!3<(9:2DR&2!K ,:92;"|52;"@"@2?'65@,:2?9c$Ld &5-,/9:,:"@2?+ #'(2b'(!+-32h0C21B~5-!+-,/219/9:2b"` (+
Cn 3.132t<MC (+ ('(2b3AQ5@+-,BD2b+@>12?9:9/2F B~|[ c2?3AJ+I76 (2b [/C[ OY~[ c>;BD,:<(+@&76 (213421'651Ka,:9 2?RO,"`5@2 ('(23AJ5@+-,:B?2
U,:'O0!21+-"@,/)#9/245@219/9:2G76 (2
T = U−1AU"`C,_55@+-,AJ' C (9:A!,/+-2h"@ (<E>?+-,/21 (+-2![pj! (+5-! &5
δ > 0!' $&>?^#'(,/5 #'(2t3AQ5@+-,BD2
$(,:A !!'MAJ9:2Dδ = diag(1, δ, δ2, · · · , δn−1)
$&2=5@219/9:28"`C+`5-287C #2*9A3AQ5-+@,BD2Tδ$(>D^#'(,:28<#A!+
Tδ = (UDδ)−1A(UDδ) = D−1
δ TDδ
0C>?+-,_^M2
Tδ =
t11 δt12 · · · δn−1t1n
0
[ [ [ [[[[[[ [ [ [ [ [ [
δtn−1n
0 · · · 0 tnn
A 0C21BT =
t11 t12 · · · t1n
0
[ [ [ [[[[[[ [ [ [ [ [ [ [[[0 · · · 0 tnn
.
4X5-AJ'65f$&C'('(>ε > 0
KO!' <M21 &5BIT(C,:"@,/+δ"@ $ "-AJ332?'65c<M2?5@,/5f<MC (+c7C #2*9:21">19/>132?'65-"2?R65-+-AI
$(,:A !!'MAJ &R=$&2Tδ"@!,:2?'65a5@+-.1" <M2?5@,/5-"\AJ #"-"@, K1<#AJ+l2DR&2?3<(9:2X<E! (+\7C #2!K;<MC (+a5-! &5
1 ≤ i ≤ n−1K
n∑
j=i+1
δj−i|tij | ≤ ε.
9/C+-"=9Wd AJ<(<(9:,B?AQ5-,/C'B → ‖(UDδ)
−1B(UDδ)‖∞21"`5] ('(2 '(C+@32"@ ()MC+-$(!'('(>12tFn76 (, $(>N
<E2?'M$ $(2A2D5εY 76 (, 0!>?+-,/^#2N)#,/21'F` [ CY~[
O C:p A ' % % Mn(C)
" $ 5 0 % 08 5 - $
=%"limi→+∞ Ai = 0
"limi→+∞ Aix = 0
#0&% 0 &%x ∈ Cn
"ρ(A) < 1
" ! 0 1 -% 0 % L -%%0 5 ‖A‖ < 1"
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5 + '()"!$! )#$ #
hg oEil1m?Q&m;knoEi 1t!'65@+-!'M"e5@! (5c$ad A!)MC+-$ 76 (2 F| Ye,/3<(9:,:76 (2F !Y~[ d ,/'(>CA!9/,/5@>
‖Aix‖ ≤ ‖Ai‖‖x‖34C'65@+-2e76 (2
limi→+∞ Aix = 0[ '#"` #,_5-2!K(FCYl,/3<(9:,:76 (2=F Y\B?AJ+;K "`,
ρ(A) ≥ 1KQAJ9:!+I"l,/9O2DR&,"|5-2
λ2D5x 6= 0
5@219:"f76 (2Ax = λx
2?5 |λ| = ρ(A)K#2D51KM<#AJ+fB?!'#"@>176 (21'651K&9A4"@ (,/5@2
Aix = λix'(2
<M21 &5f<#A!"B?!'O0!21+ C2?+ 0C2?+I"#[ 0 !332Y1F Y ,/3<(9:,:76 (2F)6Y[*21"`5 #'(28BD!'M"`>;7C #2?'#B?2N,:334>;$&,AQ5@2$&29AZX+-!<EC"@,_5-,/C'b [:![OK!,/9E'(2f+-21"`5@2f<#9/ #"e76 ad 83!'65@+-2?+ 76 (2 F6YX,/3<(9:,7C #2 F| Y~['j! (+ B?2?9A(K!'bBD!'M"`,$&.?+-2=9:A4'(C+@32=3AQ5-+@,BD,:2?9:9/28"@ ()E!+I$&!'('#>?2N5@2?9:9:2]7C #2 ‖A‖ < 1
K&2D5fC' A
‖Ai‖ ≤ ‖A‖i → 09/C+-"-76 (2
i→ +∞,BD2876 (,a3!'65@+-2876 (2
Ai 5-2?'#$ 0!21+-"#[
! )+/-/'*%('*),+3+ 67+%: % 8 % 43'!' % 0QAJ'65]$&2$&>1B?+@,:+-2%9:21"=AJ9 !C+@,/5@T(32;"N$&2+->1"@!9: &5@,:!'$&2"`P&"`5@.?32;"N9:,:'(>1A!,/+-21"1K,/9j'(C #"*nAJ &5
>?0!&76 (21+ 9:21"e<(+-!)(9:.?32;"e$(2N<(+->1B?,:"@,:!' 2?5c$(2="`5-A!)(,:9/,/5@>=$& #"cA! &R 2?+-+@21 (+I" $Ld AJ+-+@C'#$&,W[ 'h2,E2?51K$(AJ'#"* ('!+I$&,:'#AQ5-2? (+*,:9 'ad PSAG<#AC"=$&2B?AJ9BD #9:"*2DR(ACB~5-"1K2D589AG<(+->1B?,:"@,:!' 2;"|5=9:,/3,/5@>?2bB?A! #"@2$& '(C3%)(+-2S$(2S)(,/5-"b (5@,:9/,"`>;" <MC (+h+-2?<#+@>;"`21'C5-2?+ 9/2;"G'#!3%)#+@2;"G+->?219:" 0f$Ld T#AJ)(,/5@ M$&2 C I )(,/5-"bF BD2b7C #, nA!,_5t<E2? <#+@.;" $ ! % BIT(, ,E+-21"4"`, !'#,_^MB1AQ5-,_n"IY~[9enAJ &5$&!'MBnAJ,:+-25@+-.1"AQ5`5-2?'65@,:!' A! &R ,:'(>106,/5-A!)(9:21"X21+@+-2? #+-" $Ld AJ+-+-!'#$&,#2D5 89:2? #+X<(+-!<#A CAQ5-,/C'4A! BDC (+-" $Ld #' B?AJ9BD #9 [ 21"e3>D5-T(&$&21"e'O (3>?+-,7C #21"e$&2N+->1"@!9: &5-,/C'G$&28"`P&"`[email protected];" 9:,:'(>1A!,/+-21"e76 (,'ad A!3<(9/,/^#21'C5c<#AC" B?21"2?+-+@21 (+-"*"@!'65=$(,_5-21"8"`5-AJ)#9/2;"?[ X'<(+IAQ5@,76 (2!KC'S &5-,/9:,"`21+-A $&!'MB $&2;"=AJ9 !!+-,/5@T(321"=76 (, "@!'65=9:A H!," C(!O6m?m;$ O [ 0 2D5@5@2bAJ3<(9:,_^EB?AQ5-,/C'x$&2;"42?+-+@21 (+-" $&>?<E2?'M$x$&2b9A 3AJ5@+-,:B?2BD!'M"`,$&>?+->?2C[.jC (+c7C MAJ'65@,/^#2?+BD2=<(T#>?'(C34.1'(2!KOC'G,:'C5-+@&$& #,_5c9A%'(!5@,:!' $&2*B?!'#$(,_5-,/C'('(2?321'C5$Ld #'(283AQ5@+-,BD2![
hg jilknm1knoEi C:z -% % -%%0 8 5 0( 0 ‖A‖ < % 0 2 K= "=%"=B%"v2 !"# M. 5 %
A ∈ Mn(C) % 0 u 7 -% - &%* 2 M :%
cond(A) = ‖A‖.‖A−1‖.0 2?5`5-2%'(!5@,:!' $&24BD!'M$&,_5-,/C'('(213421'650QA<M21+@32?5`5@+-2 $(2]321"@ (+@21+9Wd A!3<(9/,/^MB1AQ5@,:!' $(21"f21+
+@21 (+-"$&2;"f$&C'('(>?2;"8Fn"@21B?!'#$h32?3%)#+@2=! 3AQ5-+@,BD2 YAJ +@>;"` (9/5-AJ51[
F oM oE1knm1k oMi C& A5 % % "
b 6= 06 &% 0 &"
=6" xx+ δx
- 1 % # 6$ $4 $
Ax = b A(x + δx) = b+ δb.
O -% 0 ‖δx‖‖x‖ ≤ cond(A)
‖δb‖‖b‖ .
F` [ Y
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MH )$ )($ * # ( $*+
" xx+ δx
3 - 01 % # $*6$ $4
Ax = b (A+ δA)(x + δx) = b.
O :% 0 ‖δx‖‖x+ δx‖ ≤ cond(A)
‖δA‖‖A‖ .
F` [ 6Y 7 $
89++: ; 0 !5 : "
O&Ll &C N't$&,:+-A76 ad ('(2]3AJ5@+-,:B?2821"`5)(,:2?' BDC'#$&,/5@,:!'('#>?2]"@, "`C' BDC'#$&,/5@,:!'1'#2?32?'65%2;"|5%<#+@&BIT(2G$(2 Fn"-Ab0QAJ9:2? (+ 3,/'(,:3AJ9:2;Y=2?5476 ad 2?9:9/2G21"`5%3AJ9 B?!'#$(,_5-,/C'('(>?2 "@, "@!'B?!'#$(,_5-,/C'('(2?321'C5=21"`5 !+IAJ'#$a[ B?A! #"@24$&2;"=+@>;"` #9_5IAQ5-"*$&29AX+-!<EC"@,/5@,:!' [:![: #K#2?'<(+IA5-,:76 (24,:9lnA! #$&+IAnA!,/+-2 AJ5`5@21'65@,:!'SA! &Rt2?+-+@21 (+I"$Ld AJ+-+@C'#$&,\"`,j!' +@>;"`C &5N ('"`P&"`[email protected]:,/'(>;AJ,:+@2<E! #+c ('(283AQ5-+@,BD2=3AJ9aB?!'#$&,/5@,:!'#'(>?2C[ • g oEil1m?Q&m;knoEi jC (+G3!'65-+@21+G9:2 <(+-2?3,:2?+G+->1"@ (9/5-AJ51Ke!' +-2?3AJ+I76 (2 76 (2
Aδx = δbK
2?5 $&C'#B ‖δx‖ ≤ ‖A−1‖.‖δb‖ [eN+;K !' AxAJ #"-"@, ‖b‖ ≤ ‖A‖‖x‖ KeBD2 76 (,=$&C'('(2xF` [ Y~[0 2?5`5-2G,:'(> 6AJ9:,_5-> 21"`5 C<&5@,:3AJ9:2hAJ "@2?'M"%"@ (,/0QA!'C5 0l<E! #+]5@C &5@2h3AQ5@+-,BD2AK\,:9e2DR&,:"`5@2
δb2D5
xF 7C #,j$(>?<E2?'#$&21'658$(2
AYf5-2?9"=7C #2 F` [ Y21"`5=2?' nA!,_58 ('#2 >CAJ9:,/5@>![ X'2-,2D5;KL$Ld A!<(+-.1"* ('(2
<#+@C<(+@,:>D5->8$&21"c'#!+-342;"3AQ5@+-,BD,:2?9:9/2;"f"` #)MC+-$&C'('(>121"=FH0C!,:+f9/2 2?332 [/C[ CY ,:9a2?RO,"`5@2x5-2?9
76 (2 ‖b‖ = ‖A‖‖x‖ 2D5f,:9a2?RO,"`5@2 δb 5-2?9 76 (2 ‖δx‖ = ‖A−1‖‖δb‖ [j! #+ !)&5-2?'(,:+ F| [ 6Y !' +-2?3AJ+I76 (2b7C #2Aδx + δA(x + δx) = 0
K 2D5G$&C'#B ‖δx‖ ≤‖A−1‖‖δA‖‖x + δx‖ K B?2 7C #,X,/3<(9:,7C #2 F| [ 6Y~[cjC (+]2?'$&>13!'65@+-2?+89Wd C<&5@,:3AJ9:,_5->!KaC'0 A3C'C5-+@21+X76 (2f<E! (+ 5-! &5-2c3AQ5@+-,BD2
A,:9#2?RO,"`5@2c ('(2<M21+`5- (+@)MAQ5@,:!'
δA2?5 ('"`2;BDC'#$434213%)(+-2
b<E! #+ 9:21"-7C #2?9"X,:9MPA8> 6AJ9:,_5->![ 3*+ CBD2fAJ 2?332% [:![ *,/9M2DR&,:"`5@2
y 6= 05-2?976 (2 ‖A−1y‖ =
‖A−1‖‖y‖ [ C,_5 ε (' "@B1AJ9AJ,:+@2N'(!'h'6 #9 [#N'G<EC"@2 δA = εId2D5b = (A+ δA)y
[MN'h0!>1+@,/^#2A!9/C+-"76 (2
y = y + δx2?5δx = −εA−1y
K(2D5B?!332 ‖δA‖ = |ε| !'b!)(5@,:2?'65f9 d > 6AJ9:,_5->8$(AJ'#"F` [ OY~[
21"fB?!'#$&,/5@,:!'#'(2?32?'65I"9/2;"<(9/ M"c &5@,:9:,:"@>1"21' <#+-AJ5@,7C #2="@!'65
condp(A) = ‖A‖p‖A−1‖p<E! #+
p = 1, 2,+∞,.9:21"h'(!+-321" ‖A‖p "`C'65h2DR&<(9:,BD,/5@2?321'C5t$&>?^#'(,:21"b$(AJ'M"h9:2 2?332y [:![:![cN'0C>?+-,_^#2nACBD,:9/2132?'65c ('tBD2?+@5-A!,/' '(C3%)(+-28$&28<(+-!<(+-,/>?5@>;"e$( bB?!'#$(,_5-,/C'('(2?321'C5;[
a6Q!knC &C L _8 + g*+8 26A ∈Mn(C)
> K&8 ^_& E<>
cond(A) = cond(A−1) ≥ 1 , cond(αA) = cond(A) ∀α 6= 0 ,E>+$& G*+8 241&" 24 3
, cond2(A) = µn(A)µ1(A) ,
"!µ1(A), µn(A)
!d ;/!6$&/26H8 ]*&;" N$ "Q!;$&6H8 9I4" S$"Q! j7 ;] " a!R8j " 8 )&';
A ,YE>+$& g*+8 26g L *+ " , cond2(A) = |λn(A)|
|λ1(A)| , "!|λ1(A)|, |λn(A)|
!d E//!6$&/26H8 ]*&;" N$ "Q!;$&6H8 9I4" S$"Q! j7 ;] " $& E$&';N* _ " I
A ,@Z>+$& 9 9g*+8 26g8 d8'U , cond2(U) = 1 ,
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5 + '()"!$! )#$ #
eE>;$& E 9 E/9g*+8 26g &8 d8'U , cond2(AU) = cond2(UA) = cond2(A)
>
a6Q!k ! &C C aEb" I26 E_8 H8 &*/`b" N*;8 241b8 jL8 _8 9KKh ,
K/$ : ?Z><A$& " N*+K6J LI/! K&" K&*+&9!`^ 8 !
P1
$ $ " 8 K4I & $ " 328 &,4!d
cond2(Kh) ≈ 4
π2h2.
F` [ CY
+G* E/'&7FEg" /! ] " '!;$& E$&'4!DKh
!d E/
λk = 4h−2 sin2
(
kπ
2(n+ 1)
)
1 ≤ k ≤ n,$& a/! ]29& '!+$ E$&/!
uk EK4!;$3&C" '!B26 E*1$& !H /9/!
ukj = sin
(
jkπ
n+ 1
)
1 ≤ j, k ≤ n.
O&L\ C1 d 2;"|5-,/3AQ5-,/C'F| [ CY*$& BDC'#$&,/5@,:!'('(2132?'65%$&2 9Ab3AQ5@+-,BD2 $&2+-, C,$&,_5-> Kh
"`213%)(9:2X5-+@.;" <E21"-"`,:3,:"`5@2CK10C!,:+@2 B?AQ5IA!"`5@+-!<(T#,:76 (2C[ X'%2,E2?51K 9Af3>D5-T(&$&2 $&2;" >19/>132?'65-"^#'(,"BDC'O0!2?+ !2="@,
h = 1/(n+ 1)5@2?'M$b0!2?+I"2?>1+@M[ &5-+@2132?'65N$&,/51KE$&21"c+->1"@ (9/5-AJ5-"c<(+->1B?,:"f'(2
<M21 (0!21'C5.?5@+-2NC)&5@21'6 M"e76 (28"@,L9:A 3AQ5-+@,BD2*21"`55@+-.1" !+IAJ'M$&22?5e5-+@.;"e3A!9aBDC'#$&,/5@,:!'('(>12![51 A!,:"$(AJ'#"BD2]B?AC"?K#9/2;"c,/'#>?0O,_5IAJ)(9:21"f2?+-+@21 (+-"c$ad A!+@+-!'#$(,a"` (+9/2%"@21B?!'#$h32?3 )(+@28C t"` (+9:A3AJ5@+-,:B?2+@,"@76 (21'65$Ld .D5-+@2=>1'(!+-3>?32?'65fAJ3<(9:,_^M>?21"cA! h<E!,:'65f$&2=+@21'#$&+-2N9A"`C9/ &5-,/C' $(,:"-BD+-.D5-2
uh5@+-.1"
$&, ,>?+-2?'65-2 $&24"@AG9/,:3,_5-2 <(+->1$(,_5-2![ 2(!+@5NT#2? (+-2? #"@2?321'C5;KM,:9\'ad 21' 2;"|5*+@,:2?' 21' <(+IAQ5-,:76 (24B?AJ+N9/2"`2;BD!'M$%32?3 )(+-2
bh$& "@PO"`[email protected]/,:'(>1A!,/+-2 KhUh = bh
'ad 2;"|5X<MA!"X76 (219:B?!'#76 (22D5X'#2f+@21'#$ <#A!"9/2;" ,:'(> 6AJ9:,_5->1" $&29AuX+-!<EC"@,/5@,:!'t [/C[/ =!<&5-,/3A!9/2;"?[ ,E9 d !' +-2?<#+@21'#$9:A $&>?3C'#"|5-+-AJ5@,:!' $&29 d !<&5-,/3A!9/,/5@>%$(2%BD2;"f,:'(> 6AJ9:,_5->1"1K(!' "1d A!<M21+ ?!,/576 ad 219/9:2]2;"|5C)&5@21'O (2]<E! (+f #' 0!2;B~5@21 (+
b76 (,
21"`5*0C21B~5-2? (+*<(+-!<(+-2%$&2 KhA!"-"`&BD,:>% "@AG<(9: #" !+IAJ'#$&2 0QAJ9:2? (+*<(+-!<(+-2
λn[ )%d AJ<(+-.1"N9Wd R&21+
BD,BD2b [/C[ KM ('5-2?9X0C21B~5-2? (+=<(+-!<#+@24C"-BD,:9/9:2 H!+@5@2?321'C5]"@ (+89:243A!,/9:9:A !2bFn"@21"8B?!3<M6"@A!'65@21"BIT#AJ' C2?'65$&24"`, !'(2 $Ld #'(2%3AJ,:9:9/249 d AJ (5@+-2;Y~[ ,\9/24"`2;BD!'M$b34213%)(+-2
bh21"`5<(9: #"Y|+-> C (9:,/21+h
FnBJd 21"`5$&,:+@276 ad ,/9(2;"|5 B?!3 )(,/'MAJ,"`C'%9:,/'(>;AJ,:+@2c$&2;"K<(+@213,/21+-"j0C21B~5-2? (+I"\<(+-!<(+-21"
uk $&2 KhY~K
!' <M21 &5NAJ3>19/,:!+-2?+e9/28+->1"@ (9/5-AJ5c$(289:A_X+-!<EC"@,_5-,/C'S [:![: %2?'b!)&5-2?'#A!'65‖δx‖‖x‖ ≤ C(K)
‖δb‖‖b‖ ,
C(K)
2;"|5 ('#2bB?!'#"`5-A!'65@2b,/'M$&>?<E2?'#$#AJ'65@2 $&2n[ 0 d 21"`52DR(A!BD5@213421'65B?2 76 (,fAJ+-+@,:0!2 2?'
<(+-AJ5@,76 (2!Ka2?5%BD2?5`5@2h$&2?+-'(,:.?+-24,:'(>CAJ9:,/5@>*| #"`5@,/^#2 9Wd &5-,/9:,"@AJ5@,:!'$&2 9Ab3>D5@T#O$(2 $&21"8>19/>132?'65-"^#'(,"c3AJ9 !+->N9A4<(+@>;"`21'#BD28$Ld 2?+-+-2? (+I"$Ld A!+@+-!'M$&,L$(AJ'#"9:21"cB1AJ9BD (9"c"@ (+c!+I$&,:'#AQ5-2? (+I"?[ •
! %/)$- 8 -/'!2(9 %8
g m oalb g nk kni\&m;knoEil (l; d ,$&>12*<#+@,:'#BD,:<#A!9/2=$&2=BD2D5@5@2=3>D5-T(&$&2=21"`5c$(2*"@2=+-A!3421'(2?+ 9:A +@>;"`C9/ (5@,:!'G$Ld ('b"`P&"`[email protected]
9/,:'(>1A!,/+-28$&!'65c9A 3AJ5@+-,:B?2N2;"|5c5-+@,AJ' C (9AJ,:+@2C[ X't2,E2?51K(9A4+@>;"`C9/ (5@,:!'h$ad ('t"@P&"|5-.?32=9/,:'(>1A!,/+-2!K
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M )$ )($ * # ( $*+
Tx = bK9:Ax3AQ5@+-,BD2
T21"`5h5-+@,AJ' ! (9AJ,:+@2 2D5t,/'O0C2?+I"`,:)(9:2!Ke21"`5h5-+@.;" nACBD,:9/2<#AJ+t"@,/3<(9:2
"@ ()#"`5@,/5@ (5@,:!'b+@>;BD (+I"@,/0C2![ X'b2-,2D5;K(9/2]"@P&"|5-.?32
t1,1x1+ t1,2x2+ · · · · · · t1,nxn = b1
t2,2x2+
[ [ [· · · t2,nxn = b2[ [ [ [ [ [ [[[
tn−1,n−1xn−1+ tn−1,nxn = bn−1
tn,nxn = bn
.
