(n + u) +v i= n+ (u +v).

361 Internati onal Ser ies of Numerical Mathematics.Vol. 91 © 1989 BirkhauserVerlag Basel

Second variations for domain optimization problems

Jacques Simon

Departement de Mathematiqucs Appliquees Universite Blaise Pascal

Introd uct io n

The calculation of th e first vari a tion with respect to domain variation is well known , for solut ions of p.d.e, as well as for associated fun ctio nals .

Very little is known concerning th e second variation, with the exception of some calcu­la tio ns for tor sional rigi dity by N. Fuj ii [1).

Usually the second vari ation is the variation of the first variation. For domain dependent fun cti onals this is no longer t rue. Indeed the variation of a domain n is represented by a vector field u , and the vari ation t hat result from two successive variations u an d v is not t he sum u + v , that is

(n+ u ) +v i= n + (u + v ).

for not constant v. Here we prove that, for any functional 9 depending on domain, th e second vari a tion gil

is related to the vari a tion of the first var ia tion, (g/)', by the following formula

g"(n; u, u ) = (g')' (n; u , u) - g'(n; u . \7u).

Moreover gil exist s as soon as g' and (g')' exist . With th is formula the secon d variation for solutions of p.d .e. as well as for associated

functionals is calculated based on the well known formulas for first var iations. T he outline is as follows

1. First varia tions formu las revisited 2. Second var ia tions formulas 3. T heorem on the rela tion between gil and (g')' 4. Second var iation of the dr ag of a body .

In the first two sections we give general ideas an d formul as; assumptions are specified in the las t two sections .

T he author is ha ppy to thank the organizers of the Conference in Vor a u where t his paper was written , and to thank M. Yoge1ius for fruitful d iscu ssions .

362 Second uariaiions for dom ain optim ization problem"

1. First variations fo rmulas revisited

1.1. Optimal design problem. Let n be a bounded open su bset of RN wit h a boundary an. Let yen) be th e solut ion

of a boundary value problem in n, say

A(y(n)) = f in n, B(y(n )) = h on an.

The cost fun ct ion is t he real number

.4, B and C being par tial differential operators, an d I, 11 an d d being funct ions given in all of R N .

In optim al des ign one seeks a domain no, which minimizes J(n) in some class 'Dad of ad missi ble domains.

1.2. F irst variation JI. Let u be a vector field defined in all of RN , represen ting the var iations of n; u is assumed

to be in L ipk(R N; R N ) , k ? 1, the norm of which is denoted by Illk T he new domain is defined by

n +u = [z + u(x) : x E n }.

We ar e in terested in the firs t order expansion of J(n + u) with respect to u , that is

J(n + u) = J( n ) + JI(n j u) + o(lIullk)

with a linear l ' (n ; ·). T he pro blem is to calculate the first var ia tion J/ , and to prove that the expansion holds .

R e mark . As for any optimization problem , the first vari ation yields a necessar y optimality cond it ion: if no is an opt imal domain then , denoting 'D~ d the tan gent cone to 'D ad at no ,

Moreover , based on J' we can use a gradient met hod to const ru ct a locally opt imal domain no. 1.3. First lo cal va r ia t ion y' .

To find the first varia tion of J we need the variat ion of y . That is we seek an expansion of the type

yen+ u) = yen) + y'(nj u) + o( lIullk)'

However y(D+ u) lives in the domain D+ u which depends on u, bu t yen) an d y'(D;u) live on the fixed domain n. Therefore this expansion cannot be satisfied in all of n; it can be sat isfied only in the intersect ion , for all u , of the domains n + u.

We will say that the expansion holds locally if it holds in any w cc n. T his is mean­ingfull since then we n + u for all small enough u .

The function y'( D; u) is defined in all of n by it 's restrict ions to any w. It is so-called the firs t local varia tion .

-

Simon 363

1.4. Formula for first variations of the optima l design problem. We suppose now that y(n +u ) depends on u regular ly and uniformly up to the boun dary

an + u. This assumption , wh ich will bc specified late r, is satisfied if, for t he fixed domain D,

y(D) is unique and depends regular ly 01 the coefficients of A an d B an d on the fu nctions f a n d h.

If A, B and C are linear', then , for an y u , the firs t local var ia tion y' = y' (n; u) is solution of the boundary value problem

Ay' = 0 in ll, B y' = - U ri ~ (B y - h) on aD.

Here n is an uni tary normal vector to aD, Un = U . n is t he no rmal component of u and y = y(D).

T he first varia tion of the cost is

l' (n ;u) =1(C y - d)C y' dx + ~ ~n un lCy - dI 2 d".

If .4 , B and P are differe ntiable in convenient fun cti on spaces on n, den ot ing D.4 the deriva t ive of A , y' sa tisfies more generally

DA(y;y' ) = 0 in ll , DB (y;y') = - U n :n(B (y) - h) OIl an ,

And , if J (D) = In P(y(D» dx , J' is given by

J ' (D;u ) = r DP(y;y')dx + r ll n P (y )ds . Jn Jan

T hese formulas are well known in mechanics , but suit able assumptions are no t so known .

1.5 . Total variation y ' up to the boundary. To get the boundary condition on v' , an d to get the variation of the integral of C (y ),

we need uniform dependence of yell + u) on u u p to the bo undary aD+ u, which will be obtained by mapping y(n + u) onto the fixe d domain ll .

