Isoperimetric inequalities and cavity interactions Duvan Henao and Sylvia Serfaty Laboratoire Jacques-Louis Lions Universit´ e Pierre et Marie Curie - Paris 6, CNRS May 17, 2011 Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
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Isoperimetric inequalities and cavity interactions
Duvan Henao and Sylvia Serfaty
Laboratoire Jacques-Louis LionsUniversite Pierre et Marie Curie - Paris 6, CNRS
May 17, 2011
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Motivation
[Gent & Lindley ’59]
Internal rupture of rubberunder hydrostatic tension
Gent & Lindley ’59Oberth & Bruenner ’65
Gent & Park ’84Dorfmann ’03
Bayraktar et al. ’08Cristiano et al. ’10
[Lazzeri & Bucknall ’95
Dijkstra & Gaymans ’93]
Rubber toughening of plastics(polystyrene, ABS, PMMA)
Lazzeri & Bucknall ’95Cheng et al. ’95
Steenbrink & Van der Giessen ’99Liang & Li ’00
[Petrinic et al. ’06]
Ductile fracture by voidgrowth and coalescence
Goods & Brown ’79
Tvergaard ’90
Petrinic et al. ’06
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
ΩW (Du) dx among W 1,p deformations; conditions of invertibility,
orientation preservation, incompressibility, loading at the boundary
I number of cavities, shapes, sizes, location of singularities; interactionbetween cavities; dependence on loading conditions, domain geometry,material parameters; void coalescence, alignment of cavities, crack formation
[Petrinic et al. ’06] [Xu & H. ’11][Lian & Li, preprint]
I lack of lower semicontinuity and quasiconvexity; detDu not weaklycontinuous; weak limit does not preserve incompressibility and invertibility
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Connection with Ginzburg-Landau superconductivity
Cavitation
I u(r , θ) =√A2 + r2e iθ
I minu∈W 1,p
∫Ω
|Du|p; detDu ≡ 1
I |Du|p ∼x=0
Ap
rp
Ginzburg-Landau
I u(r , θ) = e idθ
I minu∈H1
∫Ω
|Du|2 +1
ε2(1− |u|2)2
I |Du|2 ∼x=0
d2
r2
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Upper bound
Theorem: a1, a2 ∈ R2, v1 ≥ v2. For all δ ∈ [0, 1] there exists a∗ ∈ [a1, a2]and a piecewise smooth homeomorphism u : R2 \ a1, a2 → R2 such thatDetDu = 1 · L2 + v1δa1 + v2δa2 and for all R > 0∫
B(a∗,R)\(Bε(a1)∪Bε(a2))
|Du|2
2≤ C(v1 + v2 + πR2) + (v1 + v2) log
R
ε
+ C(v1 + v2)
((1− δ)
(log
R
d
)+
+ δ 4
√v2
v1 + v2log
d
ε
)
Terms in lower bound: C(v1 + v2)
(minv2
1 ,v22
(v1+v2)2 − πd2
v1+v2
)log(
min
4√
v1+v24πd2 ,
Rd, dε
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Upper bound
Theorem: a1, a2 ∈ R2, v1 ≥ v2. For all δ ∈ [0, 1] there exists a∗ ∈ [a1, a2]and a piecewise smooth homeomorphism u : R2 \ a1, a2 → R2 such thatDetDu = 1 · L2 + v1δa1 + v2δa2 and for all R > 0∫
B(a∗,R)\(Bε(a1)∪Bε(a2))
|Du|2
2≤ C(v1 + v2 + πR2) + (v1 + v2) log
R
ε
+ C(v1 + v2)
((1− δ)
(log
R
d
)+
+ δ 4
√v2
v1 + v2log
d
ε
)
Terms in lower bound: C(v1 + v2)
(minv2
1 ,v22
(v1+v2)2 − πd2
v1+v2
)log(
min
4√
v1+v24πd2 ,
Rd, dε
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Geometric construction
2d
2d1 2d2
2dδ
2d
a2a1
Ω1 Ω2
d1 d1 d2 d2
Ratio|Ω1||Ω2|
=v1
v2; u(x) ≡ λx on ∂Ω1 ∪ ∂Ω2,
λ2 − 1 :=v1 + v2
|Ω1 ∪ Ω2|=
v1
|Ω1|=
v2
|Ω2|
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Geometric construction
2d
2d1 2d2
2dδ
2d
a2a1
Ω1 Ω2
d1 d1 d2 d2
Ratio|Ω1||Ω2|
=v1
v2
; u(x) ≡ λx on ∂Ω1 ∪ ∂Ω2,
λ2 − 1 :=v1 + v2
|Ω1 ∪ Ω2|=
v1
|Ω1|=
v2
|Ω2|
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Geometric construction
2d
2d1 2d2
2dδ
2d
a2a1
Ω1 Ω2
d1 d1 d2 d2
Ratio|Ω1||Ω2|
=v1
v2; u(x) ≡ λx on ∂Ω1 ∪ ∂Ω2,
λ2 − 1 :=v1 + v2
|Ω1 ∪ Ω2|=
v1
|Ω1|=
v2
|Ω2|
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Cavity shapes
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-3 -2 -1 0 1
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
δ = 0.1 δ = 0.4 δ = 0.9
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Angle-preserving maps
a2a1
Ω1 Ω2
d1 d1 d2 d2
Ω2Ω1
a∗
u(x) = λa∗ + f (x)x− a∗
|x− a∗|, λn − 1 :=
v1 + v2
|Ω1 ∪ Ω2|=
v1
|Ω1|=
v2
|Ω2|.
