\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 36, pp.1--24. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/36\hfil Homogenization of nonlinear monotone operators] {Homogenization of nonlinear monotone operators beyond the periodic setting} \author[Gabriel Nguetseng \& Hubert Nnang \hfil EJDE--2003/36\hfilneg] {Gabriel Nguetseng \& Hubert Nnang} \address{Gabriel Nguetseng (\textit{Corresponding Author})\hfill\break University of Yaounde I, Department of Mathematics, P. O. Box 812 Yaounde, Cameroon} \email{gnguets@uycdc.uninet.cm} \address{Hubert Nnang \hfill\break University of Yaounde I, Ecole Normale Sup\'{e}rieure, P. O. Box 47 Yaounde, Cameroon} \email{hnnang@uycdc.uninet.cm} \date{} \thanks{Submitted February 25, 2003. Published April 9, 2003.} \subjclass[2000]{46J10, 35B40} \keywords{Homogenization structure, nonlinear monotone operator} \begin{abstract} We study the homogenization of nonlinear monotone operators beyond the classical periodic setting. The usual periodicity hypothesis is here replaced by an abstract assumption covering a wide range of concrete behaviours such as the periodicity, the almost periodicity, the convergence at infinity, and many more besides. Our approach is based on the recent theory of homogenization structures by the first author. The exactness of the results confirms the major role the homogenization structures are destined to play in a general deterministic homogenization theory equipped to consider the physical problems in their true perspective. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{definition}{Definition}[section] \newtheorem{remark}{Remark}[section] \newtheorem{example}{Example}[section] \numberwithin{equation}{section} \section{Introduction} Let $1
0$ such that, a.e. in $y\in \mathbb{R}^{N}$:\\ (i)\quad $(a(y,\lambda )-a(y,\mu ))\cdot (\lambda -\mu )\geq \alpha | \lambda -\mu | ^{p}$\\ (ii)\quad $| a(y,\lambda )-a(y,\mu )|\leq c| \lambda -\mu | ^{p-1}$\\ for all $\lambda ,\mu \in \mathbb{R}^{N}$, where the dot denotes the usual Euclidean inner product in $\mathbb{R}^{N}$. } \label{1.3} \end{equation} Let $\Omega $ be a bounded open set in $\mathbb{R}_{x}^{N}$ (the space $\mathbb{R}^{N}$ of variables $x=(x_{1},\cdots ,x_{N})$), and let $f\in W^{-1,p'}(\Omega ;\mathbb{R})$, $p'=\frac{p}{p-1}$. For each given real $\varepsilon >0$, we consider the boundary value problem \begin{equation} -\mathop{\rm div}a\big( \frac{x}{\varepsilon },Du_{\varepsilon }\big) =f \quad \text{in }\Omega ,\; u_{\varepsilon }\in W_{0}^{1,p}(\Omega ;\mathbb{R}), \label{1.4} \end{equation} which uniquely determines $u_{\varepsilon }$ (see Section 6 of \cite{l3}). Here $D$ denotes the usual gradient, i.e., $D=(D_{x_{i}})_{1\leq i\leq N}$, where $D_{x_{i}}=\frac{\partial }{\partial x_{i}}$. We are interested in the homogenization of (\ref{1.4}) (i.e., the analysis of the behaviour of $u_{\varepsilon }$ when $\varepsilon \to 0$) under a suitable assumption on the behaviour of $a(y,\lambda )$ in $y\in \mathbb{R}^{N}$. Such an assumption will be referred to as a \emph{structure hypothesis} \cite{n1,n2}. The common structure hypothesis is the so-called periodicity hypothesis, that is, the assumption that $a(y+k,\lambda )=a(y,\lambda )$ a.e. in $y\in \mathbb{R}^{N}$, where $k$ is any arbitrary point in $\mathbb{Z}^{N}$ ($\mathbb{Z}$ denotes the integers). The homogenization of (\ref{1.4}) under the periodicity hypothesis has been widely studied (see, e.g., \cite{c1,d1,l2,l3,w1}). However, there is no doubt that in a great number of physical situations the periodicity hypothesis is inappropriate and should be therefore substituted by a realistic structure hypothesis. A few examples of such structure hypotheses will perhaps help us to have a clear idea of what a non periodic homogenization setting (for problem (\ref{1.4})) may look like. In what follows, $a_{i}$ denotes the $i^{th}$ component of the function $(y,\lambda )\to a(y,\lambda )$. \begin{example} \rm \label{ex1.1} \rm Let $\mathcal{B}(\mathbb{R}^{N})$ denote the space of all bounded continuous complex functions on $\mathbb{R}^{N}$, and let $\mathcal{B}_{\infty }(\mathbb{R}^{N})$ be the space of all $\varphi \in \mathcal{B}(\mathbb{R}^{N})$ such that $\varphi (y)$ has a (finite) limit when $| y| \to +\infty $. We define $\mathcal{B}_{\infty ,{\rm per}}(Y)$ (with $Y=(0,1)^{N}$) to be the closure in $\mathcal{B}(\mathbb{R}^{N})$ (with the supremum norm) of the set of all functions $u\in \mathcal{B}(\mathbb{R}^{N})$ of the form \begin{equation*} u(y)=\sum_{k\in F}\varphi _{k}(y)e^{2i\pi k\cdot y}\quad (y\in \mathbb{R}^{N})\text{ with } \varphi _{k}\in \mathcal{B}_{\infty }(\mathbb{R}^{N}) \end{equation*} where $F$ is any arbitrary finite subset of $S=\mathbb{Z}^{N}$. Then one natural structure hypothesis is that \begin{equation} a_{i}(\cdot ,\lambda )\in \mathcal{B}_{\infty ,{\rm per}}(Y) \quad \text{for any } \lambda \in \mathbb{R}^{N} \quad (1\leq i\leq N). \label{1.5} \end{equation} \end{example} \begin{remark}\label{r1.2}\rm The structure hypothesis (\ref{1.5}) includes two particular cases of major interest: 1) the case where $a_{i}(\cdot ,\lambda )\in \mathcal{B}_{\infty }(\mathbb{R}^{N})$ for any $\lambda \in \mathbb{R}^{N}$ ($1\leq i\leq N$), and 2) the case of the periodicity hypothesis. However, (\ref{1.5}) does not reduce to these two particular cases (see Remark 3.1 of \cite{n3}). \end{remark} \begin{example}\label{ex1.2} \rm Let $\mathcal{C}_{\rm per}(Y')$ (with $Y'=(0,1)^{N-1}$, $N\geq 2$) be the space of all continuous complex functions $u$ on $\mathbb{R}^{N-1}$ such that $u(y'+k')=u(y')$ for all $y'=(y_{1},\cdots ,y_{N-1})\in \mathbb{R}^{N-1}$ and all $k'\in \mathbb{Z}^{N-1}$ (such a function is said to be $Y'$-periodic). Let $\mathcal{B}_{\infty }(\mathbb{R};\mathcal{C}_{\rm per}(Y'))$ denote the space of all continuous functions $u:\mathbb{R}\to \mathcal{C}_{\rm per}(Y')$ such that $u(y_{N})$ converges in $\mathcal{C}_{\rm per}(Y')$ (with the supremum norm) when $| y_{N}| \to +\infty $. Then, we may consider the homogenization of (\ref{1.4}) under the structure hypothesis \begin{equation} a_{i}(\cdot ,\lambda )\in \mathcal{B}_{\infty }(\mathbb{R};\mathcal{C} _{\rm per}(Y'))\quad \text{for any }\lambda \in \mathbb{R}^{N} \quad (1\leq i\leq N). \label{1.6} \end{equation} \end{example} \begin{example} \label{ex1.3} \rm One may as well investigate the behaviour, as $\varepsilon \to 0$, of $u_{\varepsilon }$ (the solution of (\ref{1.4})) under the structure hypothesis \begin{equation} a_{i}(\cdot ,\lambda )\in AP(\mathbb{R}^{N})\quad \text{for any } \lambda \in \mathbb{R}^{N} \quad (1\leq i\leq N) \label{1.7} \end{equation} where $AP(\mathbb{R}^{N})$ denotes the usual space of all almost periodic continuous complex functions on $\mathbb{R}^{N}$ \cite{b1,g1,l1}. Let us point out two cases of practical interest that reduce to (\ref{1.7}). \\ (1) Suppose there exists a family of networks $S_{i}$ in $\mathbb{R}^{N}$ ($1\leq i\leq N$) such that $a_{i}(\cdot ,\lambda )$ is $S_{i}$-periodic for any $\lambda \in \mathbb{R}^{N}$. Then (\ref{1.7}) is fulfilled. \\ (2) Suppose that to each $\lambda \in \mathbb{R}^{N}$ there is assigned a network $S_{\lambda }$ in $\mathbb{R}^{N}$ such that $a_{i}(\cdot ,\lambda )$ is $S_{\lambda }$-periodic ($1\leq i\leq N$). This assumption naturally leads to (\ref{1.7}), once again. \end{example} \begin{remark} \rm \label{rm1.3}% Given a function $f$ on $\mathbb{R}^{N}$ and a network $S$ in $\mathbb{R}^{N}$ ($S$ is also termed a r\'{e}seau \cite{b2,d2}), by $f$ to be $S$-periodic we mean that $f(y+k)=f(y)$ for $y\in \mathbb{R}^{N}$ and $k\in S$. In the literature of periodic homogenization one often says $f$ is $Y$-periodic in place of $f$ is $S$-periodic, $Y$ being a suitable parallelepiped attached to $S$ (see \cite{n4}). \end{remark} \begin{example} \label{ex1.4} \rm As will be seen later, it is also possible to study the homogenization of (\ref{1.4}) for $p=2$ under the more general almost periodicity hypothesis \begin{equation} a_{i}(\cdot ,\lambda )\in L_{AP}^{2}(\mathbb{R}^{N}) \quad \text{for any } \lambda \in \mathbb{R}^{N} \quad (1\leq i\leq N) \label{1.8} \end{equation} provided the following condition is fulfilled: \begin{equation} \parbox{10.5cm} {For $\Psi \in AP(\mathbb{R}^{N};\mathbb{R})^{N} =AP(\mathbb{R}^{N};\mathbb{R})\times \cdots \times AP(\mathbb{R}^{N} ;\mathbb{R})$ ($N$ times), we have $\sup_{k\in \mathbb{Z}^{N}}\int_{k+Y}| a(y-r,\Psi (y))-a(y,\Psi (y))| ^{2}dy\to 0$ as $|r| \to 0$ }\label{1.9} \end{equation} where $Y=(0,1)^{N}$. It should be recalled that $L_{AP}^{2}(\mathbb{R}^{N})$ denotes the space of all functions in $L_{\rm loc}^{2}(\mathbb{R}^{N})$ that are almost periodic in the sense of Stepanoff (cf. \cite{b1,n4}). \end{example} The novelty of this article is to study the homogenization of (\ref{1.4}) under diverse structure hypotheses such as (\ref{1.5})-(\ref{1.9}) instead of the classical periodicity hypothesis. In fact, the real scope of the present study is much wider. We investigate here the behaviour, as $% \varepsilon \to 0$, of $u_{\varepsilon }$ (the solution of (\ref{1.4}% )) under an abstract assumption on $a(\cdot ,\lambda )$ (for fixed $\lambda $% ) covering a variety of concrete structure hypotheses beyond the periodic setting. Our main tool is the recent theory of homogenization structures \cite{n1,n2} and our basic approach is an adaptation of the two-scale convergence method \cite{a1,l3}. The achieved results are quite similar to those provided by periodic homogenization theory (see, e.g., \cite{c1,l2}), which confirms, as was anticipated in \cite{n1}, that the recent homogenization approach earlier presented in \cite{n1,n2} fits nonlinear partial differential equations as well. Thus, this work falls within the scope of the new deterministic homogenization theory especially framed in \cite{n1,n2} to bridge the gap between periodic and