\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 74, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/74\hfil Sigma-convergence] {Sigma-convergence of stationary Navier-Stokes type equations} \author[G. Nguetseng, L. Signing\hfil EJDE-2009/74\hfilneg] {Gabriel Nguetseng, Lazarus Signing} % in alphabetical order \address{Gabriel Nguetseng \newline Dept. of Mathematics\\ University of Yaounde 1 \\ P.O. Box 812 Yaounde, Cameroon} \email{nguetseng@uy1.uninet.cm} \address{Lazarus Signing \newline Dept. of Mathematics and Computer Sciences\\ University of Ngaound\'{e}r\'{e} \\ P.O. Box 454 Ngaound\'{e}r\'{e}, Cameroon} \email{lsigning@uy1.uninet.cm} \thanks{Submitted February 3, 2009. Published June 5, 2009.} \subjclass[2000]{35B40, 46J10} \keywords{Homogenization; sigma-convergence, Navier-Stokes equations} \begin{abstract} In the framework of homogenization theory, the $\Sigma$-convergence method is carried out on stationary Navier-Stokes type equations on a fixed domain. Our main tools are the two-scale convergence concept and the so-called homogenization algebras. \end{abstract} \maketitle\newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} \section{Introduction} We study the homogenization of stationary Navier-Stokes type equations in a fixed bounded open subset of the $N$-dimensional numerical space. Here, the usual Laplace operator involved in the classical Navier-Stokes equations is replaced by an elliptic linear differential operator of order two, in divergence form, with variable coefficients. Let us give a detailed description of our object. Let $\Omega$ be a smooth bounded open set in $\mathbb{R}_{x}^{N}$ (the $N$-dimensional numerical space $\mathbb{R}^{N}$ of variables $x=(x_{1},\dots ,x_{N})$), where $N$ is a given positive integer; and let $\varepsilon$ be a real number with $0<\varepsilon <1$. We consider the partial differential operator $P^{\varepsilon }=-\sum_{i,j=1}^{N}\frac{\partial }{\partial x_{i}} \Big( a_{ij}^{\varepsilon }\frac{\partial }{\partial x_{j}}\Big)$ in $\Omega$, where $a_{ij}^{\varepsilon }(x)=a_{ij}( \frac{x}{\varepsilon })$ $(x\in \Omega )$, $a_{ij}\in L^{\infty }(\mathbb{R}_{y}^{N};\mathbb{R})$ $(1\leq i,j\leq N)$ with $$a_{ij}=a_{ji}, \label{eq1.1}$$ and the assumption that there is a constant $\alpha >0$ such that $$\sum_{i,j=1}^{N}a_{ij}(y)\xi _{j}\xi _{i}\geq \alpha | \xi | ^{2} \quad \text{for all }\xi =(\xi _{i})\in \mathbb{R}^{N}\text{ and for almost all } y\in \mathbb{R}^{N}, \label{eq1.2}$$ where $|\cdot|$ denotes the usual Euclidean norm in $\mathbb{R}^{N}$. The operator $P^{\varepsilon }$ acts on scalar functions, say $\varphi \in H^{1}(\Omega )=W^{1,2}(\Omega )$. However, we may as well view $P^{\varepsilon }$ as acting on vector functions $\mathbf{u}=(u^{i})\in H^{1}(\Omega )^{N}$ in a \textit{diagonal way}, i.e., $(P^{\varepsilon }\mathbf{u})^{i}=P^{\varepsilon }u^{i}\quad (i=1,\dots ,N).$ \begin{remark} \label{rem1.1} \rm For any Roman character such as $i,j$ (with $1\leq i,j\leq N$), $u^{i}$ (resp. $u^{j}$) denotes the $i$-th (resp. $j$-th) component of a vector function $\mathbf{u}$ in $L_{loc}^{1}(\Omega )^{N}$ or in $L_{loc}^{1}(\mathbb{R}_{y}^{N})^{N}$. On the other hand, for any real $0<\varepsilon <1$, we define $u^{\varepsilon }$ as $u^{\varepsilon }(x)=u(\frac{x}{\varepsilon })\quad (x\in \Omega )$ for $u\in L_{loc}^{1}(\mathbb{R}_{y}^{N})$, as is customary in homogenization theory. More generally, for $u\in L_{loc}^{1}(\Omega \times \mathbb{R}_{y}^{N})$, it is customary to put $u^{\varepsilon }(x)=u(x,\frac{x}{\varepsilon }) \quad (x\in \Omega )$ whenever the right-hand side makes sense (see, e.g., \cite{bib7,bib8}). There is no danger of confusion between the preceding notation. \end{remark} Having made these preliminaries, let $\mathbf{f}=(f^{i})\in H^{-1}(\Omega ; \mathbb{R})^{N}$. For any fixed $0<\varepsilon <1$, we consider the boundary value problem \begin{gather} P^{\varepsilon }\mathbf{u}_{\varepsilon }+\sum_{j=1}^{N}u_{\varepsilon }^{j} \frac{\partial \mathbf{u}_{\varepsilon }}{\partial x_{j}}+\mathop{\rm grad} p_{\varepsilon }=\mathbf{f}\quad \text{in }\Omega , \label{eq1.3} \\ \mathop{\rm div}\mathbf{u}_{\varepsilon }=0\quad \text{in }\Omega , \label{eq1.4} \\ \mathbf{u}_{\varepsilon }=0\quad \text{on }\partial \Omega , \label{eq1.5} \end{gather} where $\frac{\partial \mathbf{u}_{\varepsilon }}{\partial x_{j}}=\Big(\frac{ \partial u^{1}}{\partial x_{j}},\dots ,\frac{\partial u^{N}}{\partial x_{j}} \Big).$ We will later see that if $N$ is either $2$ or $3$, and if $\mathbf{f}$ is \textquotedblleft small enough", then \eqref{eq1.3}-\eqref{eq1.5} uniquely define $(\mathbf{u}_{\varepsilon },p_{\varepsilon })$ with $\mathbf{u}_{\varepsilon }=(u_{\varepsilon }^{i})\in H_{0}^{1}(\Omega ;\mathbb{R})^{N}$ and $p_{\varepsilon }\in L^{2}(\Omega ;\mathbb{R})/\mathbb{R}$, where $L^{2}(\Omega ;\mathbb{R})/\mathbb{R=}\big\{v\in L^{2}(\Omega ;\mathbb{R}): \int_{\Omega }vdx=0\big\}.