Denoting by I the ide ntity in R N, I + u maps n onto D + u, Thus the funct ion y(D. +u) 0 (I + u) lives in D, and the un iform dependence up to the boundary is given by

yell u) 0 (I + u) = y( D.) + y (D; u) + o(lIullk) in D.

T his condition, in a sui table function space on D. depending on A , B and C, imply tha t 1 has a first order expansion, tha t locally y has a first order expan sion , an d tha t F (D;u) an d y'(ll ;u) satisfy the ab ove formu las .

The reader is refer red to J . Simon [2] for precise statements and for proof of these results. The function y' (D;u) being a kind of total derivative is so-cal led the first total variation.

364 S econd variat ions for domain optim ization problem s

R emark: Respective use of local and tot al variations. T he existence of th e total variation y ' is neces sary for our proof of the existence of J' ,

However kno wing the exact value of y ' is not required . To cal culate the value of J' we use only th e value of th e local var iation v' , which is

determ ined by the above boundary value problem . This is the reaso n why we use a t the sam e time the two different ob ject s y' an d y '.

T he value of y ' may be ob tai ned by

y'(n; u ) :- y'( O;u ) + u· \7y(O).

1.6. B asi c formulas for first local variations. Now we will give formulas for the first varia tions of elementary equat ion , boundary

condit ion and integral. Let z]n+u ) be a fun ct ion defined in n+ u, which depends on u regularly and uniformly

up to the bound ary an+ u . It's local var iati on z'(O; u ) has th e following propert ies.

(1) If, V u, z (n + u) 0 m n + u then , V u, ;;' ( .11; u) :::: 0 In n.77:

(2) If, V u, zen + u) :::: 0 on 8n + u th en, V tz , z'(O; u) +]/ . V'z(.I1) = 0 on · dO .

-.:he integral K(n .!. u) . , fn +u zen+ u)dx has a firs t variation which is, "Iu ,

(3) K' (n;u ) :::: 1z' (n i !t ) + \7 ·(u ~ (n) )dx .

Here \7 = (al , . .. , aN) where 8i = aj8x i , th us u . \7z :::: BiUiaiZ an d V' . u = BiaiUj . T he regular an d uniform dep endence on u means that , in a con venien t function space

on 0, there exis ts a to t al var iation z'( Oj u). The precise statements , with assum ptio ns in Sobolev spaces, and pro ofs may be found

in [21, lemma 2.1, p. 657 an d th eorems 3.1, 3.2 and 3.3 p. 663- 664. Here we will only give rough proofs.

R emark: Explicit dependence on u". In (2), z(O) =0 on an, t hus \7z( O) = n8z(0";'f}' i and

:.:'( 0 ;u) = - u" a z(O)j an on an.

In (3) Stokes formu la yields

K'(n ;u ) = f z'(n; u )dx + f u nz (O)ds . in Jon In fact , for any quan tity g(n) depending on n , if the first varia t ion g'(n ju) exists, it

depends on u only by it 's normal component 'Un . This is proved in 13), theorem 3.1. page IIl .20.

R e m ark . The formulas for firs t var ia t ions of sect ion 1.4 follow by choosing successively z

to be A(y) - I, B(y ) - h, and ! IC(y) - d1 2 .

Simon 365

Rough p roof of (1). The local exp ansion red uces to zl(S1 ;U) + o( lIu llk) = 0 in any fixed domain w CC D. Thus z'(D;u ) = 0 in w and therefore in S1.

Rough proof of (2). Now z(S1 + u) 0 (l + u) = 0 on the fixed boundary oD, thus th e first total var iation sa t isfies z'(S1 ju) = 0 on oD. We conclude since z' (S1 ;u) = zl(Sl ;u ) + It· 'Vz(Sl).

Rough proof of (3). J3y th e chan ge of variable 1 + u we obtain an integral on th e fixed dom ain Sl:

I«D + u) = in z(S1 + u) o (I + u)J ac(J + u)dx .

T he firs t var ia tion of J ac(I + u ) is 'V . u , thus

K'(Sl;u ) = in z'(S1 ;u ) + z(Sl)'V · udx = l ;;'(S1j u) + u · 'Vz(Sl ) + z(D)'V · udx,

2. Second variations formulas

2.1. Second variation J" and y". V>le are now interested in th e second order expansion of J wit h respect to u , that is

J (S1 + u) = J(D) + J' (Sl;u) + ~ J" ( S1 ; u, u) + o((lIulld2)

with a bilinear JII(S1 ;. , .). T he problem is to calc ulate ) 11, an d to prove th a t the expansion holds. To find J" we need t he second local var ia tion y /l , that is for any w CC D, an expansion

y(S1 + u) = y(Sl ) + y'(D;u) + ~ y"(n; u,u) + o(( lIulld 2) III w .

