Reiterated homogenization of nonlinear monotone operators in a general deterministic setting

. We study reiterated homogenization of a nonlinear non-periodic elliptic diﬀerential operator in a general deterministic setting as opposed to the usual stochastic setting. Our approach proceeds from an appropriate notion of convergence termed reiterated Σ-convergence. A general deterministic homogenization theorem is proved and several concrete examples are studied under various structure hypotheses ranging from the classical periodicity hypothesis to more complicated, but realistic, structure hypotheses.


Introduction
We study the homogenization (as 0 < ε → 0 ) of the boundary value problem where Ω is a bounded open set in R N x (the N -dimensional numerical space of variables x = (x 1 , • • •, x N )), f is given in W −1,p (Ω; R) with p = p p−1 , 1 < p < ∞, D and div denote the usual gradient and divergence operators, respectively, in Ω, and finally, a is a given function (y, z, ξ) → a(y, z, ξ) from R N × R N × R N to R N (N ≥ 1 ) with the following properties : (1.2) For any arbitrary ξ ∈ R N , the function (y, z) → a(y, z, ξ) possesses the Caratheodory property, i.e., (i) for each z ∈ R N , the function y → a(y, z, ξ) is measurable from R N (with Lebesgue measure) into R N , (ii) for almost every y ∈ R N , the function z → a(y, z, ξ) maps (1.3) a(y, z, ω) = ω for almost all y ∈ R N and for all z ∈ R N , where ω denotes the origin in R N . ( There are four constants c 1 , c 2 > 0, 0 < α 1 ≤ min(1, p − 1), and α 2 ≥ max(p, 2) such that for almost every y ∈ R N and for every z ∈ R N we have for ξ 1 , ξ 2 ∈ R N , where the dot denotes the usual Euclidean inner product in R N , and |•| the associated norm.
For the sake of clearness it is well to note that the equation in (1.1) actually writes as where a ε (•, •, Du ε ) stands for the function x → a( x ε , x ε 2 , Du ε (x)) from Ω to R N .However, since the set x × R N y × R N z (for Lebesgue measure dxdydz ), it is not clear that this function is well defined.Indeed, unless the function (y, z) → a(y, z, ξ) (for fixed ξ ) is a continuous mapping of R N × R N into R N and v is taken in C(Ω; R) N = C(Ω; R)×•••×C(Ω; R) (N times), it would be naive to state that a( x ε , x ε 2 , v(x)) is the value taken by a(y, z, v(x)) at point (y = x ε , z = x ε 2 ).All that will be clarified in Section 2.
Provided the differential operator u → div a ε (•, •, Du), u ∈ W 1,p (Ω; R), is well defined and has suitable properties (see Corollary 2.1), it is a classical matter to prove an existence and uniqueness result for (1.1) (see, e.g., [16]).Thus, we have a generalized sequence (u ε ) ε>0 at our disposal, and the problem is to study, under a suitable condition on a(y, z, ξ) (for fixed ξ ) -called a structure hypothesis, the limiting behaviour of u ε as ε → 0. This lies within the class of so-called reiterated homogenization problems.Reiterated homogenization was introduced in [4] for linear operators.Multiscale convergence was first applied to reiterated homogenization in [8].The reiterated homogenization of nonlinear elliptic operators was first studied in [14,15], and latter in [16], in the usual periodic setting.
In this study we investigate the homogenization of (1.1) not under the periodicity hypothesis as in the previous references, but in a general deterministic setting including the periodicity, almost periodicity, convergence at infinity hypotheses, and others.Our approach proceeds from an appropriate notion of convergence termed reiterated Σ-convergence.A general deterministic homogenization theorem for (1.1 ) is established, and several examples considered in various concrete settings are presented by way of illustration.Reiterated Σ-convergence is likely to carry over to other settings.In particular, by a suitable adaptation of the approach carried out in [2,3], it is possible to frame, using reiterated Σ-convergence, a reiterated homogenization theory of integral functionals in a general deterministic setting similar to that which is introduced in the present study.
The study is organized as follows.Section 2 deals with preliminary notions and results about the traces a( x ε , x ε 2 , v(x)) (x ∈ Ω) and reiterated Σ-convergence.In Section 3 we study the abstract deterministic homogenization problem for (1.1).The periodicity hypothesis stated in [14,15,16] is here replaced by an abstract assumption covering a great variety of concrete structure hypotheses.Finally, Section 4 is concerned with a few concrete examples of homogenization problems for (1.1).More precisely, we consider the problem of investigating the limiting behaviour (as ε → 0) of u ε (the solution of (1.1)) under various concrete structure hypotheses ranging from the classical periodicity condition to more complicated (but realistic) structure hypotheses, and we show how each of them can reduce to the abstract hypothesis in Section 3.
Except where otherwise stated, vector spaces are considered over the complex field, C, and scalar functions assume complex values.If X and F denote a locally compact space and a Banach space, respectively, then C(X; F ) stands for the space of continuous functions of X into F , and B(X; F ) stands for those functions in C(X; F ) that are bounded.We equip B(X; F ) with the supremum norm u ∞ = sup x∈X u(x) , where • denotes the norm in F .For shortness we write C(X) = C(X; C) and B(X) = B(X; C).Likewise the spaces L p (X; F ) and L p loc (X; F ) (X provided with a positive Radon measure) are denoted by L p (X) and L p loc (X), respectively, when F = C (we refer to [6,7,9] for integration theory).Finally, it is always assumed that the numerical space R d (d a positive integer) is equipped with Lebesgue measure dx = dx 1 • • • dx d .

