Random Homogenization of an Obstacle Problem

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Random homogenization of an obstacle problem

L.A. Caffarelli

, A. Mellet

aDepartment of Mathematics and ICES, University of Texas at Austin, Austin, TX 78712, USA bDepartment of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada Received 4 March 2007; received in revised form 9 September 2007; accepted 9 September 2007

Available online 26 October 2007

Abstract

We study the homogenization of an obstacle problem in a perforated domain, when the holes are periodically distributed and have random shape and size. The main assumption concerns the capacity of the holes which is assumed to be stationary ergodic.

Crown Copyright_©2007 Published by Elsevier Masson SAS. All rights reserved.

Keywords:Homogenization; Obstacle problem; Free boundary problem

1. Introduction

Let (Ω,F,P)be a given probability space. For every ω∈Ω and every ε >0, we consider a domainDε(ω) obtained by perforating holes from a bounded domainDof Rⁿ. We denote byTε(ω)the set of the holes (we then haveDε(ω)=D\Tε(ω)). The goal of this paper is to study the asymptotic behavior asε→0 of the solution of the following obstacle problem:

min

D

1

2|∇u|²dx−

D

f u dx;u∈H₀¹(D), u0 a.e. inT_ε(ω)

(1) for some given functionf (x)inL²(D).

This is a classical homogenization problem and the asymptotic behavior of the solutions strongly depends on the properties of the setT_ε(ω). This type of problems was first studied by L. Carbone and F. Colombini [2] in periodic settings and then in more general frameworks by E. De Giorgi, G. Dal Maso and P. Longo [9], G. Dal Maso and P. Longo [7] and G. Dal Maso [6]. Our main reference will be the papers of D. Cioranescu and F. Murat [4,5], in which the particular case of a periodic repartition of holes is studied. More precisely, they takeT_ε(ω)=T_ε(independent ofω) given by

T_ε=

k∈Zⁿ

B_a^ε(εk). (2)

✩ The first author is partially supported by NSF grants. The second author is partially supported by NSERC grants.

* Corresponding author.

E-mail address:[email protected] (A. Mellet).

0294-1449/$ – see front matter Crown Copyright©2007 Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2007.09.001

It is then proved that there exists a critical radius a^εεsuch that the limiting problem is no longer an obstacle problem, but a simple elliptic boundary value problem with a new term that takes into account the effect of the holes.

More precisely, in dimension n3, whena^ε=r₀εⁿ−ⁿ2 then there exists a constantα₀>0 such thatu=limε→0u^ε solves

min

D

1

2|∇u|²+1

2α0u²₋dx−

D

f u dx; u∈H₀¹(D)

,

whereu₋=max(0,−u).

Our goal is to generalize this result to the case where the holes are still located in small neighborhoods of lattice pointsεZⁿbut have random size and shape. More precisely, we assume that for anyεandω, we can write

T_ε(ω)=

k∈Zⁿ

S_ε(k, ω)

∩D,

where the holesS_ε(k, ω)satisfy, in particular, S_ε(k, ω)⊂B_ε/2(εk) for allk∈Zⁿ.

Further assumptions concerning the setsS_ε(k, ω)will be made in the next section (we can already point out the fact that we will not exclude the case whereS_ε(k, ω)= ∅for somek, thus allowing the fact that no holes may be present at some lattice points). The proof of the main theorem, which is detailed in Section 3, relies on the construction of an appropriate corrector. This construction is detailed in Sections 4 and 5, first in the case where the holes are balls of random radii in dimensionn2, then when no assumptions are made on the shape of the holes (in dimensionn3 only).

2. Assumptions and main result

Let us now make precise the assumptions on the holesSε(k, ω). The first assumption is mainly technical:

Assumption 1.There exists a (large) constantMsuch that for allk∈Zⁿand a.e.ω∈Ωwe have S_ε(k, ω)⊂B_Mεn/(n−2)(εk) ifn3,

Sε(k, ω)⊂B_exp(−Mε−2)(εk) ifn=2 forεsmall.