The function y"( Sl ;u , u) is defined in all of D by it 's res trict ion to a ny w,

R emark. The second variat ion J" gives a su fficient cond ition for local op tim ality: if

JI(no; u) = 0 and J"( Slo;u , u ) 2: 0 V u E 'D~ d '

then S1 0 is locally optimal. Moreover base d on J" we can improve t he velocit y of gradient me tho ds . For example

by choosi ng the n + 1 approxima tion of Do to be

2.2. Formula relating J" and (J')'. Usually the second varia tion with respect to a parameter is the varia tion of the first

var iatio n . Usually means for a par ameter in a linear space. In optimal design the parameter is Sl , which is not in a linear space. T he var ia t ions of

S1 ar e represen ted by the parame ter u which is in a line ar space, however the behavior is

36l Second vari ati ons JOT dom ain optimizatio n problem»

not usu al since the var ia tion of ,11 that result from two successive varia tions u an d v is not the su m u + v. In fact ,

(1) ( ,11 + u) + v = n+ (u + v 0 (I + u)).

Indeed by defini t ion n + u = (I + u)(n ), thus th is follows from (I + v) 0 (I + u) = J + u + v o(I + u).

Assume tha t l (n + u) has a first order expansion wit h respect to u, and t hat it 's first vari a t ion l' (n + u; w) has a first order exp ansion with respect to u . T hen l(n + u) has a second ord er expansion wit h res pect to u, and the second var ia tion is given by

(2) J"(D.; 1.1 , 1.1 ) = (l ' )'(D; 1.1 , u ) - l' (n;u . \71.1 )

where u · \7u = ~ i U i 8i U .

'Wit h this relat ion the second ord er expansion 1" may be ob tained by using twice the formulas for first variat ions.

Assumptions will be sp ecified an d the whole proof will be given in sect ion 3.

Rough proof: We define a map j by j (u ) = 1 (D. + u ). T his map is defined on a linear space of vector fields u , an d the expansion of J(n + u) is Taylor' s for mu la for j at O. T hus

r(n;u ,u) = D2j (O; u , u ) , l'(n ;u ) = Dj (O;u ),

where D 2 j and Dj are the usu al derivatives :

D2 j (O; u , w ) = lim ~ (Dj ( tu ; w) - Dj (O ;w)), Dj (tu, w) = lim ~(j ( tu + sw) - j(tu )). 1- 0 t s.-0

Let us calculate th ese quantit ies. At first (1) with v = swO(I +tU)-1 yields D.+(tU+3W) = (n + tu ) + sw 0 (I + tU)- l, thus

Dj(tu; w) = lim ~ ( J ((n + tu) + sw 0 (J + tu )- l ) - J(n + tu )) 3- 0 S

=l'(D. + tu ;w 0 (I + tU)- l ).

Thus , si nce l ' (D. + tUi ' ) is linear and w 0 (I + tU)- 1 = W - tu . \7w + o(t) ,

1D2 j (0;u , w) = lim - el 'en + tu;w 0 (I + tU)- I) - JI(D. ;W))

1-0 t

= lim ~ ( l ' (n + tu;w) - l' (n ;w) 1- 0 t

1 _ + l '(n + tu; - (w o (I + tu ) I _ W))

t = (l ' )'(n ; lI , w) + J'(n; - u . \7w).

R e m a r k. T his relation between variations is sat isfied by any function of domain , and in particular by Ylw' Indeed in this cal culation we have no t used the pa rticular defi nition of J .

- - - -

Simon 367

2 .3. Bas ic formulas for second local variations. We will now deduce formulas for the second varia tions of an equation , of a boundary

condition or of an integral , from th e formulas for first variat ions by using the rela tion between o" and (g' )',

Let z(n +u) be a fun ct ion defined in n + u, which depends on u regularly and uniformly up to the boundary on + u, The second local variat ion zl/(n ;u , u) has the following properties.

(1) If, Vu , zen + u ) = 0 in n + u then , V u , zll(n;u,u) = 0 in n. If , V u, zen + u) = 0 on an + u then, V u,

(2) zll(n ;u, u) + 2u · \7z' (n; u ) + uu · \72 z(n ) = 0 on on.

The integr al ]« n + u ) = fo+" zen + u )dx has a second varia t ion which is, Vu ,

(3) ]{II(n; u, u) = i zl/(fl j u , u ) + 2\7 · (u / ( fl; u) + \7 . (u\7· (uz(n) ) - (uz(n ) · \7 )u )dx .

Here U ti · \72z = 'EjjUjUj OjZOj Z an d \7 · (u\7· (u z ) -(uz· \7 )u ) = 'EijOj(Ujaj (Uj z ) - UjZOjUj ).

The regular an d uniform dependence on u means tha t zen +u ) has a first to t al variat ion z'(n; u) for all n, an d th erefore a first local var ia tion z' (n; 10 ), an d that z' (n ;u) has a to tal variation. T hen locally, that is in any fixed domain wee n, u(n +u) has a second order expansion.

R emark: Explici t dependence on U n an d (u . \7u )" . In (3) Stokes formula yields

KI/(n ; u, u) = r z"(n; U , u )dx r 2un z' (n ; u) + u"u\7 ' zen ))(4 ) in ian

(u\7 , u - u· V'U)nz(n )ds .

The boundary condit ion (2) yields , since z' (n; u ) + u · \7z(n) and zen ) are null on on ,

(5) z" + 2u· \7z ' = - unO(/ + u · \7z )/on + (u · \7u )nOz/ on on on.

It may also be written in the following way

(6) z" = - 2uno(zl +u · \7z )/ 8n + 2(u · \lu )" oz/ on + uu - V 2 z on on.