"@2%+->1"@! (52?' B1AJ9BD (9AJ'65=$Ld A!)MC+-$xn = bn/tn,n
KM<( #,:"xn−1
KE2D58AJ,:'#"@,l$&2"@ (,_5-2f| #"-76 ad x1[
N'AJ<(<E2?9:9:24B?2 <(+-&BD>;$&> QOoEim g Fn$(A!'#"*9/2B?AC"N$Ld ('(23AQ5-+@,BD2%5@+-,:A!' ! #9:A!,/+-2],:'&H>?+-,/21 (+-2!K9:2<(+-OB?>1$&>"@,/3,:9:A!,/+-27C #, B?A!9:B? (9j9/2;"]BDC34<EC"-AJ'65-21"*$&29:Ab"@!9: &5@,:!'$&2
x1xn
2;"|5]AJ<#<M219/>\O1COim1 Y~[ c2?3A!+-76 (C'#"876 (2G9Wd C'A AJ,:'#"@,X+->1"@!9: 9/2G"@P&"|5-.?32
Tx = b"-AJ'#"%,/'O0!21+-"@2?+89A
3AJ5@+-,:B?2T[ )29Ab3 .?323AJ'(,:.?+-2!K 9:A 3>D5-T(&$&2G$Ld >?9:,/3,:'#AQ5-,/C'$&23=AJ #"-"80QA +@>;"`C #$&+-29/2
"@P&"|5-.?32Ax = b
"-AJ'M"cB?AJ9BD #9/21+e9Wd ,/'O0!21+-"@2=$&289:A43AQ5-+@,BD2A[
A34>?5@T(&$&28$Ld >?9:,:34,:'#AJ5@,:!'t$&23=AJ M"@"c"@28$&>1B?!3<M6"`2=21'G5-+@C,:">?5-AJ<E21" 0Fn,Y >19/,:3,/'#AJ5@,:!' 0LB1AJ9BD (9j$Ld ('(243AJ5@+-,:B?2
M,:'O0!2?+I"@,/)(9:285-2?9:9/276 (2
MA = T"@!,/5*5@+-,:A!'1
! (9AJ,:+-2="@ (<E>?+-,/21 (+-2!KFn,/,ZY 3,:"@28|C (+f$& t"@21BDC'#$h34213%)(+-2 0#B?A!9:B? (9a"@,/3 (9_5IAJ'(>]$(2
MbK
Fn,/,:,Yb"@ ()#"`5@,/5@ (5@,:!'20J+->1"@!9: &5@,:!'$& "@P&"|5-.?325@+-,AJ' C (9:A!,/+-2Tx = Mb
<#A!+X"`,:3<(9/2+@2134C'15@>12![
d 2?RO,"`5@2?'MBD2 $ad ('#2]5@219/9:2%3AJ5@+-,:B?2M
21"`5 6AJ+IAJ'65@,:28<#AJ+*9:2%+->1"@ (9_5IAQ5N"@ (,:0QAJ'65*$&C'65!' 0QA$(!'('(21+ ('#2*$&>13!'#"`5@+IAQ5-,/C'B?!'#"`5@+- #BD5@,:0!276 (,E'ad 21"`5e+-,/21' $Ld AJ &5-+@2*7C #29A83>D5@T#O$(2*$Ld >?9:,:34,'MAQ5@,:!'b$&23=A! #"@"1[
F QoM oE1knm1knoEi C&C A
5 % -%$% < % 0 0-B%" 0 0 % %
M 5 %
T = MA > % 0+%& %* I '
% &%"
Uh*`"r =gT,+gfd`"r2t:¡IVW8;<B=8;I&::=DRGS:¡B=E<DRGFVW@38HVW:¡@3<3:\DF@8;GW:ZS:\[CKGFVW8;B=:TDAk X 1 ≤ k ≤ n
X"S E<AGL~:TD
(k−1)I VW:=[8;TVW:=DB=EL~E<3<:=D%DFE<AGVW:T[I3L;8;:=DS:=z#VWEDDFE@3DL~K_S8~KmE<3KL;:iQpKvV[EQS8H§1BKvGW8;E<3DDF@BT
B=:=DFDF8;:=D=X E<CI3KDFDF:S :A1 = A
An = T
>A@381:=DRG9GFVW85K<3m@3L~K8HVW:DF@I&zTVW8;:=@VW:ik<<EGW: (ak
ij
)
1≤i,j≤nL~:TDz=L;z=[:T<QGWD\S :L~K[CKGFVW8;B=:Ak XY:#GE<KII&:=L;L;:I8;EGS: Ak LNM z=L;z=[:=<AG ak
kk
icYE@ VI3KDFDF:TVZS:L~K[CKGFVW8;B=:
Ak L~KC[CKGFVW8;B=: Ak+1 X"E<D=M KDFDF@ VW:ZGWE@ G_S"M KO&EVS>A@3:¡L;:ZI38;EG akkk
<M :=DRGI1KDk< @LNi ¢ M 8;LLM :=DRGX'E<I&:TVW[@GW:¡L~K
kT[:L;8;m<:K:=BZ@<3:K@ GFVW:JL;8~m<3:JI&E@VK[:=<:TV_:=<I&EDF8HGW8;E<S:ZI8;EGk@3<
z=L;z=[:=<AG<E<< @LNic@38;D_E<I VWEAB=S:JLNM z=L;8;[8~<3KGW8;E<0S:GWE@3D_L;:=D_z=L;z=[:T<QGWDkS:JL~Kk =[:¡B=EL;E<3<3:
:=<S:=DFDFE@3DoS:\L~Kk =[:L;8;m<::T<NK8;DWK<QGS:=D]B=E[ZO38;<3K8;DFE<DoL;8;<3zK8HVW:=D_S:\L~KJL;8;m<3:¡B=E@VK<AGW:¡K:=B
L5Kk =[:L;8;m<:iL;@DoI VWz=B=8;DFz=[:=<AGXE<::]":TBTGW@3:_L;:=DxEI&z#VKGW8;E<DkDF@38;K<AGW:=D=i1k<l[@3LHGW8;I3L;8;:
Ak I3KV]@3<:_[CKGFVW8;B=:S:xI&:TVW[Z@ GKGW8;E<
P k I&E@V9EO GW:=<38HV Ak = P kAk GW:=L;L;:k>A@3:xDFE<I38;EG akkk
DFE8HG9<3E<<Q@Li ¢ 8ak
kk 6= 0X
KL;EVWD_8;LDF@JG_S:ZI VW:=<1SQVW:P k = Id
i ¢ 8;<3E<X"D=M 8;L:#y8;DRGW:ak
ik 6= 0K:=B
i ≥ k + 1X'E<I&:TVW[@GW:\L~K
k =[:L~8;m<3:K:=BL~K
i =[::T<IVW:T<1K<AG
P k = (e1, ..., ek−1, ei, ek+1, ..., ei−1, ek, ei+1, ..., en)sNDF8
GWE@3D%L;:=D¤z=L;z=[:=<AGWD%S:]L~Kk =[:xB=EL;E<<3:oDFE@3D9L5K\S 85KmE<1KL;:X
akik
K:=Bi ≥ k
XQDFE<QG9<Q@3L;D=XQKL;EVWDt8~L3<M
Ecol
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5 + '()"!$! )#$ #
K¡VW8~:T<NK8HVW: w#i1@38;DxE<l[Z@3LHGW8;I3L;8;:Ak I1KvV]@3<3:k[CKGFVW8;B=: Ek X3Sz#§1<38;:I3KV
Ek =
1
0
i i iiii
1iii − akk+1,k
akk,k
1iii
iii i i i
0 − akn,k
akk,k
1
,sR= i w
BT:\>Q@8"z=L;8~[8;<:kGWE@D]L;:=D]B=EA:TJB=8;:=<AGWDoS :_L~Kk =[:kB=EL;E<<3:_:=<lS:=DFDFE@3DoS:L~KJS 8~KmE<1KL~:i'k<lI&EDF:
Ak+1 = EkAk =
a111 · · · · · · · · · · · · a1
1n
0
i i i iiiiii i i i
akk,k ak
k,k+1 · · · akk,niii
0 ak+1k+1,k+1 · · · ak+1
k+1,niii iii iii iii0 . . . 0 ak+1
n,k+1 · · · ak+1n,n
K:TBak+1
ij = akij −
aki,k
akk,k
akk,j
I&E@ Vk + 1 ≤ i, j ≤ n
ipK[CKGFVW8;B=:Ak+1 KSE<BO38;:=<0L~KldEVW[:
S z=DF8HVWz=:K:=BDF:=DkIVW:=[8;TVW:TDB=EL;E<3<3:TD\KAK<AG_@3<8~>Q@:=[:=<AGoS:=D=zTVWED_DFE@3D_L~KlS 8~KmE<1KL~:i O]IVW=D
(n − 1)zTGKI&:=D=XL~K[CKvGFVW8;B=:
An :=DRGJGFVW8~K<m@3L~K8;VW:lDF@3I&zTVW8;:=@ VW:U:TGZzTVW8H§1:An = MA
K:=BM =
En−1P n−1...E1P 1 i3cKZ[CKvGFVW8;B=: M:TDRGo8;<Q:TVWDF8;OL~:_B=KV
detP i = ±1:TG
detEi = 1i
k<I&:=@ G_[:TGFGFVW:¡P E@V\L;:JDF:TB=E<3S[:=[OVW:CsNBM :=DRGFS 8;VW:ZBKL;B=@L~:#VMb
wK@u@ V\:TG\U[:TDF@VW:>A@:¤LNM E<BKL;B=@L;:]L;:=D[CKGFVW8;B=:TD
P k :TG Ek iQk<B=E<DRGFVW@38HG@3<:tDF@38HGW:xS:tDF:=BTE<1S D[:=[ZO VW:=D (bk)1≤k≤nS zT§1<8;DtI1KVb1 = b, bk+1 = EkP kbk I&E@V 1 ≤ k ≤ n− 1,
:#G9E<CKO38;:=<bn = Mb
iAYE@ V%VWzTDFE@1SQVW:xL~:]DR DRGW=[:tL;8;<3zK8;VW:Ax = b
8~L3<3:tVW:=DRGW:xI3L;@3D%>A@M oVWz=DFE@1SQVW:L;:_DR DRGW=[:
Anx = MbE
An = T:=DRG]@<3:[CKvGFVW8~BT:kGFVW8~K<3m@3L~K8HVW:\DF@3I&z#VW8~:T@VW:i
()*+,-.)0/"2N/12N/ ¥ <DF8;DRGWE<3D_DF@ V\>A@3:=L~>A@3:TD_KDFI&:=B#GWD_IVKGW8~>A@3:TD\S:ZL~K[zTGWn3EQS :ZS"M zTL~8;[8;<1KvGW8;E<S : BK@DFD=i
i¡k<l<:BKL;B=@L;:xPRK[CK8~DM
1k<l<'M KI1KDxO&:=DFE8;<S:[@3LHGW8;I3L;8;:TVxL;:=Do[CKvGFVW8;B=:=DEk :TG P k I&E@V
BKL;B=@L;:TVMb
:#GAn i
£Ai ¢ 8A<'M :TDRGtI1KD¤8;< :TVWDF8;O3L;:XQE<K_GFVWE@3:#V¤@3<CB=EA:#B=8;:=<AGxS8~KmE<3KL'S:
An = T<Q@3L1:TG9E<<3:
K¡I1KD]I&E@3E8HVtVWz=DFE@3S VW:Tx = Mb
i1pKvV]B=E<AGFVW:LNM z=L;8;[8~<3KGW8;E<:TDRG]GWE@P E@VWDxI&EDFDF8~OL;:iQi OLNM zTGKI&:
kXE<<:][EQS 8H§1:x>A@3:tL~KI1KvVFGW8;:kS :=DL~8;m<3:=D
k+1
nB=E[I VW8;DF:]:T<QGFVW:tL;:=D%B=EL;E<3<3:TD
k + 1
ni
i9E[[:oDFE@3DR I VWEQS@38HGoS:kLM z=L;8;[8;<1KvGW8~E<S: BK@3DFD=X3E<I&:=@ GxBKL;B=@L;:TV]L~:\S zTGW:TVW[8;<1K<AG]S:L~K[CKGFVW8;B=:
Ai<:+]":TGX3E<K
detA = ±detTDF:=L;E<L;:_<3E[ZO VW:S:kI&:TVW[Z@ GKGW8;E<Dt::]":=BTGW@z=:=D=i
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M $ )$ )($ * # ( $*+
Ai_YE@V]EO GW:=<38HV]@3<:[:=8;L;L~:T@V]DRGKO8~L;8HGWz_<Q@3[zTVW8~>A@3:kS3K<3DxL;:=DxBKL;B=@L~D]DF@VxEVS8;<1KvGW:=@VWD]E<UI&:=@Bn3E8~DF8HVKDRGW@B=8;:=@3DF:T[:=<AG_L;:ZI38;EG
akkk
iYE@V_z=Q8HGW:TV\L~KUI VWEI3KmKvGW8;E<0S :=D_:TVFVW:=@ VWD_S'M KvVFVWE<1S 88;L%uK@ GZBn3E8~DF8HVL;:IL;@3D\mVK<1SI8~EG¡I&EDFDF8;O3L;::=<0KL;:=@ VZKO3DFEL~@:iK?lT[:L~EVWDW>A@3:L;:CI38;EG<3KGW@VW:TL
akkk
<M :=DRGI3KDZ<Q@LXYE<?I&:TVW[Z@ GW:CI&E@VZK[:T<3:TVJDWKI3L~KB=:l@<I3L;@3DmVK<3SI38;EGak
kk
i1k<S 8;Gt>Q@:oLNM E<NK8;Gx@<CI38;EG¤I1KVFGW8;:=L"DF8&E<UI VW:=<1SL;:I3L;@D¤mVK<1SCI38;EG¤I&EDFDF8;O3L;:_SK<3DL~K
k =[:\B=EL;E<3<::T<S:=DFDFE@3DoS:\L~KS 8~KmE<1KL~:UsNB=E[[:_E<KJNK8HGS3K<DoL~KCSzT[E<3DRGFVKvGW8;E<
B=8HS:TDFDF@3Dw#i"k<lS8HGo>A@3:LNM E<uK8HG]@3<I8~EGtGWEGKLcDF8'E<I VW:=<1SL;:_I3L;@3DxmVK<1SlI8;EG]I&EDFDF8;OL~:SK<3D_L~KlDFE@3DR [CKGFVW8;B=:8;<uz#VW8~:T@VW:CS8~KmE<3KL;:S:¡GK8;L;L~:
(n − k) × (n − k)sS3K<3DBT:JBKD\E<
I&:TVW[@GW:L;8;m<:_:TG]B=EL;E<3<3:w#i
•
g m o lb\ n (Jm;o#Qkn;&m;knoEi A%3>D5-T(&$&2 BD!'M"`,"|5-2N%nA!BD5@!+-,"`21+ 9A]3AQ5-+@,BD2
A21'G ('h<(+-O$( (,_5c$&2=$&2? (R 3AJ5@+-,:B?21"
5-+@,AJ' ! (9AJ,:+@2;"A = LU
K(L21"`5f5-+@,AJ' C (9AJ,:+@2=,:'&H>?+-,/21 (+-2 F
L<E! (+Y|9:e2?+N2?' A!' !9AJ,"-Ye2?5
U21"`5N5-+@,AJ' C (9AJ,:+@2]"@ (<E>?+-,/21 (+@2 F
U<E! #+Y` (<(<E2?+]2?'SA!' !9AJ,"IY~[/9 "1d A !,/5N21'tnAJ,/5=$( 3 .?32
A!9 C!+-,_5-T(32 76 (2BD219/ (,&$&2e9Wd >19/,:3,/'#AJ5@,:!'4$&2 3=A! #"-"l$(A!'#"\9/2B?AC" <MAJ+@5@,BD (9:,/21+l. oEi i\%jkHEo#m;;((kn [cf'(24H!,"=>D5IAJ)(9:,:249AGnA!B~5-!+-,:"-AQ5-,/C' $(2
AKL9:A +->1"@!9: &5@,:!'$& "`P&"`5@.?329:,/'(>;AJ,:+@2
Ax = b21"`5N>;7C #,/0QAJ9:2?'65-2 9AG"@,/3<(9:2]+->1"@!9: &5@,:!' $(2 $&21 &R "`P&"`5@.?32;"c5@+-,AJ' C (9:A!,/+-21"
Ly = b<# (,:"Ux = y
[
F QoM oE1knm1knoEi C1Q ' % A = (aij)1≤i,j≤n
:%-%n 5 0
$ $' % * +60 :$0 :%-%k 2 $7 :%
∆k =
a11 · · · a1k"""" " "
"""
ak1 · · · akk
,
% $ " ,5 I % % $(L,U) - U % 0+%& %
I % &% L % +0& % % &% -4% > L +60 :%0= 5
A = LU.
O&Ll &C d TOPO<M!5@T(.;"`2%$(2 9A,X+-!<EC"@,_5-,/C' [/C[/ 4'ad 21"`5N<#AC"N$&>1+-A!,:"@!'#'#AJ)(9:2![ '2-,2D5;K\219/9:2 2;"|5406+IAJ,:2G"@, K\<#A!+]2DR&2?3<(9:2!K
A2;"|5$&>D^M'(,/2h<M6"`,/5@,:0!2C[ X'x2-,2D5;K\"@,
∆k ' d 2;"|5<MA!"e,/'O0C2?+I"`,:)(9:2!K&AJ9:!+I"e,/9L2?RO,"`5@2= (' 0!21BD5@21 (+'(!' 'O (9
xk ∈ Ker∆k 2D5c2?' 9:2=B?!3<(9:>D5IAJ'65c<#AJ+$(21" ?>1+@6"*!'BDC'#"|5-+@ #,_58 ('0!2;B~5@21 (+8'(C'S'O (9
x = (xk, 0)76 (,X0!>1+@,/^#2
Ax · x = 0KaB?2 7C #,
B?!'65@+-21$(,_5c9:2][email protected]+@2*$&>D^#'#,a<M6"`,/5@,/ $&2A[ •
g oEil1m?Q&m;knoEi #<(<M6"`C'#"76 ad A! B?! (+I"e$&2=9 d >?9:,/3,:'#AQ5-,/C' $(243=A! #"-" ,:9L'ad PGAJ,/5c<#A!"e)E2N"@!,:']$&2 nAJ,:+@2 $&2 <E2?+-3% &5IAQ5-,/C'#" <E! #+lBIT#A!' !21+l$&2 <(,/0CJ5;K1BJd 21"`5$&,:+@2 7C #2X5@C #"a9:21"l<(,:0!!5-"l'#A5- (+-2?9"
akkk
"`C'C5X'#!''O (9:"1[ 9/C+-"1K!A 0!2;B 9:21"X'(J5IAQ5@,:!'M"X$&2c9ADX+@C<M6"`,/5@,:!' [/C[/ C'AAn =
En−1...E1AA 0C21B
Ek $&>?^#'(,:2]<#A!+%F| [ CY~[LN'b<M6"`2 U = An 2D5 L = (E1)−1...(En−1)−1 [
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5 + '()"!$! )#$ #%
9:!+I"c!' AA = LU
2?5,:9l+-21"`5@2%"@,:34<#9/213421'65N0!>1+@,/^#21+76 (2L21"`5N)#,/21'b5@+-,:A!' ! (9AJ,:+-28,/'&H>H
+@,:2? (+-2![(f'4B?A!9:B? (96nA!BD,:9:2 3!'65-+@2e7C #2(Ek)−1 "?d !)&5-,/21'65 nA!B?,/9:2?321'C5 N<#AJ+@5@,:+\$(2 Ek 2?'4BIT#AJ'1
!21A!'659:28"`, !'#2=$(21"c>19/>13421'65-"c"@! #"9A$&,:A !C'#AJ9:2!K&BJd 21"`5$&,:+@2=76 (2!K&<EC"-AJ'65lik = ak
i,k/akk,k
K<MC (+
k + 1 ≤ i ≤ n K(C' A
Ek =
1
0
[ [ [[[[
1[[[ −lk+1,k
[ [ [[[[ [[[ [ [ [0 −ln,k 1
, (Ek)−1 =
1
0
[ [ [[[[
1[[[+lk+1,k
[ [ [[[[ [[[ [ [ [0 +ln,k 1
.
f' AJ (5@+-2]B?AJ9BD #9 3!'65@+-2]76 (2L2;"|5f5-+@,AJ' ! (9AJ,:+@2=,:'&H>?+-,:2? (+-282D5N76 (2 "@A
k .?32]BDC9/C'('(282;"|5
9:A43 .?32]76 (28BD219/9:2]$&2(Ek)−1
L =
1 0 . . . 0
l2,1
[ [ [ [ [ [ [[[[[[ [ [ [ [ [ [
0ln,1 . . . ln,n−1 1
.
9anAJ (5f3A!,/'65-2?'#A!'C5f0C>?+-,_^M2?+f76 (2]9:21"c<#,/0CJ5-"f'(2]"?d AJ'#'6 #9/21'C5f<MA!"f"@! #"f9 d TOP6<EJ5-T(.1"@2=nAJ,/5@2]"@ (+9/2;"]3AQ5-+@,BD2;"
∆k [jN'9:2 0!>1+@,/^#2<#A!+]+->1BD #+@+-2?'#B?2![ 2 <(+-2?3,/21+]<(,:0!!5 a112;"|5 '(!''O (9eB?AJ+
> CA!9E∆1 76 (,2;"|5,/'O0C2?+I"`,:)(9:2![(N'G"` #<(<M6"`2=76 (2N5@! M" 9:21" <(,:0!!5-"l| #"-7C d %9 d !+I$&+-2 k− 1
"@!'65'(!'x'6 #9:"1[1t!'65-+@C'#" 76 (2 9:2h'(C (0!2;AJ x<(,/0CJ5
akkk
2;"|5AJ M"@"@,'(!'x'6 #9 [ 0 C3432 9/2;"k − 1<(+@213,/21+-" <#,/0CJ5-" "`C'C5X'#!''O (9"?KC!' A=<( B1AJ9BD (9:2?+ "-AJ'M"X2?'MBD!3 )(+-2e9A83AQ5-+@,BD2
Ak [ON'>;BD+-,_5AJ9:!+I"=9 d > 6AJ9:,_5->
(E1)−1...(Ek−1)−1Ak = A"@! #"89AbHC+@32 $Ld ('(2 >CA!9/,/5@>21'C5-+@2G3AJ5@+-,:B?21"
<#AJ+c)(9:&B?"(
Lk11 0
Lk21 Id
)(
Uk11 Ak
12
Ak21 Ak
22
)
=
(
∆k A12
A21 A22
)
,
A 0!21BUk
11
KLk
11
KE2D5∆k $(21"N)(9/&B1"NB1AJ+-+@>;"f$&2 5-AJ,:9:9/2 k KM2?5 Ak
22
KIdKE2D5
A22$(21"N)(9/&B1"NB1AJ+-+@>;"
$&25-A!,/9:9/2n − k [ 'AJ<(<(9:,7C MAJ'65]9:Ab+-. C9/2$&23 (9/5@,:<(9/,B?AJ5@,:!'$&2;"]3AQ5-+@,BD2;"=<#AJ+8)(9:&B?"1Ka!'!)&5-,/21'C5
Lk11U
k11 = ∆k,
Uk
11
21"`5N ('(2 3AQ5@+-,BD285-+@,AJ' C (9AJ,:+@2%"@ (<E>?+-,/21 (+-2!K#2?5Lk
11
('(2%3AJ5@+-,:B?2]5@+-,AJ' C (9:A!,/+-28,:'&H>N+@,:2? (+-28A 0!2;BN$&2;"8]"@ (+c9:A$&,A C!'#A!9/2C[MN'h21't$&>1$& #,_5N76 (289A43AQ5@+-,BD2
Uk11 = (Lk
11)−1∆k 2;"|5
,/'O0!21+-"@,:)(9/28B?!332*<#+@&$& (,/5$&283AQ5-+@,BD2;",/'O0C2?+I"`,:)(9:21"1[ !'t$&>D5-2?+-3,/'#A!'65f21"`5$&!'#B*'(C'b'6 #9 [N+
detUk11 =
k∏
i=1
akii 6= 0,
$&!'#B*9:28<(,:0!J5ak
kk
49 d >D5IAJ<E2k2;"|5f'#!' '6 #9 [
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( )$ )($ * # ( $*+
9N'(2 +-21"`5@2t<(9: #"G76 ad 0C>?+-,_^#21+9 d ('(,BD,/5@>C[ !,/5G$&2? &R$&>;BD!3<EC"@,_5-,/C'#" $&2 9A3A5-+@,BD2
A = L1U1 = L2U2[\N'2?'$&>;$& (,/5 76 (2
L−12 L1 = U2U
−11
Kl9Ab3AJ5@+-,:B?2L−1
2 L12;"|585-+@,AJ' ! (9AJ,:+@2,:'&H>?+-,:2? (+-22D5U2U
−11
21"`585@+-,:A!' ! #9:A!,/+-2"` (<E>?+-,:2? (+-22?'0!21+`5- $& 213432 [:![:(&[ 9/9:21"f"@!'65$&C'#BN5@C &5@2;"c9/2;"f$&21 &R $&,A C!'#A!9/2;"?KO2?5NBDC3432=9A$&,:A !C'#AJ9:2*$(2
L−12 L12;"|5B?!3<EC"@>?2=$&2!K&C' A
L−12 L1 = U2U
−11 = Id
[
O C T5 0 % ' % +%& % % &% " 0 % 0< 1 ! >B
% % +0& % % &% 3 , ++ 0 3$ '% % D,J +60 0%
T"
T ′ % % % +0& % % &% " 8% 0
TT ′ $ % 0+%& % I% &% 3 ++5 %:'0 6$* ++5P%
T 6
T ′ "
! #"9AJ,"-"`C'#"AJ b9/2;B~5-2? (+c9A$&>?3!'M"|5-+-AJ5@,:!' >?9:>?32?'65-A!,/+-2=$& 2?332 [/C[/O[
!\Q&m1k Lll (!m1oM k 1(m1knoEi N'<M21 &5]B?AJ9BD #9/21+*9AGnA!B~5-!+-,:"-AQ5-,/C'LUF "`,219/9:2b2DR&,:"`5@2 Y%$ad ('#2h3AQ5-+@,BD2
A<#AJ+4,$&2?'65@,/^MB1AQ5@,:!'y$&2
AA! <(+-&$& (,/5
LU[ X'x<EC"-AJ'65
A = (aij)1≤i,j≤nK(2D5
L =
1 0 . . . 0
l2,1
[ [ [ [ [ [ [[[[[[ [ [ [ [ [ [
0ln,1 . . . ln,n−1 1
, U =
u1,1 . . . . . . u1,n
0 u2,2
[[[[[[ [ [ [ [ [ [ [[[0 . . . 0 un,n
,
B?!332L21"`5c5@+-,:A!' ! #9:A!,/+-2=,/'&H>1+@,:2? #+@2=2D5
U5-+@,AJ' C (9AJ,:+@28"@ (<M>1+@,:2? #+@2CK&<MC (+
1 ≤ i, j ≤ n ,/90O,:2?'65
ai,j =
n∑
k=1
li,kuk,j =
min(i,j)∑
k=1
li,kuk,j .
'h,$&2?'65-,_^MA!'65e<#A!+!+I$&+@2*BD+-!,"@"-AJ'65 9:21"B?!9:!'('(2;" $(2AC'G2?' $&>1$( (,_5c9:21"BDC9/C'('(2;" $&2
L2D5
$(2U[ ,:'#"@, K6AJ<(+-.1"jA 0C!,:+jB1AJ9BD (9:>c9:21"
(j−1)<(+-2?3,:.?+-21" B?!9:!'('(2;" $&2
L2D5 $(2
U21'4H!'#BD5@,/C'
$(21"(j − 1)
<#+@2134,:.?+-21"cB?!9:!'('#21"$&2AK#!' 9/,/5f9:A
j .?328BDC9/C'('(28$&2
A
ai,j =
i∑
k=1
li,kuk,j ⇒ ui,j = ai,j −i−1∑
k=1
li,kuk,j<E! (+
1 ≤ i ≤ j,
ai,j =
j∑
k=1
li,kuk,j ⇒ li,j =ai,j −
∑j−1k=1 li,kuk,j
ujj
<E! (+j + 1 ≤ i ≤ n.
N'B?A!9:B? (9/2$&C'#B%9:21"j<#+@2134,:.?+-21"=B?!3<M6"@A!'65@21"*$&249:A
jW.1342B?!9:!'('(24$&2
U2?5=9:21"
n − j$(2?+-'(,/.1+@2;"XBDC3<M6"@A!'C5-21" $&29:Aj .?32fBDC9/C'('(2f$&2
L2?'H!'MB~5@,:!' $(2f9/21 (+I"
(j−1)<(+-2?3,/.1+@2;"
B?!9:!'('#21"1[(N't$&,:0O,:"@2=<#AJ+c9:28<(,:0!!5ujj
7C #,a$&!,/5$&!'MB.?5@+-2='(!'b'O (9 5
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5 + '()"!$! )#$ (
EoM knm iLl g QkL\ N'>1BD+-,/5h2?'<#"@2? #$&W9AJ' CA C2b,/'&HC+@3AJ5@,7C #2 9Wd A!9 C!+-,_5-T(32BD!+-+-21"@<MC'#$(AJ'65X89A83>D5-T(&$&2N$&2*$&>1B?!3<EC"@,_5-,/C' ][&N' 0O,/21'65 $&20C!,:+ 76 (2f9Wd C'<#AJ+IBDC (+@59:A43AQ5-+@,BD2
ABD!9:!'#'(2N<MAJ+cBDC9/C'('(2 0#%9Wd >D5-A!<M2
k!'bB?AJ9BD #9/2*9:A
k .?32=BDC9/C'('(2=$&2
LK&<( (,"
!' nA!,_5]A!<(<#AJ+IA"5@+-2%$&2;" ?>1+@6"N"@! #"*9Ah$(,:A !!'MAJ9:2%$&29:Ak .?32BD!9:!'#'(2 21' 2-,21BD5@ #A!'658$(21"
BD!3 )(,:'#AJ,"`C'#" 9:,/'#>1AJ,:+-21" $(29Ak .?32f9:, C'(2NA 0C21BcBIT#ACBD ('(2N$&21" 9/, !'(2;" $&2
k+1n[ 0 !332
9Wd >?5-AJ<E2k9:21"
k<(+-2?3,/.1+@2;"]9:, C'(21"42?549:21"
k − 1<(+@213,/.1+@2;" B?!9:!'('(2;" $(2G9A 3AQ5-+@,BD2h'(2
"`C'C5 <(9/ M" 3&$&,/^#>?2;"?KC!' <E2? &5"|5-&B,!2?+ $(AJ'#" (' 3 .?325-AJ)#9/2;AJ aKOBDC'C5-2?'#A!'65X,/'(,/5@,AJ9:2?321'C5e9:A3AJ5@+-,:B?2
AK69:21" 3AQ5-+@,BD2;"
Ak 2D5 Lk = (E1)−1...(Ek−1)−1 FHC' '(2="|5-&B,!276 (2*9/2;" >19/>132?'65-"'(!'5-+@,:0O,:A! &R$&2
Lk ]9A8<(9A!B?2N$&2;" 1>?+-C"X$&2 Ak "`C #"X9A%$&,A C!'#A!9/2f$(2*"@21" k− 1<(+-2?3,:.?+-21"
BD!9:!'#'(21"IY~[ 9A^#' B?2]5IAJ)(9:21A! BD!'65-,/21'#$&+IA9/2;"3AJ5@+-,:B?21"L2D5UFL$(A!'#""-A<#A!+`5-,/2 ,/'(H>N
+@,:2? (+-2Fn"-AJ'#"9A$&,A !C'#AJ9:2=$&2;Ye2D5U$(A!'#"c"@A4<#A!+`5-,/28"@ (<E>?+-,/21 (+@2 Y~[
=%A)8k = 1, n− 1 ←− (#4%* < k =%A8
i = k + 1, n ←− 9+: < iaik = aik
akk←− :#=%A#< < ;%= =%:): < @<
=%A8j = k + 1, n
aij = aij − aikakj ←− ;%= 3 59: 49+B0=%: @<#B 9+: < B i <6(k 9: @< 4 =%A5; < <%:
j 9: @< 4 =0A5; < <%:i 9: @< 4 =%A ; < <%:
k
oE\m1x:o# g Q&m;knoEil jC (+%321"@ (+-2?+ 9 d 2 B?ACBD,/5@>G$(2G9Wd A!9 C!+-,_5-T(32G$&2 9A $&>;BDC34<EI"`,/5@,:!' C' BDC34<(5@2%9:24'(!3 )(+@2 $Ld C<M>1+-AJ5@,:!'#"'(>1B?21"-"@A!,/+-21"G"@!' ACB?BDC3<(9/,"-"`213421'65 Fn76 (,"`21+-AS<(+-!<E!+@5@,:!'('(219c"@!'5@2134<M" $Ld 2DR&>1B? &5@,:!' "` #+ ('xC+-$(,/'#AJ5@21 (+IYD[ N'x'(2 B1AJ9BD (9:2 <#A!"2DR(A!BD5@2?321'C5cBD2='(C3%)(+-2N$ad C<M>1+-AJ5@,:!'#"1KO2D5c!' "`2=BDC'C5-2?'65@28$& <(+-2?3,:2?+ 5@2?+-328$&28"`C'h$&>10!2N9/C<(<M2132?'65*AC"`PO3<&5@!5@,76 (2]9:!+I"@76 (289A $&,:32?'#"@,/C'
n21"`5!+IAJ'M$&2![ )N2%<#9/ #"1KE<MC (+N"@,/3<(9:,/^#2?+
!' '(2=BDC3<&5@2*7C #2N9:21" 3% (9/5@,:<(9:,:B1AQ5-,/C'#" 2?5$(,/0O,"`,:!'#"*FH2?5e<MA!" 9/2;" AC$($&,/5@,:!'#"e$&!'65e9/2*'(!3 )(+-221"`5f2?' !>?'#>?+IAJ9L$& b3 .?328!+I$&+-2*$&2 C+-A!'#$&21 (+IYD[
4X9:,/3,:'#AQ5-,/C't! b$&>1B?!3<EC"@,_5-,/C' 0&9/28'(C3%)(+-28$Ld !<E>?+IAQ5@,:!'M"Nop
2;"|5
Nop =
n−1∑
j=1
n∑
i=j+1
(1 +
n∑
k=j+1
1),
76 (,WK#AJ <(+-2?3,/21+c!+I$&+-2!K($&C'('(2Nop ≈ n3/3
[ ()M"|5-,_5- &5@,:!' FHC +-2?3!'65->?2N$&2;"@B?2?'65@2"` (+ 9:21"t$&21 &R"`P&"`[email protected];"h5@+-,:A!' ! (9AJ,:+-21"IY 0c9/2'(C3%)(+-28$Ld C<M>1+-AJ5@,:!'#"
Nop2;"|5$(!'('(>=<#A!+c9:A HC+@3 (9/2
Nop = 2n∑
j=1
j,
76 (,WK#AJ <(+-2?3,/21+c!+I$&+-2!K($&C'('(2Nop ≈ n2 [ 5@!5-A!9#9A8+@>;"`C9/ (5@,:!'$Ld #'G"`P&"`[email protected]/,:'(>1A!,/+-2
Ax = b<#AJ+ 9:A83>?5@T(&$&2$(2f9:A8nA!B~5-!+-,:"-AQ5-,/C'
$(2?3AJ'#$(2Nop ≈ n3/3
!<E>?+IAQ5-,/C'#"]B1AJ+n2 21"`5%'(>!9:, C21A!)(9/2h$&2?0QAJ'65 n3 76 #AJ'#$ n 2;"|5
!+IAJ'#$L[
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( )$ )($ * # ( $*+
O&Ll &C Op N' &5@,:9:,:"@2NA! #"@"@,(9A83>D5-T(&$&2$&2f9A=nACB~5@C+@,"-AQ5@,:!' <MC (+ B1AJ9BD (9:2?+9:2S$(>D5@21+@3,:'#AJ'65 2D5b9 d ,/'O0C2?+I"`2 $Ld ('(23AQ5-+@,BD2C[Xj! #+G!)(5@2?'#,/+
A−1 K!'$(>1BDC3<M6"`2 A 2?'nACB~5-2? (+I" 2?5 !'=+->1"@! &5
n"@PO"`[email protected]"9:,/'(>;AJ,:+@2;"aA 0!2;BjBD!332X"@21BDC'#$("32?3 )(+-21"L9:21"L0C21BD5@2? #+-"
$(28)#A!"@2(ei)1≤i≤n
F4"` #)#"|5-,_5- &5@,:!'M"f<#A!+c+@>;"`C9/ &5-,/C'aK&3AJ,"9/2;"c0!2;B~5-2? (+I"c$&28)#AC"`2ei!'65$&2
'#!3%)#+@21 #"`2;" B?!3<EC"-AJ'65@2;" 'O (9:9:21"1K(BD2=7C #,L+@21'#$G3C,/'#"cBIT#.?+-2N9Wd >?5-A!<M28$&28$&2;"@B?2?'65@2=A 0!2;BLY~[
2='(!3 )(+@28$Ld !<E>?+IAQ5-,/C'#"e<MC (+B1AJ9BD (9:2?+A−1 21"`5
Nop ≈n3
3+
n∑
j=1
j2
2+ n(
n2
2) ≈ n3.
jC (+ B?A!9:B? (9/21+9/2t$&>?5@21+@3,:'#AJ'65G$&2AK !'$&>1B?!3<EC"@2
A21'nACB~5@21 (+I" 2D5 C'yB1AJ9BD (9:2
9:2t$&>D5-2?+-3,/'#A!'65$&2UFnBD219/ #,f$&2
L0QAJ &5t Y~KXBD2b76 (,c'(2b'(>;BD21"-"@,_5-2 76 (2t$&2 3% (9/5@,:<(9:,/21+9/2;"
>19/>132?'65-"*$&,A !C'#AJ (Rb$&2U2?'65-+@2 2? &RF
n − 13% (9/5@,:<(9:,:B1AQ5@,:!'M"-YD[ 2%'(C3%)(+-2%$Ld !<E>?+IAQ5-,/C'#"
<E! #+fB?AJ9BD #9/21+detA
2;"|5f$&C'#BNop ≈ n3/3
[ •g m o lb\ o nO '0 d 21"`5c ('(283>D5-T(&$&287C #,L'(28"?d AJ<(<#9/,76 (2*76 ad AJ (RG3AQ5@+-,BD21""@PO34>?5@+-,:76 (2;"e+@>12?9:9/2;"?K&$&>?^#'(,:21"
<EC"@,/5@,:0!21"1[ X9/9:2BDC'#"`,"`5@2 GnA!BD5@C+@,"`21+N #'(2 3AJ5@+-,:B?2A"`C #"N9:AGH!+-32
A = BB∗ B 2;"|5 #'(283AQ5@+-,BD2*5@+-,:A!' ! #9:A!,/+-2,:'&H>?+-,:2? (+-2FH2D5
B∗ "@!'bA!$ |C,/'65-2*C G5-+-A!'#"@<M6"`>12;Y~[
F QoM oE1knm1knoEi C&z A5 % J$4 % ,5 J% 2 " !