As mentioned in the introduction, the asymptotic behavior of theu^εstrongly depends on the size of the holes. The critical size for which interesting phenomena is observed corresponds to finite, nontrivial capacity of the setT_ε. More precisely, we assume:

Assumption 2.For allk∈Zⁿand a.e.ω∈Ω, there existsγ (k, ω)(independent ofε) such that cap

S_ε(k, ω) =εⁿγ (k, ω),

where cap(A)denote the capacity of a subsetAofRⁿ (as defined below). Moreover, we assume that there exists a constantγ >¯ 0:

γ (k, ω)γ¯ for allk∈Zⁿand a.e.ω∈Ω. (3)

We take the following definitions for the capacity of a subsetAofRⁿ: cap(A)=inf

Rⁿ

|∇h|²dx;h∈H¹(Rⁿ), h1 inA, lim

|x|→∞h(x)=0

,

in dimensionn3 and cap(A)=inf

B₁

|∇h|²dx;h∈H₀¹(B1), h1 inA

,

in dimensionn=2 and for setsA⊂B₁.

Finally, we need to make an assumption to ensure that some averaging process occurs asεgoes to zero:

Assumption 3.The processγ:Zⁿ×Ω→ [0,∞)is stationary ergodic: There exists a family of measure-preserving transformationsτk:Ω→Ω satisfying

γ (k+k, ω)=γ (k, τ_kω) for allk, k∈Zⁿandω∈Ω,

and such that ifA⊂Ω andτ_kA=Afor allk∈Zⁿ, thenP (A)=0 orP (A)=1 (the only invariant set of positive measure is the whole set).

Let us make a few remarks concerning those assumptions: First of all, we stress out the fact that the shape of the holesS_εis left unspecified and may change withε; Only the rescaled capacityγ (k, ω)is independent ofε. The first assumption, which implies that the diameters of the holes decrease faster thanε, guarantees that the capacities of neighboring sets separate at the limit (i.e. that cap(

S_ε)∼

cap(Sε)). And the choice of scaling for the capacity guarantee that cap(Tε)remains bounded as ε goes to zero (since #{Zⁿ∩ε⁻¹D}Cε⁻ⁿ). Whenn3, we have cap(Br)=c_nrⁿ⁻², and so Assumption 2 implies that ifS_ε(k, ω)is a ball, then its radius is of the formr(k, ω)εⁿ⁻ⁿ². In particular, we recover Cioranescu–Murat’s result in the periodic case. Finally, the hypothesis of stationarity is the most general extension of the notions of periodicity and almost periodicity for a function to have some self-averaging behavior.

Under those assumptions, we prove the following result:

Theorem 2.1. Assume thatn3 and thatT_ε(ω)=

k∈ZⁿS_ε(k, ω)satisfies Assumptions1,2and 3 listed above.

Then there existsα₀0such that whenεgoes to zero, the solutionu^ε(x, ω)of min

D

1

2|∇u|²dx−

D

f u dx; u∈H₀¹(D), u0a.e. inT_ε(ω)

converges weakly inH¹(D)and almost surelyω∈Ωto the solutionu(x)¯ of the following minimization problem:

min

D

1

2|∇u|²+1

2α₀u²₋−f u dx; u∈H₀¹(D)

,

whereu₋(x)=max(0,−u(x)).

Moreover, if there existsγ >0such that γ (k, ω)γ for allk∈Zⁿand a.e.ω∈Ω, thenα0>0.

The same result holds whenn=2in the case where the setsSε(k, ω)are balls.

The general result (with holes of random shape) holds also in dimensionn=2. However, because the fundamental solution of Laplace’s equation is different in that case, the proof is slightly different and more technical.

Note that the Euler–Lagrange equations for the minimization problems yield

⎧⎪

⎨

⎪⎩

−u^ε=f inDε, u^ε(x)0 forx∈T_ε, u^ε(x)=0 forx∈∂D\T_ε

(4)

and

−u¯−α₀u¯₋=f inD,

¯

u(x)=0 forx∈∂D.

As in Cioranescu–Murat [4,5], the proof of this result relies on the construction of an appropriate corrector. More precisely, the key is the following result:

Proposition 2.2.Assume thatn3and thatTε(ω)=

k∈ZⁿSε(k, ω)satisfies Assumptions1,2and3listed above.

