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J. Math. Anal. Appl. 289 (2004) 608–628

www.elsevier.com/locate/jma

Homogenization in perforated domains beyondthe periodic setting

Gabriel Nguetseng

Department of Mathematics, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon

Received 17 April 2003

Submitted by William F. Ames

Abstract

We study the homogenization of a second order linear elliptic differential operator in an opin RN with isolated holes of sizeε > 0. The classical periodicity hypothesis on the coefficients ofoperator is here substituted by an abstract assumption covering a variety of concrete behavioas the periodicity, the almost periodicity, and many more besides. Furthermore, instead of th“periodic perforation” we have here an abstract hypothesis characterizing the manner in whholes are distributed. This is illustrated by practical examples ranging from the classicalequidistrib-ution of the holes to the more complex case in which the holes are concentrated in a neighbouof the hyperplanexN = 0. Our main tool is the recent theory of homogenization structures anbasic approach follows the direct line of two-scale convergence. 2003 Elsevier Inc. All rights reserved.

Keywords: Homogenization; Perforated domains; Homogenization structures; Homogenization algebras

1. Introduction

Throughout this study,Ω denotes a smooth bounded open set inRNx (theN -dimensional

numerical space of variablesx = (x1, . . . , xN)), S denotes aninfinite subset ofZN

(Z stands for the integers), andT denotes a compact set inRNy with smooth boundary

and nonempty interior. Furthermore, it is assumed that

T ⊂ Y =(−1

2,

1

2

)N

. (1.1)

E-mail address: [email protected].

0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2003.08.045

G. Nguetseng / J. Math. Anal. Appl. 289 (2004) 608–628 609

.

f

-

:

, theren-

g the”as

in per-rchers.

ion inwhich

Now, letε > 0. We define

tε = k ∈ S: ε(k + T )⊂Ω,

T ε =⋃k∈t ε

ε(k + T )

and

Ωε =Ω \ T ε (points inΩ lying off T ε).

It is to be noted thatS andΩ can be chosen in such a way as to maketε an empty setAnyhow tε is finite, sinceΩ is bounded. HenceT ε is closed inRN

x and soΩε is open.Specifically, whenΩ andS are such thattε is nonempty, the open setΩε is what remains oΩ after the latter is deprived of its compact subsetT ε. For which reasonΩε is commonlyqualified as aperforated domain. T is called thereference hole, eachε(k+T ) is referred toas a hole of sizeε, and the familyε(k+T )k∈t ε as the holes of the perforated domainΩε.

This being so, for fixedε > 0, let

aε0uε −N∑

i,j=1

∂

∂xi

(aεij

∂uε

∂xj

)= f in Ωε,

N∑i,j=1

aεij∂uε

∂xjγi = 0 on∂T ε (boundary ofT ε),

uε = 0 on∂Ω (boundary ofΩ), (1.2)

wheref ∈ L2(Ω), γ = (γi)1iN the outer unit normal to∂T ε with respect toΩε,aε0(x) = a0(x/ε) (x ∈ Ω) with a0 ∈ L∞(RN

y ) and Rea0(y) α0 > 0 a.e. (almost every

where) inRNy , aεij (x)= aij (x/ε) (x ∈Ω) with aij ∈ L∞(RN

y ) and the ellipticity conditionthere exists a constantα > 0 such that

ReN∑

i,j=1

aij (y)ξj ξ i α|ξ |2 (ξ ∈CN) a.e. iny ∈R

N. (1.3)

For convenience we will also assume the symmetry conditionaji = aij (complex conjugateof aij ); but this is not essential.

It is a classical fact that under the preceding hypotheses (see in particular (1.3))exists one and only one functionuε ∈H 1(Ωε) satisfying (1.2). Our main purpose is to ivestigate the behaviour ofuε whenε→ 0. No doubt, this requires the setS and the familya0, aij 1i,jN to be suitably structured. The classical model consists in assumincoefficientsa0, aij to beY -periodic and the open setΩε to be “periodically perforatedin the sense thatS is chosen equal toZN . The corresponding homogenization problem wfirst studied by Cioranescu and Saint Jean Paulin [9]. Since then, homogenizationforated domains has been attracting the attention of an increasing number of reseaFor a detailed bibliography we refer, for example, to [6–8,13].

However, much yet remains to be done in this field. For example, homogenizatperforated domains needs to be released from the classical periodicity hypothesis to

610 G. Nguetseng / J. Math. Anal. Appl. 289 (2004) 608–628

to thehesis,g ap-

c set-

such assides.ristic

thealin

proachin the0] to

chiss that

s.he so-d thed out

-

e re-with

about

reference is systematically made throughout the literature, especially with regardcoefficients of the differential operators under consideration. The periodicity hypotwhich is only one of many structure hypotheses (see, e.g., [19,20]), is far from beinpropriate to any physical problem considered in its true perspective.

This paper deals with the homogenization of (1.2) in a very general deterministiting:

(i) The classical periodicity hypothesis on the coefficientsa0, aij 1i,jN is here sub-stituted by an abstract assumption covering a wide range of concrete behavioursthe periodicity, the almost periodicity, the convergence at infinity, and many more be

(ii) Instead of the usual “periodic perforation” it is assumed here that the charactefunction of the setG= RN

y \⋃k∈S(k + T ) verifies an abstract hypothesis relevant tomanner in which the holesε(k + T ) (k ∈ S) are distributed. This is illustrated by practicexamples ranging from the classicalequidistribution of holes to the more complex casewhich the holes are concentrated in a neighbourhood of the hyperplanexN = 0.

Our main tool is the recent theory of homogenization structures and our basic apfollows the direct line of two-scale convergence. Thus, the present study falls withscope of the new deterministic homogenization theory introduced earlier in [19,2bridge the gap between periodic and stochastic homogenization.

The configuration of the domainΩε clearly reveals that the latter is perforated in sua way that the holes do not intersect it outer boundary,∂Ω. It should be stressed that thentails no loss of generality, as will be seen later. Indeed, we will see that the holeintersect the outer boundary are of no effect as regards the homogenization proces

The rest of the paper is organized as follows. In Section 2 we state and study tcalled abstract model problem for the deterministic homogenization of (1.2) beyonperiodic setting. Finally, in Section 3 a few significant concrete examples are workein order to illustrate our approach and point out its wide scope.

Except where otherwise stated, all vector spaces are considered overC and scalar functions are assumed to take complex values. Given a locally compact spaceX and a BanachspaceF , we will denote byC(X;F) the space of continuous mappings ofX into F , byK(X;F) the space of functions inC(X;F) that have compact supports, and byB(X;F)

the space of bounded continuous mappings ofX into F . We will assumeB(X;F) to beequipped with the supremum norm‖u‖∞ = supx∈X ‖u(x)‖ (u ∈ B(X;F)) where‖ · ‖ de-notes the norm inF. For shortness we will writeC(X)= C(X;C), K(X) =K(X;C) andB(X) = B(X;C). For basic concepts and notations concerning integration theory wfer to [2,3,10]. Finally, the spaceRN and its open sets are assumed to be providedLebesgue measure denoted byλ or, as usual,dx = dx1 . . . dxN .