Preliminaries
whenever the right-hand side has meaning.This is obviously the case if u is continuous on Ω × R N y × R N z , since the right of (2.1) is then none other than the value of u(x, y, z) , there is no serious difficulty in verifying that the right-hand side of (2.1) still has meaning (though in a generalized sense), which determines a function Clearly where ), the function u ε is defined in the sense of Proposition 2.1 and hence we are justified in still making use of the notation in (2.1).
Let us now try to give a meaning to the notation a( ).We will need the following lemma, which is of interest in itself.
Then, the function Proof.It is clear that A z is a commutative C * -algebra with identity.Its spectrum, Δ(A z ), is a metrizable compact space admitting {δ z } z∈R N (δ z the Dirac measure on R N at z ) as a dense subset.Furthermore, the Gelfand transformation on A z , denoted below by G , is an isometric isomorphism of the C * -algebra A z onto the C * -algebra C(Δ(A z )) (see, e.g., [13] for further details).Having made these preliminaries, let us fix an arbitrary point s ∈ Δ(A z ).Let (z n ) n∈N (N denotes the nonnegative integers) be a sequence in where the brackets denote the duality pairing between A z (topological dual of A z ) and A z .It follows that the function y → G(f (y, •))(s) is measurable from R N to R, since the same is true of each of the functions y → δ zn , f(y, •) = f (y, z n ) (n ranging over N), according to property (iv).In other words, if δ s denotes the Dirac measure on Δ(A z ) at s, and g denotes the function y → G(f (y, •)) from R N to C(Δ(A z )), then the function y → δ s , g(y) is measurable from R N into R and that for any arbitrary s ∈ Δ(A z ).Therefore, given any arbitrary Radon measure ρ on Δ(A z ) with finite support, i.e., ρ of the form the function y → ρ, g(y) is measurable from R N into C, the brackets denoting this time the duality pairing between M(Δ(A z )) = C (Δ(A z )) (space of complex Radon measures on Δ(A z )) and C(Δ(A z )).With this in mind, fix freely a Radon measure η on Δ(A z ).Note that η is bounded, since Δ(A z ) is compact.Thus, we may assume without loss of generality that η lies in the closed unit ball B ⊂ M(Δ(A z )).But B with the relative weak * topology on M(Δ(A z )) is a metrizabble compact space (the compacity is classical, the metrizability follows by property (i)).Hence, recalling a classical result (see, e.g., [6, ) for all ξ ∈ R N and for almost all y ∈ R N .Unfortunately Lemma 2.1 does not apply because the space B(R N z ) is not separable.We will see that a condition such as (2.5) is nevertheless fulfilled in practice as a consequence of the concrete structure hypothesis on a i (y, z, ξ) (for fixed ξ ); see Section 4. Now, recalling (1.2)-(1.4),we see that Lemma 2.1 applies with f (y, z) = a i (y, z, ξ), where i and ξ are freely fixed.Hence )) (the verification is an easy matter).Hence, according to Proposition 2.1, we can define the function Proof.This follows in view of the proof of [19,Proposition 2.1].
As a direct consequence of this, we have the following Corollary 2.1.Let the hypotheses of Proposition 2.2 be satisfied.For Remark 2.2.It is sometimes convenient to denote the function ).However, though entirely justified above, this is merely a formal notation.