Then, there exists a nonnegative real numberα0and a functionw^ε(x, ω)such that

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

w^ε(x, ω)=1 forx∈T_ε(ω), w^ε(x, ω)=0 forx∈∂D\T_ε(ω), w^εbounded inL^∞(D)

w^ε(·, ω)→0 H¹(D)-weak for almost allω∈Ω, and

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

For all sequencesv^ε(x)satisfying:

⎧⎨

⎩

v^ε(x)0 forx∈T_ε, v^εbounded inL^∞(D), v^ε→v inH¹(D)-weak

and for anyφ∈D(D)such thatφ0, we have:

εlim→0

D

φ∇w^ε· ∇v^εdx−α0

D

φv dx with equality ifv^ε(x)=0forx∈T_ε.

(5)

The same result holds whenn=2in the case where the setsS_ε(k, ω)are balls.

Condition (5) may seem rather complicated, but it is exactly the property that will be needed to prove Theorem 2.1.

This condition is satisfied in particular if the Laplacian ofw^ε is equal to α0 outside the holes as in the following lemma:

Lemma 2.3.Letw^ε(x, ω)be a function satisfying

⎧⎪

⎨

⎪⎩

w^ε=α₀ inD_ε(ω), w^ε(x, ω)=1 forx∈Tε(ω), w^ε(x, ω)=0 forx∈∂D\T_ε(ω)

(6)

for almost allω∈Ω, and

w^ε(·, ω)→0 H¹(D)-weak a.s.ω∈Ω.

Thenw^εsatisfies(5).

The same conclusion holds if we only have

D_ε

|w^ε−α₀|dx→0 a.s.ω∈Ω.

We will strongly rely on this lemma in the sequel. In particular, in the case where the holes are all balls of random radii, we will constructw^εby showing that for a criticalα0, the unique solution of (6) converges to 0 inH¹weak for almost everyω.

Proof of Lemma 2.3. Sinceϕ=0 on∂D, we have:

−

Dε

w^εϕv^εdx=

Dε

ϕ∇w^ε· ∇v^εdx+

Dε

∇ϕ· ∇w^εv^εdx−

∂T^ε∩D

w_ν^εϕv^εdσ (x).

Sincew^εgoes to zero inH¹(D)-weak andv^εconverges tovinL²(D)-strong, it is readily seen that the second term in the right-hand side goes to zero asε→0. Furthermore, it is readily seen thatw^ε1 andw^ε=1 onT_ε, and so w^ε_ν0 on∂T_ε. Sincev^ε0 onT_εandϕ0, we deduce that

εlim→0−

D_ε

w^εϕv^εdxlim

ε→0

D_ε

ϕ∇w^ε· ∇v^εdx.

Finally, we have

εlim→0

D_ε

w^εϕv^εdx=lim

ε→0

D_ε

α0ϕv^εdx+lim

ε→0

D_ε

|w^ε−α0|ϕv^εdx

lim

ε→0

D

α₀ϕv^εdx

using the fact thatv^εis bounded inL^∞. Hence we have

εlim→0

D_ε

ϕ∇w^ε· ∇v^εdx−

D

α0ϕv dx.

It is now easy to check that all the inequalities becomes equalities wheneverv^ε=0 onT_ε. 2

The proof of Proposition 2.2 will occupy most of this paper. It will be split into two parts: In Section 4, we consider the (simpler) case when the holesS_ε(k, ω)are all balls of random radius. In Section 5, we will use this first result to treat the general case (when the holes have unspecified shapes).

Before turning to this proof, we briefly give, in the next section, the proof of the main theorem.

3. Proof of Theorem 2.1

We introduce the initial and limit energies:

J(v)=

D

1

2|∇v|²−f v dx and

Jα(v)=

D

1

2|∇v|²+1

2α₀v₋² −f v dx.

With theses notations, we have thatu^ε(x, ω)satisfies J(u^ε)= inf

v∈Kε

J(v)

withK_ε= {v∈H₀¹(D);v0 a.e. inT_ε}and standard estimates give the existence of a functionu(x, ω)¯ such that u^ε(·, ω)→ ¯u(·, ω) inH₀¹(D)-weak.

We now have to show that for almost everyω,u(¯ ·, ω)satisfies:

Jα(u)¯ = inf

v∈H₀¹(D)

Jα(v).

This will follow from two lemmas:

Lemma 3.1.For any test functionϕinD(D)we have

εlim→0

D

|∇w^ε|²ϕ dx=

D

α₀ϕ dx.