5' ,5 ' 0 % %! B
% +0& % I% &% 5 0 J $ ,3 ++:'
5 #: ! 5 % 2
A = BB∗. g oEil1m?Q&m;knoEi AJ+jAJ<(<#9/,B?AJ5@,:!'%$&2e9A^X+-!<EC"@,_5-,/C' [/C[/ &K1,/9O2DR&,"|5-2 (' ('(,76 (2eB?! (<#9/2$(2 3AQ5-+@,BD2;"
(L,U)5-2?9X76 (2
A = LUA 0!2;B
U5-+@,AJ' ! (9AJ,:+@2 "` (<E>?+-,:2? (+-2 2?5
L5@+-,AJ' C (9:A!,/+-2
,:'&H>1+@,:2? (+-28A PCA!'C5 ('(28$&,A C!'#A!9/2=$&2![MN'b'(!5@2D9:A43AQ5-+@,BD28$&,A C!'#A!9/2=$&>D^M'(,/28<MAJ+
D =diag(
√uii)
[9 2;"|5<EC"-"@,/)(9:2]$&2]<#+@21'#$&+-2*9A+IA!B?,/'(28B1AJ+-+@>8$&2;"f>?9:>?321'C5I"f$&2]9A$&,A C!'#A!9/28$&2UB1AJ+8 ('AJ+ ! #3421'658$(23% (9/5@,:<(9:,:B1AQ5@,:!'$&23AJ5@+-,:B?21"=<#A!+])(9:OB1"=3!'65@+-2 76 (2 ∏k
i=1 uii =det∆k > 0
KO∆k 21"`5e9A%"@! M"h 3AQ5@+-,BD2N$(,:A !!'MAJ9:2$Ld !+I$&+@2 k 2?R65-+-A!,_5-2N$&2 A K&$&C'#B*BIT#A!76 (2
uii2;"|5X"`5@+-,:BD5@213421'65\<EC"@,_5-,_|[CN'4<EC"@2eA!9/C+-"
B = LD2D5C = D−1U
76 (,&0!>1+@,/^#21'C5A = BC
[0 C3342
A = A∗ Kj!'21'x$&>;$& (,/5 C(B∗)−1 = B−1(C∗)[ X'0C2?+@5@ $& 213432 [/C[/
9AS3AJ5@+-,:B?2C(B∗)−1 2;"|545@+-,AJ' C (9:A!,/+-2G"@ (<E>?+-,/21 (+@2CK\5-AJ'M$&,:"76 (2 B−1C∗ 21"`545@+-,:A!' ! (9AJ,:+-2
,:'&H>1+@,:2? (+-2![ X9/9:21" "`C'65 $&!'#Be5-! &5-21" 9/2;"X$&2? (R$&,A C!'#A!9/2;"?[#)N2f<(9: #"1K!9:21" >19/>13421'65-" $(,:A !!'MAJ &R$(2
B2D5C"@!'65]9:21"%3.1321"1Kl$&C'#B9A $&,A !C'#AJ9:2$&2
B−1C∗ 'ad 21"`5 B?!'#"`5@,/5@ (>12G76 (2G$&2 !KB!d 2;"|5F I $(,/+-2 76 (2
C(B∗)−1 = B−1C∗ = IdK$&!'MB
C = B∗ [jj! (+h3!'65@+-2?+ 9Wd ('(,BD,/5@>$(2t9:A$&>;BD!3<EC"@,_5-,/C'$&2 0 T(!9:21",OP!K !'"@ (<(<EC"@2t7C d ,:9f2DR&,"|5-2 $&2? (RnA!BD5@C+@,"@AJ5@,:!'#"
A =B1B
∗1 = B2B
∗2
K $Ld B−1
2 B1 = B∗2(B∗
1 )−1 [ )N 2?332 [/C[/OK C'2?'$&>;$& (,/5 76 (2B−1
2 B1 = D = diag(d1, ..., dn)KL2D5]$&!'MB476 (2
A = B2B∗2 = B2(DD
∗)B∗2
[ 0 !332B22;"|5 ,/'O0!21+-"@,:)(9/2CKQ,/9M0O,/21'C5
D2 = Id$&!'MB
di = ±1[ON+ 5@C #"X9:21"XB?O2 BD,:2?'65-" $&,:A !C'#AJ &R$Ld ('(2
$(>1BDC3<M6"`,/5@,:!' $&2 0 T(!9:21",OP"@!'65<M6"`,/5@,/n"<#A!+eTOPO<EJ5@T#.1"@2
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5 + '()"!$! )#$
n!\ (m1kL\4l n (Jm;o#Qk 1&m;knoEi \ o 6 ' X' <(+IAQ5@,76 (2!K1!'%B1AJ9BD (9:2 9:2XnACB5@2? #+f$&2 0 T(!9:21",OP
B<#AJ+c,$&2?'65-,_^MB1AQ5-,/C' $(A!'#"9Wd >CAJ9:,/5@>
A = BB∗ [ !,/5 A = (aij)1≤i,j≤nK
B = (bij)1≤i,j≤nA 0!21B
bij = 0"`,i < j
[j! (+1 ≤ i, j ≤ n K#,/9L0O,:2?'65
aij =
n∑
k=1
bikbjk =
min(i,j)∑
k=1
bikbjk.
X' ,$&2?'65-,_^MA!'65N<#A!+NC+-$&+-2%B?+@C,:"-"@A!'659/2;"NB?!9:!'('#21"$(2AFH! "`2;"9:, !'(2;"?KBD247C #,l+-2?0O,:2?'65*A!
3.1324<# (,:"-76 (2A2;"|58"@P63>?5@+-,:76 (2 YfC' 2?'$&>1$( (,_5]9:21"=B?!9:!'('(2;"N$(2
B[ ,/'M"`,WKaAJ<(+-.1"*A 0!C,/+
B?AJ9BD #9/>N9/2;"(j − 1)
<(+-2?3,:.?+-21"eBD!9:!'#'(21" $&2B2?'GH!'#BD5@,:!' $&21"
(j − 1)<#+@2134,:.?+-21"eBDC9/C'('(2;"
$&2AK(C'h9:,/5f9:A
jW.13428B?!9:!'('(28$&2
A21' $(21"-"`C #"c$&289A4$(,:A !!'MAJ9:2
ajj =
j∑
k=1
(bjk)2 ⇒ bjj =
√
√
√
√ajj −j−1∑
k=1
(bjk)2
ai,j =
j∑
k=1
bjkbi,k ⇒ bi,j =ai,j −
∑j−1k=1 bjkbi,kbjj
<MC (+j + 1 ≤ i ≤ n.
N'hB?A!9:B? (9:2*$&C'#BN9:AjW.1342=BDC9/C'('(2=$&2
B21'GHC'#B~5-,/C'h$&2="`2;"
(j − 1)<(+-2?3,:.?+-21"eBDC9/C'('(21"1[
B?A! #"@2]$& t5@T#>?!+-.?32]<#+@>;BD>1$(2?'651KM!'t21"`5N" -(+N76 (2CKM"@,A21"`5N"@P63>?5@+-,:76 (2%$(>D^#'(,:2]<EC"@,_5-,/0C2!K
9/2;"5-2?+-321"="`C #"*9/2;"*+-ACBD,:'(21"*B?A!+@+->?2;""@!'65="`5@+-,B~5@2132?'658<M6"`,/5@,/n"?[ B?!'65@+IAJ,:+@2CKE"@,A'ad 2;"|5
<#A!"e$&>D^M'(,/2*<EC"@,_5-,/0C2!K6!' 5-+@C (0!21+-A876 (2ajj −
∑j−1k=1(bjk)2 ≤ 0
<E! #+ (' BD21+`5IAJ,:' +-A!' jK&BD2
7C #,a2?3< .1BIT(28$&2=5-2?+-34,:'(21+c9 d AJ9 !C+@,/5@T(32C[
EoM knm *iLl g QkL\ d A!9 C!+-,_5-T(32X$&2 0 T(C9/2;",OP<E2? (5\"1d >;BD+-,/+-2 $&2 nA ?!'*5@+-.1"lBDC3V<#A!BD5@22?'4 &5@,:9:,:"-AJ'65 ('"`21 (9O5-A!)(9:21AJ 76 (,(BDC'65@,:2?'65 !+-, C,/'(219/9:2?321'C5
A2D5X76 (,("@2+@2134<#9/,/5X<M21 4
<M21 $& 4nA!B~5-2? (+B[f2?3AJ+I76 (!'#" 76 ad ,:9#"` 45e$&2f"|5-&B,!2?+ 9A=3!,/5@,:>c,/'(H>?+-,/21 (+@2f$(2
A<( (,"@76 (2
A21"`5c"@PO3>D5@+-,7C #2
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( )$ )($ * # ( $*+
B?!3<(9:,:"-"@2?32?'651[ 2f'#!3%)#+@2f$&2+-ACBD,:'(21"XB1AJ+-+@>121"j2;"|5n76 (,M21"`5 '(>!9:, C21AJ)#9/2f$(A!'#" B?2BDC3<&5@2
$ad C<M>1+-AJ5@,:!'#"1[ 2#ACB~5@C+@,"-AQ5@,:!' $&2 0 T(C9/2;",OP 0&9:28'(!3 )(+-2=$ad C<M>1+-AJ5@,:!'#"
Nop21"`5
Nop =
n∑
j=1
(j − 1) +
n∑
i=j+1
j
,
76 (, KMAJ <(+@213,/21+c!+I$&+@2CK&$&!'('#2Nop ≈ n3/6
[ ()#"`5@,/5@ (5@,:!' 0l,/9 nAJ (5%2-,21BD5@ (21+ ('#2 +@213!'65@>12 2D5% #'(2h$&2;"@B?2?'65@2h"` (+%9/2;" "@P&"|5-.?321"5@+-,:A!' ! #9:A!,/+-2A!"-"`&BD,:>1"=
B2?5B∗ [ 2'(C3%)(+-2$Ld !<E>?+IAQ5@,:!'M"=21"`5%AJ <(+@213,/21+8!+I$&+@2
Nop ≈ n2 [ Ab3>D5@T#O$(2G$&2 0 T(C9/2;",OP 2;"|5 $(!'#BA!<(<(+- RO,:3AQ5-,/0C2?32?'65 lO oEknt n\bQ&jkn\ 76 (2B?2?9:9/28$&23=A! #"-"<MC (+c ('(283AQ5-+@,BD28"@PO34>?5@+-,:76 (28$&>?^#'(,:28<M6"`,/5@,:0!2C[
&m?QknCO\(i\\66m&m1 k !O4JQOl;O !+I"@76 ad ('(2 3AJ5@+-,:B?2eAN)E21AJ MBD! #<%$&2BDO2 BD,:2?'65I"l'O (9"1K;C' $&,/5 7C d 219/9:2 2;"|5 ! O\1 [ ,O9:21"
>19/>132?'65-"l'(!'%'6 #9:"j"@!'65l+->?<#A!+`5-,:"lf<(+- R&,/3,/5@>e$&2 9A$&,A C!'#A!9/2CK?C'%$&,/5j76 (2e9:A3AQ5@+-,BD2 A ('(2"`5@+- #BD5@ (+-2 \#il\ [#j! #+%BD2;"]$&2? (RS5|PO<E21"%$(2 3AQ5@+-,BD21"GFn76 (, A!<(<#A!+-A!,:"-"`21'65='MAQ5@ #+@219/9:2?32?'65$#AJ'#"c9A3>D5-T(&$&2%$&2;"f>19/>13421'65-"^#'#,:"NBD!332%$(A!'#"c9A<(9: (<#A!+`5$&2;"A! &5@+-21"c3>?5@T(&$&2;"-YDK#!'<E2? (5 AJ3>?9:,/C+@21+j9/2fB?!3<&5@2c$Ld !<E>?+IAQ5-,/C'#"j2D5X9AN5-A!,/9:9/2c$&2f"`5@&B,QA C2 '#>1BD2;"@"-AJ,:+-2 <E! (+ +@>;"`C #$&+-2 #'b"@P&"|5-.?32=9/,:'(>;AJ,:+@2C[ 0 2 6AJ,:'h2;"|5c5-+@.;",/3<E!+@5-A!'C5c21' <#+-AJ5@,7C #2![
g \i\kHm;knoEi CM q 0 % A ∈ Mn(R)
J % 086 %% 0 :%$'++ &%M6 08%*< #-%M ++ - >B
p ∈ N $JD,:% 2>
ai,j = 0#0&% |i−j| > p
" :% ++&%*6 % $ 2: -%
2p+ 1"
d ,:'65@>1+ .?5c$(21"c3AQ5-+@,BD2;"e)#AJ'#$(21"0O,/21'65$&289:A4<(+-!<#+@,:>D5->*"@ (,:0 A!'65@2![
a6Q!knC &C DG# ag" 4! c329 L8 !=H8 E!Z 6 CJ E" 4!4PvF26 E!=&]/+" 1!9 32 'b( 3I4!;*+8 26/!/>
O&Ll &C ,\9/2;"fnACB~5-!+-,:"-AQ5-,/C'#" 2D58$&2 0 T(!9:21",OPh<#+@>;"`21+@0C2?'65f9:Ah"|5-+@ MB~5@ #+@2)MAJ'#$&24$&21"*3AQ5-+@,BD2;"?KM,/9j'ad 2?'S21"`5*<MA!"*$&2%3 .?32$&249/21 (+*"`5@+- #BD5@ (+-2 B?+@21 #"`2C[ ' !>1'(>?+IAJ9WKM"@,A21"`5NBD+-2? M"`2GFn3 .?32 9Wd ,/'65@>1+@,:2? #+N$Ld ('(2 )#AJ'#$(2;Y~KE9:21"cnACB~5-2? (+I"
L2?5UKE!
B2D5B∗ "@!'65`<(9:2?,:'#"hFH9:2]BDC'65@+IAJ,:+@2*$&2]BD+-2? &R#Ye49 d ,:'C5->?+-,/21 (+f$&289A43.1328)#AJ'#$(2
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5 + '()"!$! )#$
AC"@"@!'#"4AJ yB?AC" $&2;"43AJ5@+-,:B?21"4BD+-2? #"@21"1[ R&<(9/,76 (!'M"4$ad A!)MC+-$9AS3A!'(,:.?+-2h$&2t"`5@&B,!21+BD21"=3AJ5@+-,:B?21"8BD+-2? M"`2;"*$#AJ'#"89A 3>?3!,:+@2$&2;"=!+I$&,:'#AQ5-2? (+I"?[ 0 C3342C'S'(2 "`5@&B,C276 (29:21">?9:>?32?'65-"N'(C' 'O (9:"*$&2 9:A3AJ5@+-,:B?2!KE!' C)&5@,:2?'65N #' CA!,/' $&2 <(9A!BD24AJ<(<#+@>;BD,AJ)(9:2![f! M"<(+->"`21'C5-!'#"e ('#2834>?5@T(&$&2]$(2="`5@&B,QA C2!K&AJ<#<M219/>12 1m1o (M o#Q; K&"@ (+c ('b2DR&2134<#9/28"@,/3<(9:2!K>D5-A!'65=21'65@2?'M$& 76 ad !''(29Wd &5-,/9:,"`22?'<(+IAQ5-,:76 (276 (2<MC (+8$&21"=3AJ5@+-,:B?21"=$&2 !+IAJ'#$(2%5IAJ,:9/9:2![ !,/5$&!'MBN9A43AQ5@+-,BD2
A =
9 0 −3 07 −1 0 40 5 2 01 0 −1 2
.
21"=>19/>132?'65-"8$&2A"@!'65]"`5@&B,!>;"?K9:, C'(2<#AJ+89:, C'(2!Ka$#AJ'#"= ('5IAJ)(9:21A! ('(2"@2? (9:22?'65-+@>12
[XN'$&>D^M'(,_5 ('5IAJ)(9:21A! 7C #, ,:'#$&,76 (2G9:2h$&>1)( &5$&21" 9/, !'#21"%$(2A$#AJ'#"
0j<(9: #"4<(+->1BD,"@>?32?'65 F FiY@Y]2;"|549/2 <(+-2?3,:2?+4>?9:>?32?'65'(!''O (9c$&2
9:Ab9:, C'(2i[\N'AtAJ #"-"@, )M2;"`C,/'$Ld ('5-A!)(9:21AJ :# 76 (, ,:'#$&,76 (29AtBD!9:!'#'(2$&2GBIT#AC7C #2
>?9:>?32?'65 "`5@&B,!>f$#AJ'#" 0C"@,ai,j
2;"|5e"|5-OB,C>f$(AJ'M" (k)K6AJ9:!+I",
(k) = j[
2G'#!3%)#+@2 $Ld >?9:>?321'C5I"%'#!''O (9"%$&2A2;"|54> CA!9 t9Ab5IAJ,:9/9:2h$& 0!21BD5@21 (+ # Fn! $(
0!21BD5@21 (+ Y~[ 2 '#!3%)#+@24$&29/, !'(2;"=$&2A2;"|5=> 6AJ9 9A 5IAJ,:9/9:2$& 0C21BD5@2? #+ [
)NA!'#"'(J5-+@2=2?RO213<(9/2CK(!'bA
#
9 1 1−3 3 3
7 1 6−1 2 8
4 45 22 31 1−1 3
2 4
' !>1'(>?+IAJ9WK&9/2;"cnACB~5@C+@,"-AQ5@,:!'M" 2D5*$&2 0 T(C9/2;",OPh$ad ('#2]3AQ5@+-,BD2]BD+-2? M"`2=<(+-&$& (,"`21'C5$&21"NnA!BD5@21 (+-"Y`<(9/21,/'M" A 0C21B8<E2? $Ld >19/>132?'65-"*'(C' 'O (9" Fn,/9 2?RO,"`5@2 $(21"=AJ9 !C+@,/5@T(32;"76 (,j3,'(,/3,"`21'65%BD2l`+@2134<#9/,"@"-A C2%3AJ,"=,:9:"]"@!'6583!,:'#"82 B1A!B?21"=76 (2 $(AJ'M"=9/2 B1A!"8$&2;"83AQ5@+-,BD21")#AJ'#$(21"IY~[ X'b<(+IAQ5-,:76 (2CK&9/2;"c3AQ5-+@,BD2;"cBD+-2? #"@21""@!'65"`C (0!21'65AC"@"@&BD,:>?28AJ &Rh34>?5@T(&$&2;"c,/5@>?+IA 5@,:0!21"%7C #2 '(! M"80!21+@+-!'#"8$#AJ'#"89A C #"!"@21B~5-,/C' [:![ tBD," $(21"-"`C #"?[ '2-,2D51K\9:21"8<(+-&$& (,/5-"3AJ5@+-,:B?2 0!2;B~5@21 (+c"`C'65enACBD,:9/2;"c4>?0QAJ9: (2?+fA 0C21B*BD2*5|PO<M2]$(2]"|5-OB,QA !2![
# L\kH( OilCt:o# g Q&m;knoEil6m4 Eo#QkHm \ am?Q(;1Oik<CVW:T[CKV>A@3:]>A@3:xGWE@ GW:=D¤B=:=D9[zTGWn3EQS :=D9S :]VWzTDFEL;@GW8;E<lS:]DR DRGW=[:=D9L;8;<3z=K8HVW:=DtE<AG9@3<CB=E[I GW:
S"M EI&zTVKvGW8~E<3D]S:kLM EVS VW:\S:n3 XS :[=[:>A@:kL;:_BKL;B=@LcS :kLNM 8;< :TVWDF:sNE8HVxL~K@]:=[CKV>A@:¡= i;i;%$w
E@UL~K¡[Z@3LHGW8;I3L;8;BKvGW8~E<lS :S :=@yC[CKvGFVW8~BT:=DtS'M EVS VW:ni ¥ L&<:kD=M Km8HGoI3KDxS"M @<3:]B=E<B=8~S:=<B=:B=E[[:]L;:
[E<AGFVW:tL~:xVWz=DF@L;GKvG¤DF@8~K<AG9>A@383[E<AGFVW:XAS :][CK<38;TVW:xDF@VWI VW:=<1K<AGW:XA>A@3:xLNM 8;< :TVWDF8;E<S'M @3<:x[CKGFVW8;B=:
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( )$ )($ * # ( $*+
KL~K[=[:xB=E[I3L;:Ty 8HGWz>A@3:oL~K[Z@LHGW8~IL;8~B=KGW8;E<US:_S :=@yJ[CKGFVW8;B=:=DsNO8;:=<>A@3:]LNM 8~<Q:#VWDF8~E<lDF:=[O3L;:X IVW:=[8;TVW:Q@:X @3<3:kEI&zTVKvGW8~E<O38;:=<UI3L;@3DxB=E[I3L;8~>Q@z=:vw#i
%)*0*)/ '2N/12 24/(+*-,2J /0 ( !3.&6*+268+(+,(+&K= .!$I= &62 #1= 268-."&62J /0 ( !3.&6*+268+(+, /5& = .!+! ( 8- "!#1=L(7M24&.,>! #%&; &62 :$A( 8 (+,&024*7( :$A(, 26=(7M24,&)(3$'/ .Q= 9 *-24&'!(#I "$I*= $'/ (30 (8-(+, -#+*7.&62J "/A,&)(:= :$%(,+ //1 !3<:*(0 +#+* .&62J /5, (+,&_< *+/ #I.* O(nα)
.(78α ≥ 2
K.Q= *-, /#5(+$'&_8 /A,&6*-$I2S*( $'/.Q= 9 *-24&'! (#I $'*= .$'&6*( +#+*7.&626 / 0Q /A& =L(/% !<:*7(0 +#+* .&62J /5, (+,&D.$',,-2 < *-/D#I."* O(nα) Uh*`"r =gT,+gfd`"r2 ¢ E8;G
I(n)L;:U<3E[OVW:S'M EI&zTVKGW8;E<DJI&E@ VJBKL;B=@3L;:TV
A−1 I1KVZ@3<?KL~mEVW8HGWn3[:SE<<3zXAGW:TL>A@'M 8;L&:Ty 8;DRGW:
C:#G
α ≥ 2z#VW8;§1K<AG
I(n) ≤ Cnα i5?lE<QGFVWE<3Dx>A@M 8;L&:Ty 8;DRGW:@3<KL~mEVW8HGWn3[:I&E@VBKL;B=@L~:#VL;:JI VWEQS@8;G
ABSE<AG_L;:J<E[OVW:S'M EI&zTVKvGW8;E<3D
P (n):TDRG_GW:=L>Q@'M 8~L:Ty 8;DRGW:
C′ I&E@VL~:=>Q@:=L
P (n) ≤ C′nα K:TB_L~:_[T[:k:Ty I&EDWK<AGαi"k<VW:=[CKV>A@3:k>A@3:
(
Id A 00 Id B0 0 Id
)−1
=
(
Id −A AB0 Id −B0 0 Id
)
.
pKVtB=E<3DFz>A@:=<AGXAL;:IVWEQS @38HGAB
:=DRGtEO GW:=<Q@C:=<C8~<Q:#VWDWK<AG¤@<3:][CKGFVW8;B=:_uE8;DtI3L;@3D¤mVK<3S:i'CE<3B
P (n) ≤ I(3n) ≤ C3αnα.
¢ E8HG%[CK8;<AGW:=<1K<AGP (n)
L;:x<3E[ZOVW:¤S'M EI&zTVKvGW8;E<3D%I&E@ V9BKL;B=@3L;:TVAB
I3KV%@3<KL;mEVW8HGWn[:oS E<<3zXGW:=L>Q@'M 8~L':Ty 8;DRGW:
C:TG
αzTVW8H§1K<AG
P (n) ≤ Cnα iA?lE<AGFVWE<D]>Q@'M 8~L':Ty 8;DRGW:_@3<UKL;mEVW8HGWn3[:_I&E@ V]BKL;B=@L~:#VA−1 SE<AGL;:<E[OVW:¡S'M EI&z#VKGW8;E<D I(n)
:=DRGkGW:=Lp>A@M 8;Lp:Ty 8;DRGW:C′ I&E@ VL;:>A@3:=L I(n) ≤ C′nα K:=B
L~:_[T[:k:Ty I&EDWK<AGαi"k<CVW:=[CKV>A@:_>A@3:
(
A BC D
)−1
=
(
A−1 + A−1B∆−1CA−1 −A−1B∆−1
−∆−1CA−1 ∆−1
)
.
Kv:=B∆ = D − CA−1B
sKI3I&:=L;z_I1KvVFuE8~D]B=E[I3L;z=[:=<AGxS : ¢ BnQ@V#w#i3k<l:=<lSzS @38HG
I(2n) ≤ 2I(n) + 6P (n),
DF8E<<3z=mL;8~m:\L~:TDoKS3S 8;GW8;E<3D=i&<l8HGWzTVK<AGoBT:TGFGW:uEVW[Z@3L;:I&E@Vn = 2k XE<lEOGW8;:=<AG
I(2k) ≤ 2kI(1) + 6
k−1∑
i=0
2k−i−1P (2i) ≤ C
(
2k +
k−1∑
i=0
2k−i−1+αi
)
.
9E[[:α ≥ 2
X3E<:=<S zS@8HGI(2k) ≤ C′2αk.
¢ 8n 6= 2k I&E@ VxGWE@G k
X38;L:Ty 8~DRGW:kGW:=LY>A@:
2k < n < 2k+1 i&k<U8;<3DFBTVW8HGkKL;EVWD]L~KJ[CKGFVW8;B=: ASK<3D
@3<3:kI3L;@3DtmVK<1S :_[CKGFVW8;B=:\S :kGK8;L;L;:2k+1
(
A 00 Id
)
E Id:=DRG]LNM 8~S :=<AGW8;GWz\S"M EVS VW:
2k+1 − ni&k<UEOGW8;:=<AG
I(n) ≤ C′(2k+1)α ≤ C′2αnα,
Ecol
e Po
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re.