Reiterated Σ-convergence.
Let us first state some fundamentals of homogenization structures beyond the classical two-scale setting.
Let d be a positive integer.Let H = (H ε ) ε>0 be an action of R * + (the multiplicative group of positive real numbers) on the numerical space R d , i.e., H is a family, indexed by R * + , of permutations where • and Id R d denote usual composition and the identical mapping, respectively, of R d .We assume further that : ω denote the Euclidean norm and the origin in R d , respectively.
(H ) 3 The Lebesgue measure λ on R d is quasi-invariant under H , i.e., to each ε > 0 there is attached some Remark 2.3.In view of (H ) 1 , the mapping H ε is a homeomorphism of R d onto itself and therefore the image measure H ε (λ) is well defined (see, e.g., [7]).We recall that H ε (λ) is the Radon measure on R d given by We denote by Π ∞ (R d , H), or simply Π ∞ when there is no danger of confusion, the space of those u ∈ B(R d ) for which a complex number a positive linear form on Π ∞ attaining the value 1 on the constant function 1 .Such a linear form is necessarily continuous and of norm exactly one.We call M the mean value on R d for H .
We are now in a position to introduce the notion of a homogenization structure in the present general setting.We begin by setting an underlying notion.By a structural representation on R d for the action H is meant any set Γ ⊂ B(R d ) with the properties : Next, in the collection of all structural representations on R d for H , we consider the equivalence relation ∼ defined as : Γ ∼ Γ if and only if CLS(Γ) = CLS(Γ ), where CLS(Γ) denotes the closed vector subspace of B(R d ) spanned by Γ.By an H -structure on R d for H (H stands for homogenization) we shall understand any equivalence class modulo ∼ .
The notion of an H -structure is intimately connected with that of an H -algebra.Specifically, let Σ be an H -structure on R d for H . Let A = CLS(Γ), where Γ is any equivalence class representative of Σ (such a Γ is termed a representation of Σ).The space A is a so-called H -algebra on R d for H , i.e., a closed subalgebra of B(R d ) with the features : (HA1) A with the supremum norm is separable (HA2) A contains the constants (HA3) A is stable under complex conjugation Furthermore, A depends only on Σ and not on the chosen representation Γ of Σ; so that we may set A = J (Σ) (the image of Σ), which yields a mapping Σ → J (Σ) that carries the collection of all H -structures (for H ) bijectively over the collection of all H -algebras (for H ) (see [18,Theorem 3.1]).
It is an easy matter to see that the theory of H -structures developed earlier in the particular setting of [18] carries over to the present general setting.Thus, basic notions such as the partial derivatives on Δ(A) (A a given H -algebra on R d for H ), the Sobolev spaces W 1,p (Δ(A)), the Σconvergence, etc., remain valid and hence are not worth repeating here.We refer the reader to [18,19] for further details.
In the present work we are concerned with three specific actions of R * + : the action Each of these three actions satisfies properties (H ) 1 -(H ) 3 , mutatis mutandis.Now, let Σ y be an H -structure of class C ∞ on R N y for H , and Σ z be an H -structure of class C ∞ on R N z for H . Their product Σ = Σ y × Σ z is defined exactly as in [18,Definition 3.4], and is an H -structure of class C ∞ on R N ×R N for the product action H * .It is an elementary exercise to verify that Proposition 3.2, Theorem 3.2 and Corollaries 3.1-3.2 of [18] carry over mutatis mutandis to the present context.We will put A y = J (Σ y ) (image of Σ y ), A z = J (Σ z ) and A = J (Σ), and use the same letter, G , to denote the Gelfand transformation on A y , A z and A, as well.Points in Δ(A y ) (resp.Δ(A z )) are denoted by s (resp.r ).The compact space Δ(A y ) (resp.Δ(A z )) is equipped with the so-called M -measure, β y (resp.β z ) for A y (resp.A z ).It is fundamental to recall that Δ(A) = Δ(A y ) × Δ(A z ) (cartesian product) and further the M -measure for A, with which Δ(A) is equipped, is precisely the product measure β = β y ⊗ β z (see [18]).
Before we can introduce the concept of reiterated Σ-convergence, we require one further notion.
x )-weak * as ε → 0, where ψ ε is defined (in an obvious manner) in (2.2), and M is the mean value on The set Γ is a structural representation on R N for H and H , as well.We define Σ Z N (resp.Σ Z N ) to be the unique H -structure on R N for H (resp. H ) of which Γ is one representation.Σ Z N is referred to as the periodic H -structure on R N represented by R N (see [18,Example 3.2]).We have where R y and R z are countable subgroups of R N .The set Γ y (resp.Γ z ) is a structural representation on R N for H (resp. H ). We define Σ Ry (resp.Σ Rz ) to be the unique H -structure on R N for H (resp. H ) of which Γ y (resp.Γ z ) is one representation.Σ Ry (resp.Σ Rz ) is referred to as the almost periodic H -structure on R N represented by R y (resp.R z ), see [18,Example 3.3].According to [18,Example 3.6], the product  [18,Example 3.7]), from which one can easily deduce that Σ is a reiteration H -structure. Example 2.6.Let Σ = Σ ∞,Ry ×Σ ∞,Rz where R y and R z are as above.This is clearly an H -structure on R N × R N for H * , and there is no serious difficulty in verifying that Σ is a reiteration H -stucture.
Returning now to the preceding general framework, we assume from now that the H -structure Σ = Σ y × Σ z is a reiteration H -structure.The letter Ω throughout will denote a bounded open set in R N x .Here is a fundamental result.
where u ε is defined in (2.1) and u denotes the complex function on Ω given by u Proof.Starting from the convergence property in Definition 2.1, we see immediately that the proposition follows by the density of C(Ω) ⊗ A in C(Ω; A) and by that of C(Ω; A) in L p (Ω; A).
We are now ready to introduce the concepts of reiterated weak and strong Σ-convergence.The letter E throughout will denote a family of positive real numbers admitting 0 as an accumulation point.For example E = R * + .Attention is drawn to the especial case where E = (ε n ) (integers n ≥ 0) with ε n > 0 and ε n → 0 as n → ∞; E is then referred to as a fundamental sequence.