$ Our main goal is to investigate the limiting behavior, as $\varepsilon \to 0$, of $(\mathbf{u}_{\varepsilon },p_{\varepsilon })$ under an abstract assumption on $a_{ij}$ $(1\leq i,j\leq N)$ covering a wide range of concrete behaviour beyond the classical periodicity hypothesis. The linear version of this problem (i.e., the homogenization of \eqref{eq1.3}-\eqref{eq1.5} without the term $\sum_{j=1}^{N}u_{\varepsilon }^{j}\frac{\partial \mathbf{u} _{\varepsilon }}{ \partial x_{j}}$) was first studied by Bensoussan, Lions and Papanicolaou \cite{bib2} under the periodicity hypothesis on the coefficients $a_{ij}$. These authors presented a detailed mathematical analysis of the problem by the well-known approach combining the use of asymptotic expansions with Tartar's energy method. The present study deals with a more general situation involving two major difficulties: 1) the equations are nonlinear; 2) the homogenization problem for \eqref{eq1.3}-\eqref{eq1.5} is considered not under the periodicity hypothesis, as is classical, but in the general setting characterized by an abstract assumption on $a_{ij}(y)$ covering a wide range of behaviours with respect to $y$, such as the periodicity, the almost periodicity, the convergence at infinity, and others. The motivation of the present study lies in the fact that the homogenization problem for \eqref{eq1.3}-\eqref{eq1.5} is connected with the modelling of heterogeneous fluid flows, in particular multi-phase flows, fluids with spatially \ varying viscosities, and others; see, e.g., \cite{bib16} for more details about such heterogeneous media. Our approach is the $\Sigma$-convergence method derived from two-scale convergence ideas \cite{bib1}, \cite{bib11} by means of so-called homogenization algebras \cite{bib9}, \cite{bib10}. Unless otherwise specified, vector spaces throughout are considered over the complex field, $\mathbb{C}$, and scalar functions are assumed to take complex values. Let us recall some basic notation. If $X$ and $F$ denote a locally compact space and a Banach space, respectively, then we write $\mathcal{C}(X;F)$ for the continuous mappings of $X$ into $F$, and $\mathcal{B}(X;F)$ for those mappings in $\mathcal{C}(X;F)$ that are bounded. We shall assume $\mathcal{B}(X;F)$ to be equipped with the supremum norm $\| u\| _{\infty}=\sup_{x\in X}\| u(x)\|$ ($\|\cdot\|$ denotes the norm in $F$). For shortness we will write $\mathcal{C}(X)=\mathcal{C}(X;\mathbb{C})$ and $\mathcal{B}(X)=\mathcal{B}(X;\mathbb{C})$. Likewise in the case when $F=\mathbb{C}$, the usual spaces $L^{p}(X;F)$ and $L_{\mathrm{loc}}^{p}(X;F)$ ($X$ provided with a positive Radon measure) will be denoted by $L^{p}(X)$ and $L_{\mathrm{loc}}^{p}(X)$, respectively. Finally, the numerical space $\mathbb{R}^{N}$ and its open sets are each provided with Lebesgue measure denoted by $dx=dx_{1}\dots dx_{N}$. The rest of the study is organized as follows. In Section 2 we discuss the homogenization of \eqref{eq1.3}-\eqref{eq1.5} under the periodicity hypothesis on the coefficients $a_{ij}$. In Section 3 we reconsider the homogenization of problem \eqref{eq1.3}-\eqref{eq1.5} in a more general setting. The periodicity hypothesis on the coefficients $a_{ij}$ is here replaced by an abstract assumption covering a variety of concrete behaviour including the periodicity as a particular case. A few concrete examples are worked out. \section{Periodic homogenization of stationary Navier-Stokes type equations} We assume once for all that $N$ is either $2$ or $3$. We set $Y=(-\frac{1}{2},\frac{1}{2})^{N}$, $Y$ considered as a subset of $\mathbb{R}_{y}^{N}$ (the space $\mathbb{R}^{N}$ of variables $y=(y_{1},\dots ,y_{N})$). Our purpose is to study the homogenization of \eqref{eq1.3}-\eqref{eq1.5} under the periodicity hypothesis on $a_{ij}$, i.e., under the assumption that $a_{ij}$ is $Y$-periodic. \subsection{Preliminaries} Let us first recall that a function $u\in L_{\mathrm{loc}}^{1}(\mathbb{R} _{y}^{N})$ is said to be $Y$-periodic if for each $k\in \mathbb{Z} ^{N}$ ($\mathbb{Z}$ denotes the integers), we have $u(y+k) =u(y)$ almost everywhere (a.e.) in $y\in \mathbb{R}^{N}$. If in addition $u$ is continuous, then the preceding equality holds for every $y\in \mathbb{R}^{N}$, of course. The space of all $Y$-periodic continuous complex functions on $\mathbb{R}_{y}^{N}$ is denoted by $\mathcal{C} _{\mathrm{per}}(Y)$; that of all $Y$-periodic functions in $L_{\mathrm{loc}}^{p}(\mathbb{R}_{y}^{N})$ $(1\leq p<\infty )$ is denoted by $L_{\mathrm{per}}^{p}(Y)$. $\mathcal{C}_{\mathrm{per}}( Y)$ is a Banach space under the supremum norm on $\mathbb{R}^{N}$, whereas $L_{\mathrm{per}}^{p}(Y)$ is a Banach space under the norm $\| u\| _{L^{p}(Y)}=\Big(\int_{Y}| u(y)| ^{p}dy\Big)^{1/p}\quad (u\in L_{\mathrm{per}}^{p}(Y)).$ We need the space $H_{\#}^{1}(Y)$ of $Y$-periodic functions $u\in H_{\mathrm{loc}}^{1}(\mathbb{R}_{y}^{N}) =W_{\mathrm{loc}}^{1,2}(\mathbb{R}_{y}^{N})$ such that $\int_{Y}u(y)dy=0$. Provided with the gradient norm, $\| u\| _{H_{\#}^{1}(Y)}=\Big( \int_{Y}| \nabla _{y}u| ^{2}dy\Big)^{1/2}\quad (u\in H_{\#}^{1}(Y)),$ where $\nabla _{y}u=(\frac{\partial u}{\partial y_{1}},\dots ,\frac{ \partial u}{\partial y_{N}})$, $H_{\#}^{1}(Y)$ is a Hilbert space. Before we can recall the concept of $\Sigma$-convergence in the present periodic setting, let us introduce one further notation. The letter $E$ throughout will denote a family of real numbers $0<\varepsilon <1$ admitting $0$ as an accumulation point. For example, $E$ may be the whole interval $(0,1)$; $E$ may also be an ordinary sequence $(\varepsilon _{n})_{n\in \mathbb{N}}$ with $0<\varepsilon _{n}<1$ and $\varepsilon _{n}\to 0$ as $n\to \infty$. In the latter case $E$ will be referred to as a \textit{fundamental sequence}. Let us observe that $E$ may be neither $(0,1)$ nor a fundamental sequence, of course. Let $\Omega$ be a bounded open set