Rough p r o of of (1) : T he secon d order expansion reduces to z' (n ; u) + t z"( n; tz , u) + o« lIu llk)2) in any w . T hus z' (n; u ) = zll(n ;u ,u ) = 0 in w and th erefore in n. Rough proof of (2): The boundary conditio n on z' given by the form ula (2) in section 1.6 is satisfied for an y domain , an d in particular for fl + u . T hat is

z'( n+u;w) + w · \7z (fl + u) = O on on + u.

-----,...- .-.-..•.- .-..-...- .-.------ - - - - - - ­

368 Second variat ions for dom ain opt im izat ion problems

Again by the formula (2) of section 1.6, the variation of th is new bou ndary cond ition is

(z'e ;w ) +w · 9 z)'(D;u ) + u · 9( z' (D;w ) + w · 9 z (D» = 0 on oD.

For w = u this yields

( z' )' (D; 1t , 1t ) + 2u ' V'z'(l2 ; u) + uu· 9'2z(\l ) + (u· \7u)· \7 z(n ) = 0 on an .

On the other hand the boundary condition on z' yields

z'( n; u· \7u )+(u · \7u ) · V'z(n ) = o on an.

The desired boundary condit ion on zll(n ;u , u) = ( z' )'(n ;u , u) - zl(n ;u · \7u) is ob tained by sub tracting these two equations.

Rough proof of (3): T he first varia tion of K given by the formula (3) of section 1.6 is sat isfied for any domain, an d in particular for D + u . That is

](' (D+u ;w ) = r z ' (l2 +u ;w ) + 9. (wz(D + u» dx . In+u

Again by the formuly (3) of sect ion 1.6, the vari ation of t his new in tegral is

(K')'(D; u; w) = 10(z'(. ;w) + V' . (wze ))' (D ;It) + \1 . (u (z'( D;w ) + \1 . (wz(D»)))dx.

Thus

(]{')'(n; u ;u ) = 10 (z')' (D ;u , u) + 29 . (u z' (n ;u) + 9· (u\1 · (uz(n» )dx.

T he desired valu e of K" (n ;u , u ) = (J{' )'( n ;u, u) - K'(n; u · \1u) follows since

[('(n;u· 9 u) = 10 z' (n; u · 9 u ) + \1 . « u · \7u)z(il»dx.

2.4. Formulas for second variations of the optimal design problem . We are now in position to calc ulate the second varia tion for the op t imal problem of

sec tion 1.1. T he bas ic formulas of the preceding sect ion yield the following results. We assume that the solut ion y(il + u) depen ds on u regul arly and uniformly up to the

boundary aD + u,

If A , Ban d C arc linear, the second local variation v" = yl/(n; u, u) satisfles

" f) ( , ) ( ) az '2Ay" = 0 in n, By = - 2u ,, -;- z +u ·9z + 2 u · \7 u n- +uu·9 z on on. un an

Simon 369

and the second varia tion of the cost is

1"(0 ; H, u ) : l (Cy - d)C y" + (C y' )2dx

1 I 1 2 1 2+ 2u n(C y - d)C y -; -2UnU' 'V(ICy - dl ) + - (u \7 · 1./. - u . \7U)nICy - dl ds . ao 2

T he regu lar an d uniform dependence on u means that yen +u) an d y'(n ;u) have total variations in convenient function spaces depending on t he operators A an d Ban d C . It is satisfied if, for the fixed domain n, yen) is unique and regu lar ly depend on the coefficients of A and B and on the functions f an d h .

An exam ple of opt imal design pr oblem satisfying these properties will be given in sect ion 4.

R emark. For non linear opera tors, similar formula ar e obtained by using

/1(y)" = D A(y; y") + D2 A(y;v' , y' ), A(y)' = D A(Yi y' ).

3. Theorem on the relation between gil and (9')'

3.1. Preliminary estimations fo r composed m a ps. We denote by Lipk(RN i R N ) , k being an integer 2: 1, th e space of bounded functio ns

with derivat ives of order :5 k - 1 which are uniformly Lipschitz continuous. This space coincides wit h the Sobolev space W",OO (R N;RN ), an d is provided with the norm

11 1./.11 " = Su p :Z:l; ~N IDC/(x )l· °5.1<>19

Lemma 3.1. Let U E L ip"(RN ;RN ) , k 2: 1, such that lI ulik :5 a k where a" =

(4 k N"+3)- 1/2

t. The map 1 + U invert ible, (I + U)- I =1 + u' where u · E Li pk(RN; R N) and

H. For any io E L,o"(RN; RN), W 0 (1 + u) and W 0 (I + U ) - l are in L ip k(RN ;R N),

and

IIw0 (I + u)lI " :5 ck llwllk' IIw0 (I + u)-l il k :5ckllwllk'

IIw0 (I + u) - wll k :5 CkllWIlk+l IlUlik IIw0 (I + U )- 1 - wll k :5 c" IIWIlk+l llul/k.

370 Seco nd vari atio ns fOT domain optim izat ion problems

We denote by I the identi ty of R N and by CJ; various positi ve numbers depending onl y on k and on the space dimension N .