detDu(x) =f n−1(x)∂f∂r (x)
rn−1≡ 1 ⇔ f n(x) = |x− a∗|n + A
(x− a∗
|x− a∗|
)n
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Dirichlet conditions
Reference configuration Deformed configuration
Necessary condition: π(R22 − R2
1 ) = 2−3π18 (1− δ)(v1 + v2).
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Dirichlet conditions
Reference configuration Deformed configuration
Necessary condition: π(R22 − R2
1 ) = 2−3π18 (1− δ)(v1 + v2).
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Dirichlet conditions
Theorem: Suppose πR21 , π(R2
2 − R21 ) ≥ C (v1 + v2)(1− δ); R1 ≥ 2d . Then∫
B(a∗,R)\(Bε(a1)∪Bε(a2))
|Du|2
2≤ C(v1 + v2 + πR2) + (v1 + v2) log
R
ε
+ C(v1 + v2)
((1− δ)
(log
(v1 + v2)(1− δ)
πd2
)+
+ δ 4
√v2
v1 + v2log
d
ε
),
with u|∂B(a∗,R2) radially symmetric.
Previous upper bound: C(v1 + v2)
((1− δ)
(log πR2
πd2
)+
+ δ 4
√v2
v1+v2log d
ε
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Dirichlet conditions
Theorem: Suppose πR21 , π(R2
2 − R21 ) ≥ C (v1 + v2)(1− δ); R1 ≥ 2d . Then∫
B(a∗,R)\(Bε(a1)∪Bε(a2))
|Du|2
2≤ C(v1 + v2 + πR2) + (v1 + v2) log
R
ε
+ C(v1 + v2)
((1− δ)
(log
(v1 + v2)(1− δ)
πd2
)+
+ δ 4
√v2
v1 + v2log
d
ε
),
with u|∂B(a∗,R2) radially symmetric.
Previous upper bound: C(v1 + v2)
((1− δ)
(log πR2
πd2
)+
+ δ 4
√v2
v1+v2log d
ε
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Dirichlet conditions
Corollary: Ω = BR , R ≥ 2d . For every v1 ≥ v2 there exist a1, a2 ∈ Ω with|a2 − a1| = d and a Lipschitz homeomorphism u : Ω \ a1, a2 → R2 suchthat DetDu = L2 + v1δa1 + v2δa2 , u|∂Ω ≡ λid, and∫
B(a∗,R)\(Bε(a1)∪Bε(a2))
|Du|2
2≤ C(v1 + v2 + πR2) + (v1 + v2) log
R
ε
+ C(v1 + v2) minδ∈[δ0,1]
((1− δ)
(log
(v1 + v2)(1− δ)
πd2
)+
+ δ 4
√v2
v1 + v2log
d
ε
),
with δ0 := max0, 1− |Ω|−4πd2
Cπd2 .
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)).
Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω.
Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,
DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε,
sup ‖uε‖L∞ <∞ and∫Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞
and∫Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0.
If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2
then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular;
if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2,
then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2
then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε)
and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Compactness (ε→ 0)
Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫
Ωε
|Du|2
2≤ (v1,ε + v2,ε) log
diam Ω
ε+ C(|Ω|+ v1,ε + v2,ε).
Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1
loc(Ω) and convergent subsequences.
i) if minv1, v2 = 0, the only cavity opened is circular
ii) Suppose v1 ≥ v2 > 0. If a1 6= a2 then the cavities are circular; if moreoverv1 + v2 < 4π dist(a, ∂Ω)2, then they are well separated:
π|a2 − a1|2 ≥ C(v1 + v2) exp
−4(1 + |Ω|v1+v2
)(C + log π(diam Ω)2
v1+v2)
Cv2
2(v1+v2)2
iii) if a1 = a2 then |a2,ε − a1,ε| = O(ε) and the cavities are distorted:
lim infε→0
v1D(E(a1,ε, ε))2 + v2D(E(a2,ε, ε))2
v1 + v2> C
v 22
(v1 + v2)2.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Summarizing
I Connection between cavitation and Ginzburg-Landau theory
I Role of isoperimetric inequalities in elasticity (c.f. Muller ’90)
I Relation between quantities in the reference and deformedconfiguration (c.f. Ball & Murat ’84; surface energy)
I Repulsion effect, role of incompressibility
I Void coalescence
I Explicit test maps (angle-preserving)
I Dirichlet conditions (Dacorogna-Moser flow; Riviere-Ye)
I Compactness (Struwe ’94)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions