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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
Mathematical analysis (Gent & Lindley ’59, Ball ’82, Stuart ’85, Horgan &
Abeyaratne ’86, Sivaloganathan ’86, Ertan ’88, Meynard ’92, Horgan ’92, . . . )
Stored-energy density W : Mn×n1 → R
Incompressible elastic ball
T(x) = DW(Du)DuT − p(x)1
TR(x) = DW(Du)− p(x) cof Du(x),
detDu(x) = 1 for x ∈ B(0, 1) \ 0
Radial symmetry: u(x) = u(|x|)x/|x|
Elastostatics, traction-free cavity surface
Div TR(x) = 0, x ∈ B(0, 1) \ 0
TR(x)ν(x) = Pν(x), x ∈ ∂B(0, 1)
T(εθ)θε→0−→ 0, |θ| = 1, θ ∈ Rn
[Ball ’82]
extensions allowing for compressibility, Blatz-Ko and Varga
materials, anisotropic loading, material anisotropy,
elastodynamics, plasticity, elastic membrane theory,
material inhomogeneity, . . .
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Variational approach to cavitation (Ball ’82, Ball & Murat ’84, Muller &
Spector ’95, Sivaloganathan & Spector ’00, . . . )
I minimize∫
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
Connection with Ginzburg-Landau
Distributional determinant: DetDu = 1n
Div((adjDu)u
), appearing in nonlinear
elasticity, geometric measure theory, liquid crystals, superconductivity, . . .
Cavitation
Deformation u : Ω→ Rn, detDu = 1
DetDu = 1 · Ln +M∑i=1
viδai , vi > 0
(Ball ’77, Muller & Spector ’95,
Sivaloganathan & Spector ’00)
Ginzburg-Landau
Order parameter u : Ω ⊂ R2 → S1
(when κ→∞), detDu = 0,
DetDu =M∑i=1
diδai , di ∈ Z
(Bethuel, Brezis & Helein ’94,
Sandier & Serfaty ’07)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Basic estimate
Ginzburg-Landau:
u(x) = f (x)e iϕ(x), |Du|2 = |Df |2 + f |Dϕ|2. For f ≈ 1, |Du| ≈ |Dϕ|2,∫Ω
|Du|2 ≥∫ R
ε
∫∂B(a,r)
|∂τϕ|2 ds dr ≥∫ R
ε
(∫∂B(a,r)
∂τϕ
)2dr
2πr= 2πd2 log
R
ε
Hence ∫Ω
|Du|2
2dx ≥
M∑i=1
πd2i log
Ri
εi
Ability to predict number of vortices, their vorticities and locations; energy
estimates; repulsion and confinement effects
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Radial symmetry, isoperimetric inequality
Incompressible neo-Hookean material; Ωε := Ω \⋃M
i=1 Bε(ai ) ⊂ R2:∫∂B(a,r)
|Du|2
2ds ≥
∫∂B(a,r)
|∂τu|2
2ds
≥ 1
4πr
(∫∂B(a,r)
|∂τu| ds
)2
=
(Per E(a, r)
)2
4πr
Isoperimetric inequality: (PerE)2 ≥ 4π|E |, implies∫ε<|x|<R
|Du|2 − 1
2dx ≥ (cavity volume) · log
R
ε.
Equality is attained only for radially symmetric deformations
⇒ round cavities (Sivaloganathan & Spector 2010)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Radial symmetry, isoperimetric inequality
Incompressible neo-Hookean material; Ωε := Ω \⋃M
i=1 Bε(ai ) ⊂ R2:∫∂B(a,r)
|Du|2
2ds ≥
∫∂B(a,r)
|∂τu|2
2ds
≥ 1
4πr
(∫∂B(a,r)
|∂τu| ds
)2
=
(Per E(a, r)
)2
4πr
Isoperimetric inequality: (PerE)2 ≥ 4π|E |, implies∫ε<|x|<R
|Du|2 − 1
2dx ≥ (cavity volume) · log
R
ε.
Equality is attained only for radially symmetric deformations
⇒ round cavities (Sivaloganathan & Spector 2010)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Radial symmetry, isoperimetric inequality
Incompressible neo-Hookean material; Ωε := Ω \⋃M
i=1 Bε(ai ) ⊂ R2:∫∂B(a,r)
|Du|2
2ds ≥
∫∂B(a,r)
|∂τu|2
2ds
≥ 1
4πr
(∫∂B(a,r)
|∂τu| ds
)2
=
(Per E(a, r)
)2
4πr
Isoperimetric inequality: (PerE)2 ≥ 4π|E |, implies∫ε<|x|<R
|Du|2 − 1
2dx ≥ (cavity volume) · log
R
ε.
Equality is attained only for radially symmetric deformations
⇒ round cavities
(Sivaloganathan & Spector 2010)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Radial symmetry, isoperimetric inequality
Incompressible neo-Hookean material; Ωε := Ω \⋃M
i=1 Bε(ai ) ⊂ R2:∫∂B(a,r)
|Du|2
2ds ≥
∫∂B(a,r)
|∂τu|2
2ds
≥ 1
4πr
(∫∂B(a,r)
|∂τu| ds
)2
=
(Per E(a, r)
)2
4πr
Isoperimetric inequality: (PerE)2 ≥ 4π|E |, implies∫ε<|x|<R
|Du|2 − 1
2dx ≥ (cavity volume) · log
R
ε.
Equality is attained only for radially symmetric deformations
⇒ round cavities (Sivaloganathan & Spector 2010)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Lower bound
I Ωε := Ω \⋃M
i=1 Bεi (ai )
I u ∈ H1(Ωε,R2), incompressible, DetDu = L2 +M∑i=1
(vi + πε2i )δai
If B(ai ,R) ⊂ Ω then∫Ωε
|Du|2 − 1
2dx ≥ (v1 + · · ·+ vM) log
R
2(ε1 + · · ·+ εM)
Ball construction (Jerrard, Sandier)
Does not consider individual cavity sizes or distances between cavities;
insufficient to determine optimal locations, or study cavity interactions.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Lower bound
I Ωε := Ω \⋃M
i=1 Bεi (ai )
I u ∈ H1(Ωε,R2), incompressible, DetDu = L2 +M∑i=1
(vi + πε2i )δai
If B(ai ,R) ⊂ Ω then∫Ωε
|Du|2 − 1
2dx ≥ (v1 + · · ·+ vM) log
R
2(ε1 + · · ·+ εM)
Ball construction (Jerrard, Sandier)
Does not consider individual cavity sizes or distances between cavities;
insufficient to determine optimal locations, or study cavity interactions.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Lower bound
I Ωε := Ω \⋃M
i=1 Bεi (ai )
I u ∈ H1(Ωε,R2), incompressible, DetDu = L2 +M∑i=1
(vi + πε2i )δai
If B(ai ,R) ⊂ Ω then∫Ωε
|Du|2 − 1
2dx ≥ (v1 + · · ·+ vM) log
R
2(ε1 + · · ·+ εM)
Ball construction (Jerrard, Sandier)
Does not consider individual cavity sizes or distances between cavities;
insufficient to determine optimal locations, or study cavity interactions.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Lower bound
I Ωε := Ω \⋃M
i=1 Bεi (ai )
I u ∈ H1(Ωε,R2), incompressible, DetDu = L2 +M∑i=1
(vi + πε2i )δai
If B(ai ,R) ⊂ Ω then∫Ωε
|Du|2 − 1
2dx ≥ (v1 + · · ·+ vM) log
R
2(ε1 + · · ·+ εM)
Ball construction (Jerrard, Sandier)
Does not consider individual cavity sizes or distances between cavities;
insufficient to determine optimal locations, or study cavity interactions.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Quantitiative isoperimetric inequality
(Per E )2 ≥ 4π|E |
(1 + CD(E )2
)where
D(E) := min
|E4B||E | : B ball, |B| = |E |
, (Fraenkel asymmetry of E)
which depends on the shape of E only (not on its size).
Bernstein 1905; Bonnesen ’24; Fuglede ’89; Hall, Hayman & Weitsman ’91;
Hall ’92; Fusco, Maggi & Pratelli ’08
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Quantitiative isoperimetric inequality
(Per E )2 ≥ 4π|E |(1 + CD(E )2
)where
D(E) := min
|E4B||E | : B ball, |B| = |E |
, (Fraenkel asymmetry of E)
which depends on the shape of E only (not on its size).