Definition 2.2. A sequence (u
, where v ε is defined as in (2.1), and v = G • v (i.e., v denotes the function in (ii) strongly Σ-converge reiteratively in L p (Ω) to some u 0 ∈ L p (Ω×Δ(A)) if the following condition is fulfilled : We express this by writing u ε → u 0 reiteratively in L p (Ω)-weak Σ in case (i), and u ε → u 0 reiteratively in L p (Ω)-strong Σ in case (ii).
There is no difficulty in verifying the following results. ( Also, the proof of the next proposition is a simple exercise left to the reader. The results of the Σ-convergence setting [18] carry over mutatis mutandis, together with their proofs, to the present setting.Let us state the most important of such results.Proposition 2.5.Assume that 1 < p < ∞.Given a fundamental sequence E and a sequence (u ε ) ε∈E which is bounded in L p (Ω), a subsequence E can be extracted from E such that the sequence (u ε ) ε∈E weakly Σ-converges reiteratively in L p (Ω).

Reciprocally, if p = 2 and if assertions (i)-(ii) hold, then
Proposition 2.7.Suppose the two real numbers p, q ≥ 1 are such that The notion of a W 1,p (Ω)-proper reiteration H -structure will play a fundamental role in this study.We refer to, e.g., [1] for the classical Sobolev space W 1,p (Ω), to [19] (see also [18]) for special Sobolev spaces such as W 1,p (Δ(A y )) and W 1,p # (Δ(A y )) together with the various associated derivative operators.
Definition 2.3.The reiteration H -structure Σ = Σ y × Σ z is termed W 1,p (Ω)-proper (p a given real number with p ≥ 1 ) if the following three conditions are satisfied.
, there are a subsequence E extracted from E and three functions where reit.stands for reiteratively.
Our next purpose is to present a few examples of W As a preliminary step, we have the following Then, there exists a unique couple of functions Proof.This is a simple adaptation of the proof of [19,Lemma 3.4].
We are now able to state the desired result.
Example 2.8.The aim here is to verify that the reiteration H -structure Σ = Σ Ry × Σ Rz of Example 2.2 is W 1,2 (Ω)-proper.The first step will be to frame a preliminary lemma analogous to Lemma 2.2.For u = (u i ) ∈ D(Δ(A)) N = G(A ∞ ) N , where G is the Gelfand transformation on A (we recall that G maps A ∞ isomorphically over D(Δ(A))), let div y u = G(div y G −1 (u)), and div Proof.This follows in view of the proof of [17,Lemma 4.2].This leads to the next proposition.Proposition 2.9.The reiteration H -structure Σ = Σ Ry × Σ Rz is W 1,2 (Ω)-proper.
We are now able to provide further examples of W 1,p (Ω)-proper reiteration H -structures.