Lemma 3.2.Letu^εbe any bounded sequence inH₀¹(D)such thatu^ε0inT_ε. If u^εu¯ inH¹(D)-weak,

then lim inf

ε→0 J(u^ε)Jα(u).¯

Proof of Theorem 2.1. For anyv∈D(D), the functionv+v₋w^εis nonnegative onT_ε(ω)and is thus admissible for the initial obstacle problem. In particular by definition ofu^ε, we have

J(u^ε)J(v+v₋w^ε).

Next, we can expandJ(v+v₋w^ε)as follows:

J(v+v₋w^ε)= 1

2

|∇v|²+ |∇v₋|²w^ε2+ |v₋|²|∇w^ε|² dx +

[v₋∇v₋w^ε∇w^ε+ ∇v∇v₋w^ε+ ∇vv₋∇w^ε]dx−

[f v+f v₋w^ε]dx and it is readily check that Lemma 3.1 and the weak convergence ofw^εto 0 inH¹(D)implies

εlim→0J(v+v₋w^ε)=Jα(v), as soon as|v₋|²∈D(D). We deduce

Jα(v)lim sup

ε→0

J(u^ε) and so, using Lemma 3.2, we get:

Jα(v)Jα(u)¯

for allv∈D(D)such that|v₋|²∈D(D).

Finally, by approximating separately the positive and the negative parts, one can show that every function v∈ H₀¹(D)is the limit of a sequence of functionsv_k∈D(D)such that(v_k)₋∈D(D). Theorem 2.1 follows. 2 Proof of Lemma 3.1. This is an immediate consequence of (5): Since 1−w^ε=0 inT_εwe have

εlim→0

ϕ∇w^ε· ∇(1−w^ε) dx=lim

ε→0

ϕ₊∇w^ε· ∇(1−w^ε) dx−lim

ε→0

ϕ₋∇w^ε· ∇(1−w^ε) dx

= −

α₀ϕ₊dx+

α₀ϕ₋dx and so

εlim→0−

ϕ∇w^ε· ∇w^εdx= −

α0ϕ dx. 2

Proof of Lemma 3.2. See Cioranescu and Murat [5], Proposition 3.1.

4. Proof of Proposition 2.2: Balls of random radius

Throughout this section, we assume that the setsSε(k, ω)are balls centered atεk. Since

cap(Br)=

⎧⎪

⎨

⎪⎩

n(n−2)ωnrⁿ⁻² ifn3,

− 2π

logr ifn=2.

Assumption 2 becomes in this framework:

S_ε(k, ω)=B_a^ε_(r(k,ω))(εk) for allk∈Zⁿ with

a^ε(r)=

rε^n/(n−2) ifn3, exp(−r⁻¹ε⁻²) ifn=2, and

r(k, ω)=

⎧⎨

⎩

γ (k, ω) n(n−2)ωn

1/(n−2)

ifn3, γ (k, ω)/2π ifn=2.

In particular that the process r:Zⁿ×Ω→ [0,∞) is stationary ergodic and satisfies

r(k, ω)r¯ for allk∈Zⁿand a.e.ω∈Ω (7)

for some constantr >¯ 0. Without loss of generality, we can always assume thatr <¯ 1/2 (so that there is no overlapping of the holes for anyε <1):

4.1. The auxiliary obstacle problem

After rescaling, we look for the correctorw^ε(x, ω)in the form w^ε(x, ω)=ε²v^ε(x/ε, ω)

withv^ε(y, ω)solution to

v=α, inε⁻¹D_ε,a.e.ω∈Ω, v=ε⁻² on

k∈ZⁿB_a¯ε(k,ω)(k), where

¯ a^ε(r)=

rε^2/(n−2) ifn3, ε⁻¹exp(−r⁻¹ε⁻²) ifn=2, and such that

ε²v^ε(x/ε)→0 inH¹-weak.

One of the main tool in the proof is the fundamental solution of the Laplace equation, given by:

h(x)=

⎧⎪

⎨

⎪⎩ 1 n(n−2)ωn

1

|x|ⁿ⁻² ifn3,

− 1

2π log|x| ifn=2.