2. The abstract homogenization problem for (1.2)

2.1. Fundamentals of homogenization structures

For the benefit of the reader we summarize below a few basic notions and resultsthe homogenization structures. We refer to [19,24] for further details.


-

a-tion).

nd

refer

We start with one underlying concept. We say that a setΓ ⊂ B(RNy ) is a structural

representation onRN if

(1) Γ is a group under multiplication inB(RNy );

(2) Γ is countable;(3) γ ∈ Γ impliesγ ∈ Γ (γ the complex conjugate ofγ );(4) Γ ⊂Π∞.

Here,Π∞ denotes the space of functionsu ∈ B(RNy ) such thatuε →M(u) in L∞(RN

x )-weak∗ asε→ 0 (ε > 0), whereM(u) ∈C and

uε(x)= u

(x

ε

)(x ∈R

N). (2.1)

We recall in passing that the mappingu→M(u) of Π∞ into C is a positive contin-uous linear form withM(1) = 1 andM(τhu) =M(u) (for u ∈ Π∞ andh ∈ RN ) whereτhu(y)= u(y − h) (y ∈RN ). Thus,M is a mean value (see [21] for further details).

Now, in the collection of all structural representations onRN we consider the equivalence relation∼ defined asΓ ∼ Γ ′ if and only if CLS(Γ ) = CLS(Γ ′), where CLS(Γ )

denotes the closed vector subspace ofB(RNy ) spanned byΓ . By anH -structure on RN

y

(H stands forhomogenization) is understood any equivalence class modulo∼.An H -structure is fully determined by its image. Specifically, letΣ be anH -structure

on RN . PutA= CLS(Γ ) whereΓ is any equivalence class representative ofΣ (such aΓis termed arepresentation of Σ). The spaceA is a so-calledH -algebra onRN

y , that is, a

closed subalgebra ofB(RNy ) with the properties:

(5) A with the supremum norm is separable;(6) A contains the constants;(7) If u ∈A thenu ∈A;(8) A⊂Π∞.

Furthermore,A depends only onΣ and not on the chosen representationΓ of Σ . Thus,we may setA= J (Σ) (the image of Σ). This yields a mappingΣ → J (Σ) that carriesthe collection of allH -structures bijectively over the collection of allH -algebras onRN

y

(see [19, Theorem 3.1]).Let A be anH -algebra onRN

y . ClearlyA (with the supremum norm) is a commuttive C∗-algebra with identity (the involution is here the usual one of complex conjugaWe denote by∆(A) the spectrum ofA and byG the Gelfand transformation onA. Werecall that∆(A) is the set of all nonzero multiplicative linear forms onA, andG is themapping ofA into C(∆(A)) such thatG(u)(s)= 〈s, u〉 (s ∈∆(A)), where〈 , 〉 denotes theduality betweenA′ (the topological dual ofA) andA. The topology on∆(A) is the rel-ative weak∗ topology onA′. So topologized,∆(A) is a metrizable compact space, athe Gelfand transformation is an isometric isomorphism of theC∗-algebraA onto theC∗-algebraC(∆(A)). For further details concerning the Banach algebras theory weto [16]. The basic measure on∆(A) is the so-calledM-measure forA, namely the positive


g

f

9]

z

m-

re).ions

Radon measureβ (of total mass 1) on∆(A) such thatM(u) = ∫∆(A) G(u) dβ for u ∈ A

(see [19, Proposition 2.1]).The partial derivative of indexi (1 i N ) on ∆(A) is defined to be the mappin

∂i = G Dyi G−1 (usual composition) ofD1(∆(A)) = ϕ ∈ C(∆(A)): G−1(ϕ) ∈ A1into C(∆(A)), whereA1= ψ ∈ C1(RN): ψ,Dyiψ ∈A (1 i N). Higher order deriv-atives are defined analogously (see [19]). At the present time, letA∞ be the space oψ ∈ C∞(RN

y ) such that

Dαy ψ = ∂ |α|ψ

∂yα11 . . . ∂y

αNN

∈A

for every multi-indexα = (α1, . . . , αN ) ∈NN, and letD(∆(A))= ϕ ∈ C(∆(A)): G−1(ϕ)

∈ A∞. Endowed with a suitable locally convex topology (see [19]),A∞ (respectivelyD(∆(A))) is a Fréchet space and further,G viewed as defined onA∞ is a topologicalisomorphism ofA∞ ontoD(∆(A)).

Any continuous linear form onD(∆(A)) is referred to as a distribution on∆(A). Thespace of all distributions on∆(A) is then the dual,D′(∆(A)), of D(∆(A)). We endowD′(∆(A)) with the strong dual topology. If we assume thatA∞ is dense inA (this condi-tion is always fulfilled in practice), which amounts to assuming thatD(∆(A)) is dense inC(∆(A)), thenLp(∆(A))⊂D′(∆(A)) (1 p ∞) with continuous embedding (see [1for more details). Hence we may define

H 1(∆(A))=W1,2(∆(A)

)= u ∈L2(∆(A)): ∂iu ∈L2(∆(A)

)(1 i N)

,

where the derivative∂iu is taken in the distribution sense on∆(A) (exactly as the Schwartderivative is taken in the classical case). This is a Hilbert space with the norm

‖u‖H1(∆(A)) =(‖u‖2

L2(∆(A))+

N∑i=1

‖∂iu‖2L2(∆(A))

)1/2 (u ∈H 1(∆(A)

)).

However, in practice the appropriate space is notH 1(∆(A)) but its closed subspace

H 1(∆(A))/C=

u ∈H 1(∆(A)

):∫

∆(A)

u(s) dβ(s)= 0

equipped with the seminorm

‖u‖H1(∆(A))/C =(

N∑i=1

‖∂iu‖2L2(∆(A))

)1/2 (u ∈H 1(∆(A)

)/C).

Unfortunately, the pre-Hilbert spaceH 1(∆(A))/C is in general nonseparated and noncoplete. This leads us to introduce the separated completion,H 1

# (∆(A)), of H 1(∆(A))/C,and the canonical mappingJ of H 1(∆(A))/C into its separated completion. For modetails we refer the reader to [19] (see in particular Remark 2.4 and Proposition 2.6

To a givenH -structureΣ onRN there are attached different other fundamental notsuch as weak and strongΣ-convergence inLp(Ω) (1 p <∞) for which we refer the


roachhat an

of

needr

e

e.

pecif-aviour,

reader to [19]. Also, one of the major concepts underlying the homogenization appfollowed in the present study is that of proper homogenization structures. We say tH -structure (of classC∞) Σ on RN is proper (forp = 2) if the following conditions arefulfilled:

(PR)1 Σ is total (forp = 2), i.e.,D(∆(A)) is dense inH 1(∆(A)), whereA= J (Σ).