The abstract homogenization problem
3.1.Setting of the abstract problem and preliminaries.Before we can state the abstract homogenization problem for (1.1) we need further details.The basic notation is as in Section 2. Let 1 ≤ p < ∞.We denote by Ξ p (R N y ; B(R N z )), or simply Ξ p when there is no danger of confusion, the space of those functions u ∈ L p loc (R N y ; B(R N z )) for which x .This is a Banach space with norm where B N is the open unit ball in R N x .
Remark 3.1.It should be noted that the way we defined where n is some positive integer such that U ⊂ nB N .Thus, as ε ranges over (0, 1), the mappings u → u ε are uniformly equicontinuous from ) (or X p Σ , or simply X p when there is no danger of confusion) to be the closure of A = J (Σ) in Ξ p .We provide X p Σ with the Ξ p -norm, which makes it a Banach space.Propositions 2.3-2.4 and Corollaries 2.1-2.2 of [18] carry over mutatis mutandis, together with their proofs, to the present context.Let us especially attract attention to the following two fundamental propositions.
Proposition 3.1.The mean value M on R N × R N for H * , viewed as defined on A = J (Σ), extends by continuity to a positive continuous linear form (still denoted by M ) on X p Σ .Furthermore, for each u ∈ X p Σ , we have

2)) is considered as a function in L p (Ω), and Ω is any fixed bounded open set in R N
x .

Proposition 3.2. The Gelfand transformation G : A → C(Δ(A)) extends by continuity to a (unique) continuous linear mapping, still denoted by G , of X p
Σ into L p (Δ(A)).
Remark 3.2.As a direct consequence of the preceding propositions, we have Another result worth mentioning is the following, where Ω is as in Proposition 3.1.
Proof.There is no difficulty in showing that (2.6) holds for v ∈ C(Ω) ⊗ X p ,∞ Σ .Hence the desired result follows by mere routine and use of the density of C The next result is a direct consequence of Proposition 3.3.

Corollary 3.1. Let the hypotheses be as in Proposition
We turn now to the statement of the abstract homogenization problem for (1.1).
Throughout the remainder of the present section it is assumed that 1 < p < ∞.Our main goal is to investigate the limiting behaviour, as ε → 0, of u ε (the solution of (1.1) for fixed ε > 0 ) under the abstract hypothesis where p = p p−1 and where a i (•, •, ψ) stands for the function (y, z) → a i (y, z, ψ(y, z)) from This is referred to as the homogenization problem for (1.1) in the abstract deterministic setting associated with the reiteration H -structure The resolution of this problem requires a few preliminaries.We start with one fundamental lemma.
Proof.Thanks to part (iii) of (1.4), the lemma is proved if we can check that for ψ ∈ (A R ) N and 1 But for any fixed real q ≥ 1 , we clearly have A y ⊗ A z ⊂ L q loc (R N y ; A z ), where A y = J (Σ y ) and A z = J (Σ z ), of course.Hence (3.2) follows by three facts : 1) Remark 3.3.Lemma 3.1 shows immediately that (3.1) implies (2.5).
At the present time, let the index 1 where The next proposition and corollary can be achieved using Proposition 2. x .Suppose (3.1) holds.Then the following assertions are true : ) N extends by continuity to a mapping, still denoted by b, of and Corollary 3.2.Let the hypotheses be those of Proposition 3.4.
where : with respect to Δ(A z ); the functions ψ 1 and ψ 2 are viewed as defined on . By ∂ j ϕ to be the partial derivative

The abstract homogenization result.
Let the basic notation be as above.Let 1 < p < ∞.We assume that the reiteration where and an analogous definition for W 1,p # (Δ(A z ); R).We provide F 1,p 0 with the norm where u = (u 0 , u 1 , u 2 ) is uniquely defined by (3.6).
Proof.It is an easy matter to check using Corollary 2.1 that the (generalized) sequence (u ε ) ε>0 is bounded in W 1,p 0 (Ω).Thus, given an arbitrary fundamental sequence E , appeal to the W 1,p (Ω)-properness of Σ yields a subsequence E from E and some triple u = (u 0 , u 1 , u 2 ) ∈ F 1,p 0 such that (3.7) and (3.8) hold as E ε → 0 .Therefore, recalling that the variational problem (3.6) admits at most one solution, we see that the theorem is proved if we can show that u = (u 0 , u 1 , u 2 ) verifies the variational equation in (3.6).For this purpose, let us fix freely as in Remark 3.5, and let us attach to φ the sequence (φ ε ) ε>0 , φ ε given by (3.5).It is a simple exercise to verify that The next step is to pass to the limit in (3.9) when E ε → 0. Before we can do this, however, we need to know that as ε → 0, where D xi = ∂ ∂xi .Let us verify this.Considering the obvious equality where the index i is freely fixed and the notation u ε is as in (2.1), one is immediately led to