stochastic homogenization \cite{a2,b6}. The layout of the paper is as follows: Section 2 is devoted to some preliminary results and remarks about the justification of such expressions as $a(\frac{x}{\varepsilon },\mathbf{v}(x))$. Among other things this permits us to give a rigorous meaning to the left-hand side of (\ref{1.4}). In Section 3 we recall the fundamentals of homogenization structures and point out the main results underlying our homogenization approach. Attention is drawn to the particular case of proper homogenization structures. In Section 4 we prove a homogenization result for problem (\ref{1.4}) under an abstract structure hypothesis (in place of the usual periodicity hypothesis). Finally, Section 5 deals with the homogenization of (\ref{1.4}) under various concrete structure hypotheses such as those presented in the preceding examples. Except where otherwise stated, the vector spaces throughout are assumed to be complex vector spaces, and the scalar functions are assumed to take values in $\mathbb{C}$ (the complex field). This permits us to make use of basic tools provided by the classical Banach algebras theory. For basic concepts and notations about integration theory we refer to \cite{b3,b4}. We shall always assume that the $N$-dimensional numerical space $\mathbb{R}^{N}$ and its open sets are each equipped with the Lebesgue measure. \section{Preliminaries} Throughout this section, $\Omega $ denotes a bounded open set in $\mathbb{R}_{x}^{N}$ and $\varepsilon $ is a positive real number. Let us begin by recalling a standard notation. For $u\in L_{\rm loc}^{1}(\Omega \times \mathbb{R}_{y}^{N})$, we set \begin{equation} u^{\varepsilon }(x)=u\left( x,\frac{x}{\varepsilon }\right) \quad (x\in \Omega ) \label{2.1} \end{equation} whenever the right-hand side has meaning. On this point it seems useful to recall the main cases in which (\ref{2.1}) actually determines a function (for further details see \cite{l3,n4}. The most evident two cases are when $u$ is a continuous (real or complex) function on $\Omega \times \mathbb{R}_{y}^{N}$ (or $\overline{\Omega }\times \mathbb{R}_{y}^{N}$, $\overline{\Omega }$ the closure of $\Omega $) and when $u$ lies in $L_{\rm loc}^{p}(\Omega )\otimes L_{\rm loc}^{p'}(\mathbb{R}_{y}^{N})$ ($1\leq p\leq \infty $, $\frac{1}{p'}=1-\frac{1}{p}$), i.e., is of the form \begin{equation*} u=\sum_{\rm finite}\varphi _{i}\otimes u_{i}\quad \text{with }\varphi _{i}\in L_{\rm loc}^{p}(\Omega )\text{ and } u_{i}\in L_{\rm loc}^{p'}(\mathbb{R}_{y}^{N}). \end{equation*} Next comes the less obvious case of the spaces $L^{p}(\Omega ;A)$ ($1\leq p\leq \infty $) where $A$ denotes a closed vector subspace of the space $% \mathcal{B}(\mathbb{R}_{y}^{N})$ (equipped with the supremum norm) of bounded continuous complex functions on $\mathbb{R}_{y}^{N}$. Given $u\in L^{p}(\Omega ;A)$, there clearly exists a negligible set $\mathcal{N}\subset \Omega $ such that for each $x\in \Omega \backslash \mathcal{N}$ the mapping $% y\to u(x,y)$ lies in $A$. Hence the complex number $u(x,\frac{x}{% \varepsilon })$ is well defined and so we may consider the function $% x\to u(x,\frac{x}{\varepsilon })$ defined almost everywhere in $% \Omega $ and belonging to $L^{p}(\Omega )$. Thus, $u^{\varepsilon }\in L^{p}(\Omega )$ is well defined by (\ref{2.1}) when $u\in L^{p}(\Omega ;A)$. This yields a linear mapping $u\to u^{\varepsilon }$ that sends continuously $L^{p}(\Omega ;A)$ into $L^{p}(\Omega )$ and has norm exactly $1$. We turn finally to the still less obvious case of the space $\mathcal{C}(\overline{\Omega };L^{\infty }(\mathbb{R}_{y}^{N}))$. By all probability, in the present case the right-hand side of (\ref{2.1}) cannot be apprehended directly as we did before. One can nevertheless define $u(x,\frac{x}{% \varepsilon })$, $x\in \Omega $, by extension by continuity thanks to two facts : on one hand, the space $\mathcal{C}(\overline{\Omega })\otimes L^{\infty }(\mathbb{R}_{y}^{N})$ is dense in $\mathcal{C}(\overline{\Omega }% ;L^{\infty }(\mathbb{R}_{y}^{N}))$ (cf. \cite[p.46]{b3}), on the other hand, as pointed out thereinbefore, the right-hand side of (\ref{2.1}) is well defined for $u\in \mathcal{C}(\overline{\Omega })\otimes L^{\infty }(\mathbb{R}% _{y}^{N})$. More precisely, we have the following lemma whose proof can be found in \cite{n4}. \begin{lemma} \label{lm2.1} The transformation $u\to u^{\varepsilon }$ (see (\ref{2.1})) considered as a mapping of $\mathcal{C}(\overline{\Omega })\otimes L^{\infty }(\mathbb{R}_{y}^{N})$ into $L^{\infty }(\Omega )$ extends by continuity to a linear mapping, still denoted by $u\to u^{\varepsilon }$, of $\mathcal{C}(\overline{\Omega };L^{\infty }(\mathbb{R}_{y}^{N}))$ into $L^{\infty }(\Omega )$ with \begin{equation*} \| u^{\varepsilon }\| _{L^{\infty }(\Omega )}\leq \sup_{x\in \overline{\Omega }}\| u(x)\| _{L^{\infty }(\mathbb{R}^{N})}, \quad u\in \mathcal{C}(\overline{\Omega };L^{\infty }(\mathbb{R}_{y}^{N})). \end{equation*} \end{lemma} The next result will prove to be of great interest. \begin{lemma} \label{lm2.2} Let $u\in \mathcal{C}(\overline{\Omega };L^{\infty }(\mathbb{R}_{y}^{N}))$. Suppose that for each $x\in \overline{\Omega }$ we have $u(x,y)\geq 0$ a.e in $y\in \mathbb{R}^{N}$. Then $u^{\varepsilon }(x)\geq 0$ a.e. in $x\in \Omega $, $u^{\varepsilon }$ (given by (\ref{2.1})) being defined in the sense of Lemma \ref{lm2.1}. \end{lemma} \begin{proof} Let $u_{n}$ (integers $n\geq 0$) be a sequence in $\mathcal{C}(% \overline{\Omega })\otimes L^{\infty }(\mathbb{R}_{y}^{N})$ such that $% u_{n}\to u$ in $\mathcal{C}(\overline{\Omega };L^{\infty }(\mathbb{R}% ^{N}))$ when $n\to \infty $. Set $q_{n}=\sup_{x\in \overline{\Omega }% }\| u_{n}(x)-u(x)\| _{L^{\infty }(\mathbb{R}^{N})}$ (integers $n\geq 0$). Fix an integer $n\geq 0$ and let $x\in \overline{\Omega }$. Then $% u_{n}(x,y)+q_{n}\geq 0$ a.e. in $y\in \mathbb{R}^{N}$. According to Lemma 1.2 of \cite{n4}, we deduce that there is a negligible set $\mathcal{N}% _{n}\subset \mathbb{R}_{y}^{N}$ ($\mathcal{N}_{n}$ independent of $x$) such that $u_{n}(x,y)+q_{n}\geq 0$ for all $y\in \mathbb{R}^{N}\backslash \mathcal{N}% _{n}$. Hence, letting $\mathcal{N}=\bigcup_{n\in \mathbb{N}}\mathcal{N}% _{n}$, it follows $u_{n}(x,y)+q_{n}\geq 0$ for all $y\in \mathbb{R}% ^{N}\backslash \mathcal{N}$, $x\in \overline{\Omega }$ and $n\in \mathbb{N}$. Therefore $u_{n}^{\varepsilon }(x)+q_{n}\geq 0$ for any $x\in \Omega \backslash \varepsilon \mathcal{N}$ and any integer $n\geq 0$. But, as $% n\to \infty $, we have $u_{n}^{\varepsilon }\to u^{\varepsilon }$ in $L^{\infty }(\Omega )$ (Lemma \ref{lm2.1}) and $% q_{n}\to 0$. Hence, by extraction of a suitable subsequence it follows that $u^{\varepsilon }(x)\geq 0$ a.e. in $x\in \Omega $, as claimed.% \end{proof} Now, let $G:\mathbb{R}^{N}\times \mathbb{R}^{N}\to \mathbb{R}$ be a function with the following properties : \begin{equation} \parbox{10cm}{For each $\lambda \in \mathbb{R}^{N}$, the function $G(\cdot,\lambda )$ is measurable } \label{2.2} \end{equation} \begin{equation} \parbox{10cm}{$G(y,\omega )=0$ a.e. in $y\in \mathbb{R}^{N}$} \label{2.3} \end{equation} \begin{equation} \parbox{10cm}{There is a positive constant $c$ such that $| G(y,\lambda )-G(y,\mu )| \leq c| \lambda -\mu | ^{p-1}$ for all $\lambda ,\mu \in \mathbb{R}^{N}$ and for almost all $y\in\mathbb{R}^{N}$, where $1
0$), $M(u)$ a complex number and \begin{equation} u^{\varepsilon }(x)=u\left( \frac{x}{\varepsilon }\right) \quad \text{for } x\in \mathbb{R}^{N}\quad (\varepsilon >0). \label{3.1} \end{equation} It is worth recalling that $\Pi ^{\infty }$ contains the constants and is translation invariant, and further the mapping $u\to M(u)$ of $\Pi ^{\infty }$ into $\mathbb{C}$ is a positive linear form with $M(1)=1$ and $% M(\tau _{a}u)=M(u)$ for $u\in \Pi ^{\infty }$ and $a\in \mathbb{R}^{N}$, where $% \tau _{a}u(y)=u(y-a)$ for $y\in \mathbb{R}^{N}$. See \cite{n3} for further details. Now, by an $H$-\emph{structure} on $\mathbb{R}_{y}^{N}$ ($H$ stands for \emph{% homogenization}) is meant any equivalence class modulo $\sim $, where the equivalence relation $\sim $ in the collection of all structural representations on $\mathbb{R}_{y}^{N}$ is defined as $\Gamma \sim \Gamma '$\emph{\ if and only if }$CLS(\Gamma )=CLS(\Gamma ')$, $CLS(\Gamma )$ standing for the closed vector subspace of $\mathcal{B}(\mathbb{R}_{y}^{N})$ spanned by $\Gamma $. If $\Sigma $ is a given $H$-structure on $\mathbb{R}_{y}^{N}$, we let $A=CLS(\Gamma )$, where $\Gamma $ is any equivalence class representative of $\Sigma $ (such a $\Gamma $ is termed a \emph{representation} of $\Sigma $). $A$ is a so-called $H$-algebra on $\mathbb{R}_{y}^{N}$, that is, a closed subalgebra of $\mathcal{B}(\mathbb{R}_{y}^{N})$ with the properties: \begin{itemize} \item[(HA1)] $A$ with the supremum norm is separable \item[(HA2)] $A$ contains the constants \item[(HA3)] $u\in A$ implies $\overline{u}\in A$ \item[(HA4)] $A\subset \Pi ^{\infty }$. \end{itemize} Furthermore, $A$ depends only on $\Sigma $ and not on the chosen representation $\Gamma $ of $\Sigma $. Thus, we may set $A=\mathcal{J}% (\Sigma )$ (image of $\Sigma $). This yields a mapping $\Sigma \to \mathcal{J}(\Sigma )$ that carries the collection of all $H$-structures bijectively over the collection of all $H$-algebras on $\mathbb{R}_{y}^{N}$ (see Theorem 3.1 of \cite{n1}). Given an $H$-algebra $A$ on $\mathbb{R}_{y}^{N}$, we will denote by $\Delta (A)$ the spectrum of $A$ and by $\mathcal{G}$ the Gelfand transformation on $A$, i.e., the mapping $u\to \mathcal{G}(u)$ of $A$ into $\mathcal{C}(\Delta (A))$ such that $\mathcal{G}(u)(s)=\langle s,u\rangle $ for $s\in \Delta (A)$ and $u\in A$, where $\langle ,\rangle $ stands for the duality between $A'$ (the topological dual of $A$) and $A$. It is worth noting that $\Delta (A)$ is here a metrizable compact space and $\mathcal{G}$ is an isometric