(PR)2 For any bounded open setΩ in RNx , H 1(Ω)=W1,2(Ω) is Σ-reflexive in the fol-

lowing sense: Given a fundamental sequenceE, that is, an ordinary sequencereals 0< εn 1 such thatεn→ 0 asn→∞, from any bounded sequence(uε)ε∈Ein H 1(Ω) one can extract a subsequence(uε)ε∈E′ (i.e.,E′ is a subsequence fromE)such that asE′ ε → 0, we haveuε → u0 in H 1(Ω)-weak and∂uε/∂xj →∂u0/∂xj + ∂ju1 in L2(Ω)-weakΣ (1 j N ), whereu1 ∈ L2(Ω;H 1

# (∆(A)))

(for more details see [16,24]).

2.2. Preliminaries

Before we can precisely state the abstract homogenization problem for (1.2) wea few notions and notation. Throughout the rest of this section,Σ denotes a propeH -structure onRN with A= J (Σ), and theM-measure forA is denoted byβ .

Let 1 p < ∞. Let Ξp be the space of allu ∈ Lp

loc(RNy ) for which the sequenc

(uε)0<ε1 (uε defined in (2.1)) is bounded inLp

loc(RNx ). Provided with the norm

‖u‖Ξp = sup0<ε1

( ∫BN

∣∣∣∣u(x

ε

)∣∣∣∣p

dx

)1/p

(u ∈Ξp),

whereBN denotes the open unit ball inRNx , Ξp is a Banach space. We defineX

pΣ to be

the closure ofA in Ξp . We equipXpΣ with theΞp-norm, which makes it a Banach spac

Let G=RNy \Θ with

Θ =⋃k∈S

(k + T ).

It is a classical exercise to check (using the compactness ofT ) thatΘ is closed inRNy and

thereforeG is an open set inRNy . We denote byχG the characteristic function ofG in RN

y .We are now in a position to state the abstract homogenization problem for (1.2). S

ically, our main purpose in the present section is to investigate the asymptotic behasε→ 0, of uε (the solution of (1.2)) under the hypotheses

χG ∈XpΣ, (2.2)

M(χG) > 0, (2.3)

a0, aij ∈XqΣ (1 i, j N), (2.4)

where the given positive real numbersp andq are such that

1 + 1 1. (2.5)

p q 2


lem is

t

at

al

-e (see,

meant

This is what is understood bythe abstract homogenization problem for (1.2) in a deter-ministic setting (as opposed to a stochastic setting). As will be seen later, this probquite solvable. Before we can do this, however, we need a few preliminary results.

Lemma 2.1. Under hypothesis (2.2), there exists a β-measurable set G⊂∆(A) such thatχG = χ

Ga.e. in ∆(A), where χG = G(χG), and χ

Gdenotes the characteristic function of

G in ∆(A).

Proof. First, we note thatχG ∈ X2Σ, sinceX

pΣ ⊂ X2

Σ (observe thatp 2, according to(2.5)). By [19, Corollary 2.1] it follows thatχGχG ∈ X1

Σ andG(χGχG)= G(χG)G(χG).But χGχG = χG. HenceχG(s) ∈ 0,1 a.e. ins ∈ ∆(A). So, letN be a negligible sein ∆(A) such thatχG(s) ∈ 0,1 for every s ∈ ∆(A) \ N . Put G = s ∈ ∆(A) \ N :χG(s)= 1. Clearly χG = χ

Ga.e. in∆(A). Therefore the lemma follows by the fact th

the functionχG is measurable (indeed,χG ∈ L∞(∆(A)) [19, Section 2.3]). Remark 2.1. According to Lemma 2.1, we haveχε

G → χG

in Lp(Ω)-weak Σ as

ε → 0 (see [19, Example 4.1]) whereχεG(x) = χG(x/ε) (x ∈ Ω). Furthermore,β(G) =∫

∆(A) χG(s) dβ(s)=M(χG) (see [19, Section 2.3]).

Now, letQε =Ω \ εΘ. This is an open set inRN . The next result will prove essentito our homogenization approach. The proof is a simple exercise left to the reader.

Lemma 2.2. Let K ⊂Ω be a compact set (K independent of ε). There is some ε0 > 0 suchthat (Ωε \Qε)⊂ (Ω \K) for any 0< ε ε0.

Now, letε > 0 be freely fixed. We define

Vε =v ∈H 1(Ωε): v = 0 on∂Ω

equipped with theH 1(Ωε)-norm. This makesVε a Hilbert space. The proof of the forthcoming extension result follows the same line of argument as in the classical case.g., [9]). The details are left to the reader.

Lemma 2.3. For each real ε > 0, there exists an operator Pε of Vε into H 10 (Ω) with the

following properties:

(i) Pε sends continuously and linearly Vε into H 10 (Ω).

(ii) (Pεv)|Ωε = v for all v ∈ Vε.

(iii) ‖D(Pεv)‖L2(Ω)N c‖Dv‖L2(Ωε)N for all v ∈ Vε where the constant c > 0 dependssolely on Y and T , and D denotes the usual gradient operator, i.e., D = Dxi 1iN

with Dxi = ∂/∂xi.

Before we proceed any further let us recall that by a fundamental sequence ishere any ordinary sequenceE = (εn)n∈N with 0< εn 1 andεn → 0 asn→∞.


f

Lemma 2.4. Let E be a fundamental sequence. Let (uε)ε∈E ⊂ L2(Ω) and (vε)ε∈E ⊂L∞(Ω) be two sequences such that:

(i) uε → u0 in L2(Ω)-weak Σ ;(ii) vε → v0 in L2(Ω)-strong Σ ;(iii) (vε)ε∈E is bounded in L∞(Ω).

Then uεvε → u0v0 in L2(Ω)-weak Σ.

Proof. This follows immediately by [19, Proposition 4.7 and Theorem 4.1].At the present time, let

F10 =H 1

0 (Ω)×L2(Ω;H 1#

(∆(A)

)).

This is a Hilbert space with norm

‖v‖F

10=[

N∑i=1

‖Div‖2L2(Ω×∆(A))

]1/2 (v ∈ F

10

),

where

Div= ∂v0

∂xi+ ∂iv1 for v= (v0, v1) ∈ F

10.

Furthermore,F∞0 = D(Ω)× [D(Ω)⊗ J (D(∆(A))/C)] (J the canonical mapping o

H 1(∆(A))/C into its separated completionH 1# (∆(A))) is a dense vector subspace ofF1

0.For more details see [19].