Concrete homogenization problems for (1.1)
In this section, we consider a few examples of homogenization problems for (1.1) in a concrete setting (as opposed to the abstract framework of Section 3) and we show how their study leads naturally to the abstract setting of Section 3 and so we may conclude by merely applying Theorem 3.1.

Problem I (Periodic setting).
We assume here that for each fixed ξ ∈ R N , the function (y, z) → a(y, z, ξ) satisfies the following condition, commonly known as the periodicity hypothesis : For each k ∈ Z N and each ∈ Z N , we have a(y One also expresses (4.1) by saying that a(y, z, ξ) (for fixed ξ ∈ R N ) is Yperiodic in y and Z -periodic in z , or simply that a(y, z, ξ) is Y ×Z -periodic in (y, z), where Y = (0, 1) N and Z = (0, 1) N (see Example 2.7).
Our purpose is to study the homogenization of (1.1) under the periodicity hypothesis.This problem was studied in [16] from a different point of view.Here we present an approach which is in keeping with the general pattern of deterministic homogenization theory.
It is clear that the right reiteration H -structure for this problem is the periodic Example 2.7).In this setting, (3.6) takes a rather simple form : where : Given an arbitrary real p > 1 , the aim is to show that as ε → 0, we have u ε → u 0 in W 1,p 0 (Ω)-weak and , where u ε is the solution of (1.1) (for fixed ε > 0) and u = (u 0 , u 1 , u 2 ) is (uniquely) defined by (4.2).But since Σ is W 1,p (Ω)-proper (Proposition 2.8), we see by Theorem 3.1 that the whole problem reduces to verifying that (3.1) holds.
Let 1 ≤ i ≤ N and ψ ∈ (A R ) N be freely fixed.Let z ∈ R N be arbitrarily fixed.In view of (1.2) and part (iii) of (1.4), the function h : R N × R N → R given by h(y, ξ) = a i (y, z, ξ) (y, ξ ∈ R N ) has the Caratheodory property.Hence, a classical result (see, e.g., [11, p.75]) reveals that if u is any measurable function from R N into R N , then the function y → h(y, u(y)) is measurable from R N into R. Choosing in particular u(y) = ψ(y, z) (y ∈ R N ), we see that the function y → a i (y, z, ψ(y, z)) is measurable from R N into R, and that for any arbitrary z in R N .On the other hand, by a routine calculation using [part (ii) of] (1.2) and [part (iii) of] (1.4) one can easily show that the function z → a i (y, z, ψ(y, z)) (for fixed y ) is continuous on R N .Taking account of (4.1), we deduce that, a.e. in y ∈ R N , the function z → a i (y, z, ψ(y, z)) of R N into R, denoted below by a i (y, •, ψ(y, •)), lies in C per (Z).Hence it follows by Lemma 2.1 that the function y → a i (y, •, ψ(y, •)), denoted by a i (•, •, ψ), is measurable from R N into C per (Z).From this we deduce using (1.

Problem II.
We study here the homogenization of (1.1) under the structure hypothesis Proof.First of all, we note that B ∞ (R N z ; C per (Y )) is exactly the image of the reiteration H -structure Σ = Σ Z N × Σ ∞ (Example 2.4) and further the latter is W 1,2 (Ω)-proper.Therefore, the proof is complete once we have established that To do this, let 1 According to (4.3), we may view a i as a function ξ Still calling a i the restriction of the latter function to K , we have therefore as n → ∞.Thus, (4.4) is proved if we can check that each q n (•, •, ψ) lies in A. However, it is enough to verify that we have q(•, •, ψ) ∈ A for any function q : R N Given one such q , by the Stone-Weierstrass theorem we may consider a sequence as n → ∞, where f n (ψ) stands for f n • ψ (usual composition) and χ(ψ) stands for χ • ψ .We deduce that χ(ψ) lies in A R , since the same is true of each f n (ψ) (A R being an algebra).The proposition follows thereby.