In particular, we note that

h|∂Baε (r(k,ω))_¯ (0)=

⎧⎪

⎨

⎪⎩

1

n(n−2)ωnrⁿ⁻²ε⁻² ifn3, 1

2π(log(ε)+r⁻¹ε⁻²) ifn=2,

so we expect the rescaled correctorv^ε(x, ω)to behave, near the holeB_a(k,ω)¯ (k), like the function h_k(x):=

⎧⎨

⎩

γ (k, ω)h(x−k)=r(k, ω)ⁿ⁻²

|x−k|ⁿ⁻², ifn3, γ (k, ω)h(x−k)= −r(k, ω)log|x−k| ifn=2, where

γ (k, ω)= r(k, ω) ⁿ⁻²n(n−2)ωn ifn3,

2π r(k, ω) ifn=2.

Sinceh_ksatisfies

hk= −γ (k, ω)δ(x−k), we will constructv^ε(x, ω)by solving

v=α−

k∈Zⁿ∩Aγ (k, ω)δ(x−k) inε⁻¹D,

v=0 on∂ε⁻¹D.

The main issue is thus to find the criticalαfor which the solution of the above equation has the appropriate behavior nearx=k.

Following [3], this will be done by introducing the following obstacle problem, for every open setA⊂Rⁿ and α∈R:

¯

v_α,A(x, ω)=inf

v(x);vα−

k∈Zⁿ∩A

γ (k, ω)δ(x−k), v0 inA, v=0 on∂A

. (8)

Clearly, the functionv¯_α,Ais solution of

v=α−

k∈Zⁿ∩A

γ (k, ω)δ(x−k) (9)

whenever it is positive. Note that the function h_α,k(x):= α

2n|x−k|²+h_k(x−k) (10)

=

⎧⎪

⎨

⎪⎩ α

2n|x−k|²+r(k, ω)ⁿ⁻²

|x−k|ⁿ⁻², ifn3, α

2n|x−k|²−r(k, ω)log|x−k| ifn=2, also satisfies

h_α,k(x)=α−γ (k, ω)δ(x−k).

It follows from (9) and the maximum principle that ifB1(k)⊂A, then, for allxinB1(k)and for almost everyωinΩ, we have

¯

v_α,A(x, ω)

⎧⎪

⎨

⎪⎩

h_α,k(x)− α

2n−rⁿ⁻² ifn3, hα,k(x)− α

2n ifn=2.

(11)

4.2. Criticalα

The purpose of this section is to prove that for a criticalα,v¯_α,Abehaves likeh_α,knearS_ε(k, ω). For that purpose, we introduce the following quantity, which measures the size of the contact set:

¯

m_α(A, ω)=x∈A; ¯v_α,A(x, ω)=0, where|A|denotes the Lebesgue measure of a setA.

The starting point of the proof is the following lemma:

Lemma 4.1.The random variablem¯α is subadditive, and the process T_km(A, ω)=m(k+A, ω)

has the same distribution for allk∈Zⁿ.

Proof of Lemma 4.1. Assume that the finite family of sets(A_i)_i∈I is such that A_i⊂A for alli∈I,

Ai∩Aj= ∅ for alli=j, A−

i∈I

A_i =0

thenv¯_α,Ais admissible for eachA_i, and sov¯_α,Aiv¯_α,A. It follows that {¯v_α,A=0} ∩A_i⊂ {¯v_α,Ai=0}

and so

¯

mα(A, ω)=

i∈I

{¯vα,A=0} ∩Ai

i∈I

{¯vα,A_i=0}=

i∈I

¯

mα(Ai, ω), which gives the subadditive property. Assumption 3 then yields

T_km(A, ω)=m(A, τ_kω)

which gives the last assertion of the lemma. 2

Since m¯_α(A, ω)|A|, and thanks to the ergodicity of the transformationsτ_k, it follows from the subadditive ergodic theorem (see [8] and [1]) that for eachα, there exists a constant(α)¯ such that

tlim→∞

¯

m_α(B_t(0), ω)

|Bt(0)| = ¯(α) a.s.,

whereB_t(0)denotes the ball centered at the origin with radiust. Note that the limit exists and is the same if instead ofB_t(0), we use cubes or balls centered att x₀for somex₀.

If we scale back and consider the function

¯

w^ε_α(y, ω)=ε²v¯_α,B

ε−1(ε⁻¹x0)(y/ε, ω) inB₁(x₀), we deduce

εlim→0

|{y; ¯w^ε_α(y, ω)=0}|

|B₁| = ¯(α) a.s.

The next lemma summarizes the properties of(α):¯ Lemma 4.2.