Now, by (2.4) and use of [19, Corollary 2.2] we have thata0 = G(a0) andaij = G(aij )belong toL∞(∆(A)) with

a0(s) α0 > 0 a.e. ins ∈∆(A) (2.6)

and

ReN∑

i,j=1

aij (s)ζj ζ i α|ζ |2 (ζ ∈CN) a.e. ins ∈∆(A), (2.7)

according to (1.3). Also,aji = aij (1 i, j N ).This being so, let

bΩ(u,v)=N∑

i,j=1

∫ ∫Ω×G

aij (s)Dju(x, s)Div(x, s) dx dβ(s)

+∫ ∫

ˆa0(s)u0(x)v0(x) dx dβ(s)

Ω×G


a

In

m

rcive

. Ac-gprop-

for u = (u0, u1) andv = (v0, v1) in F10, whereG is that set in Lemma 2.1. This defines

sesquilinear formbΩ( , ) on F10 × F1

0 which is Hermitian, continuous and noncoercive.

order to point out the lack of coercivity, observe thatbΩ(u,u) = 0 for u = (u0, u1) ∈ F10

such thatu0(x)= 0 a.e. inx ∈Ω and∂j u1(x, s)= 0 a.e. in(x, s) ∈Ω × G (1 j N ),and conclude by arguing thatu1 has no reason to be the zero function inL2(Ω;H 1

# (∆(A)))

(see Remark 2.2).

Remark 2.2. SupposeΣ is the periodicH -structure represented by the networkR= ZN

(see [19, Example 3.2]) and takeS = ZN . ThenH 1# (∆(A))≡ H 1

# (Y ) = w ∈ H 1loc(R

Ny ):

w is Y -periodic and∫Y w(y) dy = 0 (see [22]) withY as in (1.1), so that the syste

∂ju1(x, s) = 0 a.e. in(x, s) ∈ Ω × G (1 j N ) reduces to(∂u1/∂yj )(x, ·) = 0 inY ∗ = Y \ T for almost everyx ∈Ω (1 j N ). That is, for almost everyx ∈Ω , u1(x, ·)is constant inY ∗ (and not inY !).

It will prove crucial to have at our disposal an existence result for the noncoevariational problem

u= (u0, u1) ∈ F10 and bΩ(u,v)= B(v) for everyv= (v0, v1) ∈ F

10, (2.8)

whereB is the continuous antilinear form onF10 given by

B(v)=M(χG)

∫Ω

f (x)v0(x) dx for v= (v0, v1) ∈ F10.

Lemma 2.5. Under hypotheses (2.2)–(2.5), there exists u = (u0, u1) ∈ F10 satisfying

(2.8). Furthermore, u0 is strictly unique and u1 is unique up to an additive functiong ∈L2(Ω;H 1

# (∆(A))) such that ∂jg(x, s)= 0 a.e. in Ω × G (1 j N).

Proof. Let WG denote the vector spaceH 10 (Ω)× L2(Ω;H 1

# (∆(A))) equipped with theseminorm

NG(v)=[∫Ω

∣∣v0(x)∣∣2dx + N∑

j=1

∫ ∫Ω×G

∣∣Djv(x, s)∣∣2dx dβ(s)

]1/2

(v= (v0, v1) ∈WG

).

It is immediate thatWG is a pre-Hilbert space that is nonseparated and noncompletecordingly we defineWG as separated completion ofWG andi to be the canonical mappinof WG into its separated completion (see [4,5,10]). Let us note the following classicalerties:

(i) WG is a Hilbert space;(ii) i is linear;(iii) i(WG) is dense inWG;(iv) ‖i(v)‖WG

=NG(v) for everyv ∈WG.


rcive,

m

nearop-ome

d Re-

The following proposition is also worth recalling:

If F is a Banach space andh a continuous linear mapping ofWG into F,

then there is a unique continuous linear mappingH :WG → F

such thath=H i. (2.9)

Having made this point, let us now show that there exists a unique Hermitian, coecontinuous sesquilinear formBG( , ) on WG ×WG such that

BG

(i(u), i(v)

)= bΩ(u,v) for u,v ∈WG. (2.10)

To do this, fix freelyu ∈WG and define the mapping

ru :WG →C, ru(v)= bΩ(u,v) (v ∈WG).

Recalling that some constantc1 > 0 exists such that∣∣bΩ(u,v)∣∣ c1NG(u)NG(v) (u,v ∈WG),

we apply (2.9) withF = C andh(v) = ru(v) (complex conjugate ofru(v)) for v ∈WG,

and we get a unique continuous antilinear formRu :WG →C such that

Ru(i(v)

)= ru(v) (v ∈WG).

Bearing in mind thatRu is defined for any arbitraryu ∈WG, we next fix freelyV ∈WG

and consider the mapping

qV :WG →C, qV (u)=Ru(V ) (u ∈WG).

Again applying (2.9) withF = C andh = qV leads to a unique continuous linear forQV :WG →C such that

QV

(i(u)

)= qV (u) (u ∈WG).

Finally, we define the mapping

BΩ :WG×WG →C, BΩ(U,V )=QV (U) (U,V ∈WG).

There is no difficulty in verifying that this mapping is a Hermitian continuous sesquiliform on WG × WG satisfying (2.10). The unicity follows at once by the density prerty (iii). The coercivity follows in the same way by moreover using the fact that sconstantc2 > 0 exists such that

RebΩ(v,v) c2(NG(v)

)2(v ∈WG),

the latter inequality being a direct consequence of (2.6)–(2.7) (see also (2.3) anmark 2.1).

This being so, based on (2.9), once more, let us consider the antilinear formL on WG

such that

L(i(v)

)= B(v) (v ∈WG).

Then, by (2.10) we see immediately thatu = (u0, u1) satisfies (2.8) if and only ifi(u)satisfies


).

])

ds

i(u) ∈WG and BΩ

(i(u),V

)= L(V ) for all V ∈WG. (2.11)

But then, as is classical (see, e.g., [18, p. 216]),i(u) is uniquely determined by (2.11We deduce firstly that (2.8) admits at least one solution, secondly that ifu = (u0, u1)

andz = (z0, z1) are two solutions, theni(u) = i(z). This equality means precisely thatuandz have the same neighbourhoods inWG (see [4, Chapter II, p. 23, Proposition 12or equivalently thatNG(u − z) = 0, which amounts tou0(x) = z0(x) a.e. inx ∈ Ω and∂ju1(x, s)= ∂j z1(x, s) (1 j N ) a.e. in(x, s) ∈Ω × G. The lemma follows. Remark 2.3. Condition (2.3) is fundamental. Indeed, if we assume thatM(χG)= 0, thenthe sesquilinear formBΩ( , ) fails to be coercive.