isomorphism of the $\mathcal{C}^{*}$-algebra $A$ onto the $\mathcal{C}^{*}$% -algebra $\mathcal{C}(\Delta (A))$ (see, e.g., \cite[p. 277]{l1}). The appropriate measure on $\Delta (A)$ is the so-called $M$-measure for $A$, that is, the Radon measure $\beta $ on $\Delta (A)$ such that $M(u)=\int_{\Delta (A)}\mathcal{G}(u)d\beta $ for $u\in A$. Let $A^{1}=\{\psi \in \mathcal{C}^{1}(\mathbb{R}_{y}^{N}):$ $\psi ,D_{y_{i}}\psi \in A$ ($1\leq i\leq N$)$\}$ where $D_{y_{i}}\psi =\frac{% \partial \psi }{\partial y_{i}}$. The partial derivative of index $i$ ($% 1\leq i\leq N$) on $\Delta (A)$ is defined to be the mapping $\partial _{i}=% \mathcal{G}\circ D_{y_{i}}\circ \mathcal{G}^{-1}$ (usual composition) of $% \mathcal{D}^{1}(\Delta (A))=\{\varphi \in \mathcal{C}(\Delta (A)):$ $% \mathcal{G}^{-1}(\varphi )\in A^{1}\}$ into $\mathcal{C}(\Delta (A))$, where $\mathcal{G}^{-1}$ (the inverse of $\mathcal{G}$) is viewed as defined on $% \mathcal{D}^{1}(\Delta (A))$. Higher order derivatives are defined analogously (see \cite{n1}). Now, let $A^{\infty }$ be the space of all $% \psi \in \mathcal{C}^{\infty }(\mathbb{R}_{y}^{N})$ such that $D_{y}^{\alpha }\psi =\frac{\partial ^{| \alpha | }\psi }{\partial y_{1}^{\alpha _{1}}\cdots \partial y_{N}^{\alpha _{N}}}\in A$ for each multi-index $\alpha =(\alpha _{1},\cdots ,\alpha _{N})\in \mathbb{N}% ^{N}$. Let $\mathcal{D}(\Delta (A))=\{\varphi \in \mathcal{C}(\Delta (A)):$ $% \mathcal{G}^{-1}(\varphi )\in A^{\infty }\}$. Endowed with a suitable locally convex topology (see \cite{n1}), $A^{\infty }$ (resp. $\mathcal{D}(\Delta (A))$) is a Fr\'{e}chet space and, further, $\mathcal{G}$ viewed as defined on $A^{\infty }$ is a topological isomorphism of $A^{\infty }$ onto $\mathcal{D}(\Delta (A))$. Any continuous linear form on $\mathcal{D}(\Delta (A))$ is referred to as a \emph{distribution} on $\Delta (A)$. The space $\mathcal{D}'(\Delta (A))$ (topological dual of $\mathcal{D}(\Delta (A))$) of all distributions on $\Delta (A)$ is endowed with the strong dual topology. If we assume that $% A^{\infty }$ is dense in $A$ (this amounts to assuming that $\mathcal{D}% (\Delta (A))$ is dense in $\mathcal{C}(\Delta (A))$), then we have $% L^{p}(\Delta (A))\subset \mathcal{D}'(\Delta (A))$ ($1\leq p\leq \infty $) with continuous embedding. Hence we may define \begin{equation*} W^{1,p}(\Delta (A))=\left\{ u\in L^{p}(\Delta (A)):\text{ }\partial _{i}u\in L^{p}(\Delta (A)),\; 1\leq i\leq N\right\} \end{equation*} where the derivative $\partial _{i}u$ is taken in the distribution sense on $% \Delta (A)$ \cite{n1}. We equip $W^{1,p}(\Delta (A))$ with the norm \begin{equation*} ||u||_{W^{1,p}(\Delta (A))}=||u||_{L^{p}(\Delta (A))}+\sum_{i=1}^{N}||\partial _{i}u||_{L^{p}(\Delta (A))} \quad ( u\in W^{1,p}(\Delta (A))), \end{equation*} Which makes it a Banach space. However, we will be mostly concerned with the space \begin{equation*} W^{1,p}(\Delta (A))/\mathbb{C}=\big\{ u\in W^{1,p}(\Delta (A)):\int_{\Delta (A)}u(s)d\beta (s)=0\big\} \end{equation*} provided with the seminorm \begin{equation*} ||u||_{W^{1,p}(\Delta (A))/\mathbb{C}}=\sum_{i=1}^{N}||\partial _{i}u||_{L^{p}(\Delta (A))} \quad ( u\in W^{1,p}(\Delta (A))/\mathbb{C}) . \end{equation*} So topologized, $W^{1,p}(\Delta (A))/\mathbb{C}$ is in general nonseparated and noncomplete. We denote by $W_{\#}^{1,p}(\Delta (A))$ the separated completion of $W^{1,p}(\Delta (A))/\mathbb{C}$ and by $J$ the canonical mapping of $W^{1,p}(\Delta (A))/\mathbb{C}$ into its separated completion (see, e.g., chapter II of \cite{b5} and page 29 of \cite{e1}). $W_{\#}^{1,p}(\Delta (A))$ is a Banach space and $W_{\#}^{1,2}(\Delta (A))$ is a Hilbert space. Furthermore, as pointed out in \cite{n1}, the distribution derivative $% \partial _{i}$ viewed as a mapping of $W^{1,p}(\Delta (A))/\mathbb{C}$ into $% L^{p}(\Delta (A))$ extends to a unique continuous linear mapping, still denoted by $\partial _{i}$, of $W_{\#}^{1,p}(\Delta (A))$ into $L^{p}(\Delta (A))$ such that $\partial _{i}J(v)=\partial _{i}v$ for $v\in W^{1,p}(\Delta (A))/\mathbb{C}$ and \begin{equation*} ||u||_{W_{\#}^{1,p}(\Delta (A))}=\sum_{i=1}^{N}||\partial _{i}u||_{L^{p}(\Delta (A))}\quad \text{for }u\in W_{\#}^{1,p}(\Delta (A)). \end{equation*} To an $H$-structure $\Sigma $ on $\mathbb{R}^{N}$ there are attached the important concepts of weak and strong $\Sigma $-convergence in $L^{p}$ ($% 1\leq p<\infty $), see \cite{n1}. Likewise it is possible and even desirable to introduce the concept of weak $\Sigma $-convergence in $W^{1,p}$. Let $\Sigma $ be an $H$-structure on $\mathbb{R}^{N}$. Let $A=\mathcal{J}% (\Sigma )$. We assume that $\Sigma $ is of class $\mathcal{C}^{\infty }$ \cite{n1}, i.e., $A^{\infty }$ is dense in $A$. Let $1\leq p<\infty $, and let $\Omega $ be a bounded open set in $\mathbb{R}_{x}^{N}$. First of all, we set \begin{equation*} W^{1,p}(\Omega ;L^{p}(\Delta (A)))=\{u\in L^{p}(\Omega \times \Delta (A)):D_{x_{i}}u\in L^{p}(\Omega \times \Delta (A)),\; 1\leq i\leq N\} \end{equation*} where the derivatives $D_{x_{i}}u=\frac{\partial u}{\partial x_{i}}$ are taken in the sense of vector distributions $\mathcal{D}'(\Omega ;L^{p}(\Delta (A)))$ \cite{s2} (see also \cite{v1}), since $L^{p}(\Omega \times \Delta (A))=L^{p}(\Omega ;L^{p}(\Delta (A)))\subset \mathcal{D}% '(\Omega ;L^{p}(\Delta (A)))$. We equip $W^{1,p}(\Omega ;L^{p}(\Delta (A)))$ with the norm \begin{equation*} ||u||_{W^{1,p}(\Omega ;L^{p}\Delta (A)))}=||u||_{L^{p}(\Omega \times \Delta (A))}+\sum_{i=1}^{N}||D_{x_{i}}u||_{L^{p}(\Omega \times \Delta (A))}, \end{equation*} which makes it a Banach space. Before we can define the concept of weak $\Sigma $-convergence in $W^{1,p}$, we also need to give a meaning to $\partial _{i}u$ for $u\in L^{p}(\Omega ;W_{\#}^{1,p}(\Delta (A)))$. This is straightforward. Indeed, considering $% \partial _{i}$ ($1\leq i\leq N$) as a mapping of $W_{\#}^{1,p}(\Delta (A))$ into $L^{p}\Delta (A))$ (as seen above) and using a classical result (see Theorem 4 of page 132 in \cite{b3}), we see that $\partial _{i}\circ u\in L^{p}(\Omega \times \Delta (A))$ for $u\in L^{p}(\Omega ;W_{\#}^{1,p}(\Delta (A)))$, and the transformation $u\to \partial _{i}\circ u$ (usual composition) maps continuously and linearly $L^{p}(\Omega ;W_{\#}^{1,p}(\Delta (A)))$ into $L^{p}(\Omega \times \Delta (A))$. In the sequel we set $\partial _{i}u=\partial _{i}\circ u$ for $u\in L^{p}(\Omega ;W_{\#}^{1,p}(\Delta (A)))$. Finally, the letter $E$ throughout will denote exclusively a family of positive real numbers admitting $0$ as an accumulation point. For example $E=% \mathbb{R}_{+}^{*}$ (the positive real numbers) or $E=(\varepsilon _{n})$ (integers $n\geq 0$), where $\varepsilon _{n}>0$ with $\varepsilon _{n}\to 0$ when $n\to +\infty $. In the last case $E$ is referred to as a \emph{fundamental sequence}. \begin{definition} \label{déf3.1} \rm A sequence of functions $u_{\varepsilon }\in W^{1,p}(\Omega )$ ($\varepsilon \in E$) is said to be \textit{weakly }$% \Sigma $\textit{-convergent} in $W^{1,p}(\Omega )$ to some $u_{0}\in W^{1,p}(\Omega ;L^{p}(\Delta (A)))$ if there exists a function $u_{1}\in L^{p}(\Omega ;W_{\#}^{1,p}(\Delta (A)))$ such that as $E\ni \varepsilon \to 0$, we have: \begin{itemize} \item[(i)] $u_{\varepsilon }\to u_{0}$ in $L^{p}(\Omega )$-weak $\Sigma $ \item[(ii)] $D_{x_{i}}u_{\varepsilon }\to D_{x_{i}}u_{0}+\partial _{i}u_{1}$ in $L^{p}(\Omega )$-weak $\Sigma $, $1\leq i\leq N$. \end{itemize} We then write $u_{\varepsilon }\to u_{0}$ in $W^{1,p}(\Omega )$-weak $\Sigma $ and we refer to $u_{0}$ (which is necessarily unique) as the weak $\Sigma $-limit in $W^{1,p}$ of the sequence $(u_{\varepsilon })_{\varepsilon \in E}$. The function $u_{1}$ is called a \emph{corrector} for $(u_{\varepsilon })_{\varepsilon \in E}$. \end{definition} \begin{remark} \label{rm3.1} \rm The concept of weak $\Sigma $-convergence in $W^{1,p}(\Omega )$ is a natural generalization of the usual notion of weak convergence in $W^{1,p}(\Omega )$. Indeed, a sequence $u_{\varepsilon }\in W^{1,p}(\Omega )$ ($\varepsilon \in E$) is weakly convergent in $W^{1,p}(\Omega )$ to some $u_{0}\in W^{1,p}(\Omega )$ as $\varepsilon \to 0$ if and only if as $\varepsilon \to 0$, we have $u_{\varepsilon }\to u_{0}$ in $L^{p}(\Omega )$-weak and $D_{x_{i}}u_{\varepsilon }\to D_{x_{i}}u_{0}$ in $L^{p}(\Omega )$-weak for $i=1,\cdots ,N$. But then it amounts to saying that $u_{\varepsilon }\to u_{0}$ in $W^{1,p}(\Omega )$-weak $\Sigma _{0}$, where $\Sigma _{0}$ is the trivial $H$-structure on $\mathbb{R}^{N}$ (see Example 3.1 of \cite{n1}). \end{remark} \begin{proposition} \label{propo3.1} Suppose that $\mathcal{D}(\Delta (A))$ is dense in $W^{1,p}(\Delta (A))$. If the sequence $u_{\varepsilon }\in W^{1,p}(\Omega )$ ($\varepsilon \in E$) is weakly $\Sigma $-convergent in $W^{1,p}(\Omega )$ to some function $u_{0}\in W^{1,p}(\Omega ;$ $L^{p}(\Delta (A)))$, then $u_{\varepsilon }\to \widetilde{u}_{0}$ in $W^{1,p}(\Omega )$-weak as $E\ni \varepsilon \to 0$, where $\widetilde{u}_{0}$ is given by $\widetilde{u}% _{0}(x)=\int_{\Delta (A)}u_{0}(x,s)d\beta (s)$ for $x\in \Omega $. \end{proposition} \begin{proof} By Definition \ref{déf3.1} and use of Proposition 4.4 of \cite{n1} we have, in the weak topology in $L^{p}(\Omega )$, $u_{\varepsilon }\to \widetilde{u}_{0}$ and $D_{x_{i}}u_{\varepsilon }\to (D_{x_{i}}u_{0})^{\sim }+(\partial _{i}u_{1})^{\sim }$ ($i=1,\cdots ,N$) when $E\ni \varepsilon \to 0$. But $(D_{x_{i}}u_{0})^{% \sim }=D_{x_{i}}\widetilde{u}_{0}$ (this can be easily shown) and $(\partial _{i}u_{1})^{\sim }=0$ (this follows by the same line of reasoning as in the case of Proposition 4.8 in \cite{n1}). Hence the proposition follows. \end{proof} This proposition has two useful corollaries. \begin{corollary}\label{coro3.1} Let the hypotheses be as in Proposition \ref{propo3.1}. Assume moreover that $W^{1,p}(\Omega )$ is compactly embedded in $L^{p}(\Omega )$. Then $u_{0}\in W^{1,p}(\Omega )$. \end{corollary} \begin{proof} By Proposition \ref{propo3.1} and use of the above compactness hypothesis we have $u_{\varepsilon }\to \widetilde{u}_{0}$ in $L^{p}(\Omega )$ as $% E\ni \varepsilon \to 0$, hence $u_{\varepsilon }\to \widetilde{u}_{0}$ in $L^{p}(\Omega )$-weak $\Sigma $ (use Example 4.2 and Proposition 4.6 of \cite{n1}). Therefore $u_{0}=\widetilde{u}_{0}$, according to the unicity of the weak $\Sigma $-limit. The corollary follows. \end{proof} \begin{corollary}\label{coro3.2} Let the hypotheses be as in Proposition \ref{propo3.1}, and let us assume further that each $u_{\varepsilon }$ lies in $W_{0}^{1,p}(\Omega )$ (the closure of $\mathcal{D}(\Omega )$ in $W^{1,p}(\Omega )$). Then $u_{0}\in W_{0}^{1,p}(\Omega )$. \end{corollary} \begin{proof} Indeed, since $\Omega $ is bounded, we have that $W_{0}^{1,p}(\Omega )$ is compactly embedded in $L^{p}(\Omega )$ (this is classical). Therefore the corollary follows in the same way as above. \end{proof} \begin{remark} \label{rm3.2}\rm If $\Sigma $ is an almost periodic $H$-structure (cf.Example 3.3 of \cite{n1}) then we arrive at the conclusion of Corollary \ref{coro3.1} without assuming that $W^{1,p}(\Omega )$ is compactly embedded in $L^{p}(\Omega )$ (proceed as in the proof of Theorem 4.1 of \cite{n4}). \end{remark} \subsection{Proper $H$-structures} In what follows, $\Sigma $ denotes an $H$-structure on $\mathbb{R}^{N}$. We set $A=\mathcal{J}(\Sigma )$. We assume that $\Sigma $ is of class $\mathcal{C}% ^{\infty }$ (see Subsection 3.1). Let $1
1$ if the following two conditions are satisfied: \begin{itemize} \item[(P1)] $\Sigma $ is total (for $p$), i.e., $\mathcal{D}(\Delta (A))$ is dense in $W^{1,p}(\Delta (A))$ \item[(P2)] For each bounded open set $\Omega \subset \mathbb{R}_{x}^{N}$, $W^{1,p}(\Omega )$ is $\Sigma $-reflexive. \end{itemize} \end{definition} \begin{remark} \label{rm3.5} \rm If $\Sigma $ is total (for $p$), then $J(\mathcal{D}(\Delta (A))/\mathbb{C})$ is dense in $W_{\#}^{1,p}(\Delta (A))$, where $\mathcal{D}(\Delta (A))/\mathbb{C}=\{\varphi \in \mathcal{D}(\Delta (A)):\int_{\Delta (A)}\varphi d\beta =0\}$. Furthermore, $\int_{\Delta (A)}\partial _{i}ud\beta =0$ ($1\leq i\leq N$) for $u\in W_{\#}^{1,p}(\Delta (A))$. Indeed, this follows by the same arguments as in the proof of Proposition 4.8 in \cite{n1}. \end{remark} Several examples of proper $H$-structures for $p=2$ are available in \cite{n1,n2}. However, in the present study we need to discuss the properness for a wide range of reals $p>1$. Indeed, the more the proper $H$% -structures (for $p$) at our disposal, the wider the range of those homogenization problems (for (\ref{1.4})) that can be worked out beyond the classical periodic setting. In \cite{n1} our quest of proper $H$-structures led us to a general properness result (see Theorem 4.2 of \cite{n1}) whose practicality has been established in the case $p=2$. Of course, there is much to be gained by extending such a result to $1
1$, then so also is $\Sigma $. Before we can do this, however, we need a few notations and lemmas. To begin with, if $u\in L^{p}(\Omega ;L^{p}(\Delta (A)))$, where $\Omega $ is as thereinbefore, we set $L_{\Omega }u(x)=L(u(x))$ ($x\in \Omega $), which defines an isometric isomorphism $L_{\Omega }$ of $L^{p}(\Omega ;L^{p}(\Delta (A)))$ onto $% L^{p}(\Omega ;L^{p}(\Delta (A_{2})))$, according to (H). We will denote by $% J_{2}$ the canonical mapping of $W^{1,p}(\Delta (A_{2}))/\mathbb{C}$ into its separated completion $W_{\#}^{1,p}(\Delta (A_{2}))$, whereas $J$ denotes the canonical mapping of $W^{1,p}(\Delta (A))/\mathbb{C}$ into $W_{\#}^{1,p}(\Delta (A))$, as stated above. Each of the following three lemmas can be obtained by simple adaptation of the proof of its analog in \cite{n1}. The details are left to the reader. \begin{lemma} \label{lm3.1} Suppose $\Sigma _{2}$ is total (for $p$). Then the following assertions are true: \begin{enumerate} \item[(i)] If $u\in W^{1,p}(\Delta (A))$, then $Lu\in W^{1,p}(\Delta (A_{2}))$ and further $\partial _{i}(Lu)=L(\partial _{i}u)$ $(1\leq i\leq N)$. \item[(ii)] The restriction of the operator $L$ to $W^{1,p}(\Delta (A))$ is an isometric isomorphism of $W^{1,p}(\Delta (A))$ onto $W^{1,p}(\Delta (A_{2}))$. \item[(iii)] $\Sigma $ is total (for $p$). \end{enumerate} \end{lemma} \begin{lemma} \label{lm3.2} Suppose $\Sigma _{2}$ is total (for $p$). Then there exists an isometric isomorphism $L_{\#}:W_{\#}^{1,p}(\Delta (A))\to W_{\#}^{1,p}(\Delta (A_{2}))$ such that $L_{\#}(Jf)=J_{2}(Lf)$\ for $f\in W^{1,p}(\Delta (A))/\mathbb{C}$ and $\partial _{i}L_{\#}(u)=L(\partial _{i}u)$\ for\ $u\in W_{\#}^{1,p}(\Delta (A))$ ($1\leq i\leq N$). \end{lemma} \begin{lemma}\label{lm3.3} Suppose $E$ is a fundamental sequence, and let $(u_{\varepsilon })_{\varepsilon \in E}$ be a sequence in $L^{p}(\Omega )$ ($\Omega $ as in Definition \ref{déf3.2}) such that $u_{\varepsilon }\to v_{0}$ in $L^{p}(\Omega )$-weak $\Sigma _{2}$ as $E\ni \varepsilon \to 0$, where $v_{0}\in L^{p}(\Omega \times \Delta (A_{2}))$. Then $u_{\varepsilon}\to L_{\Omega }^{-1}v_{0}$ in $L^{p}(\Omega )$-weak $\Sigma $ ($L_{\Omega }^{-1}$ the inverse isomorphism of $L_{\Omega }$). \end{lemma} We turn now to the statement and proof of the desired result. \begin{theorem}\label{théo3.1} Suppose $\Sigma _{2}$ is proper (for $p$). Then $\Sigma $ is proper (for $p$). \end{theorem} \begin{proof} Since $\Sigma $ is total for $p$ (according to Lemma \ref{lm3.1}), the whole problem amounts to verifying that $W^{1,p}(\Omega )$ is $\Sigma$-reflexive for each given bounded open set $\Omega \subset \mathbb{R}_{x}^{N}$. To do this, let $(u_{\varepsilon })_{\varepsilon \in E}$ ($E$ a fundamental sequence) be a bounded sequence in $W^{1,p}(\Omega )$. According to the $\Sigma _{2}$-reflexivity of $W^{1,p}(\Omega )$, there exist a subsequence $E'$ extracted from $E$ and two functions $v_{0}\in W^{1,p}(\Omega ;L^{p}(\Delta (A_{2})))$ and $v_{1}\in L^{p}(\Omega ,W_{\#}^{1,p}(\Delta (A_{2})))$ such that if $E'\ni \varepsilon \to 0$, then $u_{\varepsilon }\to v_{0}$ in $L^{p}(\Omega )$-weak $\Sigma _{2}$ and $D_{x_{i}}u_{\varepsilon }\to D_{x_{i}}v_{0}+\partial _{i}v_{1}$ in $L^{p}(\Omega )$-weak $\Sigma _{2}$ ($1\leq i\leq N$). Let $u_{0}=L_{\Omega }^{-1}v_{0}$ and $u_{1}=L_{\#\Omega }^{-1}v_{1}$, where $L_{\#\Omega }$ denotes the isometric isomorphism of $L^{p}(\Omega ,W_{\#}^{1,p}(\Delta (A)))$ onto $L^{p}(\Omega ,W_{\#}^{1,p}$ $(\Delta (A_{2})))$ defined by $L_{\#\Omega }u=L_{\#}\circ u$, $u\in L^{p}(\Omega ,W_{\#}^{1,p}(\Delta (A)))$. By applying \cite[Theorem 1 of page 142]{b3} with $X=\Omega $, $F=L^{p}(\Delta (A_{2}))$, $G=L^{p}(\Delta (A))$ and $u=L^{-1}$, we see immediately that $L_{\Omega }^{-1}D_{x_{i}}v_{0}=D_{x_{i}}L_{\Omega }^{-1}v_{0}$ ($i=1,\cdots ,N$). Hence $u_{0}\in W^{1,p}(\Omega ;L^{p}(\Delta (A)))$ with $D_{x_{i}}u_{0}=L_{\Omega }^{-1}D_{x_{i}}v_{0}$ ($i=1,\cdots ,N$). On the other hand, it is clear that $u_{1}\in L^{p}(\Omega ,W_{\#}^{1,p}(\Delta (A)))$ with $\partial _{i}u_{1}=L_{\Omega }^{-1}(\partial _{i}v_{1})$ ($i=1,\cdots ,N$). Hence, recalling Lemma \ref{lm3.3}, it follows that as $E'\ni \varepsilon \to 0$, we have $u_{\varepsilon }\to u_{0}$ in $L^{p}(\Omega )$-weak $\Sigma $ and $D_{x_{i}}u_{\varepsilon }\to D_{x_{i}}u_{0}+\partial _{i}u_{1}$ ($i=1,\cdots ,N$) in $L^{p}(\Omega )$-weak $\Sigma $, which completes the proof. \end{proof} In the next subsection we will use the preceding theorem to establish the properness of some specific $H$-structures. \subsection{Examples of proper $H$-structures} We present here five basic examples of proper $H$-structures. \begin{example}\label{ex3.1} \rm Periodic $H$-structures. Let $\Sigma _{S}$ be the periodic $H$-structure on $\mathbb{R}^{N}$ represented by the network $S=\mathbb{Z}^{N}$ (cf.Example 3.2 of \cite{n1}). We intend showing that $\Sigma _{S}$ is proper for each $p>1$. We will set $Y=(0,1)^{N}$ (the open unit cube in $\mathbb{R}_{y}^{N}$). We recall that the image $\mathcal{J}(\Sigma _{S})$ is here $A=\mathcal{C}_{\rm per}(Y)$ (the space of $Y$-periodic continuous complex functions on $\mathbb{R}_{y}^{N}$). On the other hand, we have (up to an isometric isomorphism) $L^{p}(\Delta (A))\equiv L_{\rm per}^{p}(Y)=\{v\in L_{\rm loc}^{p}(\mathbb{R}_{y}^{N})$: $v$ $Y$-periodic\}, $W^{1,p}(\Delta (A))\equiv W_{\rm per}^{1,p}(Y) =\{v\in W_{\rm loc}^{1,p}(\mathbb{R}_{y}^{N})$: $v$ $Y$-periodic\} and $W_{\#}^{1,p}(\Delta (A))\equiv W_{\#}^{1,p}(Y)=\{v\in W_{\rm per}^{1,p}(Y): \int_{Y}v(y)dy=0\}$. This follows by simple arguments that can be found in \cite{n4}. We also draw attention to the fact that $\Sigma _{S}$-convergence in the present context is nothing else but two-scale convergence \cite{a1,l3}. Each of the spaces $L_{\rm per}^{p}(Y)$, $W_{\rm per}^{1,p}(Y)$ and $W_{\#}^{1,p}(Y)$ is a Banach space under a standard norm: \begin{gather*} \| u\| _{L^{p}(Y)}=\Big( \int_{Y}| u(y)| ^{p}dy\Big) ^{1/p} \quad (u\in L_{\rm per}^{p}(Y)), \\ \| u\| _{W^{1,p}(Y)}=\| u\|_{L^{p}(Y)} +\sum_{i=1}^{N}\| D_{y_{i}}u\| _{L^{p}(Y)} \quad (u\in W_{\rm per}^{1,p}(Y)) \end{gather*} and \begin{equation*} \| u\| _{W_{\#}^{1,p}(Y)}=\sum_{i=1}^{N}\| D_{y_{i}}u\| _{L^{p}(Y)} \quad (u\in W_{\#}^{1,p}(Y)). \end{equation*} Finally, we conclude this preliminary step by recalling that $\Sigma_{S}$ is of class $\mathcal{C}^{\infty }$. \end{example} Having made this point, let us begin by showing that $\Sigma _{S}$ is total for any arbitrary real $p>1$. \begin{proposition} \label{propo3.2} $\Sigma _{S}$ is total for $p>1$. \end{proposition} \begin{proof} Let $\theta \in \mathcal{D}(\mathbb{R}_{y}^{N})=\mathcal{C}% _{0}^{\infty }(\mathbb{R}_{y}^{N})$ with $\theta \geq 0$, $\int \theta (y)dy=1$, $\theta $ having support in the closed unit ball of $\mathbb{R}^{N}$. For each integer $n\geq 1$, we put $\theta _{n}(x)=n^{N}\theta (nx)$, $x\in \mathbb{R}% ^{N}$, which gives a sequence of functions $\theta _{n}\in \mathcal{D}(\mathbb{R% }_{x}^{N})$. This being so, let $u\in L_{\rm per}^{p}(Y)$, where $1