5 + '()"!$! )#$ '
BT:\>Q@8":=DRG]L~:VWzTDF@3LHGKGoSzTDF8;VWzi
k<KZL;E<mGW:=[IDtBTVW@>A@3:L~KJ[@3LHGW8;I3L;8;BKGW8;E<S:[CKvGFVW8~BT:=D]S'M EVSQVW:nsN:TG]S E<3B_L;:=@ V]8~<Q:#VWDF8~E<&w
<:\I&E@3K8HGoI1KDkDF:\NK8HVW::=<[E8;<3DoS:n3 EI&zTVKvGW8~E<3D=i ?K8;DkS:=I@38;Do@<3:¡SzTB=E@:TVFGW:¡S: ¢ GFVKDFDF:=<
:T< GQGQXYE<DWK8HG>A@3:JB=:Z<M :=DRG¡I1KD_QVK8NiY<::]":#GX ¢ GFVKDFDF:=<Kl[8;D\K@I&E8;<QG\@<KL;mEVW8HGWn[:CS:[@3LHGW8;I3L;8;BKGW8;E<S :l[CKGFVW8;B=:TDC>Q@8xS:=[CK<1S :UO&:K@3B=E@3I|[E8;<3DJS'M EI&zTVKvGW8;E<3DI&E@V
nmVK<1S"i¥ LxK
EOGW:=<Q@UI&E@VxDFE<KL;mEVW8HGWn3[:_@<lB=E[IGW:_S"M EI&zTVKGW8;E<3D
Nop(n) = O(nlog27)K:=B
log2 7 ∼ 2, 81.sRQi Hw
9:oVWz=DF@3LHGKGXQGWE@Gx_NK8;G]DF@ VWIVW:=<3K<AGXQKZS :=I3@8;D¤zTGWzK[z=L;8;EVWz E<UK¡GFVWE@3zoS'M K@GFVW:=DxKL~mEVW8HGWn3[:TD=XS :xI3L;@3D%:=<I3L;@3D%B=E[IL;85>A@z=D=XAS E<AG9L;:]<3E[ZOVW:xS"M EI&zTVKvGW8~E<3D¤B#VWE8HG¤[E8~<D%Q8HGW:]I&E@V
nmVK<1S"XA[CK8;D
E<U<'M K¡GWE@PRE@VWDtI1KD¤GFVWE@zkL;:[:=8;L;L;:=@V]KL;mEVW8HGWn[:I&EDFDF8;O3L;:JsNBM :TDRGF vS8HVW:B=:=L;@8>A@8&B=E<3S@38;DF:_K@IL;@3DI&:TGW8HG9:Ty I&EDWK<AG
αGW:TL&>A@3:
Nop(n) = O(nα)w#iQ8~S:=[[:=<AGXvI&E@V9S:tGFVW=DmVK<3S:=D[CKGFVW8;B=:TD=X
L;:¡mK8~<S:_GW:=[IDoI VWEAB=@VWz\I3KVBT:=DKL~mEVW8HGWn3[:TD:=DRGkGFVWTDKIIVWz=BT85KO3L;:sNLM KL~mEVW8HGWn3[:ZS: ¢ GFVKDFDF:=<K\::]":=B#GW8~:=[:=<AG9z#GWzk@GW8;L;8;DFzDF@VtS:=D¤DF@I&:TVF EVS 8~<3KGW:=@ VWDw#i5?KL;n3:T@VW:=@DF:=[:=<AGXAB=:=D¤KL;mEVW8;GWn[:=DtE<AGDFE@3:T<QGcS:TDcIVWEO3L;=[:=DcS:DRGKO8~L;8HGWz9<Q@[zTVW8~>Q@:tsN8;L~DYK[I3L;8H§1:T<QGL;:=DY:TVFVW:T@VWDYS"M KVFVWE<3S85wc>A@38L;8~[8HGW:=<AGL;:=@ V]@3DWKm:_IVKvGW85>A@:i
M KL;mEVW8HGWn3[:¡S: ¢ GFVKDFDF:=<UVW:=I&EDF:\DF@VxL;:_L;:=[[:oDF@38;K<AG]S'M KII1KVW:T<3B=:_O8~:T<lO&z=<8~m<3:
)*0*)/"2N/12 +3bSV1`1,fdgTY*) c)'g=,v+%=)r ( #A*7 0$'24&0 (0 (+$QM !.&6*-2J8-(+,0 * 0*(#5(+$I&,:(.2S*(.(78 !$I= &62 #%= 2J8.&62J /5, (+& .909024&626 /A,).$ = 2;(+$0 (!$I= &62 #%= 2J8.&62J /5, (+&.Q09024&62J "/A,#I.*= .3* Q=L( 5.9<:24&6$A(:=4=L(
h*`"r =g=,v+gfd`"r2jo<lDF8;[I3L;:B=KL;B=@3L[E<AGFVW:>A@:
(
a bc d
)(
α βγ δ
)
=
(
m1 + m2 −m4 + m6 m4 + m5
m6 + m7 m2 −m3 + m5 −m7
)
,
K:TBm1 = (b− d)(γ + δ) m5 = a(β − δ)m2 = (a + d)(α + δ) m6 = d(γ − α)m3 = (a− c)(α + β) m7 = (c + d)αm4 = (a + b)δk<UB=E[I GW:O38;:=<H[Z@LHGW8~IL;8~B=KGW8;E<D]:#G¡ $JKSS8HGW8;E<D=i
:I&E8;<AGZBTVW@B=8~KLtSK<DL;:U:=[[:l= i;i £:=DRGJ>Q@:L~KVWTmL;:lS :C[Z@3LHGW8;I3L;8;BKvGW8~E<|S: ¢ GFVKDFDF:=<:TDRGkK@3DFDF8YKL~KO3L;:_DF8cL;:=D]B=EA:TBT8~:T<QGWDoS:TDo[CKvGFVW8~BT:=DxDFE<AGoS3K<Do@<3:_KL;m=O VW:<E<lBTE[[Z@ GKGW8;:i<I3KVFGW8;B=@L~8;:TVX3BM :=DRGoSE<3BAVK8'I&E@ VoS:=Dt[CKvGFVW8~BT:=DxO3L;EAB=D=i
¢ E8HGZKL;EVWD¡@3<:J[CKGFVW8;B=:CS :ZGK8;L;L;:n = 2k ik<Sz=BTE@3I&:BT:TGFGW:J[CKGFVW8;B=::T<0O3L;EAB=DS:GK8;L;L~:
2k−1 Xc:TGE<?KII3L;8~>A@3:L~KVW=mL;:US: ¢ GFVKDFDF:=<i ¢ 89E<0B=E[IGW:<3E<DF:=@3L;:=[:T<QG¡L;:=D¡[Z@L;GW8;IL~8;BKvGW8;E<3D[CK8;DK@3DFDF8%L~:TDZKS3S8HGW8;E<D=XYL;:C<3E[ZOVW:JS'M EI&zTVKvGW8;E<3D
Nop(n)I&E@ V¡::]":=BTGW@:TV¡L~:CI VWEQS@38HGS:CS :=@y
[CKvGFVW8;B=:=DxzTVW8H§1:Nop(2
k) = 7Nop(2k−1) + 18(2k−1)2,
B=KVLNM KS3S 8HGW8~E<JS :tS:=@ y_[CKGFVW8;B=:TDS:%GK8;L~L;:nVW:>A@38;:TVFG
n2 KS3S 8HGW8~E<3D=ijk<:%VWz=B=@ VFVW:=<3BT:9DF8;[I3L;:¤S E<<3:
Nop(2k) = 7kNop(1) + 18
k−1∑
i=0
7i4k−1−i ≤ 7k (Nop(1) + 6) .
k<:=<USzS @38HG¤uKB=8;L;:=[:=<AG]>A@3:oL~:k<3E[OVW:]EI GW8~[CKL'S'M EI&zTVKvGW8;E<3DNop(n)
zTVW8H§1:kL5K¡O&EVW<:JsRQi Hw#i
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re.
(0$ )$ )($ * # ( $*+
! % /)-/8 ' % 2( &%(',#8 21"N34>?5@T(&$&2;"*,_5->?+IAQ5@,:0!2;""@!'65N<MAJ+@5@,BD (9:,/.1+@2132?'65,:'C5->?+-21"-"@A!'65@21"<MC (+*9/2;"f5-+@.;" !+IAJ'M$&21"
3AJ5@+-,:B?21"cC 9:21"f3AJ5@+-,:B?21"fB?+@21 #"@21"1[ X't2-,2D5;KM$(A!'#"fBD2]B1A!"c9:21"f34>?5@T(&$&2;"$&,:+-21B~5-21"c<E2? #0!2?'65A 0C!,:+G (' BD/-&5b$(2B1AJ9BD (9=2D5t$&2"`5@&B,QA !2 2?'3>134C,/+-2<(+@CT(,:)(,_5-,_ Fn"@2S+IAJ<(<E2?9:2?+b76 (29:AnACB~5-!+-,:"-AQ5-,/C' ! h$(2 0 T(C9/2;",OP$&2?3A!'#$&2=$&2N9Wd !+I$&+@2N$&2
n3 !<E>?+IAQ5-,/C'#"IY~[ 0 !332?' ?!'#"<MAJ+c ('(28B?9:AC"@"@2*5@+-.1"c"@,/3<(9:2]$&2=3>D5@T#O$(21"c,/5@>1+-AJ5@,:0!21"1[
g \i\kHm;knoEi C G A
% % "=v2 %I 5 0%! +%& % 6
A< 0+ 8 7 + B $ '`u:'( % I8 % 0 %
(M,N):
M % < % u % %061 % ,5 >B 5
A = M −N " ! #I% ;% 0 * &% 8 7 +
(M,N)$ 2 J :%
x0%0 L61
Rn,Mxk+1 = Nxk + b ∀k ≥ 1.
F` [ $CY
,#9:A8"@ (,/5@2c$&2f"@!9: &5@,:!'#" A!<(<(+-&BIT(>?2;"xkBD!'O0C2?+ !2 0C2?+I"\ ('(2c9:,/3,/5@2
x76 #AJ'M$
k5@21'#$40!2?+I"
9Wd ,/'&^M'(, K(A!9/C+-"1K6<#AJ+<#AC"@"-A C249:A 9/,:3,_5-28$(AJ'#"e9A%+-2?9AQ5-,/C'G$&2=+->1BD #+@+-2?'#B?2 F` [ $CYDKO!'GC)&5@,:2?'65
(M −N)x = Ax = b.
A!+NBDC'#"@>176 (2?'65;KM"@,l9A "@ (,_5-2 $&24"`C9/ (5@,:!'#"NAJ<(<(+-&BIT(>?2;"fBDC'O0!2?+ !2CK("@A 9/,:3,_5-2 2;"|5NH!+IBD>?321'C59A"`C9/ (5@,:!'b$& t"`P&"`5@.?32=9:,/'(>;AJ,:+@2C[
) d #'%<E!,:'C5\$&2 0O (2X<#+-AJ5@,7C #2!K?,:9CnAJ &5j"@A 0C!,:+a7C MAJ'#$=C'8<M21 &5jA!+@+/.D5-2?+L9:21"l,_5->?+IAQ5-,/C'#"?K1BJd 21"`5 $(,/+-28 76 (2?9a3C3421'65
xk2;"|5"@ $ "@A!33421'65<(+-OBIT#2]$&289:AG"`C9/ &5-,/C'b,/'#B?!'('O (2
x[ 0 C3432
C''(2fBDC'('#A"5 <#A!"xK!C' '#2f<M21 &5X<#AC"X$&>1B?,:$(2?+X$Ld AJ+-+ .?5@21+ 9/2fB1AJ9BD (9#$(.1"X76 (2 ‖x−xk‖ ≤ ε
ε21"`59A4<(+->1BD,"@,/C'h$&>;"`,:+@>12![ AJ+cBDC'65@+-2NC' B?!'('#A#":5
AxF 7C #,a0QAJ &5
bYDK&2D5c ('tBD+-,_5-.?+-2*$ad A!+@+/.D5
H+->176 (2133421'65 (5@,:9/,"`>=2;"|5 ‖b−Axk‖ ≤ ε[ 0 2?<E2?'#$(A!'651K("@,L9:A4'(C+@32=$&2
A−1 21"`5 C+-A!'#$&2=BD2B?+@,/[email protected]+@2=<E2? &5#.?5@+-2*5@+-!3<M21 (+fB?A!+
‖x− xk‖ ≤ ‖A−1‖ ‖b−Axk‖ ≤ ε‖A−1‖76 (, <M21 &5f'(28<#AC".D5@+-2=<M2?5@,/51[
g \i\kHm;knoEi C8Q v2 5 ! #I% ;% % +6 L\ L\
;oEkHm oEk \MOJm;O k ilknm1k x0 ∈ Rn K% : 01 8% 1
xk 0 % ++* % : 0 !+6 x"
N'B?!332?'#B?2G<#A!+ $&C'('(21+ ('#2hBDC'#$&,/5@,:!''#>1BD2;"@"-AJ,:+-22D5"` $ "-AJ'65@2h$&2hB?!'O0!21+ C2?'#B?2$ad ('#243>?5@T(&$&2,/5@>?+IAQ5-,/0C24 9Wd A!,:$(24$& +IA P!!'S"`<E21BD5@+IAJ9 $&29:A 3AQ5-+@,BD2$Ld ,_5->?+IAQ5-,/C'xFH0C!,:+=9:A)N>D^M'(,_5-,/C' [:
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5 + '()"!$! )#$
hg oEil1m?Q&m;knoEi N'b$(>D^#'(,/5f9Wd 21+@+-2? (+ek = xk − x
[N'bA
ek = (M−1Nxk−1 +M−1b)− (M−1Nx+M−1b) = M−1Nek−1 = (M−1N)ke0.jA!+AJ<#<(9/,B?AJ5@,:!' $& 2?332 [/C[ $#K(!'t21' $&>1$( (,_5=76 (2
ek5@21'#$b0!21+-"##KM76 (219l76 (2%"@!,/5
e0K
"`,L2?5"`21 (9/2132?'65"`,ρ(M−1N) < 1
[
C_K<D_L5KCIVKvGW8~>Q@:ZL;:ZVK=AE<DFI&:TBTGFVKLS'M @<3:Z[CKvGFVW8;B=:Z:=DRG_S8HB=8;L;:UB=KL;B=@3L;:TVCsN8;LYuK@G\B=KL;B=@3L;:TVDF:TD9KL;:=@VWDIVWEIVW:=Dw#i'oM :TDRG¤I&E@ V>A@3E8E<@ GW8~L;8;DF:oS'M K@ GFVW:=D%BTE<1S 8HGW8~E<3D%DF@DWK<QGW:TD¤S:tBTE<Q:TVWm:=<3B=:XBTE[[:k8;<3S8~>A@3zB=8HS:TDFDFE@3D=i
)*0*)/"2N/121R1 2S&A$I/1( !.&6*-2J8-( A(+*-!24&62J(+/A/ ( 0 /52;( #I ,24&62(
1 2S& $'/ (0 78 ! #I ,24&62J /
*$5= 2 +*(09(A
0 K/A2;( #I.*A = M − N
.(78M
24/(+*-,2J<=L( = *+,3= .!3.&6*-2J8-(
(M∗ + N)(+,&
5(+*+! 24&62;(+/A/1( (#%= $I, ,2 (M∗ + N)
(+,&.$',,2 0 K/A2;(#I ,24&62 ( .Q= *-,
ρ(M−1N) < 1.h*`"r =g=,v+gfd`"r2 E@GoS"M KO&EVS"X3[E<QGFVWE<3Dx>A@3:
M∗ + N:TDRGxO38;:=<n:TVW[8HGW8~:T<3<3:
(M∗ + N)∗ = M + N∗ = (A + N) + N∗ = A∗ + N∗ + N = M∗ + N.
k<S zT§1<8;GkL~K<EVW[:\:=BTGWEVW8;:=L;L~: |x|A =√
〈Ax, x〉 DF@ V Rn s>Q@8Y:=DRG_O8;:=<@3<3:¡<3EVW[:\BKVA:=DRG
S zT§1<8;:JI&EDF8HGW8;:vw#iYk<<EGW: ‖.‖ L~Kl<3EVW[:Z[CKGFVW8;B=8;:=L;L;:DF@3O&EVSE<3<3zT:C |.|A iYk<KU[E<QGFVW:#V\>A@3:‖M−1N‖ < 1
B=:\>A@8'8;[IL~8~>A@3:kL~:VWzTDF@3LHGKGoSzTDF8;VWz_mVB=:¡ZL~KJpVWEI&EDF8HGW8~E< i;i HAi'k<lB=KL;B=@3L;:
‖M−1N‖2 = max|v|A=1
|M−1Nv|2A.
oVX3S'M KIVW=DxL;:\:=[[:¡Qi~i £QX&8;L':Ty 8~DRGW:vX3SzTI&:=<1SK<AGxS:
M−1NX3GW:=L>A@3: |v|A = 1
:TG
|M−1Nv|2A = ‖M−1N‖2.9E[[:
N = M −AXE<EOGW8;:=<AGX :=<I&EDWK<AG
w = M−1AvX
|M−1Nv|2A = 〈AM−1Nv, M−1Nv〉= 〈AM−1(M −A)v,M−1(M −A)v〉= 〈(Av −AM−1Av), (I −M−1A)v〉= 〈Av, v〉 − 〈AM−1Av, v〉
+〈AM−1Av, M−1Av〉 − 〈Av, M−1Av〉= 1 − 〈M−1Av, MM−1Av〉
+〈AM−1Av, M−1Av〉 − 〈MM−1Av, M−1Av〉= 1 − 〈w, Mw〉+ 〈Aw,w〉 − 〈Mw, w〉= 1 − 〈(M∗ + N)w, w〉.
oV 〈(M∗ +N)w, w〉 > 0XABKvV
(M∗ +N):=DRGtS zT§1<8;:]I&EDF8HGW8~:XQ:TG
w 6= 0B=KV
A:TG
M8;< :TVWDF8;O3L;:=D=i
CkE<3B‖M−1N‖2 = 1− 〈(M∗ + N)w, w〉 < 1,BT:\>Q@8&GW:TVW[8;<3:L~KJS z=[E<DRGFVKGW8;E<'i
@8~DW>A@:_B=:=Dt[zTGWnEQS:=Dt8HGWzTVKvGW8~:=D]S:kVWz=DFEL;@GW8;E<S :DR DRGW=[:=DtL;8;<3zK8HVW:=DoDFE<AG]S:=DRGW8;<3zT:=D]ZTGFVW:@ GW8;L~8;DFz=:TDDF@VS:=DEVS 8~<3KGW:=@ VWDSE<AGlL;:=DUBKL;B=@L~Dl<3:DFE<AGI3KDU:TyKBTGWDU[CK8;Dl:=<AGKB#nz=DS"M :#VFVW:=@VWDS"M KVFVWE<3S8NX8;L :=DRG%<3z=B=:TDFDWK8HVW:]S:tzTVW8H§3:TV%>Q@:¤B=:=D:TVFVW:T@VWD<3:tDF:tIVWEI1Km:=<AGI1KD%K@JI&E8;<AG%S:xS zTGFVW@38HVW:L~KB=E<Q:TVWm:=<3BT:JS:TD[zTGWn3EQS :=DE@X'I38HVW:X'S :ZL;:=DNK8HVW:B=E<Q:TVWm:TV_:#VWD_S:¡NK@DFDF:=D\DFEL;@GW8;E<3D=i F3EVFGn:=@VW:T@3DF:=[:=<AGtBT:=L~KZ<M :=DRG]I1KDxL~:_B=KD]B=E[[:oL~:_[E<AGFVW:kL;:VWz=DF@3LHGKvGkDF@8~K<AGi
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)$ )($ * # ( $*+
%)*0*)/ '2N/12 1 "24&$'/ (0 8 ! #I ,24&62J / * $I= 2 +*(09(A
0 /52;(#I.*A = M − N
.(78M24/(+*-,2J<=L(
% 24& $'/,:(78 /10!(+!<:*7(
b ∈ Rn (+& = . ,+ Q= $'&62J /x ∈ Rn &)(:=4=L( :$A(
Ax = b /,$ #9#I ,:( :$ 8 5. :$A(+&;.+#5(k= .!+&I 0 ( 24&+*7."&62 ( (+,-& . (78:&( 0 $'/ (3(+*-*(+$'*
εk ∈ Rn .$ ,:(+/A, xk+1/ (+,& #I., (7M9.98:&)(+!(+/A&D0Q /A/ #I.*
!.2S, #1= $'&& #I."*
xk+1 = M−1Nxk + M−1b + εk.
/,$ #9#I ,:( :$%( ρ(M−1N) < 1(+& :$ 26=1(7M24,&)($'/ (/% *+!((8:&; "*+2;(:=S=L((+&K$'/ (8 /5,-&;."/A&)(D#I ",-2S&62 (
ε&)(:=4=L(+, :$A( #I "$I*&; "$I&
k ≥ 0
‖εk‖ ≤ ε.
= *+,26= (7M24,&)($'/ ( 8 /5,&;./A&)( K :$'2_/1( 0 )#5(+/%0 :$A(09(
M−1N&)(:=4=L( :$A(
lim supk→+∞
‖xk − x‖ ≤ Kε.
Uh*`"r =gT,+gfd`"r2k< SzT§3<38HG:=<3BTEVW:LNM :TVFVW:=@Vek = xk − x
:TGE< K[CK8;<AGW:=<3K<AGek+1 =
M−1Nek + εki1k<l:T<Sz=S@38HG
ek =(
M−1N)k
e0 +
k−1∑
i=0
(
M−1N)i
εk−i−1.sR i Gw
pKV9KI3IL;8~B=KGW8;E<CS :tL5KkVWEI&EDF8HGW8;E<Qi~i H\8;L :Ty 8~DRGW:x@<3:¤<EVW[:t[CKGFVW8;B=8;:=L;L;:xDF@3O&EVSE<<3z ‖·‖s GW:=L;L~:>Q@: ‖M−1N‖s < 1I@38;DW>A@3:
ρ(M−1N) < 1ik<<EGW:S:¡L5KC[=[:uK=E<L~KC<3EVW[:¡:=B#GWEVW8;:=L;L;:
KDFDFEAB=8;z=:ikoVGWE@ GW:=DlL;:=DU<3EVW[:TDU:TBTGWEVW8;:=L;L;:=DDF@ VRn DFE<QGUz>A@38;KL;:=<AGW:=D t8;L:Ty 8;DRGW:S E<B@<3:
B=E<DRGK<AGW:C ≥ 1
X&>A@8'<:_SzTI&:=<1SUK8;<3DF8>A@3:_S :M−1N
X3GW:=L;L;:\>A@3:
C−1‖y‖ ≤ ‖y‖s ≤ C‖y‖ ∀y ∈ Rn.
<U[CKvP EVK<AG¡sRQi Gwt8~L'Q8;:=<AG
‖ek‖s ≤ ‖M−1N‖ks‖e0‖s +
k−1∑
i=0
‖M−1N‖isCε ≤ ‖M−1N‖ks‖e0‖s +Cε
1− ‖M−1N‖sS'M E LNM E<lEOGW8;:=<AGxL;:VWz=DF@L;GKvGoKv:=B
K = C2/(1 − ‖M−1N‖s)i
9*21"`5 5@213<#"h$(2S$&C'('(2?+G9/2;" 2DR&2134<#9/2;" 9:21"h<(9/ M"GBD9A!"-"@,:76 (2;" $&2 3>?5@T(&$&2;" ,/5@>?+IAQ5-,/0C21")MA!"@>?21""@ (+c ('(2]$(>1BDC3<M6"`,/5@,:!' +@>! (9:,:.?+-2![
g \i\kHm;knoEi CMz` g m oalb\ a(!o \k A = (aij)1≤i,j≤n
"4v2 5 D =
diag(aii) ++5: 3%
A"lv>!"# ! #I%J6% ' ! #I% ;%
u " 0J
M = D, N = D −A.9L"1d A !,/5$&2*9A]<(9: #""`,:3<(9:2*$&2;"e34>?5@T(&$&2;"e,_5->?+IAQ5@,:0!2;"?[ j! (+76 ad 2?9:9/2="`C,_5e)(,:2?'h$(>D^#'(,:2!K&,:9
nA! &58)(,:2?'S" -(+876 (249:Ah3AQ5@+-,BD2$&,:A !C'#AJ9:2D"`C,_5=,/'O0C2?+I"`,:)(9:2![jA!+8AJ<(<(9:,B?AQ5-,/C' $& 213432
[:![ .&K!$(AJ'M" 9:2NB1A!" A21"`5"@PO34>?5@+-,:76 (2+@>12?9:9/2CKC9A]3>D5-T(&$&2N$(2'6A!B?!)(,EBD!'O0C2?+ !2f"@,
A2D5
2D −A "@!'65f$&>D^#'#,/2;"<M6"`,/5@,:0!2;"?[
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5 + '()"!$! )#$
hg jilknm1knoEi CDCE` g m oa\ \ (\1 & Okn\ W
A = (aij)1≤i,j≤n"4v2 '
0J A
:% A = D −E − F
D = diag(aii) / +60 - −E $
L :% % +%& % I% &% < % B −F J L :% % +0& % I '% &%6< % B 6
A"*v> ! #I% % $' % ! #I% ;%0
4u " 0J
M = D −E, N = F.
jC (+X76 (2c9:A*3>D5-T(&$&2f$&23=AJ M"@" 2?,$&2?9("@!,/5 )(,/21'4$(>D^#'(,:2!K!,/9&nA! &5 76 (29:A*3AQ5-+@,BD2D−E"`C,_5\,/'O0!21+-"@,:)(9/2CK1B!d 2;"|5 $&,:+-2X7C #2
D"@!,/5l,:'O0!2?+I"@,/)(9:2Fn9:A3AJ5@+-,:B?2
(D−E)2;"|5lnA!B?,/9:2 ,/'O0!21+-"@2?+
B?AJ+j219/9:2c21"`5j5@+-,:A!' ! (9AJ,:+-2;Y~[ A!+jA!<(<(9:,:B1AQ5-,/C'4$& 2?3328 [:![ .&K;"@,A21"`5 "@P63>?5@+-,:76 (2+->?2?9:9:2
$&>D^#'#,/28<EC"@,_5-,/0C2!K&9A434>?5@T(&$&2]$(23=AJ #"-"! 21,:$&219LBD!'O0C2?+ !2![hg jilknm1knoEi CDC` g m oa\ \bQ na&m;knoEi ?
ω ∈ R+ "4v2 ! 6 % % ) < v +- &% ! v% ) B #0&% #-% %
ω 6 ;% u 0
M =D
ω−E, N =
1− ωω
D + F
jC (+76 (2]9A3>D5@T#O$(2]$&2]+-2?9AQR(AQ5-,/C'h"@!,/5f)(,/21' $&>D^M'(,/2CK#,:9LnAJ (52?'#B?!+-287C #2]9:A3AJ5@+-,:B?2D"`C,_54,/'O0C2?+I"`,:)(9:2![\C (+
ω = 1KlC'+-2D5@+-! #0!2 9A 3>D5-T(&$&2b$&2 3=A! #"@" 2?,$&2?9W[ ,
ω < 1K
!' $&,/587C d ,:9 "?d A C,_5=$Ld #'(2 3>?5@T(&$&2$&2 "@! M"! +@219:AJR(AQ5@,:!' K&5-A!'#$&,"*76 (2ω > 1
BDC+@+-21"@<E!'#$t ('(23>D5@T#O$(2 $&2"` (+ +@219:AJR&AJ5@,:!'a[ X' !>1'(>?+IAJ9WK,/9 2?R&,:"`5@2 ('<#A!+-A!34.?5@+-2 C<&5@,:3AJ9
ωopt76 (,
34,:'(,:3,:"@2]9:28+-A PC!'h"@<E21B~5-+-A!9 $&2]9A3AQ5-+@,BD28$Ld ,_5->?+IAQ5-,/C'M−1N
K#2D5N$&C'#B876 (,a3AQR&,:3,:"@289:A06,/5@2;"@"@28$&2]BDC'O0!2?+ !21'#BD2C[
a6Q!k ! &C L _8 A
*+8 26 J& *O8 H8 & K4^h8 $& !8 H8 ] >B ;E $& E9
ω ∈ ]0, 2[ ," S*K6J _Ig" /X3H8 EF26 ]&'j>
a6Q!k ! &C aE,$& " N*K4J _Ig'&" /X&H8
, F19 '!
ρ(M−1N) ≥ |1− ω| , ∀ω 6= 0,
6 32O_U &"Q" gb$&/ 26 E/]&'j I!R80 < ω < 2
>
hg jilknm1knoEi CDC ` g m oa\ \M #lk 6imQK :%0 % K% α 6= 0
"Gv>! ! #I%0 +% / ! #I% 9%0 4u " 0J
M =1
αId
N =
(
1
αId−A
)
.
A3>D5@T#O$(2 $& !+IA!$(,/21'C5 "@2?3 )(9/2t21'#BDC+@2b<(9: #"<#+@,:3,_5-,/0C2 7C #2t9/2;"3>D5@T#O$(21" <#+@>;BD>$&2?'65@2;"?K 3A!,:"219/9:2 A ('(2t,:'C5-2?+-<(+->D5-AJ5@,:!'x2?'y5-A!'C5 76 (2t3>D5-T(&$&2 $&2b3,/'#,/3,"@AJ5@,:!'$&2b9:AH!'#BD5@,:!'
f(x) = 12Ax · x− b · x
76 (, 9/ (,l$&!'('#2* #'(28<(9: #" !+IAJ'#$(2=A!<(<(9:,:B1AJ)(,:9:,_5->
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)$ )($ * # ( $*+
O CDC8C A
% P ++5: % :&% %! %λ1 ≤ λ2 ≤
... ≤ λn"
λ1 ≤ 0 ≤ λn : -% ! #I% 0K+%" 5 % ++0 &% :&%K%
α" 8
0 < λ1 ≤ ... ≤ λn : -% ! #I% 0 +% 0 % +6 &
0 < α < 2/λn / M :%0 % α!5 :
5 ρ(M−1N) $
αopt =2
λ1 + λn
min
αρ(M−1N) =
λn − λ1
λn + λ1.