Pro of: Part i. Since lIulh ::; lIullk < 1, 11. is a con ract ion , thus I +u is inverti ble. The des ired properties of u ' are given by lemma 2.4 part i page 11.15 of F . Murat & J. Simon [3J. Part ii. Let v E L ipk(RN ;RN) such that I + v is inver ti ble and (I + V) - l - I E L ipk(R N; R N). T hen , by lem ma 2.2 part i page II .B of[3]' wo(I +v ) E L ip k(R N ,R N) and II wo(I +v) lIk ~ IIwllk(1+ ckllvllk)k. The des ired properties follows by choosing successively v = u and v = u " . P art iii. If w E L ip k+l(R N; }tN), by t he lemma 2.2 part v of [3],

T he desired properties follows by choosing successively v = u and v = u " ,

3 .2. Variations of a domain of R N. Givcn Q C R N , and a k as in lemma 3.1, a fam ily of subsets of R N is de fined by

TJ = {A = Q + 1L : U E L ipk(RN; R N), Ilulik~ ad

where Q +u = {x +u(x ) : x E Q} = (I +u)(n).

Remark. If n is open an d bounded , and if it 's boundary is LiJ/', cnen every !I. in TJ has the same properties , pro vided tha t J. :::: 2. For k = 1 this is not satisfied.

We will see now that the var iation of a dom ain Q tha t resu lt from two successive varia­tions u and v is no t u + v , bu t is u + v 0 (I + u ),

Lem rna 3 .2 . Let Q C R N , u and v in L i pk(RN ;R N ) wi th lIullk ::; ak . Th en

(n + u) + v = n + (u + v 0 (I + u»

Q '-(Jl+v ) =(n +u ) + v o(1 + u )- I.

Proof: T he first equ a t ion is given by (1 + v) 0 (I + u) =1 + u + v 0 (I + u), and the second one is obt ained by replacing v by v 0 (1 + u)- 1.

Remark. The smal l var ia tions p reserve V : if A E 'D and if v is smal l enough , then A + v E 'O.

Indeed A + v = n + u + v 0 (I + u) and, by lemm a 3.1, lIu + v 0 (I + u)lIk < (; '. if lIukilk < (ak - II t,ll d /ck. Remark. T he family 'D is a neighb orhood of n in the met ric space 'Dk defined by

vk = {A = n + u : It E Lipk(R N; R N),

1+ u is invertib le and (I + U) - l - [ E L ip k(R N ; Rf\ ; }

and provided wit h the metri c defined by F . Murat & J. Simon [3J in section 2.5 page II.2G.

Simon 371

3.3. The main result. We will see now that th e secon d var iation g/l exists as soon as the first variation g' and

it 's variation (gl)' exis t, and that gil may be calculated based on o' and on (g' )'. Now 9 is ;U1Y func t ion on V, la ter it will be chosen to be the cost funct ion J, or to be the rest riction

Y w '

Let 9 be a fun ction on V with values in a Banach space E . A""um e that [or any A E V, there exists g'(A;.) E £ (Lipk(RN ; R N ) ; E) saii,9[ying, (or

any u E Li pk(RN i RN) such th at A + u E V ,

g(A + u) = g(A) +g' (A;u) + o(lIullk) .

Assume that there exis ts (g' )' (n ; ·, ·) E £ «Lipk+ l(RN ; R N ))2:E) s a. iisiying, {or any u lind w in Lipk+1(R"'; R N) such that n + u E V,

g'(&1 +u; w) = g'(n ;w ) + (g')'(&1 ;u, w) + IIwllk+l o(llu llk+l )'

Define g"(n ; ·,· ) E £ «Lipk+l(R N ; R N))2; E) by

" (,"" ) _ ( rv rr». ) '('" '1"7)9 H , U , W - g ) <, H , U , W - 9 H,U' vW .

g(D + u) = g(&1) + g'(D; u ) + ~9 1l(&1i u , ti) +o« lIullk+2? )'

We deno te by o(Iu II k) any element of E, depe nding on u, &1, ... , su ch that o(lIullkl/ ll ulik -+ 0 us lIuli k -+ O.

3. 4. Second order ex pa ns ion in Banach spaces . We will give now, for a function G defined on a Ban ach space X, condi tions for the

second var iat ion Gil to exist , th at is for Taylor 's expansion to hold. Here Gil = (G' )' . T he theorem 3.3 will be proved by using this resu lt for G(v) = g« (I + v).

Le mma 3.4. Let X be a Banach space, z E X , a > 0 and Y = {y E X: lIy :C:: x < a}. Let G be a fun ction on Y with values in a Banach space E.

Assum e tha t for any y E Y there exis ts a function G'(y;·) E LeX; E ) sa tisfying, for any u such that y +u E Y,

G(y +u) = G(y) +G'(y ;u) +o( lIullx ).

Assum e that there exists a fun ction (G')' (x ; " .) E qX2; E ) such that , for any w E X

an d any u such that x + u E Y,

G'(x + u ; w) = G'(x;w) + (G')' (x; u, w) + Ilwllx o(lI ullx ).

372 S econd variat ions for dom ain optim ization pro blem s

Then [or any u such t l l at x + tt E Y,

G(x +u) = G(x ) + G'(x; u ) + ~ ( G')'( x ; u , u) + O«IIUli x )2).

P TO of: Let u be su ch tha t x + u E Y , and let 0 ::; t ::; 1. By th e firs t assumption on G, G(x + tu ) is d ifferentiable with respect to t in [0, 1] and it 's derivative is G'(x + tu ;u). T hus

1

G(x +u) - G(x) - G'(x ;u) =1(G'( x + ttt;tt) - G'(x ;u» dt .