in $\mathbb{R}_{x}^{N}$ and let $1\leq p<\infty$. \begin{definition} \label{def2.1} \rm A sequence $(u_{\varepsilon })_{\varepsilon \in E}\subset L^{p}(\Omega )$ is said to be: \begin{itemize} \item[(i)] weakly $\Sigma$-convergent in $L^{p}(\Omega )$ to some $u_{0}\in L^{p}(\Omega ;L_{per}^{p}(Y))$ if as $E\ni \varepsilon \rightarrow 0$, $$\int_{\Omega }u_{\varepsilon }(x)\psi ^{\varepsilon }(x)dx\rightarrow \iint_{\Omega \times Y}u_{0}(x,y)\psi (x,y)\,dx\,dy \label{eq2.1}$$ for all $\psi \in L^{p'}(\Omega ;\mathcal{C}_{per}(Y))$ ($\frac{1}{p'}=1-\frac{1}{p}$), where $\psi ^{\varepsilon }(x)=\psi (x,\frac{x }{\varepsilon })$ ($x\in \Omega$); \item[(ii)] strongly $\Sigma$-convergent in $L^{p}(\Omega )$ to some $u_{0}\in L^{p}(\Omega ;L_{per}^{p}(Y))$ if the following property is verified: Given $\eta >0$ and $v\in L^{p}(\Omega ;\mathcal{C}_{per}(Y))$ with $\Vert u_{0}-v\Vert _{L^{p}(\Omega \times Y)}\leq \frac{\eta }{2}$, there is some $\alpha >0$ such that $\Vert u_{\varepsilon }-v^{\varepsilon }\Vert _{L^{p}(\Omega )}\leq \eta$ provided $E\ni \varepsilon \leq \alpha$. \end{itemize} \end{definition} We will briefly express weak and strong $\Sigma$-convergence by writing $u_{\varepsilon }\to u_{0}$ in $L^{p}(\Omega )$-weak $\Sigma$ and $u_{\varepsilon }\to u_{0}$ in $L^{p}(\Omega )$-strong $\Sigma$, respectively. \begin{remark} \label{rem2.1} \rm It is of interest to know that if $u_{\varepsilon}\to u_{0}$ in $L^{p}(\Omega )$-weak $\Sigma$, then (\ref{eq2.1}) holds for $\psi \in \mathcal{C}(\overline{\Omega };L_{per}^{\infty }(Y))$. See \cite[Proposition 10]{bib8} for the proof. \end{remark} In the present context the concept of $\Sigma$-convergence coincides with the well-known one of two-scale convergence. Consequently, instead of repeating here the main results underlying $\Sigma$-convergence theory for periodic structures, we find it more convenient to draw the reader's attention to a few references regarding two-scale convergence, e.g., \cite {bib1}, \cite{bib6}, \cite{bib8} and \cite{bib17}. However, we recall below two fundamental results which constitute the corner stone of the two-scale convergence theory. \begin{theorem} \label{thm2.1} Assume that $10$ exists such that $$| a^{\varepsilon }(\mathbf{u,v})| \leq c_{0}\| \mathbf{u}\| _{H_{0}^{1}(\Omega ) ^{N}}\| \mathbf{v}\| _{H_{0}^{1}(\Omega )^{N}} \label{eq2.3}$$ for all $\mathbf{u}$, $\mathbf{v}\in H_{0}^{1}(\Omega ;\mathbb{R} )^{N}$ and all $0<\varepsilon <1$. We also need the trilinear form $b$ on $H_{0}^{1}(\Omega ;\mathbb{R} )^{N}\times H_{0}^{1}(\Omega ;\mathbb{R})^{N}\times H_{0}^{1}(\Omega ; \mathbb{R})^{N}$ given by $b(\mathbf{u},\mathbf{v},\mathbf{w}) =\sum_{k=_{1}}^{N}\sum_{j=1}^{N}\int_{\Omega }u^{j}\frac{\partial v^{k}}{ \partial x_{j}}w^{k}dx$ for $\mathbf{u=}(u^{k})$, $\mathbf{v=}(v^{k})$ and $\mathbf{w=}(w^{k})$ in $H_{0}^{1}(\Omega ;\mathbb{R} )^{N}$. The trilinear form $b$ has some nice properties. Let $V=\big\{ \mathbf{u}\in H_{0}^{1}(\Omega ;\mathbb{R})^{N}:\mathop{\rm div} \mathbf{u}=0\big\} .$ Then $$b(\mathbf{u},\mathbf{v},\mathbf{v})=0\quad \text{for }\mathbf{u}\in V,\; \mathbf{v}\in H_{0}^{1}(\Omega ;\mathbb{R})^{N}, \label{eq2.4}$$ and further there exists a constant $c(N)>0$ such that $$| b(\mathbf{u},\mathbf{v},\mathbf{w})| \leq c(N)\| \mathbf{u}\| _{H_{0}^{1}(\Omega )^{N}}\| \mathbf{v}\| _{H_{0}^{1}(\Omega )^{N}}\| \mathbf{ w}\| _{H_{0}^{1}(\Omega )^{N}} \label{eq2.5}$$ for all $\mathbf{u}$, $\mathbf{v}$, $\mathbf{w}\in H_{0}^{1}(\Omega ; \mathbb{R})^{N}$ (see \cite{bib5, bib15} for the proofs of these classical results). We are now in a position to verify the following result. \begin{proposition} \label{pr2.1} Suppose $\mathbf{f}$ (the right-hand side of \eqref{eq1.3}) is small enough" so that $$c(N)\| \mathbf{f}\| _{H^{-1}(\Omega )^{N}}<\alpha ^{2}, \label{eq2.6}$$ where $\alpha$ (resp. $c(N)$) is that constant in \eqref{eq1.2} (resp. \eqref{eq2.5}). Then, the boundary value problem \eqref{eq1.3}-\eqref{eq1.5} determines a unique pair $(\mathbf{u}_{\varepsilon },p_{\varepsilon })$ with $\mathbf{u}_{\varepsilon }\in H_{0}^{1}(\Omega ;\mathbb{R})^{N}$, $p_{\varepsilon }\in L^{2}(\Omega ;\mathbb{R})/ \mathbb{R}$. \end{proposition} \begin{proof} For fixed $0<\varepsilon <1$, consider the variational problem $$\begin{gathered} \mathbf{u}_{\varepsilon }\in V:\\ a^{\varepsilon }(\mathbf{u}_{\varepsilon },\mathbf{v})+b( \mathbf{u}_{\varepsilon },\mathbf{u}_{\varepsilon },\mathbf{v}) =(\mathbf{f},\mathbf{v})\quad \text{for all }\mathbf{v}=(v^{k})\in V \end{gathered} \label{eq2.7}$$ with $(\mathbf{f},\mathbf{v})=\sum_{k=1}^{N}(f^{k},v^{k}),$ where $(,)$ denotes the duality pairing between $H^{-1}(\Omega ;\mathbb{R})$ and $H_{0}^{1}(\Omega ;\mathbb{R})$ as well as between $H^{-1}(\Omega ; \mathbb{R})^{N}$ and $H_{0}^{1}(\Omega ;\mathbb{R})^{N}$. Thanks to (\ref {eq2.2})-(\ref{eq2.5}), this variational problem admits at least one solution, as is easily seen by following \cite[p.99]{bib5} or \cite[p.164] {bib15}. Let us check that \eqref{eq2.7} has at most one solution. To begin, observe that any $\mathbf{u}^{\ast }$ satisfying \eqref{eq2.7} (i.e., with $\mathbf{u}^{\ast }$ in place of $\mathbf{u}_{\varepsilon }$) verifies $$\Vert \mathbf{u}^{\ast }\Vert _{H_{0}^{1}(\Omega )^{N}}\leq \frac{1}{\alpha } \Vert \mathbf{f}\Vert _{H^{-1}(\Omega )^{N}}, \label{eq2.8}$$ as is straightforward by (\ref{eq2.2}). Now, suppose $\mathbf{u}^{\ast }$ and $\mathbf{u}^{\ast \ast }$ are two solutions of \eqref{eq2.7}. Then, letting $\mathbf{u}=\mathbf{u}^{\ast }-\mathbf{u}^{\ast \ast }$, we have in an obvious manner $a^{\varepsilon }(\mathbf{u},\mathbf{v})+b(\mathbf{u}^{\ast },\mathbf{u}, \mathbf{v})+b(\mathbf{u},\mathbf{u}^{\ast },\mathbf{v})-b(\mathbf{u},\mathbf{ u},\mathbf{v})=0$ and that for any $\mathbf{v}\in V$. By choosing in particular $\mathbf{v}= \mathbf{u}$ and recalling (\ref{eq2.4}), it follows by (\ref{eq2.2}), $\alpha \Vert \mathbf{u}\Vert _{H_{0}^{1}(\Omega )^{N}}^{2}+b(\mathbf{u}, \mathbf{u}^{\ast },\mathbf{u})\leq 0.