(i) (α)¯ is a nondecreasing functions ofα.

(ii) Ifα <0, then(α)¯ =0. Moreover, if the radiir(k, ω)are bounded from below, then(α)¯ =0for anyαsuch that α < n(n−2)infk∈Zⁿr(k, ω)ⁿ⁻²almost surely.

(iii) Ifα2ⁿn(n−2)sup_k∈Znr(k, ω)ⁿ⁻²(orα8rforn=2)almost surely, then(α) >¯ 0.

Proof. (i) The proof follows immediately from the inequality

¯

vα,Av¯_α,A for anyα, αsuch thatαα.

(ii) Ifαis negative, then the function _2n^α|x−x₀|²−_2n^α (t r)², which is a sub-solution of (9), is positive int B_r(x₀) and vanishes along∂(t B_r(x₀))for any ballB_r(x₀)and for anyt >0. We deduce:

¯

v_{α,t B}> α

2n|x−x₀|²− α

2n(t r)²>0 int B_r(x₀)

for allt >0. Thereforem_α(t B, ω)=0 for allt >0, so(α)¯ =0 for allα <0.

Furthermore, ifr(k, ω)is bounded below:

r(k, ω)r >0 for allk∈Zⁿ, a.e.ω∈Ω,

then, the function _2n^α|x−k|²+_|x−^rn_k⁻_|n²−2 −_2n^α −rⁿ⁻²is a sub-solution of (9) inB₁(k)which vanishes on∂B₁(k)and is strictly positive inB1(k)as long asα < n(n−2)rⁿ⁻². As above, we deduce thatmα(t B, ω)=0 for allt >0 and for allα < n(n−2)rⁿ⁻².

(iii) The functionh_α,k(x)=_2n^α|x−k|²+_|x−^rn_k⁻_|n²−2 is radially symmetric and reaches its minimum when

|x−k| =R(α, k):=

⎧⎪

⎪⎨

⎪⎪

⎩

n(n−2)r(k, ω)ⁿ⁻² α

1/n

whenn3, 2r(k, ω)

α 1/2

whenn=2.

(12)

In particular, for α >2ⁿn(n−2)r(k, ω)ⁿ⁻² (or n8r(k, ω)when n=2), we have R(α, k) <1/2 and so the function

g_k(x)=

h_α,k(x)−D_k inB_R(α,k)(k),

0 inRⁿ\B_R(α,k)(k)

satisfies

g_kα−γ (k, ω)δ(x−k) inC₁(k), and

g_k=0 inC₁(k)\B_1/2(k),

whereC1(k)denotes the cube of size 1 centered atk, and the constantCk is chosen in such a way thatgk and∇gk

vanish along∂BR(α,k):

D(α, k):=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ α

2n ⁿ⁻²

n

r^2(nn−2) n−2

2 ²

n

n n−2

whenn3, r

2

1−log 2r

α

whenn=2.

(13)

By definition ofv¯α,t B, we deduce that

¯

vα,t B(x)

k∈Zⁿ∩t B

gk(x) int Ba.s.

In particular, this implies thatv¯α,t B vanishes int B\

k∈ZⁿB1/2(k), and so

¯

m_α(t B, ω)

|t B| |C1| − |B1/2|

|C₁| =1−ω_n 2ⁿ a.s.

We conclude (α)¯ 1−ω_n

2ⁿ >0. 2 Using Lemma 4.2, we can define

α₀=sup

α; ¯(α)=0 .

Note thatα₀is finite under Assumption 3 (Lemma 4.2(iii)) and thatα₀0 is strictly positive as soon as ther(k, ω) are bounded from below almost surely by a positive constant (Lemma 4.2(ii)).

Our goal, in the rest of this section, is to show that the function w^ε(x, ω)=inf

w(x);wα₀inD\T_ε,w1 onT_ε∩D w=0 on∂D\T_ε

,

satisfies all the conditions of Proposition 2.2. For that purpose, we will introduce a series of intermediate functions.

4.3. Behavior ofv¯_α,ε−1Anear the holes

We fix a bounded subsetAofRⁿand we denote by

¯

v_α^ε(x, ω)= ¯v_α,ε−1A(x, ω) (14)

the solutions of (8) defined inε⁻¹A. We also introduce the rescaled function

¯

w^ε_α(y, ω)=ε²v¯_α^ε(y/ε, ω), defined inA.