Problem III.
We investigate here the limiting behaviour, as ε → 0, of u ε (defined for each ε > 0 by (1.1)) under the structure hypothesis (4.5) where AP (R N y ) is the space of almost periodic continuous complex functions on R N y [5,12,13].

3.2] that A is nothing but the image of the H
On the other hand, by a procedure similar to that followed in Example 2.9 it can be shown that Σ is W 1,p (Ω)-proper.Therefore, the proposition is proved if we can check that a i (•, •, ψ) lies in A for all ψ ∈ (A R ) N and all 1 ≤ i ≤ N which can be obtained by repeating the proof of (4.4).

Problem VII.
We assume that the following condition is satisfied : (4.9) For each bounded set Λ ⊂ R N and each real η > 0, there is some ρ > 0 such that |a(y − t, z − σ, ξ) − a(y, z, ξ)| ≤ η for all (z, ξ) ∈ R N × Λ and for almost all y ∈ R N provided |t| + |σ| ≤ ρ; and we want to investigate the homogenization of (1.1) under the structure hypothesis (4.10) which generalizes (4.3).For this purpose, let us introduce a suitable function space generalizing the usual amalgam of L p and ∞ on R N (see [10]).Let 1 ≤ p < ∞.We define (L p , ∞ )(R N y ; B(R N z )) to be the space of functions u ∈ L p loc (R N y ; B(R N z )) such that u p,∞ = sup Proof.Let (θ n ) n≥1 be a mollifier on R N y , i.e., (θ n ) n≥1 ⊂ C ∞ 0 (R N y ) with θ n ≥ 0, θ n (y)dy = 1, θ n has support in ε n B N , where B N is the closed unit ball in R N y and 0 < ε n ≤ 1 with ε n → 0 as n → ∞.Let n be freely fixed.For any arbitrary 1 ≤ i ≤ N , let q i n (y, z, ξ) = θ n (t)a i (y − t, z, ξ)dt (y, z, ξ ∈ R N ), which gives a function (y, z, ξ) → q i n (y, z, ξ) of R N × R N × R N into R with q i n (•, •, ξ) ∈ C(R N z ; C per (Y )) for any ξ ∈ R N , as it follows by (4.9).Furthermore, on letting q n = (q i n ) 1≤i≤N and recalling (1.3)-(1.4),we have for all z ∈ R N and almost all y ∈ R N , q n (y, z, ω) = 0 and |q n (y, z, ξ 1 ) − q n (y, z, ξ 2 )| ≤ c 1 (1 where ξ 1 and ξ 2 are arbitrary.The next point is to show that q i n (•, z, ξ) (where ξ is fixed) has a limit in B(R N y ) when |z| → ∞. , where c n = θ n 1 2 ∞ .Now, note that as k ranges over Z N , the sets k + Y form a covering of R N .Thus, we may consider a finite set S ⊂ Z N such that B N is contained in the union k∈S (k + Y ).Then y − B N is contained in k∈S (k + y + Y ) (where y is freely fixed in R N ).Hence, by the change of variable σ = y − t in the preceding integral and use of the Y -periodicity we arrive at , where |S| denotes the cardinality of S .This being so, let η > 0. Let ρ > 0 be such that for any z ∈ R N verifying |z| ≥ ρ.Then, clearly sup y∈R N q i n (y, z, ξ) − p i n (y, ξ) ≤ η (z ∈ R N , |z| ≥ ρ). Therefore where the integer n ≥ 1 is arbitrary.So we are justified in carrying over to the present situation the line of argument leading to (4.4).This yields (4.11) Let us next show the following assertion (4.12) Let ψ ∈ (A R ) N and 1 ≤ i ≤ N be fixed.To each η > 0 there is assigned some integer ν ≥ 1 such that q i n (•, •, ψ) − a i (•, •, ψ) 2,∞ ≤ η for any integer n ≥ ν.
First, by Holder's inequality we have q i n (y, z, ψ(y, z)) − a i (y, z, ψ(y, z)) On the other hand, given η > 0, let ρ > 0 be such that |a i (y − t, z, ξ) − a i (y, z, ξ)| ≤ η for almost all y ∈ R N and for all (z, ξ) ∈ R N × Λ provided |t| ≤ ρ, where Λ is a compact set in R N containing the range of ψ .Finally, let ν be a positive integer such that ε n ≤ ρ for any n ≥ ν .Then, recalling that integration above is actually taken over ε n B N , one quickly arrives at q i n (y, z, ψ(y, z)) − a i (y, z, ψ(y, z)) 2 ≤ η 2 for any integer n ≥ ν , for all z ∈ R N and for almost all y ∈ R N .Hence (4.2) follows in an obvious manner.Combining this with (4.11) and recalling that (L 2 , ∞ )(R N y ; B(R N z )) is continuously embedded in Ξ 2 (R N y ; B(R N z )), we obtain (3.1) with p = 2 and Σ = Σ Z N × Σ ∞ , which completes the proof.
weak * , which is very different from the convergence property in the above definition.We give below a few examples of reiteration H -structures.Example 2.1.Let Γ = {γ k : k ∈ Z N } (Z denotes the integers), where for each k ∈ R N , we write γ k for the usual exponential function on R N , i.e., γ k