$ Hence, in view of \eqref{eq2.5}, $\alpha \Vert \mathbf{u}\Vert _{H_{0}^{1}(\Omega )^{N}}^{2}\leq c(N)\Vert \mathbf{u}\Vert _{H_{0}^{1}(\Omega )^{N}}^{2}\Vert \mathbf{u}^{\ast }\Vert _{H_{0}^{1}(\Omega )^{N}}.$ By (\ref{eq2.8}) this gives $\Big(\alpha -\frac{c(N)}{\alpha }\Vert \mathbf{f}\Vert _{H^{-1}(\Omega )^{N}} \Big)\Vert \mathbf{u}\Vert _{H_{0}^{1}(\Omega )^{N}}^{2}\leq 0.$ Hence $\mathbf{u}=0$, by virtue of \eqref{eq2.6}. This shows the unicity in ( \ref{eq2.7}), and so \eqref{eq2.7} determines a unique vector function $\mathbf{u}_{\varepsilon }$. Now, by taking in \eqref{eq2.7} the particular test functions $\mathbf{v}\in \mathcal{V}$ with $\mathcal{V}=\big\{\mathbf{\varphi }\in \mathcal{D}(\Omega ;\mathbb{R})^{N}: \mathop{\rm div}\mathbf{\varphi }=0\big\}$ and using a classical argument (see, e.g., \cite[p.14]{bib15}), we get a distribution $p_{\varepsilon }\in \mathcal{D}'(\Omega )$ such that \eqref{eq1.3} holds (in the distribution sense on $\Omega$), with in addition (\ref{eq1.4})-\eqref{eq1.5}, of course. Let us show that $p_{\varepsilon }$ lies in $L^{2}(\Omega ;\mathbb{R})$. First of all, since $N=2$ or $3$, we have $H_{0}^{1}(\Omega ;\mathbb{R})\subset L^{4}(\Omega ; \mathbb{R})$ (see, e.g., \cite[pp.291, 296]{bib15}). Thus, $\mathbf{u} _{\varepsilon }\in L^{4}(\Omega ;\mathbb{R})^{N}$. Consequently, $u_{\varepsilon }^{i}u_{\varepsilon }^{j}\in L^{2}(\Omega ;\mathbb{R})$ $(1\leq i,j\leq N)$. Observing that $\sum_{j=1}^{N}u_{\varepsilon }^{j}\frac{\partial \mathbf{u}_{\varepsilon }}{ \partial x_{j}}=\sum_{j=1}^{N}\frac{\partial }{\partial x_{j}} (u_{\varepsilon }^{j}\mathbf{u}_{\varepsilon })\quad (\text{use (\ref{eq1.4}) }),$ it follows that $\sum_{j=1}^{N}u_{\varepsilon }^{j}\frac{\partial \mathbf{u} _{\varepsilon }}{\partial x_{j}}\in H^{-1}(\Omega ;\mathbb{R})^{N}$. By \eqref{eq1.3}, we deduce that $\mathop{\rm grad}p_{\varepsilon }\in H^{-1}(\Omega ;\mathbb{R})^{N}$. Therefore, thanks to a well-known result (see, e.g., \cite[p.14, Proposition 1.2]{bib15}), the distribution $p_{\varepsilon }$ is actually a function in $L^{2}(\Omega ;\mathbb{R})$, and further the said function is unique up to an additive constant; in other words, $p_{\varepsilon }$ is unique in $L^{2}(\Omega ;\mathbb{R})/\mathbb{R}$. Conversely, it is an easy exercise to verify that if $(\mathbf{u} _{\varepsilon },p_{\varepsilon })$ lies in $H^{1}(\Omega ;\mathbb{R} )^{N}\times L^{2}(\Omega ;\mathbb{R})$ and is a solution of (\ref{eq1.3})- \eqref{eq1.5}, then $\mathbf{u}_{\varepsilon }$ satisfies (\ref{eq2.7}). This completes the proof. \end{proof} \subsection{A global homogenization theorem} Before we can establish a so-called global homogenization theorem for \eqref{eq1.3}-\eqref{eq1.5}, we require a few basic notation and results. To begin, let \begin{gather*} \mathcal{V}_{Y}=\big\{\mathbf{\psi }\in \mathcal{C}_{\mathrm{per}}^{\infty }(Y;\mathbb{R})^{N}:\int_{Y}\mathbf{\psi }(y)dy=0,\;div_{y}\mathbf{\psi =}0 \big\}, \\ V_{Y}=\{\mathbf{w}\in H_{\#}^{1}(Y;\mathbb{R})^{N}:div_{y}\mathbf{w=}0\}, \end{gather*} where: $\mathcal{C}_{\mathrm{per}}^{\infty }(Y;\mathbb{R})=\mathcal{C} ^{\infty }(\mathbb{R}^{N};\mathbb{R})\cap \mathcal{C}_{\mathrm{per}}(Y)$, $\mathop{\rm div}_{y}$ denotes the divergence operator in $\mathbb{R}_{y}^{N}$. We provide $V_{Y}$ with the $H_{\#}^{1}(Y)^{N}$-norm, which makes it a Hilbert space. There is no difficulty in verifying that $\mathcal{V}_{Y}$ is dense in $V_{Y}$ (proceed as in \cite[Proposition 3.2]{bib13}). With this in mind, set $\mathbb{F}_{0}^{1}=V\times L^{2}(\Omega ;V_{Y}).$ This is a Hilbert space with norm $\Vert \mathbf{v}\Vert _{\mathbb{F}_{0}^{1}}=\Big(\Vert \mathbf{v}_{0}\Vert _{H_{0}^{1}(\Omega )^{N}}^{2}+\Vert \mathbf{v}_{1}\Vert _{L^{2}(\Omega ;V_{Y})}^{2}\Big)^{1/2},\quad \mathbf{v=}(\mathbf{v}_{0},\mathbf{v}_{1})\in \mathbb{F}_{0}^{1}.$ On the other hand, put $\mathcal{F}_{0}^{\infty }=\mathcal{V\times }[\mathcal{D}(\Omega ;\mathbb{R} )\otimes \mathcal{V}_{Y}],$ where $\mathcal{D}(\Omega ;\mathbb{R})\otimes \mathcal{V}_{Y}$ stands for the space of vector functions $\mathbf{\psi }$ on $\Omega \times \mathbb{R} _{y}^{N}$ of the form $\mathbf{\psi }(x,y)=\sum \varphi _{i}(x)\mathbf{w}_{i}(y)\quad (x\in \Omega ,\quad y\in \mathbb{R}^{N})$ with a summation of finitely many terms, $\varphi _{i}\in \mathcal{D}(\Omega ;\mathbb{R})$, $\mathbf{w}_{i}\in \mathcal{V}_{Y}$. It is clear that $\mathcal{F}_{0}^{\infty }$ is dense in $\mathbb{F}_{0}^{1}$ (see \cite[p.18] {bib15}). Now, let $\widehat{a}_{\Omega }(\mathbf{u},\mathbf{v})=\sum_{i,j,k=1}^{N}\iint_{\Omega \times Y}a_{ij}\Big(\frac{\partial u_{0}^{k}}{\partial x_{j}}+\frac{\partial u_{1}^{k}}{\partial y_{j}}\Big)\Big(\frac{\partial v_{0}^{k}}{\partial x_{i}} +\frac{\partial v_{1}^{k}}{\partial y_{i}}\Big)\,dx\,dy$ for $\mathbf{u=}(\mathbf{u}_{0},\mathbf{u}_{1})$ and $\mathbf{v=}(\mathbf{v} _{0},\mathbf{v}_{1})$ in $\mathbb{F}_{0}^{1}$. This defines a symmetric continuous bilinear form $\widehat{a}_{\Omega }$ on $\mathbb{F} _{0}^{1}\times \mathbb{F}_{0}^{1}$. Furthermore, $\widehat{a}_{\Omega }$ is $\mathbb{F}_{0}^{1}$-elliptic. Specifically, $\widehat{a}_{\Omega }(\mathbf{u},\mathbf{u})\geq \alpha \Vert \mathbf{u} \Vert _{\mathbb{F}_{0}^{1}}^{2}\quad (\mathbf{u}\in \mathbb{F}_{0}^{1})$ as is easily checked using \eqref{eq1.2} and the fact that $\int_{Y}\frac{ \partial u_{1}^{k}}{\partial y_{j}}(x,y)dy=0$. In the sequel we put \begin{gather*} b_{\Omega }(\mathbf{u},\mathbf{v},\mathbf{w})=b(\mathbf{u} _{0},\mathbf{v} _{0},\mathbf{w}_{0}), \\ L(\mathbf{v})=(\mathbf{f},\mathbf{v}_{0}) \end{gather*} for $\mathbf{u}=(\mathbf{u}_{0},\mathbf{u}_{1})$, $\mathbf{v} =(\mathbf{v} _{0},\mathbf{v}_{1})$ and $\mathbf{w}=( \mathbf{w}_{0},\mathbf{w}_{1})$ in $\mathbb{F}_{0}^{1}$, which defines a continuous trilinear form on $\mathbb{F} _{0}^{1}\times \mathbb{F}_{0}^{1} \times \mathbb{F}_{0}^{1}$ and a continuous linear form on $\mathbb{F}_{0}^{1}$, respectively, with further $b_{\Omega }(\mathbf{u},\mathbf{\ v},\mathbf{v})=0$ for $\mathbf{u}$, $\mathbf{v}\in \mathbb{F}_{0}^{1}$. Here is one fundamental lemma. \begin{lemma} \label{lem2.1} Suppose \eqref{eq2.6} holds. Then the variational problem $$\begin{gathered} \mathbf{u}\in \mathbb{F}_{0}^{1}: \\ \widehat{a}_{\Omega }(\mathbf{u},\mathbf{v})+b_{\Omega }( \mathbf{u},\mathbf{u},\mathbf{v})=L(\mathbf{v})\quad \text{for all }\mathbf{v}\in \mathbb{F}_{0}^{1} \end{gathered} \label{eq2.9}$$ has at most one solution. \end{lemma} The proof of the above lemma follows by the same line of argument as in the proof of Proposition \ref{pr2.1}; so we omit it. We are now able to prove the desired theorem. Throughout the remainder of the present section, it is assumed that $a_{ij}$ is $Y$-periodic for any $1\leq i,j\leq N$. \begin{theorem} \label{thm2.3} Suppose \eqref{eq2.6} holds. For each real $0<\varepsilon <1$, let $\mathbf{u}_{\varepsilon }=(u_{\varepsilon }^{k})\in H_{0}^{1}(\Omega ; \mathbb{R})^{N}$ be defined by \eqref{eq1.3}-\eqref{eq1.5} (or equivalently by \eqref{eq2.7}). Then, as $\varepsilon \to 0$, \begin{gather} \mathbf{u}_{\varepsilon }\to \mathbf{u}_{0}\quad \text{in } H_{0}^{1}(\Omega )^{N}\text{-weak}, \label{eq2.10} \\ \frac{\partial u_{\varepsilon }^{k}}{\partial x_{j}}\to \frac{ \partial u_{0}^{k}}{\partial x_{j}} +\frac{\partial u_{1}^{k}}{\partial y_{j}} \quad \text{in }L^{2}(\Omega )\text{-weak }\Sigma \; (1\leq j,k\leq N), \label{eq2.11} \end{gather} where $\mathbf{u}=(\mathbf{u}_{0},\mathbf{u}_{1})$ is the (unique) solution of \eqref{eq2.9}. \end{theorem} \begin{proof} Let $0<\varepsilon <1$. It is clear that $$a^{\varepsilon }(\mathbf{u}_{\varepsilon },\mathbf{v}) +b(\mathbf{u} _{\varepsilon },\mathbf{u}_{\varepsilon },\mathbf{v}) -\int_{\Omega }p_{\varepsilon }\mathop{\rm div}\mathbf{v}dx =(\mathbf{f},\mathbf{v}) \label{eq2.12}$$ for all $\mathbf{v}=(v^{k})\in H_{0}^{1}(\Omega ;\mathbb{R})^{N}$. Taking in particular $\mathbf{v}=\mathbf{u}_{\varepsilon }$ and using (\ref{eq2.2}) and (\ref{eq2.4}), it follows immediately that the sequence $(\mathbf{u} _{\varepsilon })_{0<\varepsilon <1}$ is bounded in $H_{0}^{1}(\Omega ; \mathbb{R})^{N}$. On the other hand, starting from (\ref{eq2.12}) and recalling (\ref{eq2.3}) and (\ref{eq2.5}), we see that $| (\mathop{\rm grad} p_{\varepsilon },\mathbf{v})| \leq (\| \mathbf{f}\| _{H^{-1}(\Omega )^{N}} +c(N)\| \mathbf{u}_{\varepsilon }\| _{H_{0}^{1}(\Omega )^{N}}^{2}+c_{0}\| \mathbf{u}_{\varepsilon }\| _{H_{0}^{1}(\Omega ) ^{N}})\| \mathbf{v}\| _{H_{0}^{1}(\Omega )^{N}}$ for all $\mathbf{v}\in H_{0}^{1}(\Omega ;\mathbb{R})^{N}$. In view of the preceding result, it follows that the sequence $(\mathop{\rm grad} p_{\varepsilon })_{0<\varepsilon <1}$ is bounded in $H^{-1}(\Omega ;\mathbb{R })^{N}$. Thanks to a classical argument \cite[p.15]{bib15}, we deduce that the sequence $(p_{\varepsilon })_{0<\varepsilon <1}$ is bounded in $L^{2}(\Omega ;\mathbb{R} )$. Thus, given any arbitrary fundamental sequence $E$, appeal to Theorems \ref{thm2.1}-\ref{thm2.2} yields a subsequence $E'$ from $E$ and functions $\mathbf{u}_{0}=(u_{0}^{k})\in H_{0}^{1}( \Omega ;\mathbb{R})^{N}$, $\mathbf{u}_{1}=(u_{1}^{k})\in L^{2}(\Omega ;H_{\#}^{1}(Y;\mathbb{R})^{N})$, $p\in L^{2}(\Omega ;L_{\mathrm{per}}^{2}(Y; \mathbb{R}))$ such that as $E'\ni \varepsilon \to 0$, we have (\ref {eq2.10})-(\ref{eq2.11}) and $$p_{\varepsilon }\to p\quad \text{in }L^{2}(\Omega )\text{-weak }\Sigma . \label{eq2.13}$$ Let us note at once that, according to (\ref{eq1.4}), we have $\mathop{\rm div}\mathbf{u}_{0}=0$ and $\mathop{\rm div}_{y}\mathbf{u}_{1}=0$. Therefore $\mathbf{u}=(\mathbf{u}_{0},\mathbf{u}_{1})\in \mathbb{F}_{0}^{1}$. Now, for each real $0<\varepsilon <1$, let $$\mathbf{\Phi }_{\varepsilon }=\mathbf{\psi }_{0}+\varepsilon \mathbf{\psi } _{1}^{\varepsilon }\quad \text{with }\mathbf{\psi }_{0}\in \mathcal{D} (\Omega ;\mathbb{R})^{N},\;\mathbf{\psi }_{1}\in \mathcal{D}(\Omega ;\mathbb{ R})\otimes \mathcal{V}_{Y}, \label{eq2.14}$$ i.e., $\mathbf{\Phi }_{\varepsilon }(x)=\mathbf{\psi }_{0}(x)+\varepsilon \mathbf{\psi }_{1}(x,\frac{x}{\varepsilon })$ for $x\in \Omega$. We have $\mathbf{\Phi }_{\varepsilon }\in \mathcal{D}(\Omega ;\mathbb{R})^{N}$. Thus, in view of (\ref{eq2.12}), $$a^{\varepsilon }(\mathbf{u}_{\varepsilon },\mathbf{\Phi }_{\varepsilon })+b( \mathbf{u}_{\varepsilon },\mathbf{u}_{\varepsilon },\mathbf{\Phi } _{\varepsilon })-\int_{\Omega }p_{\varepsilon }\mathop{\rm div}\mathbf{\Phi } _{\varepsilon }dx=(\mathbf{f},\mathbf{\Phi }_{\varepsilon }). \label{eq2.15}$$ The next point is to pass to the limit in (\ref{eq2.15}) as $E'\ni \varepsilon \rightarrow 0$. To this end, we note that as $E'\ni \varepsilon \rightarrow 0$, $a^{\varepsilon }(\mathbf{u}_{\varepsilon },\mathbf{\Phi }_{\varepsilon })\rightarrow \widehat{a}_{\Omega }(\mathbf{u},\mathbf{\Phi }),$ where $\mathbf{\Phi =}(\mathbf{\psi }_{0},\mathbf{\psi }_{1})$ (proceed as in the proof of the analogous result in \cite[p.179]{bib12}). On the other hand, thanks to the Rellich theorem, we have from (\ref{eq2.10}) that $\mathbf{u}_{\varepsilon }\rightarrow \mathbf{u}_{0}$ in $L^{2}(\Omega )^{N}$. Combining this with (\ref{eq2.11}), it follows by \cite[Proposition 4.7] {bib9} (see also \cite[Proposition 8]{bib8}) that as $E'\ni \varepsilon \rightarrow 0$, $b(\mathbf{u}_{\varepsilon },\mathbf{u}_{\varepsilon },\mathbf{\Phi } _{\varepsilon })\rightarrow b_{\Omega }(\mathbf{u},\mathbf{u},\mathbf{\Phi } ),$ where $\mathbf{u}$ and $\mathbf{\Phi }$ are defined above. Now, based on ( \ref{eq2.13}), there is no difficulty in showing that as $E'\ni \varepsilon \rightarrow 0$, $\int_{\Omega }p_{\varepsilon }\mathop{\rm div}\mathbf{\Phi }_{\varepsilon }dx\rightarrow \iint_{\Omega \times Y}p\mathop{\rm div}\mathbf{\psi } _{0}\,dx\,dy.$ Finally, it is an easy exercise to check that $\mathbf{\Phi }_{\varepsilon }\rightarrow \mathbf{\psi }_{0}$ in $H_{0}^{1}(\Omega )^{N}$-weak as $\varepsilon \rightarrow 0$ (this is a classical result). Having made this point, we can pass to the limit in (\ref{eq2.15}) when $E'\ni \varepsilon \rightarrow 0$, and the result is that $$\widehat{a}_{\Omega }(\mathbf{u},\mathbf{\Phi })+b_{\Omega }(\mathbf{u}, \mathbf{u},\mathbf{\Phi })-\int_{\Omega }p_{0}\mathop{\rm div}\mathbf{\psi } _{0}dx=(\mathbf{f},\mathbf{\psi }_{0}), \label{eq2.16}$$ where $p_{0}$ denotes the mean of $p$, i.e., $p_{0}\in L^{2}(\Omega ;\mathbb{ R})$ and $p_{0}(x)=\int_{Y}p(x,y)dy$ a.e. in $x\in \Omega$; and where $\mathbf{\Phi =}(\mathbf{\psi }_{0},\mathbf{\psi }_{1})$, $\mathbf{\psi }_{0}$ ranging over $\mathcal{D}(\Omega ;\mathbb{R})^{N}$ and $\mathbf{\psi }_{1}$ ranging over $\mathcal{D}(\Omega ;\mathbb{R})\otimes \mathcal{V}_{Y}$. Taking in particular $\mathbf{\psi }_{0}$ in $\mathcal{V}$ and using the density of $\mathcal{F}_{0}^{\infty }$ in $\mathbb{F}_{0}^{1}$, one quickly arrives at \eqref{eq2.9}. The unicity of $\mathbf{u}=(\mathbf{u}_{0},\mathbf{ u}_{1})$ follows by Lemma \ref{lem2.1}. Consequently, (\ref{eq2.10}) and ( \ref{eq2.11}) still hold when $E\ni \varepsilon \rightarrow 0$ (instead of $E'\ni \varepsilon \rightarrow 0$), hence when $0<\varepsilon \rightarrow 0$, by virtue of the arbitrariness of $E$. The theorem is proved. \end{proof} For further needs, we wish to give a simple representation of the vector function $\mathbf{u}_{1}$ in Theorem \ref{thm2.3} (or Lemma \ref{lem2.1}). For this purpose we introduce the bilinear form $\widehat{a}$ on $V_{Y}\times V_{Y}$ defined by $\widehat{a}(\mathbf{v},\mathbf{w})=\sum_{i,j,k=1}^{N}\int_{Y}a_{ij}\frac{ \partial v^{k}}{\partial y_{j}}\frac{\partial w^{k}}{\partial y_{i}}dy$ for $\mathbf{v}=(v^{k})$ and $\mathbf{w}=(w^{k})$ in $V_{Y}$. Next, for each pair of indices $1\leq i,k\leq N$, we consider the variational problem $$\begin{gathered} \mathbf{\chi }_{ik}\in V_{Y}: \\ \widehat{a}(\mathbf{\chi }_{ik},\mathbf{w}) =\sum_{l=1}^{N}\int_{Y}a_{li}\frac{\partial w^{k}}{\partial y_{l}}dy \quad \text{for all }\mathbf{w}=(w^{j})\text{ in }V_{Y}, \end{gathered} \label{eq2.17}$$ which determines $\mathbf{\chi }_{ik}$ in a unique manner. \begin{lemma} \label{lem2.2} Under the hypothesis and notation of Theorem \ref{thm2.3}, we have $$\mathbf{u}_{1}(x,y)=-\sum_{i,k=1}^{N}\frac{\partial u_{0}^{k}}{ \partial x_{i}}(x)\mathbf{\chi }_{ik}(y) \label{eq2.18}$$ almost everywhere in $(x,y)\in \Omega \times \mathbb{R}^{N}$. \end{lemma} \begin{proof} In \eqref{eq2.9}, choose the test functions $\mathbf{v}=(\mathbf{v}_{0}, \mathbf{v}_{1})$ such that $\mathbf{v}_{0}=0$, $\mathbf{v}_{1}(x,y)=\varphi (x)\mathbf{w}(y)$ for $(x,y)\in \Omega \times \mathbb{R}^{N}$, where $\varphi \in \mathcal{D}(\Omega ;\mathbb{R})$ and $\mathbf{w}\in V_{Y}$. Then, almost everywhere in $x\in \Omega$, we have $$\widehat{a}(\mathbf{u}_{1}(x,.),\mathbf{w}) =-\sum_{l,j,k=1}^{N}\frac{ \partial u_{0}^{k}}{\partial x_{j}}(x) \int_{Y}a_{lj}\frac{\partial w^{k}}{ \partial y_{l}}dy \quad \forall \mathbf{w}=(w^{k})\in V_{Y}. \label{eq2.19}$$ But it is clear that $\mathbf{u}_{1}(x,.)