The key properties ofv¯_α^ε are given by the following lemma:

Lemma 4.3.

(i) For everyαand for everyk∈Zⁿ, we have

¯ v_α^ε(x)

⎧⎪

⎨

⎪⎩

h_α,k(x)− α

2n−rⁿ⁻² ifn3, h_α,k(x)− α

2n ifn=2

for allx∈B₁(k)and almost everywhereω∈Ω (whereh_α,kis defined by(10)).

(ii) For everyα > α₀, we have

¯

v_α^ε(x)h_α,k(x)+o(ε⁻²)

for allx∈B_1/2(k)and almost everywhereω∈Ω. Since

h_α,k|∂Baε (r(k,ω))_¯ (0)=

⎧⎪

⎨

⎪⎩

ε⁻²+α0

2na¯^εr(k, ω) ² ifn3, ε⁻²+α₀

4 a¯^εr(k, ω) ²+r(k, ω)logε ifn=2, we deduce the following corollary:

Corollary 4.4.

(i) For everyαand everyk∈Zⁿsuch thatr(k, ω) >0, we have

¯

v_α^ε(x)ε⁻²+o(1) on∂B_a¯^ε_(r(k,ω))(k)a.e.ω∈Ω

and so

¯

w_α^ε(x)1+o(ε²) on∂Tε(ω)a.e.ω∈Ω for allα.

(ii) For everyα > α₀and everyk∈Zⁿ, we have

¯

v_α^ε(x)ε⁻²+o(ε⁻²) on∂Ba_¯^ε(r(k,ω))(k)a.e.ω∈Ω and so

¯

w_α^ε(x)1+o(1) on∂Tε(ω)a.e.ω∈Ω.

Proof of Lemma 4.3. (i) Immediate consequence of (11).

(ii)Preliminary: First of all sinceAis bounded, we have A⊂BR(x0).

Without loss of generality, we can always assume thatB_R(x₀)=B₁(0). We then introduce v^ε_α(x, ω)= ¯v_α,ε−1B1(x, ω),

the solutions of (8) inB_ε−1(0). It is readily seen thatv^ε_αis admissible for (8) and thus

¯

v^ε_α(x, ω)v_α^ε(x, ω) for allx∈ε⁻¹Aa.e.ω∈Ω.

It is thus enough to prove (ii) forv^ε_α.

We will need the following consequence of Lemma 4.1 (see [3] for the proof):

Lemma 4.5.For any ballB_r(x₀)∈B₁(0), the following limit holds, a.s. inω

εlim→0

|{v_α^ε(x, ω)=0} ∩B_ε−1r(ε⁻¹x₁)|

|B_ε−1r| = ¯(α).

Step 1:We can now start the proof: For anyδ >0, we can coverB_ε−1 by a finite numberN (Cδ⁻ⁿ) of ballsB_i with radiusδε⁻¹and centerε⁻¹xi. Sinceα > α0, we have(α) >¯ 0. By Lemma 4.5, we deduce that for everyi, there existsεi such that ifεεi, then

v^ε_α(x, ω)=0

∩B_i>0 a.s.ω.

In particular, ifεinfε_i, thenv^ε_α(y_i)=0 for somey_i inB_i a.s.ω∈Ω. We now have to show that this implies that v^ε_αremains small in eachB_ias long as we stay away from the lattice pointsk∈Zⁿ. More precisely, we want to show that

sup

Bi\

k∈ZnB1/4(k)

v^ε_αCδ²ε⁻².

Step 2:Letηbe a nonnegative function such that 0η(x)1 for allx,η(x)=1 inB_1/8andη=0 inRⁿ\B_1/4. Then the functionu=v_α^ε ηis nonnegative on 2Biand satisfies

−CuC,

whereCis a universal constant depending only onnandr. In particular, since¯ B_ihas radiusδε⁻¹, Harnack inequality yields:

sup

B_i

uCinf

Bi

u+Cα(δε⁻¹)². Step 3:We need the following lemma:

Lemma 4.6.IfvαinB_r(y₀), then 1

Br

Br(y0)

v(x) dxv(y₀)+αC(n)r²,

whereC(n)is a universal constant.