Bernstein 1905; Bonnesen ’24; Fuglede ’89; Hall, Hayman & Weitsman ’91;
Hall ’92; Fusco, Maggi & Pratelli ’08
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Two cavities
∫Ωε∩BR
|Du|2 − 1
2dx ≥ (v1 + v2) log
R
2ε
+ C
∫ d/2
ε
(v1D(E(a1, r))2 + v2D(E(a2, r))2
) dr
r
+ C(v1 + v2)
∫ R
dD(E(a, r))2 dr
r
d
Ω
Energy is minimized if ‘circles go to circles’
This is not always possible:
π(R1 + R2)2 − (πR21 + πR2
2 )
= (√v1 +
√v2)2 − v1 − v2
= 2√v1v2
Possible only if πππd2 > 2√v1v2
2√v1v2
R1 =√
v1π √
v2π
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Two cavities
∫Ωε∩BR
|Du|2 − 1
2dx ≥ (v1 + v2) log
R
2ε
+ C
∫ d/2
ε
(v1D(E(a1, r))2 + v2D(E(a2, r))2
) dr
r
+ C(v1 + v2)
∫ R
dD(E(a, r))2 dr
r
d
Ω
Energy is minimized if ‘circles go to circles’
This is not always possible:
π(R1 + R2)2 − (πR21 + πR2
2 )
= (√v1 +
√v2)2 − v1 − v2
= 2√v1v2
Possible only if πππd2 > 2√v1v2
2√v1v2
R1 =√
v1π √
v2π
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Two cavities
∫Ωε∩BR
|Du|2 − 1
2dx ≥ (v1 + v2) log
R
2ε
+ C
∫ d/2
ε
(v1D(E(a1, r))2 + v2D(E(a2, r))2
) dr
r
+ C(v1 + v2)
∫ R
dD(E(a, r))2 dr
r
d
Ω
Energy is minimized if ‘circles go to circles’
This is not always possible:
π(R1 + R2)2 − (πR21 + πR2
2 )
= (√v1 +
√v2)2 − v1 − v2
= 2√v1v2
Possible only if πππd2 > 2√v1v2
2√v1v2
R1 =√
v1π √
v2π
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Circular cavities (πd2 ≥ 2√v1v2)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.0 -0.5 0.0 0.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-3 -2 -1 0 1
-3
-2
-1
0
1
2
3
-5 -4 -3 -2 -1 0 1 2 3
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Distorted cavities
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Xu & H. ’11 Lian & Li ’11 H. & Serfaty ’11
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Vanishing volume ratio (πd2 > 2√v1v2)
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-3 -2 -1 0 1 2
-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
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
v1 = 2.5v2 v1 = 10v2 v1 = 100v2
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Scale invariance in elasticity (Ball & Murat, 1984):
The condition πd2 > 2√v1v2 is to be compared with:
Qλ Q
λQ
Qλ
Q
Q
λ
λ
λ
Q
Q
Q
Q Q
Q
Related works: Ortiz & Reina (2010), Lopez-Pamies, Idiart & Nakamura (2011)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Estimate on the distortions
∫Ωε∩BR
|Du|2 − 1
2dx− (v1 + v2) log
R
2ε
≥ C
∫ d/2
ε
(|Ea1,r |D(Ea1,r )2 + |Ea2,r |D(Ea2,r )2
) dr
r+ C
∫ R
d|Ea,r |D(Ea,r )2 dr
r
≥ C minε<r<d/2
d<r′<R
(|Ea1,r |D(Ea1,r )2 + |Ea2,r |D(Ea2,r )2 + |Ea,r′ |D(Ea,r′ )
2)
min
log
d
2ε, log
R
d
Proposition: E1 ∪ E2 ⊂ E , E1 ∩ E2 = ∅, |E1| ≥ |E2|. Then
|E |D(E)2 + |E1|D(E1)2 + |E2|D(E2)2
|E |+ |E1 ∪ E2|≥ C
(|E2|
|E1|+ |E2|
)2(
1− |E \ (E1 ∪ E2)|2√|E1||E2|
)3
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interactions
Lower bound
Theorem: Ωε := Ω \ (Bε(a1) ∪ Bε(a2)), u ∈ H1(Ωε,R2) incompressible∫Ωε
|Du|2 − 1
2≥ (v1 + v2) log
R
2ε
+ C (v1 + v2)
(minv2
1 , v22
(v1 + v2)2− πd2
v1 + v2
)log
(min
4
√v1 + v2
4πd2,R
d,d
ε
)
provided B( a1+a2
2 ,R) ⊂ Ω
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