Since (G')'( x ; ·,u ) is linear it follows that

1

G(x + u )- G(x)- G'(x; u)- ~ ( G' )'( x ; tt , u) =1(G' (x +tu ;u) - G'(x ;u )- (G')'( x; tu, u»)dt.

Now, using the second assum ption on G to bound th e norm of the right hand side, we get

II G(x + tt) - G(x) - G'(x; u) - ~ (G')' ( x; u , u)IIE ::; lIullx o( lIullx )

which pro ves th e lemma.

3. 5. Estimations for composed maps . The proof of theorem 3.3 rely on the following est imates .

Le rn rna 3 .5. Let 11 an d w in L i pk+ 2( R N ; R N ) , k ~ I , be such that lIull k+ l ::; ak+ !. TheIl

Proof: Remind tha t , if w E W k+2,P( R N ; R N ) , th en

This is proved in [3], lemma 4.4 part i i page IV-g. But it is false for p = 00 , by remark 4.3.

We assume for a moment tha t w has a com pact support. T hen w 0 (I +U )- 1 - W +u . \lw

has a compact support independent of U , thus using this equality for p ~ N an d Sobolev's theorem , we get

Now we consider s an d t such tha t 0 ::; t + s ::; 1, and we use this equali ty wit h w 0 (I + tu) - l and su 0 (I + tU)- 1 instead of w an d u. Since w a (I tU) -1 a (I + su a (I + tU)- 1)- 1 = w a (I + tu + su )- l , we obtain

Ilwa (I + tu + su ) - l - w 0 (I + t U) - 1 +5U 0 (I + tU ) - l . \l(w 0 (I + tu )- Ili k

=0( 1I 5u 0 (I + tu) - 111 k+ 2)'

- - - - - --- _ ._ ...__.....- ._ - ­

Simon 373

T hus the m ap t -+ w o(I + tU) - l is different iable from [O,lJ into L ipk(R N;RN ), and it 's derivative at t is u 0 (I + tU) - 1 . \l(w 0 (I + tU) -I) . Then, using for th is map the integral iden tity st ill used for G in the proof of lemma 3.4, we get

Therefore

IIw 0 ( I + U)-l - W + u · \lw llk ::; Suptllu 0 (I + t u )- l . \l (w 0 (I + tU)- I ) - 11' \lllllk

Using IIvwll k ::; ckllvllkllwllk an d lemma 3.1, we bou nd

IIw0 (I +u )- l - W + u · \lwll.r, ::; Suptllu 0 (I + tU)- 1 . \l(w 0 (J + tU)- 1 - w) + (u 0 (I + tU)-1 - u)· V wllk

::; Sup 'Ck llu 0 (I + tu)- l ilkilw 0 (I + tU) -1 - Wllk+l + lIu0 (I + tU )- 1 - u)lIkIlWIlk+J

s Ck llwllk+2(II U Ilk+d ·

Now th is desi red inequali ty is p roved if w has a compact support . If the suppo r t of w i,q no t com pact , we use a funct ion ip E V (RN ) such th at t.p ( x) = 1 if [z] ::; 2, an d we define, for y E R N, <f' y(x) = t.p (x - y). Since <f' yW has a compact support , it satisfies

T he left hand sid e is the norm in W k,OO(R N; R N), which is grea ter th an the norm in W k,OO(B (Yi 1); R N ), where B(y , 1) is the bal l of radius 1 cente red on y. In this ball t.py 0 (I + U ) - I == t.py == 1. Indeed, if Ix - yl ::; I , then I(I + u) - l (x) - yl = 1- u 0 (1 + u)- I (X) + X - yl ::; ak + 1 ::; 2. Therefore

Ilw 0 (l + U) - l - W + 11 ' \lw IlW" <X> ( B(y;l) ;RN) ::; ckllt.pyWIlk+2(IIUIIk+d2

::; Ckll<f'yllk+2I1 1O Ilk+2(llullk+I )2

Taking the supremum for y in R N we obtain the desired inequalites, since the no rm of t.py do no t depend on y . Now the result is proved for all w .

3.6. Proof of theorem 3.3. We have to obtain the second ord er Taylor' s expansion for G( u ) = g(Q +u) with respect

to u , for u E Lipk+ 2 . By lemma 3.4 it is enough to obtain the firs t order expansion of G(y +u) for an y y in a neighborhood of 0, and th e first order expan sion of G'(Ui 10 ).

Expansion of G(y + u ). By lemm a 3.2 we ha ve, if Ily +ullk ::; ak,

G(y + u) = g(Q + (y +u) = g« Q + y) + U 0 (I + y)-l ).

The first ass umpt ion on 9 yields

37 1 Second variations for domain optim!zo.tion prob'cm«

where R1 = o( lI u 0 (1 + y)- I II.,, ) thus, by lemma 3.1, R1 = o(llulik j. T herefore th is is th e expansion of G(y +u) , an d

G'(y; 1.1) = g'(.\1 + y; U 0 (I + y) - I ).