O&Ll &C C G ,λ1 ≤ ... ≤ λn < 0
KXAJ9:!+I"4!'A (' +@>;"` #9_5IAQ5 "`PO3>D5-+@,76 (2 $& B1A!"4$&>D^M'(, <EC"@,/5@,/2?'yBIT#AJ' C21A!'C5
α21' −α [j! (+49:2h<#A!+-A!3.D5@+-2 C<&5@,:3AJ9 αopt
K\9:2G+IA P!C'"@<E21B~5-+-A!9 $&289A3AQ5@+-,BD28$Ld ,_5->?+IAQ5@,:!'b2;"|5f ('#28H!'#BD5@,:!'tBD+-!,"-"@A!'C5-2*$( t+-A!<(<MC+`5
λn/λ1[ ,
A2;"|5AJ &5- A!$ |C,/'65-2!KCA!9/C+-" BD2+IAJ<#<MC+`5 ' d 2;"|5AJ &5-+@2*7C #2f9/2=BDC'#$&,/5@,:!'('#2?32?'65cond2(A)
$&29A3AJ5@+-,:B?2
A[ AJ+*BD!'M"`>;7C #2?'651KM<(9/ M"9A3AQ5-+@,BD2
A21"`5N)(,:2?'SBD!'M$&,_5-,/C'('(>12!KE3421,/9:9:2? (+-2]21"`5N9A
B?!'O0!21+ C2?'#B?2N$&289A43>D5@T#O$(2]$& !+IA!$&,:2?'651[ • g oEil1m?Q&m;knoEi ) d AJ<(+-.1"9:2 2?332 [/C[ (K6!'b"@A!,_5f76 (2=9A 3>?5@T(&$&2]$& C+-AC$&,/21'65e2;"|5B?!'O0!21+ C2?'65@2*"`, 2D5f"`21 (9:2?32?'65"@,
ρ(M−1N) < 1[MN+
M−1N = ( Id− αA)KE$&!'MB
ρ(M−1N) < 1⇔ |1− αλi| < 1⇔ −1 < 1− αλi < 1 , ∀i.0 2;BD,X,:3<(9:,:76 (2 76 (2
αλi > 0<MC (+=5@C &5
1 ≤ i ≤ n[ AJ+]B?!'#"@>176 (21'65*5-! &5-21"89:21"]0 A!9/21 (+I"
<#+@C<(+@2;"j$&2A$&!,:0!21'65(.?5@+-2c'(!'4'O (9/9:21"j2?5 $& 3 .?32f"`, !'#276 (2
α[ A*3>D5-T(&$&2f$& !+IA!$&,:2?'65
'#2 BD!'O0C2?+ !2 $(!'#B4<#AC"8"`,λ1 ≤ 0 ≤ λn
Ka2D5%B?21BD, 7C #2?97C #2"@!,/5α[ ,WK AJ BDC'C5-+-A!,/+-2!KC'A
0 < λ1 ≤ ... ≤ λnKXA!9/C+-" !'x2?' $&>1$& #,_5G76 ad ,/9nA! &5
0 < α < 2/λn[ \C (+B?A!9:B? (9/21+9/2
<MAJ+IAJ3.D5-+@2 !<&5-,/3AJ9αopt
KL!'S+@213AJ+I7C #2 76 (29:AGH!'MB~5@,:!'λ → |1 − αλ| 21"`5]$&>1B?+@C,:"-"-AJ'65@2"@ (+
]−∞, 1/α]<( #,:"cB?+@C,:"-"@A!'65@2="` (+
[1/α,+∞[K($&!'#B
ρ(M−1N) = max|1− αλ1|, |1− αλn|. A!+BDC'#"`>;76 (2?'651K#9/283,:'(,/3 (3 $&2]9A4H!'#BD5@,:!'
α → ρ(M−1N)2;"|5NAJ5`5-2?,:'C5*AJ b<MC,/'65N$ad ,:'
5-2?+I"`2;B~5-,/C'αopt = 2
λ1+λn
$&2]B?21"c$&21 &R $(+@C,_5-21"1[
! % /)-/ - 2( -3'*+% 9;),+
A]3>D5-T(&$&2*$( !+IA!$(,/21'C5BDC'Q| ! #>21"`5 9A]3>?5@T(&$&2*,_5->?+IAQ5-,/0C2N$&2=BIT(!,/R<MC (++@>;"`C #$&+-2$(21"*"`P&"`5@.?32;"9:,/'#>1AJ,:+-21"*$&!'65N9A 3AJ5@+-,:B?2%21"`58"`PO3>D5-+@,76 (2%+->?219/9:2 $&>?^#'(,:2 <EC"@,/5@,:0!2![*9 "?d A C,_5$ad ('#2=A!3>?9:,/C+-AJ5@,:!'G"@<M2;B~5IA!BD #9:A!,/+-2*$(2*9A43>D5@T#O$(2=$( C+-AC$&,/21'65=F "` (+@5@C &5f"`,L219/9:2=BD!3 )(,:'(>?2A 0C21B8 #' <(+->1BDC'#$&,/5@,:!'('#2?32?'651KE0!C,/+N9:A )>?^#'(,/5@,:!' [:![ Y~[j! #+*B?!'#"`5@+- (,:+@2]9A 3>?5@T(&$&2$( C+-AC$&,:2?'65B?!'Q| ! (>CK&!'b,/'65-+@&$& (,/5c9:A'#J5@,:!'b$Ld 21"@<#A!B?28$&2 *=+@PO9:Q0E[
g \i\kHm;knoEi CDC Q r0
&% 61Rn " v> !"# % %L%$4: - :
0K &%r0
Kk $'? % -% 6Rn +6 8-%3 :%
k + 1 &% r0, Ar0, ..., Akr0"
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5 + '()"!$! )#$
21"21"@<#A!B?21"$(2 *=+@PO9:Q0(Kk)k≥0
HC+@32?'65 #'(2 "@ (,/5@2%B?+@C,:"-"@A!'65@2]$&2%"@! M"! 21"@<#A!B?21"0C21B%5@!+-,:2?9"
Kk ⊂ Kk+1 ∀k ≥ 0[ 0 C3342
Kk ⊂ Rn KjBD2?5`5-2h"@ (,_5-2h$&210O,/21'C5 H!+IBD>13421'654"`5-A 5@,:!'('#A!,/+-2%<#AJ+@5@,:+ $Ld ('GB?2?+@5-A!,/'+IAJ' M[ X9: #" <(+->1BD,"@>?32?'651K6'(C #"X9AJ,"@"@!'M"X9:29:21B~5-2? (+ 0!>1+@,/^#2?+7C d ,:9a2DR&,"|5-2= ('(2]$&,:32?'#"@,/C' B?+@,/5@,76 (2
k0K#A 0C21B
0 ≤ k0 ≤ n− 1K&5@219/9:2]7C #2
dimKk = k + 1"@,
0 ≤ k ≤ k0,
dimKk = k0 + 1"`,k0 ≤ k.
X'<(+@21'#AJ'65r0 = b−Ax0
KQ,:9&21"`5jnA!B?,/9:2c$&20!!,:+ 76 (2!KQ<E! (+ 9AN34>?5@T(&$&2f$( C+-AC$&,/21'651K 9 d ,/5@>?+->?2xk
AJ<(<#A!+`5-,/21'65*h9 d 21"@<#A!B?24A-'(2[x0 + Kk−1]
F $&>D^M'(,XBDC343249 d 2?'M"`213%)(9:24$(21"*0!2;B~5@21 (+I"x5@2?9"=76 (2
(x − x0) ∈ Kk−1Y~KLBD276 (,j,/3<(9:,:76 (276 (249/24+@>;"`,$&
rk = b − AxkAJ<#<#AJ+@5@,:2?'65*
9 d 21"@<#A!B?28$&2 *=+@PO9:Q0Kk
F A!"-"`&B?,/>8AJ +->1"@,:$& ,:'(,_5-,:A!9r0Y~[
jC (+ A!3>?9:,/C+@21+ 9:A34>?5@T(&$&2$& C+-AC$&,:2?'651K !' $&>;BD,$&2S$&2S|3,:2? &R BIT(C,:"@,:+hxk
$#AJ'#"9 d 21"@<#A!B?2]A#'(2
[x0 +Kk−1][ A3>D5-T(&$&2%$& C+-AC$&,/21'65cB?!'Q| ! (>%B?!'#"@,"|5-2!K#<#A!+`5-,/+f$ad ('
0!21BD5@21 (+,/'#,_5-,:A!9x0 ∈ Rn 2?5G$&2 "@!'y+@>;"`,$&
r0 = b − Ax0K B?!'#"`5@+- (,:+@2b ('(2 "@ (,/5@2 $&2
0!21BD5@21 (+-"xk ∈ [x0 +Kk−1]
5-2?9"7C #2rk = b−Axk
2;"|5!+@5@T# !C'#AJ9MAJ h"@! M"! 21"@<#A!B?2Kk−1
K<MC (+
k ≥ 1[
O CDC A 0 % J$4 % ,5 M%! 3 2 . -%:%
n"
x0 ∈ Rn r0 = b − Ax0 (Kk)k≥0 % #6 $% L%$4: - u
r0"
6L0 +-% 1 +0 $ M 2 &%k ≥ 1 #-%
xk ∈ [x0 +Kk−1]
rk = b−Axk⊥Kk−1.F| [: 6Y
&% 0k ≥ 1
! ,5 3 &%xk
%0 #-%L<>= "= B0"5 8 7 6 0 % ++M%7 - 0L$4 M %Ax = b
0 n ;% "
O&L\ CDC 2 213432 [:![ S3!'65@+-2 76 (2 9Wd A!9 C!+-,_5-T(32 $& !+IA!$(,/21'C5GB?!' | ! (>h7C d C'AtB?!' ? BDC3432 ('#2 3>D5@T#O$(2 ,_5->?+IAQ5-,/0C221"`5]2?'nAJ,/5% ('#2 3>D5@T#O$(2G$&,:+@2;B~5@2<( (,"@76 ad ,/9*BDC'60C2?+ !2 2?' (''#!3%)#+@2b^#'#,N$Ld ,_5->?+IAQ5-,/C'#" FH2?R(A!B~5-2?32?'65
k0 + 1
k021"`5 9:A
$&,/321'#"`,:!'B?+@,/5@,76 (2 $&2 *=+-P69:Q0(Y~[ 0 2?<E2?'M$(AJ'651Kl$(AJ'#"]9:A <(+IAQ5-,:76 (2C'9Wd &5@,:9/,"@2 BD!332 ('(234>?5@T(&$&2G,_5->?+IAQ5@,:0!2 76 (,eBDC'60C2?+ !2Z|'O (3>1+@,76 (2?32?'6521')M2;AJ #B?! (<3!,:'#"]$&2
k0 + 1,_5->
+-AJ5@,:!'#"1['C5- (,/5@,:0!2?321'C58,:9 21"`5=nA!BD,:9:2$(240C!,:+=<MC (+-76 (C,j9/2 !+IA!$(,/21'C58BDC'Q| C (>4A!3>?9:,/C+@249/2!+IA!$&,:2?'65f"`,:3<(9:2![ X't2,E2?51KM9/28+->1"@,:$(
rk21"`5f!+@5@T( !!'MAJ9L4 (' "`C #" W2;"`<#ACBD2
Kk$&2]<(9: #"c21'
<(9/ M" !+IAJ'#$a[ •hg oEil1m?Q&m;knoEi 1t!'65@+-!'M"]5-! &5$Ld AJ)E!+I$76 ad ,:92DR&,"|5-2h)(,:2?'x (' #'(,:76 (2
xk76 (,c"@AJ5@,"!
nAJ,/59/2;"fTOPO<M!5@T(.;"`2;"?[ 0 !332A2;"|5N$(>D^#'(,:2]<EC"@,_5-,/0C2!KM!'t<E2? &5N$(>D^#'(,:+9:2%<#+@&$& (,/5N"-B?A!9:A!,/+-2
〈x, y〉A = Ax · y "@ (+ Rn [N' BIT(21+-BIT(2 xk"@! #"9:AHC+@32
xk = x0 + ykA 0C21B
yk ∈ Kk−1K
2D5f9A4B?!'#$(,_5-,/C' $Ld !+@5@T# !C'#AJ9:,/5@>=$&2rk$&2106,:2?'65
〈A−1r0 − yk, y〉A = 0 ∀ y ∈ Kk−1,
BD2f76 (,('ad 21"`5X+-,/21'$ad A! &5@+-2f76 (2c9:A=B1AJ+IA!B~5->?+-,:"-AQ5-,/C' $&2ykB?!332c<(+@!|21BD5@,:!'4!+@5@T( !!'MAJ9:2e$&2
A−1r0"` (+ 9:2"@! #" 21"@<#A!B?2
Kk−1Fn<MC (+ 9:2c<(+-O$( (,_5e"@B1AJ9AJ,:+@2 〈, 〉A
Y~[ 0 21B?,(<(+-! (0C2e9Wd 2DR&,:"`5@21'#BD22D5f9Wd ('#,:B?,_5->]$&2
xk[
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)$ )($ * # ( $*+
C,_5k09A $&,:3421'#"@,/C'BD+-,_5-,:76 (2h$&21" 21"@<#A!B?21" $&2 *=+-P69:Q0K BJd 21"`5 $&,:+@2 76 (2CKl<E! #+%5-! &5
k ≥ k0K
dimKk = k0 + 1[ X'<#AJ+@5@,BD (9:,:2?+;K
AKk0 ⊂ Kk0+1 = Kk0
Ka$&!'MBrk0+1 = b −
Axk0+1 = r0−Ayk0+1AJ<(<#A!+`5-,/21'65
Kk0
5@C &5 21' 9/ #,#>D5IAJ'65 C+`5-T( C!'#A!9 [ AJ+eBD!'M"`>;7C #2?'651Krk0+1 = 0
2D5xk0+1
21"`5f9:A"@!9: &5@,:!' 2DR(A!BD5@28$& t"@PO"`[email protected]*9:,:'(>1A!,/+-2![
O&Ll &C CMp AS3>D5-T(&$&2 $& !+IA!$&,:2?'65BD!'J| ! (>tAS>D5->t<(+@>;"`21'65@>?2bBDC3342b ('<#+@&BD>;$&>=$Ld C+`5-T( C!'#A!9/,"-AQ5@,:!'G<#AJ++IAJ<(<E!+@549Wd 2;"`<MA!BD2=$&2+*=+-PO9/Q0[ f! M"0!2?+-+-!'#" <(9: #"9/C,/'76 (2CK1$&2 3A!'(,:.?+-2X>176 (,:0QAJ9:2?'65@2CK?!'8<E2? (5l9Wd ,:'65@+-O$( (,/+-2XBDC3342 (']<(+-!)(9:.?32 $(2X3,/'(,:3,:"-AQ5-,/C'a[ 9/ #"f<(+@>;BD,"`>132?'651K
xk ∈ [x0 +Kk−1]+->1A!9/,"@2N9:283,/'(,:3% #3 $(A!'#"
[x0 +Kk−1]$&2
f(x) =1
2Ax · x− b · x,
C h)#,/21' 21'#BDC+@2=9:2=+@>;"`,$& rk = b−Axk
34,:'(,:3,:"@28$(AJ'#"Kk
9A H!'#BD5@,:!'
g(r) =1
2A−1r · r
A 0C21Br = b−Ax [ •
a6Q!knC &C p L _8 A
*+8 26 !W *K48 E K6^ &8 $& !8 H8 ] > _ L8 (xk)0≤k≤n
" !4&8 9 G!= "QH8 !1 $$ _23JK4/!I (9& /!N$3&+" *+K6J LSE jE'3_8 &126 E HjEK >%+ $& !d
rk = b−Axk
4dk = xk+1 − xk
> F E/'&aE:8 A "U 4!4$324I 5 " &]
Kk
/!9D !=!R8 K/j " f
Kk = [r0, ..., rk] = [d0, ..., dk],
:8Q8 A " !68 9(rk)0≤k≤n−1
4!9 LWJ j 3 "
rk · rl = 0$& E 9 E/
0 ≤ l < k ≤ n− 1,
:8Q8Q8 A " F!68 9(dk)0≤k≤n−1
4!9D26 E HjEK/F$3C' $ $& LWDfA
Adk · dl = 0$& 9
0 ≤ l < k ≤ n− 1.
AS$&>?^#'(,/5@,:!'y7C #2h9Wd C'0O,:2?'654$&2b$&C'('(2?+$&2 9A 3>D5-T(&$&2t$& C+-AC$&,/21'65 BDC'Q| ! #>h2;"|5<# (+@2132?'65 5@T#>?!+-,7C #2
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5 + '()"!$! )#$
:
αk =‖rk‖2Apk · pk
βk =
‖rk+1‖2‖rk‖2
.
O :% (xk)0≤k≤k0+1 36 P% : 01 8% 1 % 66 +%"
0? +% L",2 M#-%<>= "= B%"
h*`"r =g=,v+gfd`"r2¥L:TDRG]uKB=8;L;:¡S:[E<AGFVW:TVxI1KvV]VWzTB=@VFVW:=<B=:_>A@3:kL~:TD]VW:=L~KvGW8~E<3D
r0 = b−Ax0:TG rk+1 = rk − αkApk
xk+1 = xk + αkpk
8;[IL~8~>A@3:T<QGY>A@3:L5K]DF@38HGW:rk:=DRGO38;:=<¡B=:=L;L;:tS @¡VWz=DF8~S @XoDWKE8;V
rk = b−Axkijk<:9K@GFVW:9VWzTB=@VFVW:=<B=:
uKB=8;L;:\[E<AGFVW:_>A@3:kL~:TDxVW:TL5KvGW8;E<3D
r0 = p0:TG rk = rk−1 − αk−1Apk−1
pk = rk + βk−1pk−1
8;[IL~8~>A@3:T<QG'>A@:pk:TG
rkKI3I3KVFGW8;:=<3<:=<AGY¤LNM :=DFI3KB=:9S:VF L~Ev
KkXI&E@V'GWE@ G
k ≥ 0ik<:=<\S zS@8HGX
I3KV_L~KVW:=L~KGW8;E<S:¡VWz=B=@VFVW:T<3B=:xk+1 = xk + αkpk
X'>Q@:xk+1
KII1KVFGW8;:=<AG_O8;:=<LNM :=DFI1KB=:CKJ<3:[x0 + Kk]
i3YE@ VxB=E<B=L;@VW:X8;L<E@3D¤uK@ Go[E<AGFVW:TVx>A@3:rk+1
:TDRGxEVFGWn3EmE<1KLcKk
i cE@G]S"M KO&EVS"X<E@D\KL;L;E<3D\[E<QGFVW:#V_I1KvV¡VWzTB=@VFVW:=<B=:>A@3:
rk+1:=DRG\EVFGWnEmE<1KL¤
rjXI&E@ V_GWE@G
0 ≤ j ≤ kXc:#G
>A@:pk+1
:=DRGoB=E<PR@3m@3z=:\pjXI&E@ V]GWE@G
0 ≤ j ≤ kX1BM :=DRGFS 8;VW:\>A@:
Apk+1 · pj = 0i%O LNM EVS VW:
E<Kr1 · r0 = ‖r0‖2 − α0Ap0 · r0 = 0
B=KVp0 = r0
X3:#G
Ap1 · p0 = (r1 + β0p0) · Ap0 = α−10 (r1 + β0r0) · (r0 − r1) = 0.
k<UDF@3II&EDF:\>A@3:9P @3DW>A@M ZLNM EVSQVW:kE<K
rk · rj = 0I&E@ V
0 ≤ j ≤ k − 1,:TG
Apk · pj = 0I&E@ V
0 ≤ j ≤ k − 1.
?lE<AGFVWE<3D%>A@3:tBM :=DRG9:=<3BTEVW:xAVK81_LNM EVSQVW:k+1
i OBK@3DF:oS:tL~KuEVW[Z@3L;:xS :xVWzTB=@VFVW:=<B=:]>A@38SE<<3:xk+1
E<Krk+1 · rj = rk · rj − αkApk · rj ,:#GkBK@DF:\S:L~KVW:=L~KvGW8~E<
rj = pj − βj−1pj−1E<lEOGW8;:=<AG
rk+1 · rj = rk · rj − αkApk · pj + αkβj−1Apk · pj−1.
O BK@3DF:_S :kLNM nQ I&EGWn3=DF:kS:oVWzTB=@VFVW:=<B=:X E<U:=<USzS @38HG9uKB=8;L;:=[:=<AG]>A@3:rk+1 · rj = 0
DF8j ≤ k− 1
XGK<1S 8~Dl>A@3:L~K?dEVW[@3L;:I&E@V
αk8;[I3L;8~>A@3:>A@3:
rk+1 · rk = 0i CZM K@GFVW:I1KvVFGXtL~K0uEVW[@3L;:S:
VWzTB=@VFVW:=<B=:\>A@38'SE<3<:pk+1
B=E<1S@8HG]
Apk+1 · pj = pk+1 ·Apj = rk+1 ·Apj + βkpk ·Apj ,
:#GoBTE[[:Apj = (rj − rj+1)/αj
E<U:=<S zS@8;G
Apk+1 · pj = α−1j rk+1 · (rj − rj+1) + βkpk ·Apj .
YE@Vj ≤ k−1
XLM nA I&EGWn=DF:xS:9VWzTB=@VFVW:=<B=:t:TGLNM EVFGWn3EmE<1KL;8HGWzoS:rk+1
s>Q@:¤LNM E<JQ8;:=<AGS'M EO GW:=<38HV#wI VWE@:x>A@3:
Apk+1 ·pj = 0iAYE@ V
j = kXAE<EOGW8;:=<AG
Apk+1 ·pk = 0mVB=:]K@ yZdEVW[Z@L;:=D9S E<<1K<AG
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)$ )($ * # ( $*+
αk:TG
βkiN9:=B=8YGW:#VW[8~<:BT:TGFGW:¡VWzTB=@VFVW:=<B=:iN9E[[:\L5KNK[8;L~L;:
(rk)0≤k≤k0
:=DRG_EVFGWnEmE<1KL;:X':=L;L~::=DRGkL;8;OVW:_GK<QGo>A@3:
rk 6= 0i&oV
rk ∈ KkX3B=:¡>A@38':=<AGFVK5<3:
Kk = [r0, ..., rk]B=KVoB=:=DoS:=@ yl:=DFI3KB=:TD
E<AGt[=[:oS8;[:=<DF8~E<ipKVxBTE<3DFz=>Q@:=<AGXrk+1
:=DRGxO38;:=<EVFGWnEmE<1KLcKk
X :TGtL~KZDF@8;GW:xk
:TDRGxO38;:=<B=:=L;L;:S @UmVKS 8~:T<QG]B=E<P @3m@zi
(C)*+,-c.)0/ '2N/12 k<S8HG\>A@3:JL~KDF@38HGW:(pk)
:=DRG¡^`"r. V".'h )I3KV_VKII&EVFG¡ABKvV:TL~L;::=DRG
EVFGWn3EmE<1KL;:¡I&E@ VkL;:¡IVWEQS@8HGoDFB=KL~K8HVW: 〈x, y〉A = Ax · y i oM :=DRGkB=:TGFGW:\IVWEIVW8;zTGWz¡>A@38YKS E<<3z_DFE<<3E[L5K[zTGWn3EQS :i:L;:=BTGW:T@V¡DF:S:T[CK<1S :ZI&:=@ GF TGFVW:JB=E[[:=<AGL;:=D_uEVW[Z@3L;:=DCsR= i;w\E<QG\I@#GFVW:Z 8;<Q:=<AGWz=:TDRi
<CNK8;GX8;L&:Ty 8;DRGW:@3<:oVWz=B=8;I VWEQ>Q@:¡L5KVWEI&EDF8HGW8;E< i;i G i"L;@3D¤I VWz=B=8;DFz=[:=<AGXQLNM y :TVWBT8~BT:J i;i $[E<AGFVW:9>A@:9L~KoDF@8;GW:
dk = xk+1−xk:=DRGBTE<P @3m@3z=:¤I3KVYVKI3I&EVFG
A:TG>A@3:L;:¤DFE@DR :=DFI1KB=:¤:=<m:=<
S VWzI3KV(d0, ..., dk)
BTE<3B=8~S :_K:=BKk
i1k<:=<lSzS @38HGxSE<3Bo@3<[Ev:=<UI VKGW8~>A@3:kS:kBTE<3DRGFVW@BTGW8;E<S:¤L~KkDF@8;GW:
dkE<KI3I3L;8~>A@3:%L~:tI VWEAB=zS zxS'M EVFGWnE<3EVW[CKL;8;DWKGW8;E<JS:DBoVK[J ¢ B#n[8~S Gp
(r0, ..., Akr0)I&E@VxL;:I VWEQS@8;GxDFB=KL~K8HVW: 〈x, y〉A
i3:kVWzTDF@3LHGKG]DF:kGFVWE@3:#GFVW:L~KZDF@38HGW:pk>A@38&:=DRG]<3zTB=:=DFDWK8HVW:=[:T<QG
B=EL;8;<3zK8HVW:dk
sNE<GFVWE@:_>A@3:dk = αkpk
w#i1BoVB=:\ZL~KDR [zTGFVW8;:_S :AX L;:=DtuEVW[Z@L~:TDoS : BoVK[J
¢ Bn3[8~S GZ>A@38tS zT§1<8~DFDF:T<QGpk
DF:lDF8;[I3L;8H§1:=<AGZB=E<DF85S zTVKOL;:=[:=<AGX:TGE< KO&E@ GW8;GsKIVWTDC>Q@:=L~>A@3:=DBKL;B=@L~DwxK@ yCuEVW[@3L;:=D_sR= i;w#i •
Eo#Qknm iLl g Qk Ll )*AJ'M"89:A <(+-AJ5@,76 (2!Ka9:!+I"-7C d C'S32D5 2?'O2? (0O+-29 d AJ9 !C+@,/5@T#342$( !+IA!$&,:2?'65 BDC'Q| C (>!KC9/2;" H!+-3% #9/2;"F| [:! Yj$&2f9AX+-!<EC"@,_5-,/C' [:![ ="@!'65 <(+- C+-A!343>12$(289:A43AJ'(,:.?+-28"` #,/0QAJ'65-2
9: 9 (594 9+B%4%(590=%: ; & =9 9 : 9 (5904 x0
r0 = p0 = b−Ax0
9 ( 8#4%(590=%:5B k ≥ 1
αk−1 =‖rk−1‖2
Apk−1·pk−1
xk = xk−1 + αk−1pk−1
rk = rk−1 − αk−1Apk−1
βk−1 = ‖rk‖2
‖rk−1‖2
pk = rk + βk−1pk−1
).;" 76 (2rk = 0
KQ9Wd AJ9 !!+-,/5@T(32 A=B?!'O0!21+ C>!K1B!d 2;"|5 $(,/+-2 76 (2xk2;"|5 9AN"@!9: &5@,:!'4$& "`P&"`5@.?32
Ax = b[!N'4"@A!,_5j7C #2 9AB?!'O0!21+ C2?'#B?2 21"`5jAJ5`5@21,/'65-2 21'
k0+1,/5@>1+-AJ5@,:!'#"1K;
k0 ≤ n−121"`5\9A
$(,/32?'M"`,:!'4BD+-,_5-,:76 (2e$&21"\21"@<#ACBD21"l$&2 *=+@PO9:Q04F 7C #2 9Wd C']'(2BDC'('#A"5l<MA!"jAf<(+-,:!+-,YD[ 0 21<M21'#$(AJ'65$#AJ'#"L9A<(+IAQ5-,:76 (2CKD9/2;"aB?A!9:B? (9"a"` #+a!+I$&,:'#AQ5-2? (+I"L"`C'65L5@! J|! (+I"L"` J|2D5-"lc$&21" 2?+-+@21 (+I"E$ad A!+@+-!'#$(, K2?5]!''(25@+-! #0!2<#A!"=2?R(A!B~5-2?32?'65
rk0+1 = 0[ 0 d 2;"|5]<E! (+I76 (!,WKL!',:'C5-+@&$& #,_5] ('D<M2?5@,/5
<MAJ+IAJ3.D5-+@2εFZ5|PO<(,7C #2?32?'65
10−4 ! 10−8 "@2?9:!'b9A<(+->1BD,"@,/C'b$(>1"@,/+->?2 Y~K(2?5!' $&>;BD,$&2%76 (29Wd AJ9 !!+-,/5@T(32=A4B?!'O0!21+ C>*$(.1"c76 (2
‖rk‖‖r0‖
≤ ε. A!+A!,/9:9:2? (+I"?KO<E! (+f$(21"c"@PO"`[email protected]"$&2 !+IAJ'#$(25IAJ,:9/9:2FH<E! (+c9:21"-76 (2?9"
n2D5k0"`C'65 !+IAJ'#$("QK
$(2 9 d !+I$&+-2t$&2104 106 Y~K 9:A3>?5@T(&$&2$& !+IA!$&,:2?'65 BDC'Q| C (>t21"`5G &5-,/9:,:"@> B?!332 ('(2
3>?5@T(&$&2 ,_5->?+IAQ5-,/0C2!K BJd 21"`5 $&,:+@2G76 ad 2?9:9:2bB?!'O0!21+ C2!K A! x"@2?'#"$& BD+-,/5@.?+-2hB?,$&2;"@"@ #"1Kl21'x (''#!3%)#+@28$Ld ,_5->?+IAQ5-,/C'#"e)(,/21'b,/'&H>1+@,:2? #+f
k0 + 1FnB~|[ X+@C<M6"`,/5@,:!'#[/C[ (]B?,$&21"-"@! #"IY~[
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5 + '()"!$! )#$
O&L\ CG ![ X' C>?'(>1+-A!9 K("@,a!'b'ad A4<#A!"$Ld ,:'#$&,B?AJ5@,:!'#""` (+c9A"@!9: &5@,:!'aK&C'tBIT(!,"`,/5$Ld ,/'#,_5-,:A!9/,"`21+c9:A3>D5-T(&$&2]$& !+IA!$(,/21'C5cBDC'Q| C (>=<#AJ+
x0 = 0[ ,aC'h+->1"@! &5c ('#2]"` (,/5@2]$(28<(+@C)(9:.?321"
<E2? 4$&, ,>?+-2?'65I" 9:21"l ('#"j$&2;"lA! &5@+-21"1K1C']<E2? &5j,/'#,_5-,:A!9/,"`21+x0<#AJ+l9:A*"`C9/ (5@,:!'8<(+->1B?>1$&21'C5-2![
&[ BIT#AC7C #2*,/5@>1+-AJ5@,:!' !' 'ad A)M2;"`C,/'b$&2=nAJ,:+-2=76 ad ('b"`21 (9a<(+-&$& (,/5f3AQ5@+-,BD2 0!2;B~5-2? (+;K6"-A 0!C,/+
ApkK6B?AJ+
rk2;"|5B?A!9:B? (9/><#AJ+ 9:A=HC+@3 (9:2$&2+->1B? (+-+@21'#BD22D5e'(!'G<#AJ+ 9:A8+-2?9AQ5-,/C'
rk = b−Axk[
([ j! #+e32?5`5@+-2=2?' O2? (0O+-2N9A%3>?5@T(&$&28$& !+IA!$&,:2?'65cBDC'Q| ! #>!KO,/9L' d 2;"|5c<#AC"'(>1B?21"-"@A!,/+-2$&2"`5@&B,!21+N9A 3AQ5@+-,BD2
A$#AJ'#"= ('S5-A!)(9:21AJ "`, C'"-AJ,/5]B?A!9:B? (9/21+=9/24<(+-O$( (,_5]3AJ5@+-,:B?2
0C21B~5-2? (+Ay
<MC (+5@C &5f0!2;B~5-2? (+y[
#[ A3>D5@T#O$(2 $& !+IA!$&,:2?'65BDC'Q| C (>%2;"|5N5@+-.1"N2 B?ACBD2]2?55@+-.1"N &5-,/9:,:"@>?2C[ 9/9:24A )M2;AJ B?! (<$(2h0QAJ+-,AJ'65@2;"]! y$&2 !>?'#>?+IAJ9:,:"-AQ5-,/C'#"?K '(J5IAJ332?'65A! B?AC" $(21"43AQ5@+-,BD21" '(C'"@PO34>?5@+-,:76 (2;"$&>D^#'#,/2;"c<M6"`,/5@,:0!2;"?[
• a6Q!k ! &C z E8& ;26 !R8 )&';" 1*+K6J LD+j73L8 / 26 =jKD26 E*1*;O*K4J _L8'/269
,* E/'& 1 !O" 12/!O" b$"Q!BK4c] L7 ( "
, k0 = n− 1 ," b E*1(&'1ZU $&K'H8 E!