1$. We need one basic lemma. \begin{lemma} \label{lm3.4} Let $\mathbf{f}=(f_{i})\in L_{\rm per}^{p}(Y)^{N}$. Suppose \begin{equation} \int_{Y}\mathbf{f}\cdot \Psi dy\equiv \sum_{i=1}^{N}\int_{Y}f_{i}\psi _{i}dy=0 \quad (\Psi \in \mathcal{V}_{\rm per}) \label{3.8} \end{equation} where $\mathcal{V}_{\rm per}=\{\Psi =(\psi _{i})\in \mathcal{C}_{\rm per}^{\infty}(Y)^{N}: \mathop{\rm div}_{y}\Psi =0\}$. Then there is a unique $q\in W_{\#}^{1,p}(Y)$ such that \begin{equation} D_{y}q=\mathbf{f}. \label{3.9} \end{equation} \end{lemma} \begin{proof} Consider again the sequence $(\theta _{n})$ in the proof of Proposition \ref{propo3.2}. Let $\mathbf{\varphi }_{n}=\mathbf{f}*\theta _{n} $ (integers $n\geq 1$), and bear in mind that $\mathbf{\varphi }_{n}\in \mathcal{C}_{\rm per}^{\infty }(Y)^{N}$ and $\mathbf{\varphi }_{n}\to \mathbf{f}$ in $L_{\rm per}^{p}(Y)^{N}$ when $n\to \infty $. Now, for $% \Psi \in \mathcal{V}_{\rm per}$, note that \begin{equation*} \int_{Y}\mathbf{\varphi }_{n}\cdot \Psi dy=\int \theta _{n}(x)\left( \int_{Y-x}\mathbf{f}(y)\cdot \Psi _{x}(y)dy\right) dx \end{equation*} where $\Psi _{x}(y)=\Psi (y+x)$. But \begin{equation*} \int_{Y-x}\mathbf{f}(y)\cdot \Psi _{x}(y)dy=\int_{Y}\mathbf{f}(y)\cdot \Psi _{x}(y)dy=0\text{ (use (\ref{3.8})), since }\Psi _{x}\in \mathcal{V}_{\rm per}. \end{equation*} Hence (\ref{3.8}) still holds when $\mathbf{f}$ is replaced by $\mathbf{% \varphi }_{n}$. Consequently, by a well-known result there exists $q_{n}\in W_{\#}^{1,2}(Y)$ such that $D_{y}q_{n}=\mathbf{\varphi }_{n}$. Furthermore, thanks to a classical regularity result (see \cite[page 61]{s1}), $q_{n}$ belongs to $\mathcal{C}_{\rm per}^{1}(Y)=\mathcal{C}^{1}(\mathbb{R}^{N})\cap \mathcal{C}_{\rm per}(Y)$, thus $q_{n}\in W_{\#}^{1,p}(Y)$ and that for any integer $n\geq 1$. But then the sequence $(q_{n})$ is Cauchy in $% W_{\#}^{1,p}(Y)$. Therefore $q_{n}\to q$ in $W_{\#}^{1,p}(Y)$ as $% n\to \infty $, and it is clear that $q$ is the sole function in $% W_{\#}^{1,p}(Y)$ satisfying (\ref{3.9}). \end{proof} This leads to the claimed result. \begin{proposition} \label{propo3.3} For each $\Omega \subset \mathbb{R}_{x}^{N}$ as above, $W^{1,p}(\Omega )$ is $\Sigma _{S}$-reflexive for any arbitrary real $p>1$. \end{proposition} \begin{proof} Once we have Lemma \ref{lm3.4} at our disposal, the proposition follows by a classical way (see, e.g., \cite[Theorem 13]{l3}). \end{proof} We are now justified in stating the final conclusion. \begin{proposition} \label{propo3.4} $\Sigma _{S}$ is proper for any arbitrary $p>1$. \end{proposition} \begin{example} \label{ex3.2} \rm Any almost periodic $H$-structure on $\mathbb{R}^{N}$ is proper for $p=2$ (cf. Theorem 4.1 and Proposition 4.3 of \cite{n4}). \end{example} \begin{example} \label{ex3.3} \rm Let $\Sigma _{\infty ,S}$ ($S=\mathbb{Z}^{N}$) be the $H$-structure (of class $\mathcal{C}^{\infty }$) on $\mathbb{R}^{N}$ defined by $\mathcal{J}(\Sigma _{\infty ,S})=\mathcal{B}_{\infty ,S}(\mathbb{R}^{N})$, where $\mathcal{B}_{\infty ,S}(\mathbb{R}^{N})$ denotes the closure in $\mathcal{B}(\mathbb{R}^{N})$ of the space of all finite sums \begin{equation*} \sum_{\rm finite}\varphi _{i}u_{i}\quad \text{with }\varphi _{i}\in A_{1}=\mathcal{B}_{\infty }(\mathbb{R}^{N})\text{ and } u_{i}\in A_{2}=\mathcal{C}_{\rm per}(Y)\quad (Y=(0,1)^{N}). \end{equation*} The $H$-structure $\Sigma _{\infty ,S}$ is proper for $p=2$ (cf. \cite[Corollary 4.2]{n1}). Our purpose is to show that $\Sigma _{\infty ,S}$ is actually proper for any $p>1$. Let $\mathcal{V}=\mathcal{B}_{0}(\mathbb{R}^{N})\oplus \mathcal{C}_{\rm per}(Y)$ (direct sum), where $\mathcal{B}_{0}(\mathbb{R}^{N})$ denotes the space of those functions in $\mathcal{C}(\mathbb{R}^{N})$ that vanish at infinity. It is worth noting that $\mathcal{V}$ is dense in $A=\mathcal{B}_{\infty ,S}(\mathbb{R}^{N})$ (cf.\cite{n1}). In the sequel we use the same $\mathcal{G}$ to denote the Gelfand transformation on $A$ and on $A_{2}$, as well. This being so, define the mapping $\ell _{2}:\mathcal{V}\to \mathcal{V}$ as being the projection on $A_{2}$ along $\mathcal{B}_{0}(\mathbb{R}^{N})$. Thus, each $\psi \in \mathcal{V}$ admits the unique decomposition $\psi =\psi _{0}+\ell _{2}(\psi )$ with $\psi _{0}\in \mathcal{B}_{0}(\mathbb{R}^{N})$. Let $L_{2}$ be the mapping of $\mathcal{G}(\mathcal{V})$ into $\mathcal{C}(\Delta (A_{2}))$ defined by $L_{2}(\mathcal{G}(\psi ))=\mathcal{G}(\ell _{2}(\psi ))$ for $\psi \in \mathcal{V}$. In view of Proposition \ref{propo3.4}, the desired properness result will follow by Theorem \ref{théo3.1} exactly in the same way as Corollary 4.2 of \cite{n1} was obtained by Theorem 4.2 of the same reference provided we check that $L_{2}$ extends by continuity to an isometric isomorphism $L$ of $L^{p}(\Delta (A))$ onto $L^{p}(\Delta (A_{2}))$. This is straightforward by the classical inequality $| | a| ^{p}-| b| ^{p}| \leq p| a-b| (| a| +| b| )^{p-1}$ ($a,b\in \mathbb{C}$). Indeed, if $\psi \in \mathcal{V}$ then by choosing $a=\psi (y)$ and $b=\ell _{2}(\psi )(y)$ for fixed $y\in \mathbb{R}^{N}$, we get $| M(| \psi |^{p})-M(| \ell _{2}(\psi )| ^{p})| \leq CM(| \psi _{0}| )$, where $C$ denotes the supremum of the function $p(| \psi | +| \ell _{2}(\psi )| )^{p-1}$. Since $M(| \psi _{0}| )=0$, we deduce $M(| \psi | ^{p})=M(| \ell _{2}(\psi )| ^{p})$, i.e., $\| L_{2}(\widehat{\psi })\| _{L^{p}(\Delta (A_{2}))}=\| \widehat{\psi }\| _{L^{p}(\Delta (A))}$ with $\widehat{\psi }=\mathcal{G}(\psi )$, and that for all $\psi \in \mathcal{V}$. Therefore, thanks to the density of $\mathcal{G}(\mathcal{V})$ in $L^{p}(\Delta (A))$, the isometric isomorphism $L$ follows. \end{example} \begin{example} \label{ex3.4} \rm For any countable subgroup $\mathcal{R}$ of $\mathbb{R}^{N}$, the $H$-structure (of class $\mathcal{C}^{\infty }$) $\Sigma _{\infty ,\mathcal{R}}$ on $\mathbb{R}^{N}$ \cite[Example 3.5]{n1} is proper for $p=2$ (cf. \cite[Corollary 4.2]{n1}). In particular the $H$-structure $\Sigma _{\infty }$ \cite[Example 3.4]{n1} is proper for $p=2$, since it coincides with $\Sigma _{\infty ,\mathcal{R}}$ for $\mathcal{R}=\{\omega \}$ ($\omega $ the origin of $\mathbb{R}^{N}$). \end{example} \begin{example}\label{ex3.5} \rm Let $\Sigma _{\infty }$ be the $H$-structure of the convergence at infinity on $\mathbb{R}$, and let $\Sigma _{\mathcal{R}'}$ be the almost periodic $H$-structure on $\mathbb{R}^{N-1}$ represented by a countable subgroup $\mathcal{R}'$ of $\mathbb{R}^{N-1}$. Then the product $H$-structure $\Sigma =\Sigma _{\mathcal{R}'}\times \Sigma _{\infty }$ on $\mathbb{R}^{N}$ is proper for $p=2$ (see \cite[Example 4.4]{n1}). \end{example} \section{The abstract homogenization problem} Throughout this section, $\Sigma $ denotes an $H$-structure of class $% \mathcal{C}^{\infty }$ on $\mathbb{R}^{N}$. We put $A=\mathcal{J}(\Sigma )$ and we denote by $\mathcal{G}$ the Gelfand transformation on $A$ and by $\beta $ the $M$-measure (on $\Delta (A)$) for $A$. Finally, $\Omega $ denotes a bounded open set in $\mathbb{R}_{x}^{N}$. \subsection{Preliminaries} Let $1\leq p<\infty $. We begin by introducing the space $\Xi ^{p}$ of all $u\in L_{\rm loc}^{p}(\mathbb{R}_{y}^{N})$ for which the sequence $(u^{\varepsilon})_{0<\varepsilon \leq 1}$ is bounded in $L_{\rm loc}^{p}(\mathbb{R}_{x}^{N})$ ($% u^{\varepsilon }$ defined in (\ref{2.1})). We provide $\Xi ^{p}$ with the norm \begin{equation*} \|u\|_{\Xi ^{p}}=\sup_{o<\varepsilon \leq 1}\Big( \int_{B_{N}}| u\big( \frac{x}{\varepsilon }\big) | ^{p}dx\Big) ^{1/p} \quad (u\in \Xi ^{p}) \end{equation*} where $B_{N}$ denotes the open unit ball in $\mathbb{R}_{x}^{N}$, which makes it a Banach space. This being so, we define $\mathfrak{X}_{\Sigma }^{p}$ to be the closure of $A$ in $\Xi ^{p}$. Provided with the $\Xi ^{p}$-norm, $\mathfrak{X}_{\Sigma }^{p}$ is a Banach space. Furthermore, the Gelfand transformation on $A$ extends by continuity to a continuous linear mapping, still denoted by $\mathcal{G}$, of $\mathfrak{X}_{\Sigma }^{p}$ into $L^{p}(\Delta (A))$ (cf. \cite{n1}). This is referred to as the \textit{canonical mapping} of $\mathfrak{X}_{\Sigma }^{p}$ into $L^{p}(\Delta (A))$. Given a locally compact space $X$ (equipped with a positive Radon measure), we will most of the time put $L_{\mathbb{R}}^{p}(X)=L^{p}(X;\mathbb{R})$, $\mathcal{C}_{\mathbb{R}}(X)=\mathcal{C}(X;\mathbb{R})$. In particular we will write $\mathcal{D}_{\mathbb{R}}(\Omega )=\mathcal{D}(\Omega ;\mathbb{R})$. Likewise we will put $A_{\mathbb{R}}=A\cap \mathcal{C}_{\mathbb{R}}(\mathbb{R}^{N})$ and $A_{\mathbb{R}}^{\infty }=A^{\infty }\cap \mathcal{C}_{\mathbb{R}}(\mathbb{R}% ^{N})$. The main purpose of this section is to investigate the behaviour, as $\varepsilon \to 0$, of $u_{\varepsilon }$ (the solution of (\ref{1.4})) under the abstract structure hypothesis \begin{equation} a_{i}(\cdot ,\Psi )\in \mathfrak{X}_{\Sigma }^{p'}\quad \text{for all } \Psi \in (A_{\mathbb{R}})^{N}\quad (1\leq i\leq N) \label{4.1} \end{equation} where $p'=\frac{p}{p-1}$ with $1