Expansion of G'(u; w). T he second assum ption on 9 yields

(1) g' (n + u;W0 (I +U)- I) = g' (.\1 ;W0 (I +U)-I) + (g')'( .\1 ;U,W0 (I +U)-I ) + R2

where R2 = IIw 0 (I + u) -II1k+ l o(lIullk+d . Thus by lemma 3.1

R2 = IIwllk+Jo(lIullk+I ) '

We linear ize now the terms in th e righ t hand side of (1). Since g'(.\1 ,') is linear t he first te r m is equal to

g'( .\1;w 0 (I + ll)- l ) = g'(.\1;w) - g'(n ;u · \7w ) + R3

where R3 = g'( .\1 ;W 0 (I + u) - J - W + u, \7w). By lemma 3.5

IIR3 11 E ~ 1119'(.\1 ;·)lll lIw0 (I -I- U)-I - W + u . \7w ll k

~ Ck lllg' (n; ·) lll lI w llk+2(lIullk+d2•

Since (g')'(n ;u,· ) is linear, the second te rm in th e right hand side of (1) is equal to

(g')' (.\1 ;u,w 0 (I + U)- I) = (g')'( .\1;u;w) + R4

where R j = (g')'(.\1 ;U j W 0 (I + U ) - 1 - w). By lemma 3.1

II~ IIE ~ III(g' )' ( .\1; ·, ·) lI l l1uIlHdlw 0 (I + u) - J - Wllk+l

::; ck+J.2 III(g')'(.\1; ·,·)lll llw llk+2(lIullk+d ·

Finally (1) yields

g'(.\1 -I- U ; W 0 (I +U)-I) = g'( .\1;w) + (g' )'(.\1; u , w) - g'(.\1 ; ll ' \7w) + II w ll k+ 2o( II UIl k+ 2)'

This is G'(u ;w) =G'(O; w) + (C')'(O; u, w) :iwllk+20(lluIIH2)

with (G')'(O;u,w ) = (9')'(.\1;u;w) - g'(.\1; u · \7w).

Conclusion . We proved tha t th e assum ptions of lemma 3.4 are satisfied for G(ll) g(.\1 + u), x = 0 and X = LipH 2(R N; R N). Therefore the second order Taylor' s expansion holds for G, an d yields

Simon

4. Second variation of t he dr-ag of a body

4.1. Drag of a body. We are in terested in a motion less body 8 in a viscou s incompress ible fluid moving at <:

uni form veloci ty h on the boundary of the experiment region 1\. T he domain occupied by the fluid is n = 1\\8, an d we assume B CC 1\.

T he velocity Y = (YI, Y2, Y3 ) and the p ressure p sa t isfy

-p.6 y + y . "Vy = - "Vp In it

'\1. y = 0 in it

y = 0 on 08 , y = h on all. .

The energy dissipated by the flu id is

J = ~ ( ILvl2dx 2 in

where (LY);j = OiYj + OjYi , Oi = O/ OXi, an d the drag is J/lhl. Lemma 4 .1. t . Th ere exis ts R(O ) > 0 such that , if Ih l/p. < R(n), tbere exists a uniqu e solu tion

Y E (H 1 (D))3, '\1p E (H - I(n ))3.

If on is locally the graph of a L ip": function, and if Ihl/p. < R(n), then

Proof: T he existence of v is given by J .L. Lions [5) th eorem 7.3 p. 102, the existence of p an d the regulari ty of u an d p follow from the theorems 1 an d 2' p. 28 an d 74 of a .A. Lad yzhenskaya [6), an d the existence of R(n ) is given by F. Mur at & J . Simon [4), theorem 2.1 p.2.3.

Rernark . Without ;ila it on h , th e uniqueness of y and th erefo re of J is not necessary «nown.

4. 2 . Second order ex pansio n of the d r ag . Now we will give the expansion of J (B + u ) with respect to a Lip4 vari a t ion It of B .

T heo r e m 4 .2 . Assume that

an is locally the graph of a Lip4 fun ct ion,

Ih llit < R(n) .

J(B + It) = J(13 ) + J'(8; u) + ~J" (B ; u,u) + o((lI u Il 4)2)

375 Second variations for dom ain optim ization problems

where

l '(B ;u) = f Ly · L y' dx = ~ r u n lL y l2ds Jn 2 Jan y' E (H3(D» J being the unique solution, su ch tha.t pi E HI(n), of

_pf::" y' + y' . \7y + y . \7y' = - f::"p' in n \7. y' = 0 in D

y' = - u n oy j 8n on an and

1"(B ;tz, u ) =

r L y. Lyli + ILy'12dx + r 2u nL y · Ly' + ~ U n tL . 'V(IL yI2) + ~(u\7 . u - u · V'u) nILyI2ds In Jan 2 2

y" E ( H 2(D. » 3 being the unique soluti on, such tlJat p" E L 2(Sl), of

_ pf::"y" + y" . 'Vy + s : V'y" + 2y' . \7yl = - \7p" In D

\7. y" = 0 in D.

y" = - 2un o(y' + u · \7y)jon +2(u· 'Vu)n8yj8n ' ;- il U ' -y2y or: oD.

Remark By lemma 4.1,

4 .3. Main lines of the proof. Fir st variation . We assume for a moment that there exis ts a linear rnr-p y·(B;·) such that , for u in Li p2(R 3 ; R 3 ) ,

(H 2(D)?(1) y(B + 11 ) 0 (I + u) = y(B ) + y'(B ;11 ) + o( lIu Il2) in

Then by the lemma 2.1 p . 657 of J. Simon [2] y(B +u) has a firs t order local expansion in n. By the theorems 3.1 and 3.2 p . 663 an d 664 of [2J , t he local vari at ion y' sat isfies the desired boundary value problem . T he equation invo lving pi is obtained in t he firs t st ep in a variationa l form as in [2], proof p . 684, an d in a second step pi is obtai ned by the theorem 1 p . 28 of [6J.