2.3. The abstract homogenization result

Our goal in this section is to prove the following

Theorem 2.6. Suppose (2.2)–(2.5)hold and further the H -structure Σ is proper ( forp = 2). For each real ε > 0, let uε ∈ H 1(Ωε) be uniquely defined by (1.2), and Pε bethe extension operator of Lemma 2.3. Then, as ε→ 0,

Pεuε → u0 in H 10 (Ω)-weak, (2.12)

where u0 is the unique function in H 10 (Ω) with the following property:

(P)

There is some u1 ∈L2(Ω;H 1

# (∆(A))) such that thecouple u= (u0, u1) is a solution of (2.8).

Proof. Clearly for fixedε > 0, uε lies inVε and

∫Ωε

aε0uεv dx +N∑

i,j=1

∫Ωε

aεij∂uε

∂xj

∂v

∂xidx =

∫Ωε

f v dx (2.13)

for anyv ∈ Vε. By choosing in particularv = uε , we see immediately that

supε>0

‖uε‖H1(Ωε) <∞.

By Lemma 2.3 it follows that the sequence(Pεuε)ε>0 is bounded inH 10 (Ω). Hence, given

an arbitrary fundamental sequenceE, theΣ-reflexivity of H 1(Ω) (Section 2.1) yields asubsequenceE′ and two functionsu0 ∈H 1

0 (Ω) andu1 ∈ L2(Ω;H 1# (∆(A))) such that as

E′ ε→ 0, we have on one hand (2.12), on the other hand

∂Pεuε

∂xj→Dju= ∂u0

∂xj+ ∂ju1 in L2(Ω)-weakΣ (1 j N). (2.14)

Thus, if we can establish thatu = (u0, u1) satisfies (2.8), then not onlyu0 will have therequired property, viz.(P), but also, according to Lemma 2.5, it will turn out thatu0 is thesole function inH 1(Ω) with that property. Hence it will follow that (2.12) actually hol
0


t,

-.2,

that

ude

whenE ε→ 0 (ε ∈ E instead ofε ∈ E′) and further when 0< ε→ 0 by reason of thearbitrariness ofE (see, e.g., [19, Remark 4.1]).

So the whole problem reduces to showing that the coupleu= (u0, u1) ∈ F10 verifies the

variational equation in (2.8). To accomplish this, let

Φ = (Ψ0, J(Ψ1)) ∈ F∞

0 with Ψ1 = G Ψ1 andJ(Ψ1)= J Ψ1,

where

Ψ0 ∈D(Ω) and Ψ1 ∈D(Ω)⊗ (A∞/C),

A∞/C being the space of functionsΨ ∈A∞ with M(Ψ )= 0. Let us note in passing thain case of need,Ψ1 (respectivelyΨ1) may be viewed as a function ofΩ intoA (respectivelyH 1(∆(A))/C), which allows us to defineΨ1 (respectivelyJ (Ψ1)). Consider now

Φε = Ψ0+ εΨ ε1 ,

i.e.,

Φε(x)= Ψ0(x)+ εΨ1

(x,

x

ε

)(x ∈Ω).

ClearlyΦε lies inD(Ω) and further all the functionsΦε (ε > 0) have their supports contained in a fixed compact setK ⊂Ω . Let us then keep in mind that, thanks to Lemma 2there is someε0 > 0 such that

Φε = 0 inΩε \Qε (0< ε ε0).

Having done this, let us return to (2.13) and choose therev =Φε|Ωε (restriction ofΦε toΩε) with 0< ε ε0. By the decompositionΩε =Qε∪(Ωε \Qε) and use ofQε =Ω∩εGwe are quickly led to

∫Ω

aε0χεGΦεPεuε dx +

N∑i,j=1

∫Ω

aεij∂Pεuε

∂xj

∂Φε

∂xiχεG dx =

∫Ω

fΦεχεG dx

for 0< ε ε0. The next point is to pass to the limit whenE′ ∈ ε→ 0.First, letting 1/r = 1/p+ 1/q , by (2.2) and (2.4)–(2.5) we see thata0χG ∈Xr

Σ ⊂X2Σ .

Hence, by [19, Corollary 2.1 and Example 4.1] and use of Lemma 2.1, it followsaε0χ

εG → a0χG in L2(Ω)-weakΣ asε→ 0. On the other hand, it is clear that (1)Φε →Ψ0

in B(Ω) whenε→ 0, and (2)Pεuε → u0 in L2(Ω) whenE′ ε→ 0, as is deduced from(2.12) (whenE′ ε→ 0) by the Rellich theorem. Hence(Pεuε)Φε → u0Ψ 0 in L2(Ω)-strongΣ asE′ ε → 0 (see [19, Example 4.2]). By [19, Proposition 4.7] we conclthat, asE′ ε→ 0,∫

Ω

aε0χεGΦεPεuε dx→

∫ ∫Ω×G

a0(s)u0(x)Ψ 0(x) dx dβ(s).

Now, it is not a difficult matter to see that for fixed 1 i N, the sequence(∂Φε/∂xi)ε∈E is bounded inL∞(Ω) and further


e have

ve

ation

t since

func-

∂Φε

∂xi→DiΦ = ∂Ψ0

∂xi+ ∂iΨ1 in L2(Ω)-strongΣ.

Therefore, recalling (2.14) (whenE′ ε → 0), it follows by Lemma 2.4 that, asE′ ε→ 0,

∂Pεuε

∂xj

∂Φε

∂xi→DjuDiΦ in L2(Ω)-weakΣ.

On the other hand, by an obvious argument based on (2.2) and (2.4)–(2.5), waij χG ∈ X2

Σ ∩ L∞, henceaij χG ∈ C(Ω;X2Σ ∩ L∞) (X2

Σ ∩ L∞ equipped with theL∞-norm). Therefore, asE′ ε→ 0, [19, Proposition 4.5] and Lemma 2.1 lead us to∫

Ω

aεij∂Pεuε

∂xj

∂Φε

∂xiχεG dx→

∫ ∫Ω×G

aij (s)Dju(x, s)DiΦ(x, s) dx dβ(s).

Finally, by repeating a previous argument we quickly see that, asε→ 0,∫Ω

fΦεχεG dx→M(χG)

∫Ω

f (x)Ψ 0(x) dx (use Remark 2.1).

Consequently, by passing to the limit asE′ ε→ 0 in the preceding equation, we arriat

bΩ(u,Φ)= B(Φ),

whereΦ is arbitrary inF∞0 . Hence (2.8) follows by the density ofF∞

0 in F10. This com-

pletes the proof.

3. Some concrete examples

Our goal here is to work out, by way of illustration, a few concrete homogenizproblems for (1.2).

3.1. Preliminaries

The basic notation and hypotheses are as in Section 2. We begin by noting thathe setsk + T (k ∈ S) are pairwise disjoint, the characteristic function,χΘ , of the setΘ isgiven byχΘ =∑k∈S χk+T (a locally finite sum) or more suitably

χΘ =∑k∈ZN

θ(k)χk+T , (3.1)

whereχk+T is the characteristic function ofk+T in RNy andθ is that ofS in Z

N . We shallrefer toθ as thedistribution function of the holes.