$ (for fixed $x\in \Omega$) is the sole function in $V_{Y}$ solving the variational equation (\ref{eq2.19}). On the other hand, it is an easy matter to check that the function of $y$ on the right of (\ref{eq2.18}) solves the same variational equation. Hence the lemma follows immediately. \end{proof} \subsection{Macroscopic homogenized equations} Our goal here is to derive a well-posed boundary value problem for $(\mathbf{ u}_{0},p_{0})$. To begin, for $1\leq i,j,k,h\leq N$, let $q_{ijkh}=\delta _{kh}\int_{Y}a_{ij}(y) dy-\sum_{l=1}^{N}\int_{Y}a_{il}(y) \frac{\partial \mathcal{\chi } _{jh}^{k}}{\partial y_{l}}(y)dy,$ where: $\delta _{kh}$ is the Kronecker symbol, $\mathbf{\chi }_{jh}=( \mathcal{\chi }_{jh}^{k})$ is defined exactly as in (\ref{eq2.17}). To the coefficients $q_{ijkh}$ we attach the differential operator $\mathcal{Q}$ on $\Omega$ mapping $\mathcal{D}'(\Omega )^{N}$ into $\mathcal{D} '(\Omega )^{N}$ ($\mathcal{D}'(\Omega )$ is the usual space of complex distributions on $\Omega$) as $$(\mathcal{Q}\mathbf{z})^{k}=-\sum_{i,j,h=1}^{N}q_{ijkh}\frac{ \partial ^{2}z^{h}}{\partial x_{i}\partial x_{^{j}}}\quad (1\leq k\leq N)\quad \text{ for }\mathbf{z=}(z^{h}), z^{h}\in \mathcal{D} '(\Omega ). \label{eq2.20}$$ $\mathcal{Q}$ is the so-called homogenized operator associated to $P^{\varepsilon }$ $(0<\varepsilon <1)$. We consider now the boundary value problem \begin{gather} \mathcal{Q}\mathbf{u}_{0}+\sum_{j=1}^{N}u_{0}^{j} \frac{\partial \mathbf{u} _{0}}{\partial x_{j}}+\mathop{\rm grad} p_{0} =\mathbf{f}\quad \text{in } \Omega , \label{eq2.21} \\ \mathop{\rm div}\mathbf{u}_{0}=0\quad \text{in }\Omega , \label{eq2.22} \\ \mathbf{u}_{0}=0\quad \text{on }\partial \Omega . \label{eq2.23} \end{gather} \begin{lemma} \label{lem2.3} Suppose \eqref{eq2.6} holds. Then, the boundary value problem \eqref{eq2.21}-\eqref{eq2.23} admits at most one weak solution $(\mathbf{u} _{0},p_{0})$ with $\mathbf{u}_{0}\in H_{0}^{1}(\Omega;\mathbb{R})^{N}$, $p_{0}\in L^{2}(\Omega ;\mathbb{R})/ \mathbb{R}$. \end{lemma} \begin{proof} It can be proved without the slightest difficulty that if a pair $(\mathbf{u} _{0},p_{0})\in H_{0}^{1}(\Omega ;\mathbb{R}) ^{N}\times L^{2}(\Omega ; \mathbb{R})$ verifies \eqref{eq2.21}-\eqref{eq2.23}, then the vector function $\mathbf{u}=(\mathbf{u}_{0},\mathbf{u}_{1})$ [with $\mathbf{u}_{1}$ given by \eqref{eq2.18}] satisfies \eqref{eq2.9} (use (\ref{eq2.16})). Hence the unicity in \eqref{eq2.21}-\eqref{eq2.23} follows by Lemma \ref{lem2.1}. \end{proof} This leads us to the following theorem. \begin{theorem} \label{thm2.4} Suppose \eqref{eq2.6} holds. For each real $0<\varepsilon <1$, let $(\mathbf{u}_{\varepsilon },p_{\varepsilon })\in H_{0}^{1}(\Omega ; \mathbb{R})^{N}\times (L^{2}( \Omega ;\mathbb{R})/ \mathbb{R})$ be defined by \eqref{eq1.3}-\eqref{eq1.5}. Then, as $\varepsilon \to 0$, we have $\mathbf{u}_{\varepsilon }\to \mathbf{u}_{0}$ in $H_{0}^{1}(\Omega )^{N}$-weak and $p_{\varepsilon }\to p_{0}$ in $L^{2}(\Omega )$-weak, where the pair $(\mathbf{u}_{0},p_{0})$ lies in $H_{0}^{1}(\Omega ;\mathbb{R} )^{N}\times (L^{2} (\Omega ;\mathbb{R})/ \mathbb{R})$ and is the unique weak solution of \eqref{eq2.21}-\eqref{eq2.23}. \end{theorem} \begin{proof} A quick review of the proof of Theorem \ref{thm2.3} reveals that from any given fundamental sequence $E$ one can extract a subsequence $E'$ such that as \noindent $E'\ni \varepsilon \to 0$, we have ( \ref {eq2.10})-(\ref{eq2.11}) and $p_{\varepsilon }\to p_{0}$ in $L^{2}(\Omega )$-weak (use (\ref{eq2.13}) if necessary), and further (\ref{eq2.16}) holds for all $\mathbf{\Phi =}(\mathbf{\psi }_{0},\mathbf{\psi }_{1})\in \mathcal{D }(\Omega ;\mathbb{R} )^{N}\times \left[ \mathcal{D}(\Omega ;\mathbb{R}) \otimes \mathcal{V}_{Y}\right]$, where $\mathbf{u}=(\mathbf{u}_{0}, \mathbf{ u}_{1})\in \mathbb{F}_{0}^{1}$. Now, substituting (\ref{eq2.18}) in (\ref {eq2.16}) and then choosing therein the $\mathbf{\Phi }$'s such that $\mathbf{\psi }_{1}=0$, a simple computation leads to (\ref{eq2.21}) with evidently (\ref{eq2.22})-\eqref{eq2.23}. Hence the theorem follows by Lemma \ref{lem2.2} and use of an obvious argument. \end{proof} \begin{remark} \label{rem2.2} \rm The operator $\mathcal{Q}$ is elliptic, i.e., there is some $\alpha _{0}>0$ such that $\sum_{i,j,k,h=1}^{N}q_{ijkh}\xi _{ik}\xi _{jh}\geq \alpha _{0}\sum_{k,h=1}^{N}| \xi _{kh}| ^{2}$ for all $\xi =(\xi _{kh})$, $\xi _{kh}\in \mathbb{R}$. Indeed, by following a classical line of argument (see, e.g., \cite{bib2}), we can give a suitable expression of $q_{ijkh}$, viz. $q_{ijkh}=\widehat{a}(\mathbf{\chi }_{ik}-\mathbf{\pi }_{ik},\mathbf{\ \chi } _{jh}-\mathbf{\pi }_{jh}),$ where, for each pair of indices $1\leq i,k\leq N$, the vector function $\mathbf{\pi }_{ik}=(\pi _{ik}^{1},\dots ,\pi _{ik}^{N}):\mathbb{R} _{y}^{N}\to \mathbb{R}$ is given by $\pi _{ik}^{r}(y)=y_{i}\delta _{kr}$ $(r=1,\dots ,N)$ for $y=(y_{1},\dots ,y_{N})\in \mathbb{R}^{N}$. Hence, the above ellipticity property follows in a classical fashion (see \cite{bib2}). \end{remark} \section{General deterministic homogenization of stationary Navier-Stokes type equations} Our purpose here is to extend the results of Section 2 to a more general setting beyond the periodic framework. The basic notation and hypotheses (except the periodicity assumption) stated before are still valid. In particular $N$ is either $2$ or $3$, and $\Omega$ denotes a bounded open set in $\mathbb{R}_{x}^{N}$. \subsection{Preliminaries and statement of the homogenization problem} We recall that $\mathcal{B}(\mathbb{R}_{y}^{N})$ denotes the space of bounded continuous complex functions on $\mathbb{R}_{y}^{N}$. It is well known that $\mathcal{B}(\mathbb{R}_{y}^{N})$ with the supremum norm and the usual algebra operations is a commutative $\mathcal{C}^{\ast }$-algebra with identity (the involution is here the usual one of complex conjugation). Throughout the present Section 3, $A$ denotes a separable closed subalgebra of the Banach algebra $\mathcal{B}(\mathbb{R}_{y}^{N})$. Furthermore, we assume that $A$ contains the constants, $A$ is stable under complex conjugation (i.e., the complex conjugate, $\overline{u}$, of any $u\in A$ still