Moreover by theorem 3.3 p. 664 of [2], J (B + u) has a first order expansion and J has the desi red value.

Second variation . Assume now tha t the re exists a biline ar map y'·(B; ·, ·) such tha t , for u an d w in Li p3(R J ; R 3 ) ,

(2) y'(B + u; w) 0 (I + u) = y'(B; w) + y" (B;u, w) + o(lIuIlJ) in (H 2 (D))3.

T hen by our theorem 3.3 ap plied to the restr ict ion Yl oo for w cc D, y(B + u) has a second order loca l expansion in n with res pect to u in Lip4 (RJ ; R 3). At tha t po int the

377

rough calculat ions in section 2.3 of the boundary value problem sa tis fied by u" are justified by using twice the theorems 3.1 and 3.2 of [2J and our theorem 3.3 (the equat ion involving p" is obt ained in two steps, as we did for the equa tion involving pi in the firs t var ia tion).

Moreover by th eorem 3.3 p . 664 of [2J, l' (B + Uj w) has a first expansion an d t herefore , by our theorem 3.3, l(B +u ) has a second order expansion with respect t o u in Lip4 (R 3 ; R 3 ) .

Now the rough calculations in sect ion 2.3 of the III are justified by using twice of theorems 3.3 of [2J an d our theor em 3.3. Thus F' has the desired value.

Proof of (I): T he boundar y value problem sa tisfied by y(B + u) in D + u yields, by the map 1 + u , the boundary value problem satisfied by y(B + u) 0 (I + tt) in D. T his new boun dary value probl em has 1.1 depending coefficien ts, and may be writ ten in the form ,

F (u ; y(B + u ) 0 (I + u )) = 0

where F maps L ip2(R 3j R 3) X (H 2(D» 3 into (L2(D)? The assumption (1), which is th e differentiability of 1.1 ....... y(B + u) 0 (I + u) at 0, is

obtained by the theorem of differen tiation of the solution of an implicit equation . Indeed F (O ;·) is invertible (this is the uni queness of y). and F(D;· ) an d F(·;y(B» are

differ ent iable (this is easy in wri ting F by using (OJf) 0 (I +1.1) = Ej aji(u )OJ(f 0 (I +u» , where the matrix {aij (u)} is the inverse of {Dij + Oj U j } ).

T he det ails of a similar calculus for another optimal design problem may be found in [2], proof of theorem 6.1 pages 681-682.

Proof of (2): The outline is the same. Now y'(B + tt;w ) satis fies a boundary value problem in D +u with coefficients depending on y(B + u). By the map 1 + u , it yield s the boun dary valu e problem sa tisfied by y/(B + u;w ) 0 (I + u) in rl. T he coefficients depe nd on u and on y(B -- u) 0 (I + u) , thus it may be writ ten in th e form

H(u;y(B + u ) 0 (I + u); yi(B +u ; w) 0 (I + u) = ° where H maps Lip3(R 3;R 3

) x (H2(D» 2 x (H2(D)? into (L 2 (rlW . T he assumption (2), which is the different iability of u ....... y'(B + u ; w) 0 (I + tt) a t 0, is

obtained by the th eorem of differen tiat ion of the solution of an implicit equ a tion. Indeed H (O, y(B);·) is invert ible (this is the un iqueness of yl), an d H (O , z;· ) an d

Ht- , y(B + .) 0 (I + -): y'(B» are different iable (the last one followin g from t he differ­entiabi lity of H( ·, z; y'( B)) and H( lL , ·; y'(B» an d from (1)).

REfE1U:.\'C£S

(1 ) Fujii , N ., Second vari ation and i t 's app licati on m dom am optimIzatiOn probl em, in "Con trol of d istribu ted parameter systems," P roceedings of t he 4t h IFAC Sy m posium , Per ga mon P ress , 1987 .

[2) S imo n , J ., Differen tiation with respect to the dom ain in boundary valu e problem3, Num er ica l Func­t ional Analys is and Optim ization 2 (1980) , 649- 687.

[3] Mu rat , F . a.nd S imon , J ., Sur Ie con trOle par un dom aine geometrique, Research rep ort of th e Labora toire d 'Analyse Num erique , University of P ari s 6 (1976). 1- 222.

[4J Murat , F . and Simon , J _. Que/que re3ultai.9 sur le con trole par un domaine geometri que , Research repor t of th e Laboratoire d 'Analyse Num erique , University of Paris 6 ( 1974). 1-46 .

[5J Lions , J .L ., "Q uelques methodes de ..,sol ution d es p rob lernes aux lim ites non lineaires ," DunocJ, P ar is. 1969.

378 Second vari ations fOT dom ain optim izat ion problems

(6J Lad yzhe nskaya, a .A., "T he mat hem at ical t heory of viscous incompressible flows ," Gordon & Breach, 1963.

J acques S imon Departement de Ma t hemat iques Appliquees Univers ite Blaise Pascal (Clermont-Ferrand 2) F-631 77 Au biere Cedex France