0$. Letting $\varepsilon \to 0$, it follows by \cite[Corollary 4.1]{n1} and use of Remark \ref{rm4.1} below that \begin{equation} | b(\widehat{\mathbf{f}})-b(\widehat{\mathbf{g}})| \leq c| \widehat{\mathbf{f}}-\widehat{\mathbf{g}}| ^{p-1}\quad\text{a.e. in }% \Omega \times \Delta (A) \label{4.6} \end{equation} and \begin{equation} [ b(\widehat{\mathbf{f}})-b(\widehat{\mathbf{g}})]\cdot (\widehat{% \mathbf{f}}-\widehat{\mathbf{g}})\geq \alpha | \widehat{\mathbf{f}}-% \widehat{\mathbf{g}}| ^{p}\quad\text{a.e. in }\Omega \times \Delta (A) \label{4.7} \end{equation} where $b(\widehat{\mathbf{f}})=(b_{i}\circ \widehat{\mathbf{f}})_{1\leq i\leq N}$. Immediately (\ref{4.6}) yields \begin{equation*} \| b(\widehat{\mathbf{f}})-b(\widehat{\mathbf{g}})\| _{L^{p'}(\Omega \times \Delta (A))^{N}}\leq c\| \widehat{% \mathbf{f}}-\widehat{\mathbf{g}}\| _{L^{p}(\Omega ;L^{p}(\Delta (A))^{N})}^{p-1} \end{equation*} and that for any $\mathbf{f},\mathbf{g}\in \mathcal{C}(\overline{\Omega }% ;(A_{\mathbb{R}})^{N})$. Hence, thanks to the fact that $\mathcal{C}(\overline{% \Omega };\mathcal{C}_{\mathbb{R}}(\Delta (A))^{N})$ is (identifiable with) a dense subspace of $L^{p}(\Omega ;L_{\mathbb{R}}^{p}(\Delta (A))^{N})$, it follows that $b$ extends by continuity to a mapping, still denoted $b$, of $% L^{p}(\Omega ;L_{\mathbb{R}}^{p}(\Delta (A))^{N})$ into $L^{p'}(\Omega \times \Delta (A))^{N}$ such that (\ref{4.4}) holds for all $\mathbf{u},\mathbf{v}\in L^{p}(\Omega ;L_{\mathbb{R}}^{p}(\Delta (A))^{N})$. Finally, (\ref{4.5}) follows from (\ref{4.7}) by the above density argument combined with the continuity of $b:L^{p}(\Omega ;L_{\mathbb{R}}^{p}(\Delta (A))^{N})\to L^{p'}(\Omega \times \Delta (A))^{N}$ and use of the fact that $L^{p}(\Omega ;L_{\mathbb{R}}^{p}(\Delta (A))^{N})$ may be identified with $L_{\mathbb{R}}^{p}(\Omega \times \Delta (A))^{N}$. This completes the proof. \end{proof} \begin{remark} \label{rm4.1} \rm If $v_{\varepsilon }\in L^{p}(\Omega )$ ($\varepsilon >0$) with $v_{\varepsilon }\to v_{0}$ in $L^{p}(\Omega )$-weak $\Sigma $ (as $\varepsilon \to 0$) and if for each $\varepsilon >0$ we have $v_{\varepsilon }\geq 0$ a.e. in $\Omega $, then $v_{0}\geq 0$ a.e. in $\Omega \times \Delta (A)$. \end{remark} The preceding proposition has an important corollary. \begin{corollary} \label{coro4.1} Let $\phi _{\varepsilon }=\psi _{0}+\varepsilon \psi _{1}^{\varepsilon }$, i.e., $\phi _{\varepsilon }(x)=\psi _{0}(x)+\varepsilon \psi _{1}(x,\frac{x}{% \varepsilon })$, $x\in \Omega $, where $\psi _{0}\in \mathcal{D}_{\mathbb{R}% }(\Omega )$ and $\psi _{1}\in \mathcal{D}_{\mathbb{R}}(\Omega )\otimes A_{\mathbb{R% }}^{\infty }$. Then, when $\varepsilon \to 0$, \begin{equation} a_{i}^{\varepsilon }(\cdot ,D\phi _{\varepsilon })\to b_{i}(D_{x}\psi _{0}+\partial \widehat{\psi }_{1})\text{ in } L^{p'}(\Omega )\text{-weak }\Sigma \quad (1\leq i\leq N) \label{4.8} \end{equation} where $\partial \widehat{\psi }_{1}=(\partial _{i}\widehat{\psi }% _{1})_{1\leq i\leq N}$. Furthermore, if $(v_{\varepsilon })_{\varepsilon \in E}$ is a sequence in $L^{p}(\Omega )$ such that $v_{\varepsilon }\to v_{0}$ in $L^{p}(\Omega )$-weak $\Sigma $ as $E\ni \varepsilon \to 0$, then, as $E\ni \varepsilon \to 0$, \begin{equation*} \int_{\Omega }a_{i}^{\varepsilon }(\cdot ,D\phi _{\varepsilon })v_{\varepsilon }dx\to \int \int_{\Omega \times \Delta (A)}b_{i}(D_{x}\psi _{0}+\partial \widehat{\psi }_{1})v_{0}dxd\beta \quad (1\leq i\leq N). \end{equation*} \end{corollary} \begin{proof} Since $D\phi _{\varepsilon }=D_{x}\psi _{0}+\varepsilon (D_{x}\psi _{1})^{\varepsilon }+(D_{y}\psi _{1})^{\varepsilon }$, it is clear that \begin{equation*} \| a^{\varepsilon }(\cdot ,D\phi _{\varepsilon })-a^{\varepsilon }(\cdot ,D_{x}\psi _{0}+(D_{y}\psi _{1})^{\varepsilon })\| _{L^{p'}(\Omega )}\leq c\varepsilon ^{p-1}\| D_{x}\psi _{1}\| _{L^{p}(\Omega ;\mathcal{B}(\mathbb{R}^{N}))}^{p-1}. \end{equation*} Therefore (\ref{4.8}) follows by Proposition \ref{propo4.1} (see especially (% \ref{4.3})). Finally, recalling that for $\Psi =D_{x}\psi _{0}+D_{y}\psi _{1} $ the function $x\to a_{i}(\cdot ,\Psi (x,\cdot ))$ lies in $\mathcal{C}(\overline{\Omega };\mathfrak{X}_{\Sigma }^{p',\infty })$, we see that the last part of the corollary follows immediately by \cite[Proposition 4.5]{n1}. \end{proof} \subsection{The abstract homogenization result} The main purpose of this subsection is to state and prove the following theorem. \begin{theorem} \label{theo4.1} Let $1
0$, let $u_{\varepsilon }$ be the solution of (\ref{1.4}). As $\varepsilon \to 0$, we have \begin{equation} u_{\varepsilon }\to u_{0}\quad \text{in }W_{0}^{1,p}(\Omega )\text{-weak} \label{4.9} \end{equation} and \begin{equation} D_{x_{i}}u_{\varepsilon }\to D_{x_{i}}u_{0}+\partial _{i}u_{1}\text{ in }L^{p}(\Omega )\text{-weak }\Sigma \quad(1\leq i\leq N), \label{4.10} \end{equation} where $\mathbf{u}=(u_{0},u_{1})$ is uniquely defined by $\mathbf{u}=(u_{0},u_{1})\in \mathbb{F}_{0}^{1,p}$ and \begin{equation} \iint_{\Omega \times \Delta (A)}b(D_{x}u_{0}+\partial u_{1})\cdot (D_{x}v_{0}+\partial v_{1})dxd\beta =\left\langle f,v_{0}\right\rangle \label{4.11} \end{equation} for all $\mathbf{v}=(v_{0},v_{1})\in \mathbb{F}_{0}^{1,p}$. \end{theorem} \begin{proof} It is a routine exercise to verify that the sequence $% (u_{\varepsilon })_{\varepsilon >0}$ is bounded in $W_{0}^{1,p}(\Omega )$. Therefore, given an arbitrary fundamental sequence $E$, the properness of $% \Sigma $ and the compactness of the embedding $W_{0}^{1,p}(\Omega )\to L^{p}(\Omega )$ yield a subsequence $E'$ from $E$ and a couple $\mathbf{u}=(u_{0},u_{1})\in \mathbb{F}_{0}^{1,p}$ such that (\ref{4.9}% ) and (\ref{4.10}) hold when $E'\ni \varepsilon \to 0$ (use Corollary \ref{coro3.2}). If we prove that $\mathbf{u}=(u_{0},u_{1})$ verifies the variational equation in (\ref{4.11}), since such an equation admits at most one solution by virtue of (\ref{4.5}), then it will turn out that (\ref{4.9}) and (\ref{4.10}) hold not only when $E\ni \varepsilon \to 0$ but merely when $0<\varepsilon \to 0$. To this end let $\Phi \in F_{0}^{\infty }$, that is, let $\Phi =(\psi _{0},J(\widehat{% \psi }_{1}))$ with $\psi _{0}\in \mathcal{D}_{\mathbb{R}}(\Omega )$, $\psi _{1}\in \mathcal{D}_{\mathbb{R}}(\Omega )\otimes (A_{\mathbb{R}}^{\infty }/\mathbb{C}% ) $ and $J(\widehat{\psi }_{1})=J\circ \widehat{\psi }_{1}$. Define $\phi _{\varepsilon }$ as in Corollary \ref{coro4.1}. Clearly \begin{equation} 0\leq \left\langle f,u_{\varepsilon }-\phi _{\varepsilon }\right\rangle -\int_{\Omega }a^{\varepsilon }(\cdot ,D\phi _{\varepsilon })\cdot (Du_{\varepsilon }-D\phi _{\varepsilon })dx. \label{4.12} \end{equation} Indeed, the right-hand side is equal to \begin{equation*} \int_{\Omega }[a^{\varepsilon }(\cdot ,Du_{\varepsilon })-a^{\varepsilon }(\cdot ,D\phi _{\varepsilon })]\cdot (Du_{\varepsilon }-D\phi _{\varepsilon })dx\text{ } \end{equation*} and the latter is nonnegative. Now, noting that $\phi _{\varepsilon }\to \psi _{0}$ in $W_{0}^{1,p}(\Omega )$-weak as $\varepsilon \to 0$, we next pass to the limit (as $E'\ni \varepsilon \to 0$) in (\ref{4.12}) using (\ref{4.9}), (\ref{4.10}) and Corollary \ref{coro4.1}, and we obtain \begin{equation} 0\leq \left\langle f,u_{0}-\psi _{0}\right\rangle -\iint_{\Omega \times \Delta (A)}b(\mathbb{D}\Phi )\cdot \mathbb{D}(\mathbf{u}-\Phi )\,dx\,d\beta \label{4.13} \end{equation} where $\mathbb{D}\Phi =D_{x}\psi _{0}+\partial \widehat{\psi }_{1}$ and $\mathbb{D}% (\mathbf{u}-\Phi )=D_{x}(u_{0}-\psi _{0})+\partial (u_{1}-\widehat{\psi }% _{1})$. Thanks to the density of $F_{0}^{\infty }$ in $\mathbb{F}_{0}^{1,p}$, (% \ref{4.13}) still holds for any $\Phi \in \mathbb{F}_{0}^{1,p}$. Finally, take in (\ref{4.13}) the particular $\Phi =\mathbf{u}-t\mathbf{v}$ with $t>0$ and $\mathbf{v}=(v_{0},v_{1})\in \mathbb{F}_{0}^{1,p}$, then divide both sides of the resultant inequality by $t$ and, letting $t\to 0$, pass to the limit using (\ref{4.4}). Hence (\ref{4.11}) follows at once. \end{proof} The variational problem (\ref{4.11}) is referred to as the \emph{global homogenized problem for} (\ref{1.4}) under the structure hypothesis (\ref {4.1}) (where $\Sigma $ is assumed to be proper for $p$). It is immediate that the variational equation in (\ref{4.11}) is equivalent to the system of the two equations : \begin{equation} \iint_{\Omega \times \Delta (A)}b(D_{x}u_{0}+\partial u_{1}) \cdot \partial v_{1}\,dxd\beta =0 \quad \text{for all }v_{1}\in L^{p}(\Omega ;W_{\#}^{1,p}(\Delta (A);\mathbb{R})) \label{4.12'} \end{equation} and \begin{equation} \int_{\Omega }\Big[ \int_{\Delta (A)}b(D_{x}u_{0}+\partial u_{1})d\beta \Big] \cdot D_{x}v_{0}dx=\langle f,v_{0}\rangle \quad \text{for all }v_{0}\in W_{0}^{1,p}(\Omega ;\mathbb{R}). \label{4.13'} \end{equation} The next point is to derive the \emph{macroscopic homogenized problem for} (% \ref{1.4}) under hypothesis (\ref{4.1}). To this end, let $r\in \mathbb{R}^{N}$ be freely fixed, and let $\pi (r)$ be defined by the so-called \emph{cell problem}: $\pi (r)\in W_{\#}^{1,p}(\Delta (A);\mathbb{R})$ and \begin{equation} \int_{\Delta (A)}b(r+\partial \pi (r))\cdot \partial \theta \,d\beta =0\quad \text{ for all }\theta \in W_{\#}^{1,p}(\Delta (A);\mathbb{R}). \label{4.14} \end{equation} Thanks to (\ref{4.4}) and (\ref{4.5}), the existence and unicity of $\pi (r)$ follow by adaptation of a classical line of argument (see, e.g., \cite{h1}). This yields a mapping $\pi $ of $\mathbb{R}^{N}$ into $W_{\#}^{1,p}(\Delta (A);% \mathbb{R})$. Now, in (\ref{4.12'}) choose $v_{1}$ of the form $v_{1}(x)=\varphi (x)\theta $ ($x\in \Omega $) with $\varphi \in \mathcal{D}_{\mathbb{R}}(\Omega )$ and $% \theta \in W_{\#}^{1,p}(\Delta (A);$ $\mathbb{R})$. Then, almost everywhere in $% x\in \Omega ,$% \begin{equation*} \int_{\Delta (A)}b(Du_{0}(x)+\partial u_{1}(x))\cdot \partial \theta \text{ }d\beta =0\quad \text{for all }\theta \in W_{\#}^{1,p}(\Delta (A);\mathbb{R}). \end{equation*} Comparing with (\ref{4.14}) for $r=Du_{0}(x)$ ($x$ arbitrarily fixed), it follows (by the unicity argument) that $u_{1}=\pi (Du_{0})$, where the right-hand side stands for the function $x\to \pi (Du_{0}(x))$ of $% \Omega $ into $W_{\#}^{1,p}(\Delta (A);\mathbb{R})$. Hence, letting \begin{equation*} q(r)=\int_{\Delta (A)}b(r+\partial \pi (r))d\beta \quad (r\in \mathbb{R}% ^{N}), \end{equation*} we see by (\ref{4.13'}) that $u_{0}$ is a solution of the boundary value problem \begin{equation} -\mathop{\rm div}q(Du_{0})=f\quad \text{in }\Omega , \quad u_{0}\in W_{0}^{1,p}(\Omega ;\mathbb{R}). \label{4.15} \end{equation} \begin{remark} \label{rm4.2} \rm By a method similar to that which is commonly followed in the periodic case (see, e.g., \cite{l2}), it is possible to show that the homogenized operator, i.e., the operator $v\to \mathop{\rm div}q(Dv)$ of $W_{0}^{1,p}(\Omega )$ into $W_{0}^{-1,p'}(\Omega )$, is lipschitz continuous and strictly monotone as in (\ref{1.3}), so that $u_{0}$ is uniquely defined by (\ref{4.15}). \end{remark} \noindent\textbf{Application.} By way of illustration, let us suppose that $a(y,\lambda )$ verifies the classical periodicity hypothesis (see Section 1). We want to show that the abstract hypothesis (\ref{4.1}) is then verified so that Theorem \ref{theo4.1} holds. Let $1