In most practical casesθ will belong to the space of essential functions onZN , ES(ZN)

(see [21]). The constant functions, the periodic functions and the almost periodictions, onZN , are classical examples of essential functions onZN . In this connection, theusefulness of the following proposition will come to light in the next sections.


e

3].

nlosure

s. For

Proposition 3.1. Let Σ be an H -structure on RN with image A = J (Σ). Suppose thedistribution function of the holes belongs to ES(ZN). On the other hand, assume that forevery ϕ ∈K(Y ), the function

∑k∈ZN θ(k)τkϕ (where τkϕ(y)= ϕ(y − k) for y ∈R

N) liesin A. Then χΘ ∈X

pΣ (1 p <∞) and further

M(χΘ)=M(θ)λ(T ), (3.2)

where λ is the Lebesgue measure on RN and M(θ) the essential mean of θ.

Proof. We first show thatχΘ ∈ XpΣ, where the realp 1 is arbitrarily fixed. Letη > 0.

By consideringχT as a function inLp(Y ) and using the fact thatK(Y ) is a dense subspacof the latter, we get someϕ ∈K(Y ) such that‖χT − ϕ‖Lp(Y ) η. On letting

ψ =∑k∈ZN

θ(k)τkϕ (a locally finite sum) (3.3)

and bearing in mind thatψ ∈A, we deduce at once

‖χΘ −ψ‖p,∞ = supk∈ZN

[ ∫k+Y

∣∣χΘ(y)−ψ(y)∣∣p dy

]1/p

η,

where we recall that‖ · ‖p,∞ is the norm in the amalgam space(Lp, B∞)(RN) [11,19,22].But the latter space is continuously embedded inΞp , as is easily seen by [22, Lemma 1.HenceχΘ ∈X

pΣ , as claimed.

The next point is to prove (3.2). To begin with, let us note that

λ(T )M(χΘ)= limε→0

εN∑

k∈iε(T )

∫k+Y

χΘ(y) dy, (3.4)

whereiε(T ) denotes the set of allk ∈ ZN such thatεk ∈ T (observe thatiε(T ) is finitebecauseT is compact). Indeed, (3.4) is originally true whenχΘ is replaced by any functioin Π∞ (see [21, Theorem 4.3]), and then by a routine procedure this extends to the cof A in (Lp, B∞)(RN) (it is to be noted that the said closure is a subspace ofX

pΣ ).

Next, by substituting (3.1) in (3.4) we obtain immediately

M(χΘ)= limε→0

εN∑

k∈iε(T )

θ(k)

and so the problem reduces to computing the right side. This will proceed in two stepthe sake of convenience we will use the notation

Mε(a)= εN∑

k∈iε(T )

a(k).

Step 1. Fix freelyl ∈ ZN . Letη > 0. Our purpose here is to show that there existsε0 > 0

such that∣∣Mε(τlθ)−Mε(θ)∣∣ η (0< ε ε0), (3.5)


ged ifme

e of

whereτlθ(k)= θ(k − l) for k ∈ ZN . To this end let us observe that

θ(k)= 1

λ(T )

∫k+Y

χΘ(y) dy (k ∈ ZN).

Therefore

Mε(τlθ)−Mε(θ)=∫

I ε(T )

uε dx(uε defined in (2.1)

),

where

u(y)= 1

λ(T )

(χΘ(y − l)− χΘ(y)

)for y ∈R

N

and

I ε(T )=⋃

k∈iε(T )

ε(k + Y ).

Hence, by mere routine,

∣∣Mε(τlθ)−Mε(θ)∣∣ ‖u‖∞

[λ(J ε(T ) \ I ε(T ))+ λ

(J ε(T ) \ T )]+ ∣∣∣∣

∫uεχT dx

∣∣∣∣,whereJ ε(T )=⋃k∈jε(T ) ε(k+Y ), jε(T ) being the set of pointsk ∈ ZN such thatT meets

ε(k + Y). But each of the two setsJ ε(T ) \ I ε(T ) andJ ε(T ) \ T is contained inJ ε(∂T )

(this follows by [21, Proposition 4.2]) and furtherλ(J ε(∂T ))→ λ(∂T )= 0 asε→ 0 (thisclassical result is due to the regularity of the measureλ). On the other hand,

∫uεχT dx→ 0

asε→ 0 (indeed, in [21] the convergence result (2.4) and its proof remain unchanu is taken inL∞(RN) instead ofB(RN)). From all that we deduce the existence of soε0 > 0 such that (3.5) holds.

Step 2. Letη > 0. According to [21, Lemma 3.1] a finite familyBi1im ⊂ ZN exits

such that∣∣∣∣∣ 1

m

m∑i=1

θ(k − Bi)−M(θ)

∣∣∣∣∣ η (k ∈ ZN).

It follows∣∣∣∣∣ 1

m

m∑i=1

Mε(τBi θ)−M(θ)εN∣∣iε(T )∣∣

∣∣∣∣∣ ηεN∣∣iε(T )∣∣ (ε > 0),

where|iε(T )| stands for the cardinality ofiε(T ). On the other hand, as a consequencstep 1, there exists someε0 > 0 such that∣∣∣∣∣ 1

m

m∑i=1

Mε(τBi θ)−Mε(θ)

∣∣∣∣∣ 1

m

m∑i=1

∣∣Mε(τBi θ)−Mε(θ)∣∣ η

for any 0< ε ε0. Therefore


t.2)

e

-

riousa

nd,

e., the]

n the

nimple

)d

)ion to

∣∣Mε(θ)−M(θ)εN∣∣iε(T )∣∣∣∣ η

(1+ εN

∣∣iε(T )∣∣) (0< ε ε0).

But, by the regularity ofλ, we haveλ(J ε(T )) → λ(T ) as ε → 0. We deduce thaλ(Iε(T ))→ λ(T ) asε→ 0 (this follows by an argument included in step 1). Hence (3follows, sinceεN |iε(T )| = λ(Iε(T )). The proof is complete. Corollary 3.2. Let the hypotheses be those of Proposition 3.1. Then (2.2) and (2.3) holdtrue.

Proof. In view of the equalityχG = 1− χΘ, the corollary follows by (3.2) and use of thinequalitiesM(θ) 1 andλ(T ) < λ(Y )= 1 (see (1.1)). 3.2. The holes are equidistributed

Throughout this section, we assume thatθ(k) = 1 for all k ∈ ZN, which means con

cretely thatS = ZN . We think it appropriates to say in this case thatthe holes are equidis-tributed. In this context we intend to discuss the homogenization of (1.2) under vaconcrete structure hypotheses on the coefficientsa0, aij . Let us point out beforehandfew basic results and remarks. LetΣZN be the periodicH -structure onRN representedby the networkR = ZN (see [19, Examples 3.2 and 4.3]).ΣZN is a typical properH -structure (we refer to [19,24] for more details about the properness ofΣZN ). The image ofΣZN is the spaceCper(Y ) of Y -periodic continuous complex functions onR

Ny , and there

is no difficulty in showing thatXrΣ

ZN= Lr

per(Y ) (1 r <∞), whereLrper(Y ) denotes the

space ofY -periodic functions inLrloc(R

Ny ) (see [22, Proposition 2.9]). On the other ha

Proposition 3.1 and its corollary reveal that

χG ∈XpΣ

ZN(1 p <∞) (3.6)

with (2.3). Before we proceed any further it should be noted that the typical case (i.holes are equidistributed and the coefficientsa0, aij areY -periodic) investigated in [9falls under the framework of Section 2 withΣ =ΣZN (see Remark 2.2).