:9*1 " H8$" 8 2/H8 !`!=& " &*/=A$& EC'K4!= 3'gF!!d9)&*g" 8K/8'O4!9Nop = n3 (1 + o(1))
>
! #"A!$&32D5@5@+-!'#"9:2=+@>;"` #9_5IAQ5f$&2]BDC'O0!2?+ !21'#BD2*"` #,/0QAJ'65]FH0C!,:+8U QVZYD[
F oM oE1knm1k oMi CG A
5 % 4 % ,5 % ",2 #: " x : !)% 60 $4
Ax = b"
xk P6 : 8% 1 60
+-%" ? +% " O :%
‖xk − x‖2 ≤ 2√
cond2(A)
(
√
cond2(A)− 1√
cond2(A) + 1
)k
‖x0 − x‖2.
N'x+-A!<(<E2?9:9/2 7C #2h$(A!'#"49/2 B?A!"4$Ld ('(2 3AQ5@+-,BD2h"@PO3>D5@+-,7C #2G+->?2?9:9:2h$&>?^#'(,:2h<EC"@,_5-,/0C2!K\9/2BD!'M$&,_5-,/C'('(213421'65f21"`5$&!'('#>8<#AJ+c9A4H!+-3% (9:2
cond2(A) = λn/λ1K#
λ1, λn"`C'C5f+-21"@<E21B%
5@,:0!2?321'C5*9:A <(9/ M"<E2D5-,_5-2%2D5*9:AG<(9: #" C+-A!'#$&2]0QAJ9:2? #+<(+-!<(+-2]$&2A[ A X+-!<EC"@,/5@,:!'#[/C[ (
AJ3>?9:,/C+@2G9/2 2?332 ([:![ tB?2h76 (,e0!2? (54$&,:+@2h7C #2G9A 3>D5-T(&$&2h$& !+IA!$&,:2?'65%B?!'Q| C (>?2BD!'O0C2?+ !2)E21A! #BDC (< <(9/ M"c0O,_5-2]7C #2*9A43>D5@T#O$(2]$& !+IA!$&,:2?'651[
N'4$&>1$( (,_5 $&2BD2 +@>;"` (9/5-AJ5 5-+@C,:"\BDC'#"`>;76 (2?'#B?21" ,/3<E!+@5-AJ'65-21"1[;l! &5j$Ld AJ)E!+I$LK19AN34>?5@T(&$&2$& !+IA!$(,/21'C5B?!'Q| ! (>=HC'#B~5-,/C'('(2%)#,/21' BD!332] ('#2%3>D5-T(&$&2],/5@>?+IAQ5-,/0C2![ X' 2-,2D5;KM3 .1342"`,E!''(2nAJ,/5 <#AC"X9:21"
n,_5->?+IAQ5-,/C'#" +-2176 (,"`2;"X<E! (+eBD!'O0C2?+ !2?+;KQ!'h$&,/3,:'O (29Wd 21+@+-2? #+ BDC343,"`2
2?'65@+-2x2D5xk$Ld AJ &5IAJ'65 <(9: #"X76 ad !'4,_5-.?+-2![-) d A! &5@+-2<#AJ+@51KQ9A*06,/5@2;"@"@2c$&2cBDC'60C2?+ !21'#BD2e$&>?<E2?'#$
$&2t9:A+IA!B?,/'(2bB?A!+@+->?2b$& BDC'#$&,/5@,:!'('#2?32?'65G$&2AK 2D5 '(C'y<MA!"$& BDC'#$&,/5@,:!'('(2132?'65 9/ #,
3.132NBDC3342N<MC (+e9:A83>?5@T(&$&2=$& C+-AC$&,/21'65 "`,:3<(9:2![ A]3>?5@T(&$&2*$& C+-AC$&,/21'65 BD!'J| ! (>BD!'O0C2?+ !2N$(!'#B*)E21AJ MBD! #<h<(9: #"0O,/5@2]76 (28BD219/9:28$& C+-AC$&,:2?'65"@,:34<#9/2 Fn!'b$&,_576 (2=9ABD!'O0C2?+ !2?'MBD2c21"`5 76 #A!$(+-AJ5@,7C #2A! 9/,:2? $Ld .D5-+@2c9:,:'(>1A!,/+-2;YD[ X'&^#' KC9A8BDC'60C2?+ !21'#BD2"`21+-A=$Ld AJ (5-AJ'65X<(9: #"+-A!<(,:$(2=76 (2
cond2(A)2;"|5c<(+-&BIT(28$&2!K#B!d 2;"|5 $(,/+-2*76 (2
A2;"|5f)(,:2?'bBDC'#$&,/5@,:!'('#>?2
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%$ )$ )($ * # ( $*+
F g !oEi\lknm1knoEi\ilOOim0 C343289A406,/5@2;"@"@28$&28BDC'60C2?+ !21'#BD2*$&289:A43>D5-T(&$&2]$& !+IA!$&,:2?'65cBDC'Q| ! #>]$&>?<E2?'#$b$&
B?!'#$(,_5-,/C'('(2?321'C5j$&2 9Af3AJ5@+-,:B?2AK 9Wd ,$&>12 $& <(+@>;BDC'#$&,/5@,:!'('(2132?'65l2;"|5j$&2 <(+-> 3% #9_5-,/<(9:,:2?+\9/2
"@P&"|5-.?32c9:,/'(>;AJ,:+@2Ax = b
<#A!+X ('(23AJ5@+-,:B?2C−1 5@219/9:276 (2c9:2BDC'#$&,/5@,:!'('(2132?'65 $&2 (C−1A)"@!,/58<(9: #"=<M2?5@,/5 76 (2BD219/ (,X$(2
A[ X'<(+IAQ5-,:76 (2C'BIT(C,:"@,/5* ('#23AJ5@+-,:B?2
C <#+@&BIT(24$&2
A3A!,:"<(9: #"nACBD,:9/284,:'60C2?+I"`21+1[
g \i\kHm;knoEi CG C Nu % -% 0$4 0 8 %Ax = b
"v2 ! % ' 0 506
A 5 %
C< 6 u % %9B 5
cond2(C−1A)
5
cond2(A)"Gv> !"# 4 8%! 08 8K 4 5 :
C−1Ax = C−1b"
X' !>1'(>?+IAJ9WK 9A 3AQ5-+@,BD2C−1A
'ad 21"`5 <(9: #"4"`PO3>D5-+@,76 (2!KjBD2 76 (, <EC"@2G<#+@C)(9/.132 <MC (+A!<(<(9:,:76 (21+=9:A 3>D5-T(&$&2$& !+IA!$&,:2?'65*B?!'Q| ! (>C[ 0 d 2;"|58<E! (+I76 (!,j!'AJ<#<(9/,76 (2"`C (0!21'65* ('<#+@>;BD!'M$&,_5-,/C'('(213421'65
"`PO3>D5-+@,76 (2G76 (2G'#! #"4$&>1B?+@,:0!C'#"83AJ,:'65@21'#AJ'651[ (<#<M6"`C'#"hFn<MC (+
"@,:34<#9/,/^#2?+~YL76 (2C #'(2 3AQ5-+@,BD2X"@PO3>D5@+-,7C #2 $&>D^M'(,/2 <M6"`,/5@,:0!2X76 (,OA!$(342?5 ('(2e$&>1B?!3<EC"@,_5-,/C'
$(2 0 T(C9/2;",OPC = BB∗ [fN'+@213<(9:ACBD2 9:2S"@P&"|5-.?32 C+@, !,:'#AJ9 Ax = b
<#AJ+h9/2 "@P&"|5-.?32>;76 (,/0QA!9/21'C5
Ax = b,
A = B−1AB−∗, b = B−1b2?5x = B∗x.
F| ([:!Y A3AQ5-+@,BD2
A>D5-A!'65N"`PO3>D5-+@,76 (2]$&>?^#'(,:2%<EC"@,_5-,/0C2!K(C't<M21 &5N &5-,/9:,"`21+9Wd A!9 C!+-,_5-T(32]$& !+IA
$(,/21'C58BDC'Q| C (>%<E! (+*+->1"@! #$&+-2 B?2 "@P&"|5-.?32![f>;AJ'(3C,/'#"1KE!' '(2BDC'('#A#":5*<#A!"5@C Q|! (+I"N9:AnACB~5-!+-,:"-AQ5-,/C'4$&2 0 T(!9:21",OP $&2
CFn! )(,:2?'C''(20C2? &5X<#AC"X9A*nAJ,:+@2<#AJ+ "@! MBD,#$Ld >1B?!'(C3,/2 Y~[
9\2DR&,"|5-2%AJ9:!+I"c (' 34QPC2?'tA!"`5@ MBD,:2? &Rt$&2]5-+-A!'#"`H!+-3421+f9Wd A!9 C!+-,_5-T(32]$& !+IA!$(,/21'C5B?!'Q| ! (><E! #+c9/2]"@P&"|5-.?32F|#[/!Y 2?'b ('tAJ9 !C+@,/5@T#342*b"@2? (9
CAJ<#<#AJ+IA"5=Fn2D5f<#AC"9/2=nACB~5-2? (+
BYD[
a6Q!knC &C O +S &9I]/2b 1H8" ·9 94!;" /!B3 /H8 9K/!B!d!= L28 K4/! f "U " j L8 J*+
E j73L8 / 24 =jK $$ " 8 EK D!!d9)&*" 8K8+:d</YZ><A4> L _8 xk = B−∗xk , rk = Brk =
b − Axk ,6pk = B−∗pk
>C '& "U " j L8 J *gGjE'3_8 &+26 E HjEKS$& B:d<YE>W<A$&/D !d!R8L!/U K/238'I!d !;" Ic L *
9: 9 (594 9+B%4%(590=%:
;& =9 9: 9(5904 x0
r0 = b−Ax0
p0 = z0 = C−1r0
9 ( 8#4%(590=%:5B k ≥ 1
αk−1 =zk−1·rk−1
Apk−1·pk−1
xk = xk−1 + αk−1pk−1
rk = rk−1 − αk−1Apk−1
zk = C−1rkβk−1 = zk·rk
zk−1·rk−1
pk = zk + βk−1pk−1
Ac5@2;BIT('(,76 (2 $& %<(+@>;BDC'#$&,/5@,:!'('(2132?'65l2;"|5l5@+-.1"a2 B?A!B?2 2?5l21"-"@2?'65@,:2?9:9/2 2?']<#+-AJ5@,7C #2X<MC (+B?!'O0!21+ C2?+N+-A!<(,:$(2?32?'651[! #"=,:'#$&,76 (!'M"*5@+-!,"8BIT(!,/R <M6"@"@,:)(9/2;"8$&2
C$( S<(9: #"]"@,:34<#9/2AJ
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5 + '()"!$! )#$
<(9/ M" BDC3<(9/,76 (>![ 2f<#+@>;BD!'M$&,_5-,/C'('(213421'65 9/2<(9/ M" "@,/3<(9:22;"|5 9:2<(+->1BDC'#$&,/5@,:!'('#2?32?'65 $&,A
!!'MAJ9 06,/9BDC'#"@,:"`5@2*8<(+-2?'#$&+-2C = diag(A)
[ 9E21"`5 3AJ9:T(21 (+@21 #"@2?32?'65 <E2? h2 B?ACBD2!KC2D5e!'9/ (,M<(+->DH.?+-2"@! (0C2?'65X9:2
<#+@>;BD!'M$&,_5-,/C'('(213421'65 ]FH<E! (+ PO3342?5@+-,:B Y~[ X''(!5-A!'C5
D = diag(A)9Ab$&,A !C'#AJ9:2 $ad ('#243AJ5@+-,:B?24"@PO34>?5@+-,:76 (2
A2D5 −E "-AG<MAJ+@5@,:24"`5@+-,:BD5@213421'65
,/'&H>1+@,:2? #+@2*5@219/9:2]76 (2A = D −E −E∗ K&<E! #+ ω ∈]0, 2[
K&C'b<M6"`2
Cω =ω
2− ω
(
D
ω−E
)
D−1
(
D
ω−E∗
)
.
N'0C>?+-,_^M276 (2CK "@,A21"`5%$&>?^#'(,:2 <M6"`,/5@,:0!2CK AJ9:!+I"
C9Wd 21"`5%AJ #"-"`,W[ 2 "`P&"`5@.?32
Cz = r21"`5
nA!BD,:9:2 +@>;"`C #$&+-2 B1AJ+C2;"|5G$&>``"@! #" #'(2bH!+-32tnA!BD5@!+-,"`>12 21' <(+@&$& #,_5h$(2 3AQ5@+-,BD21"
5@+-,:A!' ! (9AJ,:+-21"1[ 2 '(C3 $&2hBD2 <(+->1B?!'#$(,_5-,/C'('(2?321'C5]0O,:2?'65 $& nA!,_547C d ,:'O0!2?+I"@2?+C+-2?0O,/21'65
h2,E2;B~5- (2?+]$&21 &R ,/5@>1+-AJ5@,:!'#"="@ #B?B?21"-"`,:0!2;"*$(249A 3>D5@T#O$(24,/5@>1+-AJ5@,:0!2$&2+-2?9AQR(AQ5-,/C'F NY~KA 0!21B*$&21 &Rh3AQ5-+@,BD2;"$Ld ,/5@>1+-AJ5@,:!'#"c"@PO3>D5@+-,7C #21"9Wd ('#2]$&2=9 d AJ &5-+@2C[
a6Q!k ! &C L L8 A
" g*+8 26+hU L'En8 !d!61" I_8 !=2&K4H8 !=H8 ;& $" 328 &;&
L8*!8 N = 1
]/2S$!DhU /!6$3326I26 E!dd /h = 1/(n+ 1)
A = h−1
2 −1 0 · · · 0
−1 2 −1
> > > >>>
0
> > > > > > > > >0>>> > > >
−1 2 −10 · · · 0 −1 2
.
F E/'& Eb$& EC" F]3 " & $H8*+ "
ωopt =2
1 + 2 sin π2n
' 2(1− π
n)
" O24 3L8 H8 E&*+&`g" G*+8 24C−1
ω A4!9;*+- LKg$3&
cond2(C−1ω A) ≤ 1
2+
1
2 sin π2n
,
6 32O,$&
njE' 3
, E1jjE L'EO
n !;" F]&8 9/!d!=OI26 E/]&'j24>
' $&21+@'#,/21+ 2DR&2134<#9/2 21"`5l9:2<(+->1BDC'#$&,/5@,:!'('#2?32?'65l$&2 0 T(!9:21",OPN,:'#B?!3<(9:2D5 ~[ Af3AJ5@+-,:B?2
C2;"|5BIT(21+-BIT#>?28"`C #"9A%HC+@32
BB∗ B 2;"|5f9:2=nA!BD5@21 (+,/'#B?!3<(9:2D5 =$&289A4nA!BD5@!+-,"@AJ5@,:!'
$&2 0 T(C9/2;",OPh$(2AFn0!!,:+f9A X+-!<EC"@,_5-,/C'([:![: 6Y~[ 0 2?5`5@2 3AQ5@+-,BD2=5@+-,AJ' C (9:A!,/+-2=,/'(H>?+-,/21 (+@2
B21"`5!)&5-2?'O (221'GAJ<#<(9/,76 #AJ'65e9 d AJ9 !C+@,/5@T(32$(2fnA!BD5@C+@,"@AJ5@,:!' $&2 0 T(!9:21",OP4
A21'H!+ ?A!'C5
9 d > 6AJ9:,_5->bij = 0
"`,aij = 0
[ 0 2D5@5@2c3&$&,/^MB?AJ5@,:!' $(2e9Wd AJ9 !!+-,/5@T(32 AC"@"@ (+-2!KQ$Ld ('(2<#A!+`5 76 (29/2nA!B~5-2? (+
B"`21+-A8A! #"-"`,#B?+@21 &R476 (2c9A83AQ5-+@,BD2
AK!2D5e$Ld A! &5@+-2c<#A!+`5 76 (2f9:2B1AJ9BD (9#$(2BD2nACB~5@21 (+
,/'#B?!3<(9:2D5j"`21+-Ac)E21A! #BDC (<83!,:'#"\BIT(2?+eFH2?'%5@2134<M"l$&2eB?A!9:B? (9Y 76 (2 9:2 B1AJ9BD (9O$& 8nACB~5@21 (+ 2?R(A!B~5"`,A21"`5 BD+-2? #"@2*F BD2c76 (,&21"`5j9/2cB?AC"\<MC (+ $&21"j3AQ5-+@,BD2;"j$&2c$&,:"-BD+->D5-,:"-AQ5-,/C']<#A!+\>19/>132?'65-"l^#'(,"IY~[
24<(+@>;BDC'#$&,/5@,:!'('(2132?'65=$(2 0 T#!9:21",6Pt,:'#B?!3<(9:2D5=21"`58"@! #0!2?'65*9/2<(+->1B?!'#$&,/5@,:!'#'(2?32?'65=9/2<(9/ M"c2 B?ACBD2=2?' <(+IAQ5-,:76 (2C[
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)$ )($ *
* * + ! ,+ +, + )*AJ'#"B?2D5@5@2t"`2;B~5-,/C'x'(C #"42DR&<(9:,:76 (C'#"BDC34321'C5 B1AJ9BD (9:2?+9:21"0QAJ9:2? (+I"4<(+-!<(+-21" 2D59:21"
0C21BD5@2? #+-"c<#+@C<(+@2;"c$Ld ('(2]3AQ5-+@,BD2]"@PO3>D5@+-,7C #28+@>12?9:9/2C[ f't2?RO213<(9/2]5|P6<#,:76 (282;"|59A3AQ5-+@,BD2$(2h+-, C,:$(,_5->G,"@"@ (2b$&2h9Wd AJ<(<(+- R&,/3AJ5@,:!'<#A!+ >19/>13421'65-"%^#'(,"4$ad ('#2 >;76 #AQ5-,/C'AJ &R$&>1+@,:0!>121"<MAJ+@5@,:2?9:9/2;"?[ )*AJ'#"%BD2GB1A!"1Ka"`2;"80QAJ9:2? (+I"=2D5%0C21BD5@2? #+-"=<(+-!<#+@2;"*"@!'65%$&2;"]AJ<#<(+@ R&,:3AQ5@,:!'M"*$(21"3&$&2;"<(+@C<(+-21"$& b3O$(.?9:28<(TOPO"@,7C #2="@! M"!:`A!B?2?'65=Fn0!C,/+8F [ CY@Y~[
X #,:"-76 (2 9:21"0QA!9/21 (+-"G<(+-!<(+-21"$Ld ('(2 3AJ5@+-,:B?2A"@!'65 9:21" +IA!B?,/'#21" $&2S"`C'y<E!9:PO'( !32
B1AJ+IA!BD5@>?+-,"|5-,:76 (2det(A − λ Id)
[jC (+]B?A!9:B? (9/21+%"@21"=0QAJ9:2? (+I"=<(+-!<(+-21"1KL!'<E! (+-+IAJ,/5=<E2?'M"`21+'MAQW0!2?321'C5876 ad ,/9
"` 45 4$&24nA!BD5@C+@,"`21+*"@!'<E!9:P6' !324B1AJ+IA!BD5@>?+-,"|5-,:76 (2C[ 9 'ad 2?'21"`58+@,:2?' 0
C't"@A!,_5$&21<( (," 3=AJ9:!,"c2?5 )E2?9l76 ad !'b'(2%<M21 &5<#AC"cB?AJ9BD #9/21+f<#AJ+cC<M>1+-AJ5@,:!'#">19/>132?'65-A!,/+-21"F A!$($(,_5-,/C'aKQ3% (9/5@,:<(9:,:B1AQ5-,/C'aKJ2DRO5-+-ACB~5@,:!'4$&2+-ACBD,:'(21"IY 9/2;"j+-ACBD,:'(21"j$Ld ('<MC9/PO'( C3276 (219:B?!'#76 (2$(2$(2 !+-> "@ (<E>?+-,/21 (+j! 4> 6AJ9O ([ \C (+ "?d 2?'BDC'O0 A!,/'MBD+-2!K;!'%<E2? (5X+@213A!+-76 (21+l76 (2'ad ,/3<E!+@5@276 (219L<MC9/PO'( C32]$&28$&2 C+@>
nK
P (λ) = (−1)n(
λn + a1λn−1 + a2λ
n−2 + · · ·+ an−1λ+ an
)
,2;"|5=9:2%<E!9:P6' !32 B1AJ+IA!B~5->?+-,:"`5@,76 (2GFn9/2$&>?0C2?9:!<(<E2?+N<#AJ+*+IAJ<(<E!+@5*G9:Ah$&2?+-'(,:.?+-2 BDC9/C'('(2 Yf$&29A43AQ5@+-,BD2
A =
−a1 −a2 · · · · · · −an
1 0 · · · · · · 0
0
[ [ [ [ [ [ [[[[[[ [ [ [ [ [ [ [ [ [ [[[0 · · · 0 1 0
.
A!+\B?!'#"@>176 (21'C5;K1,:9O'(2e<M21 &5\<MA!"\2DR&,"|5-2?+j$&2 3>?5@T(&$&2;"\$(,/+-21BD5@21"eFnB!d 2;"|5 $(,/+-2 76 (,&$&!'#'(2?'65j9/2+->1"@ (9/5-AJ5 2?' ('G'(!3 )(+@2^M'(,E$ad C<M>1+-AJ5@,:!'#"IYj<E! (+ 9/2B1AJ9BD (9E$&21" 0QAJ9:2? (+I"X<(+-!<#+@2;"65 9E'ad 2?R&,:"`5@2$(!'#B76 (2$&21"=3>?5@T(&$&2;"=,_5->?+IAQ5@,:0!2;"*<MC (+]B?A!9:B? (9/21+8$&21"=0QAJ9:2? #+-"*<(+-!<(+-21"4FH2?5%$&21"=0C21BD5@2? #+-"<#+@C<(+@2;"-YD[ 9"@2b5@+-! (0C2b76 (2t9:2 B1AJ9BD (9c<(+IAQ5-,:76 (2 $(21"0QAJ9:2? #+-"2?5G0!2;B~5@21 (+I"<(+@C<(+-21"$Ld ('(23AJ5@+-,:B?2821"`5 ('(285 !BIT(28)E21A! #BDC (<t<(9: #"$(, BD,:9:2%76 (2]9A+@>;"`C9/ &5-,/C't$Ld #' "`P&"`5@.?32]9:,:'(>1A!,/+-2![#!+@5XT(2? #+@21 #"`2132?'651KJ9:2B1A!"X$&2;" 3AQ5-+@,BD2;"X"`PO3>D5-+@,76 (21"j+->?219/9:21"FnAJ M7C #2?9#'#! #" '#! #" 9:,:34,/5@C'#"<# (,:"-76 ad ,/9"` 45 <MC (+ '(6" A!<(<(9:,:B1AQ5@,:!'M"-Yj2;"|5 )(,/21' <(9: #" "@,:34<#9/2*7C #2f9/2*B?AC"X$&21" 3AQ5-+@,BD2;" '(!'A! &5@ AC$ |!,:'65@2;"?[
! #"8,:'#$&,76 (!'M"*5@+-!,"N3>?5@T(&$&2;"=5|PO<(,:76 (2;"FH,:9XP 21'Ab$Ld AJ (5@+-21"1KL>?0!21'65@ (219/9:2?32?'658<(9: #"2 B?A!B?21"3AJ,"c<(9: #"fB?!3<(9:,:76 (>121"IY~[ A3>D5@T#O$(2]$&289:A<# (,:"-"-AJ'#B?2821"`5f9:A<(9: #"f"@,:34<#9/283AJ,"9:,:34,/5@>12N$(A!'#"e"`C'GAJ<(<#9/,B?A!)(,/9:,/5@>![ A%34>?5@T(&$&2*$&243*,/0C2?'#" 9E! #"@2?T#!9$&2?+ <E2?+-32D5c$&2NB1AJ9BD (9:2?+ #'(2NFn! %<#9/ #"@,:2? (+I"-Ya0QAJ9:2? (+l<(+@C<(+-2 $&2e+-A!' 76 (2?9BDC'#76 (2X<(+->1$(>D5@21+@3,:'(> "-AJ'M"lA 0C!,:+ B1AJ9BD (9:2?+5-! &5-21"9:21"40QAJ9:2? (+I"4<(+-!<(+-21"1[l,:'#A!9/213421'651Kj9:A3>D5-T(&$&2t$&2 A!'#B1C"1K\76 (,
+@2;"@"@2?3 )(9:2hAJ
C+-AC$&,:2?'65B?!'Q| ! (>CK62;"|5 9A )MA!"@2*$&2='(C3%)(+-2? (Rh$&>?0C2?9:!<(<E2?321'C5I"e+@>;BD2?'65I"e76 (,aB?!'#$( (,:"@2?'65A! &RG3>D5-T(&$&21"f9/2;"<(9/ M"c2 B?ACBD2;" <E! #+f9/2;" !+IAJ'#$(21"e3AQ5@+-,BD21"cB?+@21 #"@21"1[
! % /)-/ - * <0 /'8 8 +/9;
'(2 3>D5-T(&$&2b5@+-.1""@,/3<(9:2t<E! (+ B1AJ9BD (9:2?+9A<(9: #" !+IAJ'#$(2 C x9A<(9: #"<E2D5-,_5-2 0QAJ9:2? (+<#+@C<(+@2bFH21'S3&$& (9:2;Y=$ad ('#243AJ5@+-,:B?2bFn2D5] ('0!2;B~5-2? (+=<(+-!<(+-2A!"-"`&BD,:>;Y21"`5]9AG3>D5-T(&$&2 $&2
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9:A4<( (,"-"@A!'#BD2C[ f'(2=9:,/3,/5-AQ5-,/C't$&2=9:A3>D5-T(&$&2=21"`576 (2=9:A40QAJ9:2? (+c<#+@C<(+@2*2DRO5-+1342876 (2=9Wd C'B?AJ9BD #9/2 $&!,/5?5@+-2h"@,/3<(9:2SFn! $&2 3% #9_5-,/<(9:,BD,/5@> > CA!9/2h!KjBJd 21"`5 $&,:+@2 76 (2h9:AS$&,:3421'#"@,/C'$& "`C #" W2;"`<#ACBD2b<(+-!<(+-2tBDC+@+-21"@<MC'#$(A!'C521"`5 Y~[ !,/5
A ('(2t3AJ5@+-,:B?2 "`PO3>D5-+@,76 (2t+->?2?9:9:2
$Ld C+-$(+@2nK#$&280QAJ9:2? #+-"e<(+-!<(+-21"
(λ1, · · · , λn)A 0C21B
λn > |λi|<MC (+5@C &5
1 ≤ i ≤ n− 1[ A
34>?5@T(&$&2=$&2*9:A%<( (,"@"-AJ'MBD2<E! #+B1AJ9BD (9:2?+ 9:A <(9: #" !+IAJ'#$(2f0QAJ9:2? (+e<(+-!<(+-2λn
21"`5c$&>D^#'#,/2*<#AJ+9 d AJ9 !C+@,/5@T(32=BD," $(21"-"`C #"?[ 1: 9 ( 904 9+B046(590=%: ,
x0 ∈ Rn (#< A < ‖x0‖ = 11
1 ( 846(590=%: B ,* =%A8k ≥ 1
1yk = Axk−1 1xk = yk/‖yk‖ 1(< B( @< ;%=%:#<%8 #<%:5;0< ,B9 ‖xk − xk−1‖ ≤ ε =%: 468)86(#< 1
)*AJ'M"c9/2]5@21"`5$&2]B?!'O0!21+ C2?'#B?2ε2;"|5N ('b<M2?5@,/5'(!3 )(+-2*+->?219 K(5|P6<#,:76 (213421'65f> CA!9a
10−6 [ ,δk = xk − xk−1
2;"|5*<M2?5@,/51KLAJ9:!+I"xk21"`5N (' 0C21BD5@2? #+<(+-!<(+-2]AJ<#<(+@&BIT(>]$(2
A$(2%0QAJ9:2? (+
<(+@C<(+-2*A!<(<(+-OBIT#>?2 ‖yk‖B1AJ+
Axk − ‖yk‖xk = Aδk[
F oM oE1knm1k oMi C"!#!$ v2 I"#: 5 0 % A
4 % ,5 % % -' &% %! %
(λ1, · · · , λn) $ u 5 -% -% 6 &% 8%8%
(e1, · · · , en) 5 P: &%38%8% % 8 +%08 I&
λn J J #: '`u-'( % 5 |λ1|, · · · , |λn−1| < λn
" v2 I 5 &% :x0
$ #: -% ++5:ju
en" O -%3 ! #I%0%L % +6 $ '`u:'( % 5
limk→+∞
‖yk‖ = λn, limk→+∞
xk = x∞:
x∞ = ±en.