1$ (Example \ref{ex3.3}). Thus, the solution of the homogenization problem under consideration is provided by Theorem \ref {theo4.1} if we can check that (\ref{4.1}) holds with $\Sigma =\Sigma _{\infty ,S}$. In fact, we want to show a better result namely \begin{proposition} \label{propo5.1} Suppose (\ref{1.5}) holds. Then, for all $\Psi \in (A_{\mathbb{R}})^{N}$, \begin{equation} a_{i}(\cdot ,\Psi )\in A=\mathcal{B}_{\infty ,S}(\mathbb{R}^{N})\quad (1\leq i\leq N). \label{5.1} \end{equation} \end{proposition} \begin{proof} We begin by proving (\ref{5.1}) for $\Psi $ of the form \begin{equation} \Psi =\Psi _{1}+\Psi _{2}\quad \text{with }\Psi _{1}\in (A_{1})^{N} \text{ and }\Psi _{2}\in (A_{2})^{N} \label{5.2} \end{equation} where $A_{1}=\mathcal{B}_{\infty }(\mathbb{R}^{N})\cap \mathcal{C}_{\mathbb{R}}(% \mathbb{R}^{N})$ and $A_{2}=\mathcal{C}_{\rm per}(Y)\cap \mathcal{C}_{\mathbb{R}}(\mathbb{% R}^{N})$. This will be done in two steps. 1) For fixed $1\leq i\leq N$, let us assume that $a_{i}$ is of the form \begin{equation} a_{i}(y,\lambda )=\chi (\lambda )\varphi (y)\text{ \thinspace \thinspace \thinspace \thinspace \thinspace (}y,\lambda \in \mathbb{R}^{N}) \label{5.3} \end{equation} where $\varphi \in A_{\mathbb{R}}$ and $\chi \in \mathcal{C}_{\mathbb{R}}(\mathbb{R}% ^{N})$ (see (\ref{1.5})). Let $r_{1}=\lim_{| y| \to \infty }\Psi _{1}(y)$. Clearly $a_{i}(\cdot ,r_{1}+\Psi _{2})\in A$. Next, since $\Psi _{1}$ and $\Psi _{2}$ are bounded, we may consider a compact set $K\subset \mathbb{R}^{N}$ such that $\Psi (y)=\Psi _{1}(y)+\Psi _{2}(y)\in K$ for all $y\in \mathbb{R}^{N}$. On the other hand, let $c_{1}>0$ be a constant such that $\| \varphi \| _{\infty }\leq c_{1}$. Finally, let $% \eta >0$. Since the function $\chi $ is uniformly continuous on $K$, there is some $\gamma >0$ such that $|\chi (\lambda )-\chi (\mu )|\leq \frac{\eta }{c_{1}}$ for all $\lambda ,\mu \in K$ with $| \lambda-\mu | \leq \gamma $. Furthermore, let $\rho >0$ be such that $|\Psi _{1}(y)-r_{1}| \leq \gamma $ for $y\in \mathbb{R}^{N}$ with $|y| \geq \rho $. Then \begin{equation*} | a_{i}(y,\Psi (y))-a_{i}(y,r_{1}+\Psi _{2}(y))| \leq \eta \text{ for }y\in \mathbb{R}^{N}\text{ with }| y| \geq \rho . \end{equation*} Therefore $a_{i}(\cdot ,\Psi )=a_{i}(\cdot ,r_{1}+\Psi _{2})+\phi $ with $% \phi \in \mathcal{B}_{0}(\mathbb{R}^{N})$, which shows (\ref{5.1}). But then, clearly (\ref{5.1}) still holds true for $a_{i}$ in $\mathcal{C}_{\mathbb{R}}(% \mathbb{R}^{N})\otimes A_{\mathbb{R}}$ (the finite sums $\sum \chi _{j}\otimes \varphi _{j}$ with $\varphi _{j}\in A_{\mathbb{R}}$ and $\chi _{j}\in \mathcal{C% }_{\mathbb{R}}(\mathbb{R}^{N})$). 2) Now, we consider the general case where we know merely that $a$ (in addition to (\ref{1.1})-(\ref{1.3})) satisfies (\ref{1.5}). Let $\Psi $ be given with (\ref{5.2}), and let $K$ be as above. For convenience we still denote the restriction $a_{i}\big| _{K}$ by $a_{i}$. Then, clearly $a_{i}\in \mathcal{C}(K;A_{\mathbb{R}})$. Thanks to the density of $\mathcal{C}_{\mathbb{R}}(K)\otimes A_{\mathbb{R}}$ in $\mathcal{C}(K;A_{\mathbb{R}})$ (see, e.g., page 46 of \cite{b3}), there exists a sequence of functions $\zeta _{n}\in \mathcal{C}_{\mathbb{R}}(K)\otimes A_{\mathbb{R}}$ (integers $n\geq 0$) such that \begin{equation*} \| \zeta _{n}-a_{i}\| _{\mathcal{C}(K;A)}\equiv \sup_{y\in \mathbb{R}^{N},\; \lambda \in K} | \zeta _{n}(y,\lambda )-a_{i}(y,\lambda )| \to 0\quad \text{as }n\to \infty . \end{equation*} It follows immediately that $\zeta _{n}(\cdot ,\Psi )\to a_{i}(\cdot ,\Psi )$ in $\mathcal{B}(\mathbb{R}^{N})$ (with the supremum norm) as $% n\to \infty $. But $\zeta _{n}(\cdot ,\Psi )\in A$, according to step 1). Hence (\ref{5.1}) follows for $\Psi $ of the form (\ref{5.2}). This completes step 2). Finally, let $\Psi \in (A_{\mathbb{R}})^{N}$ be arbitrarily fixed. Remarking that $\mathcal{B}_{\infty }(\mathbb{R}^{N})+\mathcal{C}_{\rm per}(Y)$ is dense in $% A $, we see that $(A_{1})^{N}+(A_{2})^{N}$ is dense in $(A_{\mathbb{R}})^{N}$ so that we may consider a sequence of functions $\mathbf{f}_{n}$ of the form (\ref{5.2}) such that $\mathbf{f}_{n}\to \Psi $ in $\mathcal{B}(\mathbb{% R}^{N})^{N}$ as $n\to \infty $. Therefore (\ref{5.1}) follows by $% \| a_{i}(\cdot ,\Psi )-a_{i}(\cdot ,\mathbf{f}_{n})\| _{\infty }\leq c\| \Psi -\mathbf{f}_{n}\| _{\infty }^{p-1}$ (according to (% \ref{1.3})) and use of the fact that $a_{i}(\cdot ,\mathbf{f}_{n})\in A$, as established previously. \end{proof} \subsection{Problem III} In this subsection we assume that the family $\{a(\cdot ,\lambda )\}_{\lambda \in \mathbb{R}^{N}}$ is uniformly equicontinuous, i.e., \begin{itemize} \item[(UE)] Given $\eta >0$, there exists $\rho >0$ such that $| a(y-r,\lambda )-a(y,\lambda )| \leq \eta $ for all $y,\lambda \in \mathbb{% R}^{N}$ provided $| r| \leq \rho $. \end{itemize} Our purpose is to investigate the behaviour (as $\varepsilon \to 0$) of $u_{\varepsilon }$ (the solution of (\ref{1.4})) under the structure hypothesis (\ref{1.6}). To this end we first note that $\mathcal{B}_{\infty }(\mathbb{R};\mathcal{C}_{\rm per}(Y'))$ is the image of the product $H$% -structure $\Sigma =\Sigma _{\mathcal{R}'}\times \Sigma _{\infty }$ ($\mathcal{R}'=\mathbb{Z}^{N-1}$) on $\mathbb{R}^{N}=\mathbb{R}^{N-1}\times \mathbb{R}$ and further that the latter is proper for $p=2$ (Example \ref{ex3.5}% ). Thus, for $p=2$, the homogenization problem under consideration will have been solved through Theorem \ref{theo4.1} if we can show that (\ref{4.1}) holds (with $p=2$). But this will be a direct consequence of a stronger result. \begin{proposition} \label{propo5.2} Suppose (\ref{1.6}) holds. Then \begin{equation} a_{i}(\cdot ,\Psi )\in A=\mathcal{B}_{\infty }(\mathbb{R};\mathcal{C} _{\rm per}(Y'))\quad \text{ for all }\Psi \in (A_{\mathbb{R}})^{N} \quad(1\leq i\leq N). \label{5.4} \end{equation} \end{proposition} \begin{proof} Let $\Psi \in (A_{\mathbb{R}})^{N}$. Let $\mathbf{\theta }=\lim_{| y_{N}| \to \infty }\Psi (\cdot ,y_{N})$, where the limit is taken in $\mathcal{B}(\mathbb{R}^{N-1})$ (with the supremum norm) and where $\Psi (\cdot ,y_{N})$ stands for the function $y'=(y_{1},\cdots ,y_{N-1})\to \Psi (y',y_{N})$ of $\mathbb{R}^{N-1}$ into $\mathbb{R}$. Let us begin by verifying that the function $% y\to a_{i}(y,\mathbf{\theta }(y'))$ of $\mathbb{R}^{N}$ into $\mathbb{R}$, denoted by $a_{i}(\cdot ,\mathbf{\theta }')$, lies in $A$. For fixed $y_{N}\in \mathbb{R}$, let $a_{i}(\cdot ,y_{N},\mathbf{\theta })$ denote the function $y'\to a_{i}(y',y_{N},\mathbf{\theta }(y'))$ of $\mathbb{R}^{N-1}$ into $\mathbb{R}$. It is an easy matter to check, using (UE), that the mapping $y_{N}\to a_{i}(\cdot,y_{N},\mathbf{\theta })$ sends continuously $\mathbb{R}$ into $\mathcal{C}_{\rm per}(Y')$. Thus, we will claim that $a_{i}(\cdot ,\mathbf{\theta }')$ lies in $A$ if we have shown that $a_{i}(\cdot ,y_{N},\mathbf{\theta })$ has a limit in $\mathcal{B}(\mathbb{R}^{N-1})$ when $|y_{N}| \to \infty $. We proceed in two steps: 1) Suppose $a_{i}$ is of the form (\ref{5.3}). Let $h\in \mathcal{C}_{\rm per}(Y')$ be the limit of $\varphi (\cdot ,y_{N})$ in $\mathcal{B}(\mathbb{R}^{N-1})$ as $| y_{N}| \to \infty $. The mapping $y'\to h(y')\chi (\mathbf{\theta }(y'))$ of $\mathbb{R}^{N-1}$ into $\mathbb{R}$ lies in $\mathcal{C}_{\rm per}(Y')$ and \begin{equation*} | a_{i}(y,\mathbf{\theta }(y'))-h(y')\chi (\mathbf{\theta }(y'))| \leq C| \varphi (y)-h(y')| \quad (y\in \mathbb{R}^{N}) \end{equation*} where $C=\max_{\lambda \in K}| \chi (\lambda )| $, $K$ being a compact set in $\mathbb{R}^{N}$ such that $\mathbf{\theta }(y')\in K$ for all $y'\in \mathbb{R}^{N-1}$. This shows that when $| y_{N}| \to \infty $, $a_{i}(\cdot ,y_{N},\mathbf{\theta })$ converges in $\mathcal{B}(\mathbb{R}^{N-1})$ to the preceding function. 2) We now consider the general case where we know merely that, in addition to (\ref{1.1})-(\ref{1.3}) and (UE), $a$ satisfies (\ref{1.6}). For convenience we still write $a_{i}$ for $a_{i}\big| _{K}$, $K$ being as above. Then $a_{i}$ belongs to $\mathcal{C}(K;A_{\mathbb{R}})$. By the same line of argument as followed in proving Proposition \ref{propo5.1}, we are immediately led to the desired result. We are now in a position to show that $a_{i}(\cdot ,\Psi )$ lies in $A$. This is straightforward. Indeed, by \begin{equation*} | a_{i}(y,\Psi (y))-a_{i}(y,\mathbf{\theta }(y'))| \leq c| \Psi (y)-\mathbf{\theta }(y')| ^{p-1} \quad (y\in \mathbb{R}^{N}) \end{equation*} we are quickly led to $a_{i}(\cdot ,\Psi )=a_{i}(\cdot ,\mathbf{\theta }% ')+\phi $, where $\phi \in \mathcal{B}_{0}(\mathbb{R};\mathcal{C}% _{\rm per}(Y'))$ [it is an easy matter to show, using (\ref{1.3})(ii) and (UE), that $a_{i}(\cdot ,\Psi )\in \mathcal{C}(\mathbb{R};\mathcal{C}% _{\rm per}(Y'))$]. Hence (\ref{5.4}) follows. \end{proof} \subsection{Problem IV} The matter in hand is the homogenization of (\ref{1.4}) under the structure hypothesis (\ref{1.7}). For this purpose, starting from (\ref{1.7}) we begin by considering (exactly as in Lemma 5.1 of \cite{n4}) a countable subgroup $% \mathcal{R}$ of $\mathbb{R}^{N}$ such that \cite{n1,n4} \begin{equation*} a_{i}(\cdot ,\lambda )\in A=AP_{\mathcal{R}}(\mathbb{R}^{N}) \quad \text{for any }\lambda \in \mathbb{R}^{N}\; (1\leq i\leq N). \end{equation*} Let $\Sigma _{\mathcal{R}}$ be the almost