Example 3.1. Under the hypothesis that the holes are equidistributed, our goal ipresent example is to study the homogenization of (1.2) with

a0, aij ∈ Lq

AP

(RNy

)(1 i, j N) for some given 2< q <∞, (3.7)

whereLq

AP(RNy ) denotes the space of functions inLq

loc(RNy ) that are almost periodic i

the sense of Stepanoff (see [1,11,22]). To do this we begin by noting that by a sargument (see [22, Lemma 5.1]) we may consider a countable subgroupR of RN suchthat a0, aij ∈ X

qΣR

(1 i, j N ), whereΣR is the almost periodicH -structure onRN

represented byR (see [19, Example 3.3]). Next, letΣ =ΣR whereR is the (countablesubgroup ofRN spanned by the unionR ∪ZN . This is a properH -structure (see [19]) anfurtherΣR Σ (i.e.,J (ΣR)⊂ J (Σ)) andΣZN Σ . ThusX

qΣR

⊂XqΣ andX

pΣ

ZN⊂X

pΣ

(1 p <∞). Hence, fixing a realp > 2 such that 1/p + 1/q 1/2 and recalling (3.6(with (2.3)), we see at once that conditions (2.2)–(2.5) are fulfilled and so the solutthe homogenization problem before us is furnished by Theorem 2.6.


t (3.7)

e

nclu-

e

in

.eding

Example 3.2. The problem to be worked out here states as in Example 3.1 except thais replaced by the structure hypothesis

a0, aij ∈ Lq∞,per(Y ) (1 i, j N) for some given 2< q <∞, (3.8)

whereLq∞,per(Y ) denotes the closure in(Lq, B∞)(RNy ) of the space of finite sums

∑finite

ϕiui(ϕi ∈ B∞

(RNy

), ui ∈ Cper(Y )

),

B∞(RNy ) being the space of continuous complex functions onR

Ny converging (finitely) at

infinity. Let Σ∞,ZN be theH -structure whose image is the closure inB(RNy ) of the space

of the preceding finite sums. TheH -structureΣ∞,ZN is proper [19, Corollary 4.2] and whave

a0, aij ∈XqΣ (1 i, j N)

(see [19, Proposition 5.2]) and

XpΣ

ZN⊂X

pΣ (1 p <∞)

with Σ =Σ∞,ZN . This leads us immediately to (2.2)–(2.5) and therefore the same cosion as above follows.

Remark 3.1. The structure hypothesisa0, aij ∈ Lq(RNy )+ L

qper(Y ) (1 i, j N ) is in-

cluded in (3.8); see [19].

Example 3.3. We assume here that the coefficientsa0, aij are constant in each cellk+Y ,that is, they are of the form

a0 =∑k∈ZN r0(k)χk+Y (a locally finite sum),

aij =∑k∈ZN rij (k)χk+Y (1 i, j N),

where the complex functionsk → r0(k) andk → rij (k) (defined onZN ) are given. Ourgoal is then to investigate the behaviour, asε→ 0, of uε (the solution of (1.2)) under thstructure hypothesis

lim|k|→∞∫k+Y a0(y) dy = ζ0 ∈C,

lim|k|→∞∫k+Y aij (y) dy = ζij ∈C (1 i, j N),

where|k| denotes the Euclidean norm ofk in RN. By adaptation of the line of argument[20, Theorem 3.1 and Example 3.4] one easily arrives at (2.2)–(2.5) withΣ =Σ0∞ +ΣZN

[20, Proposition 3.4], whereΣ0∞ = Σ0S

with S = B∞(ZN) (see [20, Definition 3.4])SinceΣ is proper [20, Proposition 3.5], the conclusion is the same as in the precexamples.


2])m-

t

e,m,

e

d

rious

e line

t

3.3. The holes are periodically distributed

We assume here that the functionθ is periodic, that is, there exists a networkR in RN

with R ⊂ ZN (e.g.,R = Ni=1miZ with mi ∈ N, mi 1) such thatθ(k + r) = θ(k) for

all k ∈ ZN and allr ∈R. Let ΣR be the periodicH -structure onRN represented byR.Noting thatθ is an essential function onZN (use [21, Remark 2.2 and Proposition 3.and using Corollary 3.2 lead at once toχG ∈X

pΣR

, hence (3.6), and (2.3). Therefore Exaples 3.1–3.3 carry over without the slightest change to the present situation.

3.4. The holes are distributed in an almost periodic fashion

In the present section we assume thatθ is almost periodic, i.e., the translatesτhθ(h ∈ Z

N ) form a relatively compact set inB∞(ZN) (we refer to, e.g., [12,16] for almosperiodic functions on locally compact groups).

Now, asϕ ranges overK(Y ), the functions of the form (3.3) lie in the space AP(RN) ofalmost periodic continuous complex functions onRN (see [22, Lemma 5.4]). Furthermorlet D be a dense countable set inK(Y ) equipped for a while with the supremum norinstead of the usual inductive limit topology (D does exist becauseC(Y ) is separableY being compact), and letF denote the set of functions of the form (3.3) asϕ rangesoverD. Thanks to [22, Lemma 5.1], there exists a countable subgroupR0 of R

N such thatF ⊂ APR0(R

N), where it should be recalled that APR0(RN) denotes precisely the imag

of the almost periodicH -structureΣR0 on RN (see [19]). Hence, it follows that eachψof the form (3.3) (withϕ ∈ K(Y )) belongs to APR0(R

N). Therefore, Proposition 3.1 anCorollary 3.2 apply and we at once arrive at

χG ∈XpΣR0

(1 p <∞) with M(χG) > 0.

In this framework it is possible to work out the homogenization of (1.2) under vastructure hypotheses on the coefficientsa0, aij .

By way of illustration if we consider the structure hypothesis (3.7), then the samof argument as followed in Example 3.1 leads immediately to (2.2)–(2.5) whereΣ =ΣRis a suitable almost periodicH -structure onRN .

Likewise Example 3.2 carries over mutatis mutandis (R0 replacingZN ) to the presencontext.

3.5. The holes are concentrated in a neighbourhood of the hyperplane of equation xN = 0

We use the decompositionRN =RN−1×R (with N 2). Thus, eachy ∈RN writes asy = (y ′, yN) with y ′ = (y1, . . . , yN−1) ∈RN−1 andyN ∈R. AccordinglyZN = ZN−1×Z,

and so eachk ∈ ZN will write ask = (k′, kN).Having done this, we will in the sequel concentrate on one concrete case.