Proof. We note that the functionv(x)−_2n^α|x−y₀|²is super-harmonic inB_r(y₀). The lemma follows from the mean value formula. 2

Now, we recall thatv^ε_α(y_i)=0 andv_α^εαinB_1/4(y_i). So 1

B_1/4

B_1/4(y_i)

v_α^ε(x) dxv^ε_α(y_i)+αC(n).

In particular, we have u(y_i)

B1/4(yi)

v^ε_α(x) dxC(α, n).

Step 4:Steps 2 and 3 yield sup

Bi

uC(α, n)

1+α(δε⁻¹)² and sincev_α^ε0 inB_i\

k∈Zⁿ{k}, we have:

v_α^ε(y) 1 B_1/8

B1/8(y)

v_α^ε(x) dxCu(y)

for ally∈B_i\

k∈ZⁿB1/4(k).

It follows that for everyδand forεsmall enough, we have:

sup

B_ε−1\

k∈ZnB_1/4(k)

v_α^εCδ²ε⁻².

The definition ofv^ε_αand the fact thathα,k0 on∂B1/2implies that v_α^ε(x)h_α,k(x)+Cδ²ε⁻² inB_1/2(k)

for allk∈Zⁿ. 2

4.4. Approximated corrector

We now want to use the functionv¯^ε_α, solution of the obstacle problem (8), to study the properties of the free solution w^ε₀of

w₀^ε=α₀−

k∈Zⁿ∩Dγ (k, ω)δ(x−εk) inD,

w₀^ε=0 on∂D.

We defineh^ε_α,k(x)=ε²h_α,k(x/ε)forn3 andh^ε_α,k(x)=ε²h_α,k(x/ε)+rε²logεforn=2. We have:

h^ε_α,k(x):=

⎧⎪

⎪⎨

⎪⎪

⎩ α0

2n|x−εk|²+εⁿr(k, ω)ⁿ⁻²

|x−εk|ⁿ⁻² ifn3, α₀

2n|x−εk|²−r(k, ω)ε²log|x−εk| ifn=2.

We then prove:

Lemma 4.7.For everyk∈Zⁿ,w₀^εsatisfies

h^ε_α,k(x)−o(1)w^ε₀(x)h^ε_α,k(x)+o(1) ∀x∈B_ε/2(εk)∩Da.e.ω∈Ω. (15) In particular:

w^ε₀(x)=1+o(1) on∂T_ε∩D. (16)

Proof. For everyα, we denote byw¯^ε_α the function

¯

w_α^ε(x)=ε²v¯_α,ε−1D(x/ε),

defined inDand satisfyingw¯_α^ε=0 on∂D.

1. For everyα > α₀, we have (w^ε₀−w_α^ε)α₀−α

andw^ε₀−w_α^ε=0 on∂D. This implies w₀^ε(x₀)−w_α^ε(x₀)

D

G(x₀, x)(α₀−α) dx,

whereG(·,·)is the Green function onD(G=δx₀ andG=0 on∂D). Note that we have G(x₀, x)−h(x−x₀) ∀x,x₀∈D,

and so

w₀^ε(x₀)−w_α^ε(x₀)(α−α₀)

D

h(x−x₀) dx.

We deduce sup

D

(w^ε₀−w^ε_α)

C|D|^1/(n−1)ρ_D|α−α₀| ifn3, C|D|ρ_Dlogρ_D|α−α₀| ifn=2, with

ρ_D=inf{ρ;D⊂B_ρ}. Hence we have

w₀^εw^ε_α+O(α−α₀).

Using Lemma 4.3(ii) (sinceα > α₀), we deduce:

w₀^εh^ε_α,k(x)+O(α−α₀)+o(1) ∀x∈B_ε/2(εk)a.e.ω∈Ω which gives the second inequality in (15).

2. Similarly, we observe that for everyαα₀, we have (w^ε_α−w₀^ε)α−α0−α1_{w_α^ε=0}.

Proceeding as before, we deduce that forn3, sup

D

(w^ε_α−w^ε₀)CρD

|D|^1/(n−1)(α0−α)+Cα{w_α^ε=0}^1/(n−1)

and a similar inequality forn=2. Using Lemma 4.3(i), we get w₀^εh^ε_α,k−o(ε²)−O(α0−α)−Cα{w^ε_α=0}^1/(n−1). Finally, since

εlim→0

{w_α^ε=0}=0

for allαα₀, and (15) follows. 2