periodic $H$-structure on $\mathbb{R}^{N}$ represented by $\mathcal{R}$. We have $\mathcal{J}(\Sigma _{\mathcal{R}})=A=AP_{\mathcal{R}}(\mathbb{R}^{N})$ and $\Sigma _{\mathcal{R}}$ is proper for $p=2$, as pointed out in Example \ref{ex3.2}. Therefore, if we show that \begin{equation} a_{i}(\cdot ,\Psi )\in A\text{ for all }\Psi \in (A_{\mathbb{R}})^{N}\text{ (}% 1\leq i\leq N\text{), \thinspace \thinspace \thinspace } \label{5.5} \end{equation} which implies (\ref{4.1}) with $\Sigma =\Sigma _{\mathcal{R}}$, then for $p=2 $ the solution of the homogenization problem under examination follows at once by Subsection 4.2. To do this, let us fix freely $\Psi \in (A_{\mathbb{R}})^{N}$. Let $K$ be a compact set in $\mathbb{R}^{N}$ such that $\Psi (y)\in K$ for all $y\in \mathbb{R}% ^{N}$. For an obvious reason we may here view $a_{i}$ as belonging to $\mathcal{C}(K;A_{\mathbb{R}})$. Let us first assume that $a_{i}$ is of the form \begin{equation*} a_{i}(\lambda )=\chi (\lambda )\varphi \quad (\lambda \in K) \text{ with }\chi \in \mathcal{C}_{\mathbb{R}}(K) \text{ and }\varphi \in A_{\mathbb{R}}. \end{equation*} By the Stone-Weierstrass theorem (see chapter X, page 37 of \cite{b2}) there is a sequence $(f_{n})$ of polynomials in $\lambda =(\lambda _{1},\cdots ,\lambda _{N})$ such that $f_{n}\to \chi $ in $\mathcal{C% }(K)$ when $n\to \infty $. Thus, $f_{n}(\Psi )\to \chi (\Psi)$ in $\mathcal{B}(\mathbb{R}^{N})$ as $n\to \infty $. It follows that $\chi (\Psi )\in A_{\mathbb{R}}$, since $f_{n}(\Psi )\in A_{\mathbb{R}}$. Hence $a_{i}(\cdot ,\Psi )\in A$. But then this is still manifestly true if $a_{i}(\lambda )=\sum \chi (\lambda )\varphi $ for $\lambda \in K$, where $\chi $ (resp. $\varphi $) ranges over a finite subset of $\mathcal{C}_{\mathbb{R}}(K)$ (resp. $A_{\mathbb{R}}$). Hence the same routine argument as used in Step 2) of the proof of Proposition \ref{propo5.1} leads us to (\ref{5.5}). \subsection{Problem V} We assume here that $a(y,\lambda )$ satisfies (\ref{1.8}) and (\ref{1.9}), and we want to study the homogenization of (\ref{1.4}) for $p=2$. As is now well known, this problem reduces to the abstract problem of Section 4 if for some proper $H$-structure $\Sigma $, condition (\ref{4.1}) is fulfilled. To achieve this, let $\theta \in \mathcal{D}(\mathbb{R}^{N})$ with $\theta \geq 0$, $Supp\theta \subset $ $\overset{\_}{B}_{N}$ (closed unit ball of $\mathbb{R}^{N} $) and $\int \theta (y)dy=1$. For each integer $n\geq 1$, set $\theta_{n}(y)=n^{N}\theta (ny)$ ($y\in \mathbb{R}^{N}$), which gives a sequence of functions $\theta _{n}\in \mathcal{D}(\mathbb{R}^{N})$. Now, define \begin{equation*} \zeta _{n}^{i}(y,\lambda )=\int \theta _{n}(r)a_{i}(y-r,\lambda )dr \quad\text{for }y,\lambda \in \mathbb{R}^{N}\quad (1\leq i\leq N). \end{equation*} It is easily checked that $\zeta _{n}^{i}(\cdot ,\lambda )\in AP(\mathbb{R}% _{y}^{N})$ for $\lambda \in \mathbb{R}^{N}$ and further \begin{equation} | \mathbf{\zeta }_{n}(y,\lambda )-\mathbf{\zeta }_{n}(y,\mu )| \leq c| \lambda -\mu | ^{p-1}\quad(\lambda ,\mu ,y\in \mathbb{R}^{N}) \label{5.6} \end{equation} where $\mathbf{\zeta }_{n}=(\zeta _{n}^{i})_{1\leq i\leq N}$. Therefore, according to Subsection 5.5, there is a countable subgroup $\mathcal{R}$ of $\mathbb{R}^{N}$ such that $\zeta _{n}^{i}(\cdot ,\Psi )\in A=AP_{\mathcal{R}}(\mathbb{R}_{y}^{N})$ for $\Psi \in (A_{\mathbb{R}})^{N}$, $1\leq i\leq N$, and all integers $n\geq 1$. Furthermore, given $\Psi \in (A_{\mathbb{R}})^{N}$ and $1\leq i\leq N$, a quick calculation reveals that \begin{multline} \| \zeta _{n}^{i}(\cdot ,\Psi )-a_{i}(\cdot ,\Psi )\| _{2,\infty}^{2}\\ \leq \int_{\frac{1}{n}B_{N}}\theta _{n}(r)\Big[ \sup_{k\in \mathbb{Z} ^{N}}\int_{k+Y}| a_{i}(y-r,\Psi (y))-a_{i}(y,\Psi (y))| ^{2}dy\Big] dr \label{5.7} \end{multline} for any integer $n\geq 1$, where $\| \cdot \| _{2,\infty }$ is the norm in the amalgam space $(L^{2},\ell ^{\infty })(\mathbb{R}^{N})$ \cite{f1,n4} (see also \cite[Example 5.4]{n1}). Hence, if $\eta >0$ is given, then by (\ref{1.9}) we are led to some integer $\gamma \geq 1$ such that $\| \zeta _{n}^{i}(\cdot ,\Psi )-a_{i}(\cdot ,\Psi )\| _{2,\infty }\leq \eta $ for any $n\geq \gamma $. Since $(L^{2},\ell ^{\infty })(\mathbb{R}^{N})$ is continuously embedded in $\Xi ^{2}(\mathbb{R}^{N})$ \cite{n1} (use \cite[Lemma 1.3]{n4}), we deduce that (\ref{4.1}) follows with $\Sigma =\Sigma _{\mathcal{R}}$ (as in Subsection 5.5) and $p=2$. \begin{remark} \label{rm5.1} \rm Condition (\ref{1.9}) is fulfilled if the following holds : Given $\eta >0$, there exists $\rho >0$ such that $|a(y-r,\lambda )-a(y,\lambda )| \leq \eta $ for all $\lambda \in \mathbb{R}^{N}$ and for almost all $y\in \mathbb{R}^{N}$ provided $| r| \leq \rho $. \end{remark} \subsection{Problem VI} This subsection is intended to study the homogenization of (\ref{1.4}) for $p=2$ under the following hypotheses: \begin{equation} a_{i}(\cdot ,\lambda )\in \mathcal{B}_{\infty }(\mathbb{R};L_{\rm per}^{\infty }(Y'))\quad \text{ for each }\lambda \in \mathbb{R}^{N}\;(1\leq i\leq N) \label{5.8} \end{equation} \begin{equation} \parbox{10cm}{Given $\eta >0$, a real $\rho >0$ exists such that $| a(y-r,\lambda )-a(y,\lambda )| \leq \eta$ for all $\lambda \in \mathbb{R}^{N}$ and for almost all $y\in \mathbb{R}^{N}$ provided $| r| \leq \rho$} \label{5.9} \end{equation} where $Y'$ is as in (\ref{1.6}). As we are now familiar with the approach, the whole problem reduces to verifying that (\ref{4.1}) holds with $p=2$ and $\Sigma =\Sigma _{\mathcal{R}'=\mathbb{Z}^{N-1}}\times \Sigma _{\infty }$, hence $A=\mathcal{B}_{\infty }(\mathbb{R};\mathcal{C}% _{\rm per}(Y'))$. This proceeds by adaptation of what we did earlier in Subsection 5.6. Let $(\mathbf{\zeta }_{n})_{n\geq 1}$ with $\mathbf{\zeta }_{n}=(\zeta _{n}^{i})_{1\leq i\leq N}$ for each integer $n\geq 1$, where $\zeta _{n}^{i}$ is defined in Subsection 5.6. Let $\lambda \in \mathbb{R}^{N}$ be fixed. Clearly \begin{equation*} a_{i}(y,\lambda )=t_{i}(y',\lambda )+g_{i}(y,\lambda )\quad \text{for }y\in \mathbb{R}^{N}\; (1\leq i\leq N) \end{equation*} where the functions $y'=(y_{1},\cdots ,y_{N-1})\to t_{i}(y',\lambda )$ from $\mathbb{R}^{N-1}$ to $\mathbb{R}$ and $y\to g_{i}(y,\lambda )$ from $\mathbb{R}^{N}$ to $\mathbb{R}$ lie in $L_{\rm per}^{\infty }(Y')$ and $\mathcal{B}_{0}(\mathbb{R};L_{\rm per}^{\infty }(Y'))$ (those functions in $\mathcal{B}_{\infty }(\mathbb{R};L_{\rm per}^{\infty }(Y'))$ that vanish at infinity), respectively. Hence \begin{equation*} \zeta _{n}^{i}(y,\lambda )=\varphi _{n}^{i}(y',\lambda )+\gamma _{n}^{i}(y,\lambda )\text{ for }y\in \mathbb{R}^{N}\quad (1\leq i\leq N) \end{equation*} where \begin{equation*} \varphi _{n}^{i}(y',\lambda )=\int \theta _{n}(r)t_{i}(y-r,\lambda )dr \quad (y\in \mathbb{R}^{N}) \end{equation*} and \begin{equation} \gamma _{n}^{i}(y,\lambda )=\int \theta _{n}(r)g_{i}(y-r,\lambda )dr \quad (y\in \mathbb{R}^{N}), \label{5.10} \end{equation} $t_{i}(\cdot ,\lambda )$ being considered as a function in $L^{\infty }(\mathbb{R}^{N})$ independent of the $N^{th}$ variable. It is immediate that the function $y'\to \varphi _{n}^{i}(y',\lambda )$ lies in $\mathcal{C}_{\rm per}(Y')$. On the other hand, whereas $\gamma _{n}^{i}(\cdot ,\lambda )$, considered as a function of the variable $y_{N}$, belongs to $\mathcal{B}(\mathbb{R};\mathcal{C}_{\rm per}(Y'))$ (this follows at once by classical properties of the convolution), it is not clear, however, that the latter function vanishes at infinity. To see this, let $\eta >0$. Let $\rho >0$ be such that \begin{equation} \| g_{i}(z_{N},\lambda )\| _{L^{\infty }(\mathbb{R}^{N-1})}\leq \eta \label{5.11} \end{equation} for all $z_{N}\in \mathbb{R}$ verifying $| z_{N}| \geq \rho $, where $g_{i}(\cdot ,\lambda )$ is viewed as a function of $z_{N}\in \mathbb{R}$ with values in $L_{\rm per}^{\infty }(Y')$, of course. Fix $y=(y',y_{N})\in \mathbb{R}^{N}=\mathbb{R}^{N-1}\times \mathbb{R}$ where $y'$ is arbitrary and $y_{N}$ is subject to the condition $| y_{N}| \geq 1+\rho $. Since the variable $r=(r_{1},\cdots ,r_{N})$ in (\ref {5.10}) actually runs through $\frac{1}{n}\overset{-}{B}_{N}$, it follows from (\ref{5.11}) that \begin{equation*} \| g_{i}(y_{N}-r_{N},\lambda )\| _{L^{\infty }(\mathbb{R}^{N-1})}\leq \eta \end{equation*} for $| y_{N}| \geq 1+\rho $ provided $r$ lies in $\frac{1}{n}% \overset{-}{B}_{N}$. Based on (\ref{5.10}), we deduce that $\| \gamma _{n}^{i}(y_{N},\lambda )\| _{\infty }\leq \eta $ for $|y_{N}| \geq 1+\rho $ and so $\gamma _{n}^{i}(\cdot ,\lambda )$ lies in $\mathcal{B}_{0}(\mathbb{R};\mathcal{C}_{\rm per}(Y'))$. Hence $\zeta_{n}^{i}(\cdot ,\lambda )\in A=\mathcal{B}_{\infty }(\mathbb{R}; \mathcal{C}_{\rm per}(Y'))$. On the other hand, it is clear that (\ref{5.6}) holds true. Finally, for each fixed integer $n\geq 1$, the family $\{\mathbf{\zeta }_{n}(\cdot,\lambda )\}_{\lambda \in \mathbb{R}^{N}}$ is uniformly equicontinuous (see Subsection 5.4), as is easily seen by using (\ref{5.9}). Consequently, we are justified in replacing $a(\cdot ,\lambda )$ by $\mathbf{\zeta }_{n}(\cdot ,\lambda )$ in Proposition \ref{propo5.2}, so that $\zeta_{n}^{i}(\cdot ,\Psi )\in A$ ($1\leq i\leq N$) for all $\Psi \in (A_{\mathbb{R}})^{N}$. With this in mind, fix freely $\Psi \in (A_{\mathbb{R}})^{N}$. Clearly (\ref{5.7}) holds true for $1\leq i\leq N$. On the other hand, by (\ref{5.9}) one easily gets \begin{equation*} \sup_{k\in \mathbb{Z}^{N}}\int_{k+Y}| a(y-r,\Psi (y))-a(y,\Psi (y))| ^{2}dy\to 0\text{ as }| r| \to 0. \end{equation*} Therefore, given $\eta >0$, there is an integer $\gamma \geq 1$ such that $\| \zeta _{n}^{i}(\cdot ,\Psi )-a_{i}(\cdot ,\Psi )\| _{2,\infty}\leq \eta $ for any $n\geq \gamma $. Hence the desired conclusion follows in the same manner as in Subsection 5.6. \begin{remark} \label{rm5.2} \rm Hypothesis (\ref{5.8}) generalizes (\ref{1.6}). 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