Example 2 . 5 .
Let Σ = Σ Ry × Σ ∞,Rz where R y , R z and Σ Ry are as in Example 2.2, and Σ ∞,Rz is the H -structure on R N for H defined in[18, Example 3.5].This is an H -structure on R N × R N for H * .It can be shown that Σ is a reiteration H -structure.

Example 2 . 10 .
Let R y be a countable subgroup of R N y .The reiteration H -structure Σ = Σ Ry ×Σ ∞ (Example 2.4) is W 1,2(Ω)-proper.This follows by the same argument as used to prove [18, Corollary 4.4 and Example 4.4].The details are left to the reader.

Proposition 3 . 4 .
2 and Corollary 3.1 and reasoning as in the proof of [19, Proposition 4.1 and Corollary 4.1].Let 1 < p < ∞.Let Ω be a bounded open set in R N
3), part (iii) of (1.4), and (4.1), that a i (•, •, ψ) lies in L ∞ per (Y ; C per (Z)).But L ∞ per (Y ; C per (Z)) ⊂ L p per (Y ; C per (Z)).Hence (3.1) follows by the facts that A = C per (Y × Z) is dense in L p per (Y ; C per (Z)) and the latter space is continuously embedded in Ξ p (R N y ; B(R N z )).
This is a vector space over C and further • p,∞ is a norm under which(L p , ∞ )(R N y ; B(R N z )) is a Banach space.One fundamental result is that (L p , ∞ )(R N y ; B(R N z )) is continuously embedded in Ξ p (R N y ; B(R N z )) (this is immediate by [20, inequality (4.1)]).Let us turn now to the proof of Proposition 4.6.Let (4.9)-(4.10)hold.Then (3.1) is satisfied with p = 2 and Σ= Σ Z N × Σ ∞ (W 1,2 (Ω)-proper reiteration H -stucture on R N × R N for H * ),so that the conclusion of Theorem 3.1 holds for p = 2.
and further there is no serious difficulty in verifying that Σ is a reiteration H -stucture.Let Σ ∞ be the so-called H -structure of the convergence at infinity on R N [18,Example 3.4].This is an H -structure on R N for H and H , as well.The product Σ Our goal here is to show that the reiterationH -structure Σ = Σ Z N × Σ Z N of Example 2.1 is W 1,p (Ω)-proper for any arbitrary real p > 1 .Let us first recall that when dealing with periodic Hstructures, it is possible to do without the Gelfand representation theory (see, e.g., [19, Example 3.1]).So, let Y = (0, 1) N (the open unit cube in R N y ) and Z = (0, 1) N (the open unit cube in R N z ).We denote by C per see, e.g.,[6, p.46]), we see that we may consider a sequence (q n To this end, let us introduce the mappingξ → a * i (•, ξ) of R N into L 2 per (Y ) such that a i (•, z, ξ) → a * i (•, ξ) in L 2 (Y ) as |z| → ∞ (see (4.10)), where ξ is arbitrarily fixed.Putp i n (y, ξ) = θ n (t)a * i (y − t, ξ)dt (y, ξ ∈ R N ).This defines a function (y, ξ)→ p i n (y, ξ) of R N × R N into R with p i n (•, ξ) ∈ C per (Y ) for each fixed ξ .Let us show that q i n (•, z, ξ) → p i n (•, ξ) as |z| → ∞.To do this, we start from q i n (y, z, ξ) − p i n (y, ξ) = θ n (t)[a i (y − t, z, ξ) − a * i (y − t, ξ)]dt.We apply Holder's inequality to getq i n (y, z, ξ) − p i n (y, ξ) ≤ c n BN |a i (y − t, z, ξ) − a * i (y − t, ξ)| 2 dt 1 2