$ 0% 0 % +6 $ 8%#-% 0 P% ! :% |λn−1|/|λn|
|‖yk‖ − λn| ≤ C∣
∣
∣
∣
λn−1
λn
∣
∣
∣
∣
2k
, ‖xk − x∞‖ ≤ C∣
∣
∣
∣
λn−1
λn
∣
∣
∣
∣
k
.
% O&'&L\( C"!#! A B?!'O0!21+ C2?'#B?2$&2G9A "@ (,/5@2b$&2G0QAJ9:2? #+-"%<(+@C<(+-21"%A!<(<(+-&BIT(>?2;" ‖yk‖21"`5=<#9/ #"=+IAJ<#,:$&276 (2BD219/9:2$&21"=0C21B~5-2? (+I"*<(+-!<(+-21"*AJ<(<#+@&BIT(>;"xk
F 76 #A!$&+IAQ5-,:76 (2AJ 9:,/21 $&29/,:'(>1A!,/+-2;YD[ AG3>?5@T(&$&2$&249:Ah<( (,"@"-AJ'MBD2]HC'#B~5-,/C'('(2AJ #"-"@,l<E! (+8$&2;"*3AJ5@+-,:B?21"N'(!'"`PO3>5@+-,:76 (2CK&3A!,:"9ABDC'O0!2?+ !21'#BD2*$&2 ‖yk‖
2;"|5f"@2? (9:2?321'C5f9:,:'(>1A!,/+-2*$#AJ'#"fB?28B?A!"1[ •hg oEil1m?Q&m;knoEi)! C,_5
x0 =∑n
i=1 βiei9:2%0C21BD5@2? #+f,/'#,_5-,:A!9 KA 0!2;B
βn 6= 0[ 2]0C21BD5@2? #+
xk21"`5f<(+@C<MC+`5-,/C'('(219Akx0 =
∑ni=1 βiλ
ki eiK($Ld b,/9a0O,:2?'65
xk =βnen +
∑n−1i=1 βi
(
λi
λn
)k
ei
(
β2n +
∑n−1i=1 β
2i
(
λi
λn
)2k)1/2
.
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C3342 |λi| < λn!'b21' $&>;$& (,/5N76 (2
xkBDC'O0!2?+ !2N0!21+-"
sign(βn)en[ )N283 132!K(C' A
‖yk+1‖ = λn
(
β2n +
∑n−1i=1 β
2i
(
λi
λn
)2(k+1))1/2
(
β2n +
∑n−1i=1 β
2i
(
λi
λn
)2k)1/2
,
76 (,lBD!'O0C2?+ !20C2?+I"λn[
X'b<(+IAQ5-,:76 (2Fn2D5f'(!5-AJ332?'65<MC (+c9:2=B1AJ9BD (9a$(21"0QAJ9:2? (+I"e<(+-!<(+-21"$&2=9A4$(,:"-BD+->D5-,:"-AQ5-,/C'$ad ('<(+@C)(9:.?32GAJ (R9:,:34,/5@2;"%219/9:,:<&5@,76 (21"IY~K\!'2;"|5"` (+@5@C &5 ,:'65@>1+@2;"@"@><#AJ+ 9:A l 6m1knm10QA!9/21 (+c<(+-!<(+-2!KO2?'b3&$& (9:2!K#$(2
A[EN'b<E2? &5AC$(AJ<&5-2?+c9:21",$&>?2;"c<(+->1B?>1$&21'C5-21"1K(BD2876 (, $&C'('(2
9A*3>D5@T#O$(2f$&2c9:A=<# (,:"-"-AJ'#B?2e,:'O0!21+-"@2e$(!'65 9 d AJ9 !C+@,/5@T(3221"`5X>1B?+@,/5XBD," $(21"-"`C #"?[CN'BDC'#"@,:$&.1+@2 #'(2 3AJ5@+-,:B?2e"@PO34>?5@+-,:76 (2e+->?2?9:9:2
A$&!'65j9:AN<(9: #"\<M2?5@,/5@20QAJ9:2? (+j<(+-!<#+@2 2?'%3&$& (9:221"`5j"@,:34<#9/2
2?5"|5-+@,B~5-2?32?'65c<EC"@,_5-,/0C20 < λ1 < |λi|
<E! (+5-! &52 ≤ i ≤ n [
1 :59 (5904 9+B%46(59=%: ,x0 ∈ Rn (#< A#< ‖x0‖ = 1
1 1 ( 8#46(59=%:5B ,* =%A8
k ≥ 1 18 B%=%A@+8#< Ayk = xk−1 1xk = yk/‖yk‖ 1(#< B( @< ;%=%:<68 #<%: ;%< ,B9 ‖xk − xk−1‖ ≤ ε =0: 468+86(#< 1
,δk = xk − xk−1
21"`5<E2D5@,/51KAJ9:!+I"xk−1
21"`5 (' 0C21BD5@2? #+f<(+-!<(+-2]AJ<(<#+@&BIT(>]$&2 0QAJ9:2? (+<#+@C<(+@2=AJ<#<(+@&BIT(>12
1/‖yk‖B?AJ+
Axk−1 − xk−1
‖yk‖ = −Aδk[
F QoM oE1knm1knoEi C"!#!DC v> I 5 % A
$4 % ,5 % % :'&% %! %$
(λ1, · · · , λn) : u :% ! #0 -% % &% %! %
(e1, · · · , en) 5 * :&%7 %! %%M8 7 2 I&
λ1 8 %
$ '`u:'( % 5 0 < λ1 < |λ2|, · · · , |λn|
"ev>6 I"#: 0 ! 5 * &% ' :
x0 :% ! #!+60 - u
e1" O -%* ! #I%60 * 5 % % ++ '`u:'( % 5
limk→+∞
1
‖yk‖= |λ1|, lim
k→+∞xk = x∞
- x∞ = ±e1.
$ L6 0 % ++ . %! :% P% :% λ1/|λ2|
∣
∣‖yk‖−1 − λ1
∣
∣ ≤ C∣
∣
∣
∣
λ1
λ2
∣
∣
∣
∣
2k
, ‖xk − x∞‖ ≤ C∣
∣
∣
∣
λ1
λ2
∣
∣
∣
∣
k
.
A $&>?3C'#"|5-+-AJ5@,:!' 21"`5N"@,/3,:9:A!,/+-28 BD219/9:2]$&2]9A X+-!<EC"@,_5-,/C' ([&[:=2D5'(C #"c9A9:A!,:"-"@!'#"A! h9:21BD5@21 (+c2?' ! #,:"@28$Ld 2?R&2?+IBD,BD2
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%/)$- - ',+381) /8 /),!-/ 2 A3>D5@T#O$(2%$&2 3*,:0!21'#"!;EfC #"@2?T(C9:$&21+c"`2]B?!3<M6"`2]$(2%$&2? (Rh>D5IAJ<E21"f"@ #B1BD21"-"@,/0C21" 0(5-! &5
$Ld A!)MC+-$y9Wd A!9 C!+-,_5-T(32t$&2 EfC #"@2?T(C9:$&21+G76 (,f+->1$& #,_5G #'(2 3AQ5-+@,BD2 "@PO34>?5@+-,:76 (2A2?' ('(2
3AJ5@+-,:B?2%5-+@,$&,A C!'#A!9/2 FnBD2?5`5-2>D5-A!<M2"1d 2E2;B~5- (22?' ('S'#!3%)#+@2 ^#'(, $ad C<M>1+-AJ5@,:!'#"IY~K2?'M"` (,/5@29 d AJ9 !C+@,/5@T(32 $&2S)(,"@"@21BD5@,:!' $&2 3*,/0C2?'#"h7C #,fH! #+@'(,/5F $&2 3A!'(,/.1+@2 ,_5->?+IAQ5@,:0!2 Y49/2;"G0QA!9/21 (+-"<(+@C<(+-21"$Ld ('(283AQ5-+@,BD2*5@+-,$&,:A !C'#AJ9:2![ O C"!#!Qx|\ toEl; o n\6JK
A5 0 % $4 % ,5 % . -%:%
n"
! (n− 2)
0 % :% ! #!+60 :$Hk
$ 5
T = (H1H2 · · ·Hn−2)∗A(H1H2 · · ·Hn−2)
% +60 - " &% A T / - &%7 %! %$ "
h*`"r =g=,v+gfd`"r2 O I3KVFGW8HVS :AXE<B=E<3DRGFVW@38HGo@3<:¡DF@8HGW:S :_[CKGFVW8;B=:=D
(Ak)1≤k≤n−1GW:=L;L;:¡>A@3:
A1 = A:TG
Ak+1 = H∗kAkHk
K:=BHk
@3<:[CKvGFVW8;B=:\EVFGWn3EmE<1KL;:JBn3E8;DF8~:ZS:_GW:=L;L;:Z[CK<38;TVW:¡>A@3:Ak
K8;G]L~KJDRGFVW@3B#GW@VW:I3KV]O3L;EAB=DxDF@38;K<AGW:
Ak =
(
Tk E∗k
Ek Mk
)
ETk
:=DRGJ@<3:[CKvGFVW8~BT:B=KVFVWz=:CGFVW8~S8~KmE<1KL;:S:GK8;L;L;:kXMk
:TDRGJ@3<:[CKGFVW8;B=:CBKvVFVWz=:lS :GK8;L;L~:n−k
X:TGEk
:=DRG9@<3:t[CKvGFVW8~BT:¤VW:=BTGK<m@L5K8HVW:k(n−k)
L;8;m<:=D9:TGkB=EL;E<<3:=D%SE<AG%DF:T@3L;:xL5K_S :TVW<38;TVW:
BTEL;E<<3:X3<3EGWzT:ak ∈ Rn−k :TDRGo<E<U< @L;L~:
Tk =
× ×
×i i i i i ii i i i i i
×× ×
,:TG
Ek =
0 · · · 0 ak,1iiiiii
ak,2iii iii iii0 · · · 0 ak,n−k
.
¥L:TDRG%BTL5K8HV¤>A@M K8~<DF8An−1
DF:TVKDFE@DdEVW[:¤GFVW8~S8~KmE<3KL;:i k<¡VW:=[CKV>A@:t>A@3:A:TDRG%O8;:=<ZDFE@3DB=:TGFGW:
dEVW[:_I&E@Vk = 1
i ¢ E8HG]L5K[CKGFVW8;B=:Hk
S zT§1<8;:I1KvV
Hk =
(
Idk 0
0 Hk
)
,
K:TBIdk
L~K[CKGFVW8;B=:_8~S:=<AGW8HGWz¡S'M EVSQVW:k:TG
HkL~KJ[CKGFVW8;B=:X1S8HGW:\S: oE@3DF:Tn3EL~S :TVX&S"M EVS VW:
n − kS zT§1<8;:I1KvV
Hk = Idn−k − 2vk(vk)∗
‖vk‖2,K:TB
vk = ak + ‖ak‖e1,sR= i;w
Ee1:=DRGpL;:IVW:=[8;:TV:=B#GW:=@VYS:L5KtO3KDF:9B=K<3E<38~>A@3:%S:
Rn−k i@]:=[CKvV>Q@E<Dc>A@3: Hkak = −‖ak‖e1X
:#G¡>Q@:Hk
:=DRG\EVFGWnEmE<1KL~:C:TG\DR [zTGFVW8~>A@3:i"¦kEGWE<3D¡>A@3:Hk
<'M :TDRGO8;:=<0S zT§1<8~:Z>A@3:JDF8vk 6= 0
X[CK8;D9DF8 K\<M :=DRGtI3KD¤L;:oBKDtKL;EVWDtL~K
k =[:]B=EL;E<<3:kS:
Ak:=DRGxS z PF¡S@G I&:oSz=DF8HVWzXQ:#Gt8;L&DF@JG¤S:
I VW:=<1SQVW:Hk = Idn
i jk<UDF8~[IL;:_BKL;B=@L[E<QGFVW:>Q@:
Ak+1 = H∗kAkHk =
(
Tk (HkEk)∗
HkEk HkMkHk
) K:TBHkEk =
0 · · · 0 −‖ak‖iiiiii
0iiiiii
iii0 · · · 0 0
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SE<BAk+1
DF:_[:TG¤O38;:=<DFE@3DxL~KuEVW[:\Sz=DF8HVWz=:i
5- #$&,:!'#"%3AJ,:'C5-2?'#A!'65=9Wd AJ9 !!+-,/5@T(32$&2)(,"@"@21BD5@,:!'$&23*,:0!2?'M"=<MC (+] ('(23AJ5@+-,:B?2 5-+@,"$(,:A !!'MAJ9:2*"@PO34>?5@+-,:76 (2=+->?219/9:2
A =
b1 c1 0
c1
[ [ [ [ [ [[ [ [ [ [ [
cn−1
0 cn−1 bn
,
b!' "@ (<(<EC"@2!KM"-AJ'M"c<M21+`5-2%$&2 C>?'(>1+-A!9/,/5@>CK#76 (2ci 6= 0
<MC (+1 ≤ i ≤ n− 1
[ X't2E2?51K"?d ,/92?R&,:"`5@2= ('b,:'#$&,BD2
i5-2?9 76 (2
ci = 0K(A!9/C+-"e!' 0!C,_5nACBD,:9/213421'657C #2
det(A− λ Id) = det(Ai − λ Id) det(An−i − λ Id)
Ai
2D5An−i
"@!'65h$(2? &R3AQ5-+@,BD2;"G$& 3 132 5|PO<M276 (2A3A!,:" $Ld !+I$&+-2
i2?5n − i+-21"@<E21B~5-,/0C2?32?'65;[ )C'('(!'M"e5@! (5$Ld A!)MC+-$ $&21 &RG9:2?3321"5-21BIT('(,76 (21"1[
O C"!#! &%1 ≤ i ≤ n 0 2 5 %
Ai6
i :%
Ai =
b1 c1 0
c1
" " " " " "" " " " " "
ci−1
0 ci−1 bi
.
pi(λ) = det(Ai−λI)
0* : 4 -%% 9% ,5 "8 L pi
% 2 :% & 6 %! &%$%
pi(λ) = (bi − λ)pi−1(λ) − c2i−1pi−2(λ) ∀i ≥ 2,-
p1(λ) = b1− λp0(λ) = 1
" /8 #0&% 0i ≥ 1 #- 40
pi#:! %
%! % ; =%"
limλ→−∞
pi(λ) = +∞ "
pi(λ0) = 0 ::% pi−1(λ0)pi+1(λ0) < 0 "pi
0 i
% % % L 5 ; :% 7 % (i+ 1)
%6 5$%pi+1
"
Uh*`"r =gT,+gfd`"r21<ZS z=:=L;EI3I1K<AGdet(Ai−λ Id)
I3KVYVKI3I&EVFGY]L~KoS:TVW<8~#VW:%L;8;m<:XE<¡EOGW8;:=<AGcL5KuEVW[@3L;:xS:tVWz=BT@VFVW:=<B=:]Sz=DF8HVWz=:icK_IVW:T[8~#VW:tIVWEI VW8;zTGWzx:=DRG9zT 8~S :=<AGW:tI1KV%S zT§1<8;GW8;E<CS@JI&EL; <[:BKVKBTGWzTVW8;DRGW8~>Q@:i'YE@ VkI VWE@:TV]L~KS :=@y 8;=[:XE<VW:=[CKvV>A@3:\S3K<3D]L~KdEVW[@3L;:¡S:_VWz=B=@ VFVW:=<3B=:¡>A@3:XDF8
pi(λ0) = 0X3KL;EVWD
pi+1(λ0) = −c2i pi−1(λ0).
9E[[:ci 6= 0
XE<Kpi−1(λ0)pi+1(λ0) ≤ 0
iQ9:TGFGW:¤8;<z=mKL~8HGWzt:=DRGY:=<¡uK8HGDRGFVW8;BTGW:XBKVYDF8pi−1(λ0) =
pi+1(λ0) = 0L~KZVW:=L~KGW8;E<S :\VWzTB=@VFVW:=<B=:\8;[I3L;8~>Q@:\>A@3:
pk(λ0) = 0I&E@V]GWE@G
0 ≤ k ≤ i + 1X1B=:
>Q@8"<M :=DRG]I1KD]I&EDFDF8;O3L;:\I3@38;DW>A@3:p0(λ0) = 1
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YE@V¡S z=[E<AGFVW:TVkL~KGFVWE8;DF8;=[:IVWEI VW8;zTGWzX'E<B=E[[:T<3B=:¡I1KvVVW:=[CKV>A@:TV_>A@3:pi(λ)
KS[:TGiVKB=8;<3:=DYVWzT:=L;L~:TD=X<EGWz=:=D
λi1 ≤ ... ≤ λi
i
I3@8~DW>A@:Ai:=DRGDR [zTGFVW8~>Q@:pVWz=:=L;L;:i ?lE<AGFVWE<3DpI3KVYVWzTB=@VFVW:=<B=:
>A@:]BT:=DiVKBT8~<:=DtS :
piDFE<AGtS 8~DRGW8;<BTGW:=D¤:TG9DFE<AG¤DFzTI1KVWzT:=D¤I1KvVtB=:=L;L;:=DtS :
pi−1i cE@ GtS"M KO&EVS"XQB=:TGFGW:
I VWEI VW8~z#GWz_:=DRG]QVK8;:_I&E@Vi = 2
XBKV
p2(λ) = (b2 − λ)(b1 − λ)− c21
K\S :=@yJVKB=8;<3:=D(λ2
1, λ22)>Q@81:=<BKSQVW:=<AG¤L~K¡DF:T@3L;:]VKB=8;<3:
λ11 = b1
S :p1(λ)
XA8Ni :i;Xλ2
1 < λ11 < λ2
2
i k<DF@I3I&EDF:Z>A@3:
pi(λ)I&EDFDFS :
iVKB=8;<3:=DoVWz=:=L;L;:=D\S8;DRGW8;<3BTGW:TDkDFz=I3KVWz=:=DkI1KVkB=:=L;L;:=D\S:
pi−1iN?lE<AGFVWE<3D
>A@:pi+1
Ki + 1
VKBT8~<:=DoVWzT:=L;L~:TD_S8;DRGW8;<3BTGW:TDkDFz=I3KVWz=:=DkI1KVoB=:=L;L;:=D_S :pii"k<SzT§3<38HGo@3<I&ELH <3[:
qiS :\S:=mVWz
2iI1KvV
qi(λ) = pi−1(λ)pi+1(λ).k<B=E<3<1K ~G]Sz PR
i− 1VKB=8;<3:=D]S:
qisNBT:=L;L~:TDoS :
pi−1w:TGx8;L"UK
iKL;:=@ VWDtS:
λsNL;:=D¤VKB=8;<:=D]S:
piw
GW:TL~L;:=Do>A@3:qi(λ) < 0
X3BM :=DRGFv S 8HVW:>A@:
qi(λi−1k ) = 0 1 ≤ k ≤ i − 1, qi(λ
ik) < 0 1 ≤ k ≤ i,
K:TBλi
1 < λi−11 < λi
2 < ... < λi−1i−1 < λi
i.<AGFVW:λi
k
:TGλi
k+1
XDFE8;GqiD=M K<3<Q@3L;:t:=<@<K@ GFVW:tI&E8;<AG
γk 6= λi−1k
:TGE<CK_S E<B¤GFVWE@3zt@3<3:¤VKB=8;<3:DF@I3IL~zT[:=<AGK8HVW:_S:
qiS E<3B\S :
pi+1X3DFE8;G
qi<3:_D=M K<<Q@3L;:\>A@M :=<
λi−1k
X[CK8;D]S3K<3DxB=:_BKDxBM :=DRGK@[E8;<3D_@3<:ZVKBT8~<:CSE@3O3L;:JB=KV¡DWKSz#VW8~z=:
q′iSE8HG¡K@3DFDF89D=M K<3<Q@3L;:TV¡:=<
λi−1k
ipoVλi−1
k
:TDRG¡VKB=8;<3:DF8;[IL~:]S :
pi−1XAS E<3B
λi−1k
:=DRG¤K@3DFDF8VKB=8;<3:]S :pi+1
i'?K8;D=X BK@DF:kS :]L~KkVW:=L~KvGW8~E<lS :¤VWz=B=@ VFVW:=<3B=:XBT:=L~K¡I VWE@3:TVK8HGt>A@:
λi−1k
:=DRGt@<3:tVKBT8~<:oI&E@V%GWE@3D%L;:=D¤I&ELH <3[:=DpjK:=B
0 ≤ j ≤ i+1XAB=:o>A@38
<'M :=DRG_I1KDI&EDFDF8;OL~:ZBKVp0 = 1
i'pKV\B=E<3DFz>A@3:T<QGX1E< 8;:=<AGS:Z[E<AGFVW:TV>A@'M :T<QGFVW:¡Bn1K>A@:ZB=E@3I3L;:λi
k, λik+1
8~L9:Ty 8~DRGW:@3<:K@GFVW:VKB=8;<3:γk 6= λi−1
k
S@0I&EL; <[:qiS E<3BS :
pi+1iKO]@GWEGKLXE<
Q8;:=<AGS:JGFVWE@3:TV(i − 1)
VKBT8~<:=DJS 8;DRGW8~<BTGW:=DS:pi+1
>A@38:=<3B=KS VW:T<QGB=:=L;L;:=DZS:piippKvVJK8;L~L;:=@ VWD=X
qi(λi1) < 0
:#Gqi(λ
ii) < 0
X GK<1S 8;Do>A@:
limλ→±∞
qi(λ) = +∞.
k<U:=<S zS@8HGtLNM :#y8;DRGW:=<B=:\S:S:=@ yVKB=8;<3:TD]DF@I3IL~zT[:=<AGK8HVW:=DxS8;DRGW8;<3B#GW:=D]S:qiX3SE<3B_S :
pi+1X>A@38
:T<3BKS VW:=<AGxB=:=L;L;:=DoS:pii
O C"!#! &% µ ∈ R 2
sgnpi(µ) =
+05L6pi(µ)
pi(µ) 6= 0, +05L6
pi−1(µ) pi(µ) = 0.
N(i, µ)
> %% # +6 ,76/ +05> % 01 !/%7#0 :%6
E(i, µ) = +1, sgnp1(µ), sgnp2(µ), ..., sgnpi(µ) " O :% N(i, µ) 0L'
%0%*%6 5$*%pi
5 0 % % &% $%uµ"
h*`"r =g=,v+gfd`"r2k<VW:=[CKvV>A@3:oS'M KO&EVSU>A@3:sgnpi(µ)
:=DRGxS zT§1<8HGtDWK<DxK[ZO8;m@ NGWz]I3@8~DW>A@:XQDF8pi(µ) = 0
XAKL;EVWDpi−1(µ) 6= 0
_BK@3DF:]S@I&E8~<AG¤£S@':=[[:k= i £Qi i3k<JI VWEAB=S :oI3KVVWz=B=@VFVW:T<3B=:DF@ V
ii3YE@ V
i = 1X3E<zTVW8H§3:L;:VWz=DF@3LHGKG
µ ≤ b1 ⇒ E(1, µ) = +1, +1 ⇒ N(1, µ) = 0,µ > b1 ⇒ E(1, µ) = +1,−1 ⇒ N(1, µ) = 1.
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)
k<DF@I3I&EDF:tL;:9VWzTDF@3LHGKGAVK8vPR@DW>Q@'M oLNM EVS VW:ii ¢ E8;G
(λik)1≤k≤i
L;:=DpVKB=8;<3:=DS:pi:TG
(λi+1k )1≤k≤i+1B=:=L;L;:=DkS :
pi+1XVK<3mzT:=DoI3KVxEVS VW:_BTVWE8~DFDWK<AGi"k<K
λi1 < ... < λi
N(i,µ) < µ ≤ λiN(i,µ)+1 < ... < λi
i,
:TGλi
N(i,µ) < λi+1N(i,µ)+1 < λi
N(i,µ)+1,:=<l:#VFGW@S@UI&E8;<AG]JS@l:=[[:¡Qi £Ai i&¥L'UKGFVWE8~DxBKDoI&EDFDF8;O3L;:=D=i/12 ¢ 8
λiN(i,µ) < µ ≤ λi+1
N(i,µ)+1
X3E<Ksgnpi+1(µ) = sgnpi(µ)
X3SE<3BN(i + 1, µ) = N(i, µ)
i'2 ¢ 8
λi+1N(i,µ)+1
< µ < λiN(i,µ)+1
XE<¡Ksgnpi+1(µ) = − sgnpi(µ)
XvSE<3BN(i+1, µ) = N(i, µ)+1
i'2 ¢ 8
µ = λiN(i,µ)+1
X¤E< Ksgnpi(µ) = sgnpi−1(µ) = − sgnpi+1(µ)
XtS E<BN(i + 1, µ) =
N(i, µ) + 1X3ZBK@3DF:\S @S:T@y 8;=[:kI&E8~<AGoS@l:=[[:\Qi £Qi i
C_K<DGWE@3DL;:=D9BKDN(i+1, µ)
:=DRG%O38;:=<JL;:]<3E[OVW:tS :tVKB=8;<3:TD9S :pi+1
8;<dzTVW8;:=@VW:TD9DRGFVW8;BTGW:=[:=<AG
µi
Eo#Qknm G\Q&m1k &LlSl knMOi\ ! N'b'(J5-2λ1 ≤ · · · ≤ λn
9/2;"0QAJ9:2? (+I"<(+-!<(+-21"$&2A+IAJ' !>?2;"<#AJ+N!+I$&+-2]BD+-!,"@"-AJ'651[ jC (+NB1AJ9BD (9:2?+f'O (3>1+@,76 (2?32?'659A
i .?32]0QAJ9:2? (+N<(+-!<(+-2
λiK
C'S<#+@21'#$ #',:'65@21+@0QAJ9:9:2[a0, b0]
$#AJ'#"=9:2176 (219 !'21"`5%".-(+876 (2λi"`25-+@C (0!2bFH<MAJ+82DR&2134<#9/2
−a0 = b0 = ‖A‖2Y~[JN'8B?AJ9BD #9/2 A!9/C+-"9:2X'(C3%)(+-2
N(n, a0+b02 )
$&>?^#'(,O$(AJ'M"L9/2 2?332f ([&[
Fn9/2;"80QAJ9:2? (+I"8$&29Ab"` (,/5@2pj(
a0+b02 )
Ka<E! (+1 ≤ j ≤ n
K "@!'65]B?A!9:B? (9:>?21"=<MAJ+89A H!+-3% (9:2$&2+->1B? (+-+@21'#BD2 $& 213342([&[ CYD[ d ,/96"`2j5-+@C (0!2 76 (2
N(n, a0+b02 ) ≥ i K1AJ9:!+I"!'=2?']$&>;$& (,/5l76 (2
λiAJ<#<#AJ+@5@,:2?'65c 9Wd ,/'65@21+@0QA!9/9:2
[a0,a0+b0
2 [[ ,aAJ bBD!'65-+-A!,/+-2C'G5-+@C (0!2=76 (2
N(n, a0+b02 ) < i
KA!9/C+-"
λiAJ<#<#AJ+@5@,:2?'65% 9 d AJ (5@+-2,/'65@21+@0QA!9/9:2
[a0+b02 , b0]
[ )*AJ'#"85@! M"89/2;"8B?AC"8!'At$&,:06,"@><#AJ+$(2? &RG9Wd ,/'65@21+@0QA!9/9:2=,/'(,/5@,AJ9a76 (,aB?!'65@,:2?'65
λi[ AJ+f$(,:BIT(!5@C34,:2!K&B!d 2;"|5 $&,:+-2N21'h+->?<E>D5IAJ'65fBD2D5@5@2
<#+@&BD>;$& (+-2*$(2*$&,:0O,:"@,:!'b$&2N9Wd ,/'65@21+@0QA!9/9:28BD!'65-2?'#A!'C5λiKO!'bAJ<#<(+@&BIT(2N9:A40QAJ9:2? #+e2?R(A!B~5-2*$(2
λiA 0C21BN9:A4<(+->1B?,:"@,:!'b$&>1"@,/+->?2C[
% /)-/ - +/9 )8 A 3>D5-T(&$&24$(2 AJ'#B ?C"<M21+@32?5=$&24B?A!9:B? (9/21+N9:21"N0QAJ9:2? (+I"N<(+@C<(+-21"$ad ('#2 3AQ5-+@,BD24"`P
3>?5@+-,:76 (2 +@>12?9:9/2 2?'x &5@,:9:,:"-AJ'659:AS'(!5@,:!'x$ad 2;"`<#ACBD2 $&2 *=+@PO9:Q0EK\$&>|`,:'65@+-O$( (,_5-2 S<(+-!<EC"$(2%9Wd A!9 C!+-,_5-T(32 $& C+-AC$&,:2?'65*B?!'Q| C (>![ 2D5@5@243>D5@T#O$(2hFH2?5="@21"N'(C3%)(+-2? #"@21"!>1'(>?+IAJ9:,:"-A 5-,/C'#"IYf2;"|5N5@+-.1"N2 B?ACBD2%<E! #+*9:21"N3AQ5-+@,BD2;"N$&2 !+IAJ'M$&285-A!,/9:9:2![ 9\"1d A !,/5N<#9/ #"*,BD, $&2$&!'#'(2?+9:28<(+-,/'MBD,:<M28$&2]B?2D5`5-283>D5@T#O$(2]7C #2*9:21"$&>D5IAJ,:9:"c$&2]"@!' ,:34<#9/>13421'65-AQ5-,/C'h'O (3>1+@,76 (2![
)*AJ'#"4BD2h7C #,e"@ (,/51K\!''(!5@2A ('(2h3AJ5@+-,:B?2G"@P63>?5@+-,:76 (2 +->?219/9:2G$Ld !+I$&+-2
nKr0 6= 0 ∈
Rn (' 0!2;B~5-2? (+ $&C'('(>b'(C'y'O (9WKX2?5Kk
9Wd 21"@<#A!B?2b$(2 *=+@PO9:Q0A!"-"`&BD,:>!Kj21' !21'#$&+-> <MAJ+9:21"0C21BD5@2? #+-" r0, Ar0, · · · , Akr0
[fA!<(<M219/C'#"G76 ad ,:9*2?R&,:"`5@2 #'2?'65-,/21+k0 ≤ n − 1
KAJ<(<E2?9:>$(,/32?'M"`,:!' BD+-,_5-,:76 (2b$&2 *=+-P69:Q0K 5-2?9f76 (2CKX"`,
k ≤ k0K 9:ASnAJ3,:9/9:2
(r0, Ar0, · · · , Akr0)2;"|5
9:,:)(+@2=2?5dimKk = k + 1
KO5-A!'#$&,"c76 (2]"`,k > k0
!'bAKk = Kk0
[ d A!9 C!+-,_5-T(32]$&2 A!'#B1C"cBDC'#"@,:"`5@2]B?!'#"`5@+- (,:+@28 ('#2%"` #,_5-2]$&2]0!2;B~5-2? (+I"
(vj)1≤j≤k0+1<MAJ+c9A%HC+@3 (9:2=$(28+@>;BD (+-+@21'#BD2*"` #,/0QAJ'65-2!K(<E! (+2 ≤ j ≤ k0 + 1
K
vj = Avj−1 − (Avj−1 · vj−1)vj−1 − ‖vj−1‖vj−2 , vj =vj
‖vj‖,
F| ([:6Y
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A 0!21Bv0 = 0
2D5v1 = r0/‖r0‖
[\C (+f5-! &5=2?'65-,/21+k ≤ k0 + 1
KE!' $&>?^#'(,/5* #'(2%3AQ5-+@,BD2Vk$&25-A!,/9:9/2
n × k $(!'6589/2;"]BDC9/C'('(21"="@!'6589/2;"80!2;B~5-2? (+I" (v1, · · · , vk)K\AJ,:'#"`,X76 ad ('(23AJ5@+-,:B?2
"`PO3>D5-+@,76 (2 m?Qkn\kn EoEil Tk$(2N5IAJ,:9/9:2
k × k $&C'C5c9:21">19/>13421'65-"f"@!'65
(Tk)i,i = Avi · vi, (Tk)i,i+1 = (Tk)i+1,i = ‖vi+1‖, (Tk)i,j = 0"@, |i− j| ≥ 2.