Example 3.4 (Equidistributed concentration). We take here

S = ZN−1 × 0. (3.9)

Thus, the distribution functionθ :ZN →R is given by


llny ar-

y

.

st

θ(k′, kN)=

1 if kN = 0 (k′ ∈ ZN−1),

0 if kN != 0 (k′ ∈ ZN−1).

Let Cper(Y′) (with Y ′ = (−1/2,1/2)N−1) be the usual Banach space ofY ′-periodic

continuous complex functions onRN−1. We introduce theH -algebra [19, Example 3.7]

A= B∞(R;Cper(Y

′))

of continuous functionsψ :RN =RN−1×R→C such that:

(1) The mappingyN →ψ(·, yN ) sends continuouslyR into Cper(Y′).

(2) As |yN | →∞, ψ(·, yN ) converges inCper(Y′) (with the supremum norm).

Let Σ be theH -structure onRN of whichA is the image. It is fundamental to recathatΣ is proper (see [19, Example 4.4]). Our first purpose is to check that (2.2) (for abitrary 1 p <∞) and (2.3) hold in this context. To this end letϕ ∈K(Y ) be freely fixed,and letψ be given by (3.3). Clearlyψ =∑k′∈ZN−1 τ(k′,0)ϕ. Thereforeψ ∈ B(R;Cper(Y

′)).Next, lety = (y ′, yN) be arbitrarily fixed inRN. Since the familyk + Y k∈ZN is a cover-ing of RN , we may consider somel = (l′, lN ) ∈ ZN such thaty ∈ l + Y . The occurrencethaty lies on the boundary ofl+ Y leads toψ(y)= 0, since the (compact) support ofϕ iscontained inY . So, in the sequel we assume thaty ∈ l + Y . Then

ψ(y ′, yN)= θ(l′, lN )ϕ(y ′ − l′, yN − lN ).

But if lN != 0, thenθ(l′, lN )= 0 and thereforeψ(y ′, yN)= 0, once more. Thus, the onlsignificant occurrence is wheny lies in l+ Y with l = (l′,0). Thenψ(y ′, yN)= θ(l′,0)×ϕ(y ′ − l′, yN). Hence

supz∈RN−1

∣∣ψ(z, yN)∣∣ sup

z∈RN−1

∣∣ϕ(z, yN)∣∣and that for any arbitraryyN ∈R. Sinceϕ may be viewed as a function inK(R;B(RN−1)),we deduce that

ψ ∈K(R;Cper(Y

′)),

which impliesψ ∈ A. By proceeding as in the proof of Proposition 3.1 we arrive atχΘ ∈XpΣ for any realp 1. Therefore (2.2) (for any realp 1) follows. Let us next verify (2.3)

We actually wish to show thatM(χG)= 1, or equivalently thatM(χΘ)= 0. But, as seenabove,χΘ lies precisely in the closure ofK(R;Cper(Y

′)) in XpΣ . Thus the problem reduce

to showing thatM(ψ)= 0 forψ ∈K(R;Cper(Y′)). So, fix one suchψ . LetK be a compac

set inRNx such that

λ(K)= 1 and infx=(x ′,xN )∈K

|xN |> 0.

Let η > 0. Considerε0 > 0 such that|ψ(x/ε)| η for x ∈K and 0< ε ε0. Then∣∣∣∣∣∫K

ψ

(x

ε

)dx

∣∣∣∣∣ η

for any 0< ε ε0. On lettingε→ 0, we deduce at onceM(ψ)= 0, as claimed.


ncrete

s (3.9)

d.(3.10)

ted by

pace

), theeneralto then theriodiclly dis-

nd theholesr (1.2)

ck toikova

ntst studyriable

butedon, for

Having made these preliminaries on the perforation, let us precisely state two cohomogenization problems to be discussed in the present framework.

Problem 1. We assume that the open setΩ intersects the hyperplane of equationxN = 0,and we intend to study the homogenization of (1.2) under the perforation hypothesiand the structure hypothesis

a0, aij ∈ Lq(R;Lq

per(Y′))

(1 i, j N) for some given 2< q <∞, (3.10)

whereLqper(Y

′) is the usual space ofY ′-periodic functions inLqloc(R

N−1). According to theabove preliminaries, we have (2.2)–(2.3) where the realp > 2 is such that (2.5) is satisfieThus, thanks to Theorem 2.6, the problem is solved once we have verified thatleads to (2.4). This is straightforward:K(R;Cper(Y

′)) is contained inA= J (Σ) and is adense subspace ofLq(R;Lq

per(Y′)). On the other hand,Lq(R;Lq

per(Y′)) is continuously

embedded in(Lq, B∞)(RN), and the latter space is itself continuously embedded inΞq .Hence (2.4) follows and so Theorem 2.6 applies.

Problem 2. The statement is the same as in Problem 1 except that (3.10) is substitu

a0, aij ∈ B∞(R;Lq

per(Y′))

(1 i, j N) for some given 2< q <∞. (3.11)

The conclusion and the line of argument are the same as above.

Remark 3.2. The preceding result remains valid if in (3.11) we replace the sB∞(R;Lq

per(Y′)) by the closure ofA in (Lq, B∞).

3.6. Concluding remarks

When the holes are distributed in an almost periodic fashion (see Section 3.4homogenization of (1.2) under the structure hypothesis in (3.7) falls under the gframework of almost periodic homogenization theory for which we refer the readerpioneering papers of Kozlov [14,15]. However, attention is drawn to the fact that iframework of Section 3.4, it is not any structure hypothesis that leads to almost pehomogenization. For example, though the holes are assumed to be almost periodicatributed, the homogenization of (1.2) under the structure hypothesis in (3.8) is beyoscope of almost periodic homogenization. Likewise a periodic distribution of the(see Section 3.3) does not imply that the corresponding homogenization problems fonecessarily lie within the scope of periodic homogenization theory.

The study of the distribution of holes considered in Section 3.5 actually goes bathe early eighties of last century (see, e.g., [17,23]). Later, Oleinik and Shaposhn[25] investigated the more general case when the hyperplanexN = 0 is replaced with amanifold of small dimensiond < N . However, only differential operators with constacoefficients were considered in these works. The present paper seems to be the firin which such a distribution of holes is associated to a differential operator with vacoefficients.

Finally, in Section 3.5 we have purposely restricted the study to an equidistriconcentration of holes. One may as well consider a nonequidistributed concentrati


hbour-

work.

rated

ndary

ouverts

(1979)

nals,

iodic

ath.

reach,

, 1968.107.

uation,

(1985)

setting,

perfo-

example a periodic concentration, an almost periodic concentration, etc., in a neighood of the hyperplanexN = 0.

Acknowledgment

The author very much appreciates the reviewer’s remarks and suggestions that improved our original

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