0C21B=BD2;"'(J5IAQ5@,:!'M"?K&9A4+@>;BD (+-+@21'#BD2=$&2 AJ'#B ?C"e0C>?+-,_^#2=$&2;"c<(+-!<(+-,/>?5@>;" +-2?3A!+-76 #A!)(9/2;"?[ O C"!#!:p L
(vj)1≤j≤k0+1 ",2 / :%<>= "= B :% ‖vj‖ 6= 0
#0&%0
1 ≤ j ≤ k0+1 08 5 vk0+2 = 0
"6 &%1 ≤ k ≤ k0+1
(v1, · · · , vk+1) %- L :% ! #0 :% %Kk
01 % M#-%* J % 6 7% 0' ) ! ,5 4u
(r0, Ar0, · · · , Akr0)" M8 &%
1 ≤ k ≤ k0 + 1
AVk = VkTk + vk+1e∗k,
F| ([: CY
ek$ /
k' &% 6 0 ,5 L6
Rk
V ∗k AVk = Tk
V ∗
k Vk = Idk,F| ([:6Y
Idk
/ 0 % % 90% k × k "
h*`"r =g=,v+gfd`"r2kk@O3L;8;E<3D_I&E@V¡LNM 8~<DRGK<AG¡L~KSzT§3<38HGW8;E< sR i;TAw\S:L~KlDF@38HGW:(vj)1≤j≤k0+1
:TGVW:T[I3L~K=E<3DtL~KI3KVxL~KZ<E@3:=L;L~:kSzT§3<38HGW8;E<sSE<AGtE<l[E<QGFVW:#VK>A@M :=L;L;::=DRGxz=>Q@8;KL~:T<QGW:kUsR= i;=wFw
vj = Avj−1 −j−1∑
i=1
(Avj−1 · vi)vi , vj =vj
‖vj‖, j ≥ 2,
sR= i; Hw
K:TBv1 = r0/‖r0‖
iY6%8~:T<0DFVX%sRQi~Hw_<M KS :JDF:=<3D¡>A@3:ZDF8 ‖vj‖ 6= 0i ¢ 8 ‖vj‖ = 0
XcE<S8HG¡>A@3:LNM KL;mEVW8HGWn3[:oDRGWEI3I&:]_LNM 8~<3S8;B=:
ji pKV¤SzT§3<38HGW8;E<X
vj:=DRG%EVFGWnEmE<1KL"
viI&E@V
1 ≤ i ≤ j−1ipKV
VWzTB=@VFVW:=<B=:XE<zTVW8H§1:%NKBT8~L;:=[:T<QG>A@:vj ∈ Kj−1
i99E[[:9L~K]DF@38HGW:xS:=Dp:TDFI1KBT:=DS: VF L;EvKj
:=DRGDRGFVW8;BTGW:T[:=<AGYBTVWE8~DFDWK<AGW:tI&E@V
j ≤ k0+1XvE<Z:=<¡Sz=S@38HG>A@:X=GK<AG>A@:%LNM KL;mEVW8HGWn3[:t<M KxI1KDYDRGWEII&zX
L;:=Dk:=BTGW:=@ VWD(v1, · · · , vj)
uEVW[:=<AG@<3:¡O1KDF:EVFGWnE<3EVW[CKL;:S:Kj−1
i'pKvV_B=E<DFz>A@3:=<AGXvjzTGK<AG
EVFGWn3EmE<3KL(v1, · · · , vj−1)
:=DRGpK@3DFDF8 EVFGWnEmE<1KL Kj−2
i<¡I1KvVFGW8~BT@3L;8;:TVXL~KxNK[8;L;L;:(v1, · · · , vj)
XS zT§1<8;:UI3KVsR= i; Hw#XB=E<3BT85S :lK:TBUL~KO1KDF:UEVFGWnE<3EVW[z=:US:
Kj−1B=E<DRGFVW@38HGW:lI3KVZL~:UIVWEAB=z=Sz
S :3BoVK[J ¢ Bn3[8~S GoKII3L;8~>A@3zL~KuK[8;L;L;:(r0, Ar0, · · · , Aj−1r0)
iN9:TB=8IVWE@:>Q@:LNM KL;mEVW8HGWn3[:DRGWEI3I&:_I VWz=B=8;DFz=[:=<AG]ZL~KS8;[:=<DF8;E<lB#VW8;GW8~>A@3:\S : VF L~Ev
k0XBM :=DRGFS8HVW:¡>A@3: ‖vj‖ 6= 0
GK<AG]>A@3:j ≤ k0 + 1
:TGvk0+2 = 0
i?lE<AGFVWE<Dp[CK8;<AGW:=<1K<QGc>A@3:L;:=DpSz#§1<38HGW8;E<D¤sRQi;=Awc:#G¤sRQi; HwpS :%L~KxDF@38HGW:
(vj)DFE<AGp8~S:=<AGW8~>A@3:=D=i
9E[[:A:TDRGoDR [zTGFVW8~>A@3:XQE<K
Avj−1 · vi = vj−1 ·Avi = vj−1 · vi+1 +
i∑
k=1
(Avi · vk)(vj−1 · vk).
BoVB=:K@yUIVWEIVW8;zTGWz=DkS'M EVFGWnE<EVW[CKL;8HGWz¡S:=D(vk)
X1E<:=<SzS @38HGo>A@3:Avj−1 · vi = 0
DF81 ≤ i ≤
j − 3X3:TG]>A@3:
Avj−1 · vj−2 = ‖vj−1‖iACE<3B_L;:=D]Sz#§1<38HGW8;E<D_sRQi~TAwx:TG\sRQi~HwtB=E<B=8~S:=<AGi
FY8~<3KL;:=[:=<AGXvL~KtVW:=L~KGW8;E<UsR i;w#XIVW8;DF:9B=EL;E<3<:9I3KVpB=EL;E<3<:Xv<M :=DRGcVW8~:T<ZS"M K@GFVW:9>Q@'M @3<3:pVWz=zTBTVW8HGW@ VW:S :JsRQi~TAw¤:=<Uz=L;8~[8;<3K<AG
vjicK¡IVWEI VW8;zTGWz
V ∗k Vk = Idk
S z=B=E@L;:\S@BKVKBTGWTVW:_EVFGWnE<EVW[CKLS :L~K¡uK[8;L;L~:
(v1, · · · , vk)XQGK<1S 8;Dx>Q@:L~K¡VW:=L~KvGW8~E<
V ∗k AVk = Tk
D=M EO GW8;:=<AGxDF8~[IL;:=[:=<AG¤:=<U[Z@L;GW8HIL;85K<AGsR= i; w]mK@B#n:_I1KvV
V ∗k
BKVV ∗
k vk+1 = 0i
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6$ )
% O&'&Ll &C#! !z d A!9 C!+-,_5-T(32c$&2 A!'#B1C"jAJ<(<MAJ+IA"5 B?!332c ('(2c3>D5-T(&$&2f$&2f+->1$( #B%5-,/C'"@! M"HC+@32 5@+-,:$(,:A !!'MAJ9:2 BDC33429Wd A!9 C!+-,_5-T(324$(2E! #"@2?T#!9$&2?+*0O S<(+->1B?>1$&2133421'651[>1A!'(3!,:'#"?K!9 d AJ9 !C+@,/5@T(32c$&2 AJ'#B ?C"j' d 2;"|5X<#AC" &5-,/9:,:"@>c2?'<#+-AJ5@,7C #2B?!332c ('(2f3>?5@T(&$&2$(2 5@+-,$&,:A !C'#AJ9:,:"-AQ5-,/C'yB1AJ+;K <E! (+
n!+IAJ'#$aK 9:21"h2?+-+@21 (+-"G$Ld A!+@+-!'M$&,$&>?5@+- (,:"@2?'65 2?'<#A!+`5-,/2
9Wd !+@5@T( !!'MAJ9:,_5->*$&2;"c$&2?+-'(,:2?+I"e0!21BD5@21 (+-"vj<#A!+c+-A!<(<MC+`5cAJ (RG<(+-2?3,/21+-"1[ •
! #"AJ9:9/C'#"t3AJ,:'65@21'#AJ'65SB?!3<#AJ+-2?+ 9/2;" 0QAJ9:2? (+I" <#+@C<(+@2;"t2D50!2;B~5@21 (+I"t<(+-!<(+-21" $(21"3AJ5@+-,:B?21"
A2?5Tk0+1
Fn76 (,N'(2 "@!'65 <#A!"G$&2 3 ?32t5IAJ,:9/9:2 2?'+-. C9/2 C>?'(>1+-A!9/2 Y~[ N''(!5@2λ1 < λ2 < · · · < λm
9:21"h0 A!9/21 (+I" <(+-!<(+-21"G$&,:"`5@,:'#BD5@21" $&2AFnA 0C21B
1 ≤ m ≤ nYDK 2?5
P1, · · · , Pm9:21"f3AJ5@+-,:B?21"f$(2]<(+-J|21BD5@,:!'#"C+`5-T( C!'#A!9/2;""` (+9/2;"f"`C #" W2;"`<#ACBD2;" <#+@C<(+@2;"cBDC+
+-21"@<E!'#$(A!'65-"$&2A[N'h+IAJ<(<E2?9:9:2=76 (2
A =m∑
i=1
λiPi, Id =m∑
i=1
Pi,2D5PiPj = 0
"`,i 6= j.
F`#[/$CY
O C"!#!$& $ - &%8%8%K6Tk0+1
L 8 0 - &% %! %6
A" 8% 5 I"#: 5
r0:% 2>
Pir0 6= 0#0&%0
1 ≤i ≤ m ::%0 $: &%38%8% %
A ' - &% %! %6
Tk0+1
k0 + 1 = m"
% O&'&Ll &C#! ! )*AJ'#"89:2 B?AC"8Pir0 6= 0
<E! (+]5@! (5iKa9/2;"83AQ5-+@,BD2;"
A2D5Tk0+1C'65 2DR(ACB~5@2132?'65e9:21" 3 ?321"e0QAJ9:2? #+-" <(+-!<(+-21"1KC3A!,:"eA 0!2;Bf ('(2*3% #9_5-,/<(9:,BD,/5@>*>?0!21'65@ (219/9:2?32?'65
$(, >?+-2?'65@2b<( #,:"-76 (2b9/2;"0 A!9/21 (+I" <#+@C<(+@2;"$&2Tk0+1
"@!'65"@,:34<#9/2;"?[eN'x0!21+@+IAS$#AJ'#"49A$(>3C'#"|5-+-AJ5@,:!'G$( 2133424#[ ([/ 876 ad ,:9L2DR&,"|5-2NA! #"@"@,E ('h9:,:2?'G21'65@+-29:21"e0!2;B~5@21 (+I"X<(+-!<(+-21"e$&2A2?5Tk0+1
[ AhB?!'#$(,_5-,/C'S$&213A!'#$&>12"@ (+r0<E! #+*9A +@>;BD,:<(+-O76 (24$&2BD29:2?33242;"|58)(,:2?'
'#>1BD2;"@"-AJ,:+-2![ ' 2 2D5;KO"@,r021"`5 ('G0!21BD5@21 (+ <#+@C<(+@2N$&2
AKOA!9/C+-"
k0 = 02D59A]3AQ5-+@,BD2
Tk0+1AC$&32D5f<E! (+c #'(,:76 (280QA!9/21 (+<(+@C<(+-2=B?2?9:9/2876 (,a2;"|5fAC"@"@&BD,:>?28r0[ •
Uh*`"r =gT,+gfd`"r2 ¢ E8HGλ:TG
y ∈ Rk0+1 @3<:KL;:=@VIVWEIVW::TG@<:=B#GW:=@VIVWEIVW:GW:=L;D>A@3:Tk0+1y = λy
i 9E[[:vk0+2 = 0
XYL~KlVW:=L~KGW8;E< sR= i; wS :=Q8~:T<QGXI&E@ Vk = k0 + 1
XAVk0+1 =
Vk0+1Tk0+1X&:TGSE<3BX&I3KVKI3IL;8~B=KGW8;E<S@:=B#GW:=@V
yX&E<EO GW8;:=<AG
A (Vk0+1y) = λ (Vk0+1y)i':
:=BTGW:=@ VVk0+1y
<'M :TDRGoI3KD]<Q@3L"BKVy 6= 0
:TG]L~:TDoB=EL;E<3<:=DoS:Vk0+1
DFE<AGoL;8;OVW:TD=i&pKV]B=E<3DFz>A@:=<AGXVk0+1y
:=DRG]O38;:=<l@3<:TBTGW:=@VxI VWEI VW:\S:AKDFDFEAB=8;zZJL~KJKL;:=@V]I VWEIVW:
λ>A@8:=DRGoS E<3BdEVWB=zT[:=<AG
z=mKL~:¡Z@<S :=Dλii
@ozTB=8;IVWEQ>A@3:=[:T<QGX=E<\8;<QGFVWEQS @38HGpL;:9DFE@3DR :=DFI3KB=:¤:=BTGWEVW8;:=LEm
S:Rn :=<3m:=<1SQVWz9I1KvVpL;:=DY:TBTGW:=@VWD
(P1r0, · · · , Pmr0)i ¢ 8
Pir0 6= 0I&E@ VYGWE@ G
1 ≤ i ≤ mXB=:TDp:=BTGW:=@ VWDYDFE<AGL;8;OVW:=DBKvVL;:=DpIVWEvPR:=B#GW8~E<3D
PiDFE<AG\S :=@yCS:=@ yEVFGWnEmE<1KL~:TD=icpKvV_B=E<3DFz>A@3:T<QGX1L5KS8;[:=<DF8;E<S:
Em:TDRG_:TyKBTGW:T[:=<AG
mi
¦kE@D9KL;L~E<3D%[E<AGFVW:TV%>A@3:xS3K<3DB=:]BKDE<CKm = k0 +1
iApKVosR i; $w%E<CKAkr0 =
∑m
i=1λk
i Pir0X
BM :=DRGFS8HVW:>A@3:Akr0 ∈ Em
X'S E<BJL;:=D_:=DFI1KB=:=D¡S: VF L~Evz#VW8;§3:=<AGKk ⊂ Em
I&E@V_GWE@Gk ≥
0ix< I1KvVFGW8~BT@3L;8;:TVXxB=:=B=8k8~[IL;85>A@:>Q@:
dimKk0 = k0 + 1 ≤ mixpKvVK8;L;L;:=@VWD=X]SK<DUL~KO1KDF:
(P1r0, · · · , Pmr0)S:
EmXQL;:=D¤B=EAEVSE<3<3zT:=DtS@C:TBTGW:=@V
Akr0DFE<AG
(λk1 , · · · , λk
m)iIOo@GFVW:T[:=<AG9S 8HGX
L5KZuK[8;L;L;:(r0, Ar0, . . . , A
m−1r0)S :
Em:=DRGxVW:=IVWz=DF:T<QGWzT:SK<D]L~KZO3KDF:
(P1r0, · · · , Pmr0)I1KvVoL~K
[CKGFVW8;B=:M
SzT§3<38;:I1KvV
M =
1 λ1 λ21 · · · λm−1
1iiiiii
1 λm λ2m · · · λm−1
m
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K[CKGFVW8;B=:M
:=DRG_@3<:\[CKGFVW8;B=:S:9K<C:TV ?lE<1S :ZS"M EVS VW:m>Q@8c:=DRGk8;< :TVWDF8;O3L;:B=KVGWE@3DkL;:=D
λiDFE<AGZS8;DRGW8;<3BTGWD=i CE<BL5KuK[8;L;L~:
(r0, Ar0, . . . , Am−1r0)
:=DRGL~8;O VW:I@38;DW>A@3:(P1r0, · · · , Pmr0)LNM :=DRGiN9:TB=8Y8;[I3L;8~>A@3:\>A@3:
dimKm−1 = mX1SE<3B
m− 1 ≤ k0i1CM ELNM E<SzS @38HG]§1<1KL;:=[:=<AG]>A@3:
m = k0 + 1:TG
Em = Kk0
iBoVB=:L~KuEVW[Z@L~:sRQi;%$wE<K
A(Pir0) = λi(Pir0)i 9E[[:
Pir0:=DRG_<3E<<Q@LX"8;LpD=M Km8HG
O8;:=<S'M @3<:=B#GW:=@VkIVWEI VW:JS :AKDFDFEAB=8;zCCL~KKL;:=@ V_IVWEIVW:
λii 9E[[:
Em = Kk0
:TG_>Q@:L;:=DBTEL;E<<3:=DS:
Vk0+1uEVW[:=<AG@<3:JO3KDF:S:
Kk0
XYE<:=<Sz=S@38HG¡>A@M 8;L:Ty 8~DRGW:C@<:=B#GW:=@V¡<3E<<Q@3Lyi ∈ Rm GW:TLt>A@3:
Pir0 = Vk0+1yiik<[@3LHGW8;I3L;8;:L5KI VW:=[8;TVW:Cz=mKL~8HGWzlS:sRQi~wI3KV
yiI&E@V
EOGW:=<8HV
Tk0+1yi = V ∗k0+1AVk0+1yi = V ∗
k0+1APir0 = λiV∗k0+1Pir0 = λiV
∗k0+1Vk0+1yi = λiyi,
K@GFVW:=[:T<QG_S8HGXyi:=DRG¡:=BTGW:=@ V_IVWEIVW:CS :
Tk0+1I&E@V¡L~KlKL;:=@ V\IVWEI VW:
λii 9:C>A@8GW:#VW[8~<:ZL~K
S z=[E<DRGFVKGW8;E<'i
2=+@>;"` (9/5-AJ5f$& 2?332#[ ([/ 8<MC (+-+-A!,_59:A!,:"-"@2?+cBD+-!,:+@2=76 ad ,/9nA! &5fAJ<(<(9:,7C #2?+c9A%+->1B? (+ +@21'#BD2$(2 A!'#B1C"a| #"@76 ad 89Wd ,/5@>1+-AJ5@,:!' 3AJRO,:3AJ9:2
k0 +1K!<( (,"eB?AJ9BD #9/21+ 9:21"X0QA!9/21 (+-" <(+-!<(+-21"
$&2Tk0+1
AJ^#'t$Ld 21't$&>1$& #,/+-2=9/2;"c0QAJ9:2? (+I"<(+-!<(+-21"e2D5f9:21"0C21BD5@2? #+-"<(+-!<#+@2;"e$(2A[ 219:A+-2?'
$&+-A!,_5X9A=3>D5-T(&$&2f$&2 A!'#B1C" B?!3<#AJ+IAJ)#9/2c8BD219/9:2f$&2 3*,:0!21'#"!;EfC #"@2?T(C9:$&21+cFH2?' C>?'(>1+-A!9k021"`5 $&2f9Wd C+-$&+-2c$&2
nK!B?2f7C #,#+-2?'#$49:2fBD!3<&5-2f$Ld !<E>?+IAQ5@,:!'M" "`,:3,/9AJ,:+@2f$#AJ'#"j9:21"X$(2? &RB?AC"-YD[ )2
<(9/ M"?K A!<(<(9:,:76 (>12 AJ,:'#"@,X9:Ab3>?5@T(&$&2 $&2 AJ'MB?6"8"`21+-A!,_58,:'#"`5-AJ)#9/2'O (3>?+-,7C #2?32?'65%tB?A! #"@2$&2f9A8<M21+`5-2$Ld !+@5@T( !C'#AJ9:,_5->c$&21" 0C21BD5@2? #+-"
vjB1AJ #"@>?2c<MAJ+X9:21" ,/'(>10O,_5IAJ)(9:21" 21+@+-2? #+-" $Ld AJ+-+-!'#$&,
FH0!C,/+9A c2?3A!+-76 (24#[ ([ 6Y~[#!+@5]T(2? #+@21 #"`2132?'651Ka9:2+@>;"` #9_5IAQ5%"@ (,/0QA!'C5],:'#$&,76 (2 76 ad ,/9X' d 2;"|5]<#AC"='(>1B?21"-"@A!,/+-2$&2nAJ,:+-2
)M2;AJ #B?! (<G$ad ,/5@>1+-AJ5@,:!'#" $(AJ'#" 9A8+@>;BD (+-+-2?'#B?2$&2 AJ'#B ?6"X<E! (+e!)&5-2?'(,:+ $(21" 0 A!9/21 (+I"X<(+-!<(+-21"$&2Tk76 (,E"@!,:2?'65 $&2N)MC'('(2;" AJ<#<(+@ R&,:3AQ5@,:!'M" $&2NB?2?9:9/2;" $&2
AFnA 0C21B
k)M2;AJ #B?! (<<#9/ #" <E2D5@,/5
7C #2k0C
nYD[
F oM oE1knm1k oMi C"!#!$1 %1 ≤ k ≤ k0 + 1
" λ5 : &%/8%8%0%
Tky ∈ Rk P &% %! % 5 & " ! 5 - &% %! %
λi%A
5
|λ− λi| ≤ ‖vk+1‖|ek · y|‖y‖ ≤ ‖vk+1‖,
ek
$ / k' &% 6 0 ,5 L6
Rk "
% O&'&L\( C"!#!$&C A<(+-2?3,:.?+-2SBDC'#BD9: #"@,/C'$&29:A X+-!<EC"@,_5-,/C' #[ ([/ 21"`5b76 (2CKc"`,‖vk+1‖
21"`5%<E2D5-,_5;K\A!9/C+-"=9:21"%0 A!9/21 (+I"8<(+-!<(+-21"8$&2Tk
"@!'65 $&2G)MC'('(21"%AJ<(<(+- R&,/3AJ5@,:!'#"8$&2BD2?+@5-A!,/'#21"0QAJ9:2? (+I"<(+-!<(+-21"$(2
A[ A$&2? &R&,:.?32 BDC'#BD9: #"@,/C'x2;"|5 9A<(9: #",:3<MC+`5IAJ'65@2t21'
<(+-AJ5@,76 (2 0X"@,9:AS$&2?+-'(,:.?+-2hBDC3<M6"@A!'C5-2h$Ld ('y0!2;B~5@21 (+4<(+@C<(+-2h$&2Tk
21"`5<M2?5@,/5@2!KXA!9/C+-"%9:A0 A!9/21 (+ <#+@C<(+@2fB?!+-+@2;"`<E!'M$(AJ'65@22;"|5 #'(2)E!'('#2AJ<(<#+@ R&,:3AJ5@,:!'$Ld ('(2f0QA!9/21 (+ <(+-!<#+@2f$&2
A[
•h*`"r =g=,v+gfd`"r2 ¢ E8HG@<:=BTGW:T@VJI VWEI VW:U<E<?<Q@3L
y ∈ Rk GW:=L¤>Q@:Tky = λy
i?l@LHGW8~IL;85K<AGsR= i; wxI3KV
yE<EOGW8;:=<AG
AVky = VkTky + (ek · y)vk+1,
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)
S'M E LNM E<SzS @38HGA(Vky)− λ(Vky) = (ek · y)vk+1.
sR= i;Gwk<SzTB=E[I&EDF:_KL;EVWD
VkyDF@ VoL~KZO3KDF:\S :=Dx:=BTGW:=@ VWDtIVWEIVW:=D]S:
A
Vky =
m∑
i=1
Pi(Vky).
k<lIVW:T<1SUL;:_I VWEQS@38HGxDFBKL~K8HVW:ZS :JsRQi;GwxK:=BVky
X :TGoZLNM K8~S :\S:=DtVW:=L~KGW8;E<3D¡sR i; $wtE<K
m∑
i=1
(λi − λ) |Pi(Vky)|2 = (ek · y) (vk+1 · Vky).sR i £ w
<[8~<EVK<QG\L;:JGW:TVW[:ZS:SQVWE8HGW::TG¡:=<[CKvP EVK<AG¡B=:=L;@89S :JmK@3Bn3:JI3KV¡LNM 8;<z=mKL~8HGWzS: ¤K@3BnA ¢ Bn¤KVX8;LQ8;:=<AG
min1≤i≤m
|λi − λ| ‖Vky‖2 ≤ ‖y‖ ‖vk+1‖ ‖Vky‖.9E[[:ZL;:=D\B=EL;E<3<:=D¡S:
VkDFE<AG\EVFGWnE<3EVW[z=:=D=XE<K ‖Vky‖ = ‖y‖ X:TG\I1KvV\DF8;[I3L;8H§1BKvGW8~E<0E<EOGW8;:=<AG
min1≤i≤m
|λi − λ| ≤ ‖vk+1‖.9:TGFGW:k8;<3zTmKL;8HGWz_I&:=@ GtTGFVW:K[z=L;8;EVWz=:DF8&E<l<'M KI3IL~8~>A@3:]I1KD ¤K@B#nAQ ¢ Bn¤KVoK@CGW:TVW[:
< ek, y >S3K<D_sRQi £ w#i1CK<3DxB=:kBKD]E<GFVWE@3:
min1≤i≤m
|λi − λ| ≤ ‖vk+1‖ |ek · y|‖y‖ ,
B=:\>A@38&GW:TVW[8;<3:L~KJS z=[E<DRGFVKGW8;E<'i
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U !V ) T [/K ) ! " [/K c e[:K % -% K%8%" $ ,5 AJ5@T(>13AJ5@,76 (21"%2D5 <(<(9:,:B1AQ5@,:!'M" (K <#+@,:' !21+1K AJ+-,:"=F 6Y~[
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