lies in $A$), and finally, $A$ has the following property: For any $u\in A$, we have $u^{\varepsilon }\to M(u)$ in $L^{\infty }(\mathbb{R} _{x}^{N})$-weak $\ast$ as $\varepsilon \to 0$ $(\varepsilon >0)$, where: $u^{\varepsilon }(x)=u(\frac{x}{\varepsilon })\quad (x\in \mathbb{R}^{N}),$ the mapping $u\to M(u)$ of $A$ into $\mathbb{C}$, denoted by $M$, being a positive continuous linear form on $A$ with $M(1)=1$ (see \cite{bib9}). $A$ is called an $H$-algebra ($H$ stands for \textit{homogenization}). It is clear that $A$ is a commutative $\mathcal{C}^{\ast }$-algebra with identity. We denote by $\Delta (A)$ the spectrum of $A$ and by $\mathcal{G}$ the Gelfand transformation on $A$. For the benefit of the reader it is worth recalling that $\Delta (A)$ is the set of all nonzero multiplicative linear forms on $A$, and $\mathcal{G}$ is the mapping of $A$ into $\mathcal{C} (\Delta (A))$ such that $\mathcal{\ G}(u)(s)= \langle s,u \rangle$ $(s\in \Delta (A))$, where $\langle ,\rangle$ denotes the duality pairing between $A'$ (the topological dual of $A$) and $A$. The appropriate topology on $\Delta (A)$ is the relative weak $\ast$ topology on $A'$. So topologized, $\Delta (A)$ is a metrizable compact space, and the Gelfand transformation is an isometric isomorphism of the $\mathcal{C}^{\ast }$ -algebra $A$ onto the $\mathcal{C}^{\ast }$-algebra $\mathcal{C}( \Delta (A))$. See, e.g., \cite{bib4} for further details concerning the Banach algebras theory. The appropriate measure on $\Delta (A)$ is the so-called $M$-measure, namely the positive Radon measure $\beta$ (of total mass $1$) on $\Delta (A)$ such that $M(u)=\int_{\Delta (A)}\mathcal{G}(u) d\beta$ for $u\in A$ (see \cite[Proposition 2.1]{bib9}). 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 $A^{1}=\{ \psi \in \mathcal{C}^{1}( \mathbb{R}_{y}^{N}):\psi ,\quad D_{y_{i}}\psi \in A\quad (1\leq i\leq N)\}$, $D_{y_{i}}=\frac{\partial }{\partial y_{i}}$. Higher order derivatives can be defined analogously (see \cite{bib9}). Now, let $A^{\infty }$ be the space of $\psi \in \mathcal{C}^{\infty }(\mathbb{R}_{y}^{N})$ such that $D_{y}^{\alpha }\psi =\frac{\partial ^{| \alpha | }\psi }{ \partial y_{1}^{\alpha _{1}}\dots \partial y_{N}^{\alpha _{N}}}\in A$ for every multi-index $\alpha =(\alpha _{1},\dots ,\alpha _{N})\in \mathbb{N} ^{N}$, and 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 for example \cite{bib9} ), $A^{\infty }$ (respectively $\mathcal{D}(\Delta (A))$) is a Fr\'{e}chet space and further, $\mathcal{G}$ viewed as defined on $A^{\infty }$ is a topological isomorphism of $A^{\infty }$ onto $\mathcal{D}(\Delta (A))$. By a distribution on $\Delta (A)$ is understood any continuous linear form on $\mathcal{D}(\Delta (A))$. The space of all distributions on $\Delta (A)$ is then the dual, $\mathcal{D}'(\Delta (A))$, of $\mathcal{D} (\Delta (A))$. We endow $\mathcal{D}'(\Delta (A))$ with the strong dual topology. In the sequel it is assumed that $A^{\infty }$ is dense in $A$ (this is always verified in practice), which amounts to assuming that $\mathcal{D}(\Delta (A))$ is dense in $\mathcal{C}(\Delta (A))$. Then $L^{p}(\Delta (A)) \subset \mathcal{D}'(\Delta (A))$ $(1\leq p\leq \infty )$ with continuous embedding (see \cite{bib9} for more details). Hence we may define $H^{1}(\Delta (A))=\{ u\in L^{2}(\Delta (A)):\partial _{i}u\in L^{2}(\Delta ( A))\quad (1\leq i\leq N)\} ,$ where the derivative $\partial _{i}u$ is taken in the distribution sense on $\Delta (A)$ (exactly as the Schwartz derivative is defined in the classical case). This is a Hilbert space with norm $\| u\| _{H^{1}(\Delta (A)) }=\Big(\| u\| _{L^{2}(\Delta (A) )}^{2}+\sum_{i=1}^{N}\| \partial _{i}u\| _{L^{2}(\Delta (A))}^{2}\Big)^{1/2} \quad (u\in H^{1}(\Delta (A))).$ However, in practice the appropriate space is not $H^{1}(\Delta (A))$ but its closed subspace $H^{1}(\Delta (A))/\mathbb{C}=\big\{u\in H^{1}(\Delta (A)):\int_{\Delta (A)}u(s)d\beta (s)=0\big\}$ equipped with the seminorm $\Vert u\Vert _{H^{1}(\Delta (A))/\mathbb{C}}=\Big(\sum_{i=1}^{N}\Vert \partial _{i}u\Vert _{L^{2}(\Delta (A))}^{2}\Big)^{1/2}\quad (u\in H^{1}(\Delta (A))/\mathbb{C}).$ Unfortunately, the pre-Hilbert space $H^{1}(\Delta (A))/\mathbb{C}$ is in general nonseparated and noncomplete. We introduce the separated completion, $H_{\#}^{1}(\Delta (A))$, of $H^{1}(\Delta (A))/\mathbb{C}$, and the canonical mapping $J$ of $H^{1}(\Delta (A))/\mathbb{C}$ into its separated completion. See \cite{bib9} (and in particular Remark 2.4 and Proposition 2.6 there) for more details. We will now recall the notion of $\Sigma$-convergence in the present context. Let $1\leq p<\infty$, and let $E$ be as in Section 2. \begin{definition} \label{def3.1} \rm A sequence $(u_{\varepsilon })_{\varepsilon \in E}\subset L^{p}(\Omega )$ is said to be: \noindent (i) weakly $\Sigma$-convergent in $L^{p}(\Omega )$ to some $u_{0}\in L^{p}(\Omega \times \Delta (A))=L^{p}(\Omega ;L^{p}(\Delta (A)))$ if as $E\ni \varepsilon \rightarrow 0$, $\int_{\Omega }u_{\varepsilon }(x)\psi ^{\varepsilon }(x)dx\rightarrow \iint_{\Omega \times \Delta (A)}u_{0}(x,s)\widehat{\psi }(x,s)dxd\beta (s)$ for all $\psi \in L^{p'}(\Omega ;A)$ $(\frac{1}{p'}=1- \frac{1}{p})$, where $\psi ^{\varepsilon }$ is as in Definition \ref{def2.1} , and where $\widehat{\psi }(x,.)=\mathcal{G}(\psi (x,.))$ a.e. in $x\in \Omega$; \noindent (ii) strongly $\Sigma$-convergent in $L^{p}(\Omega )$ to some $u_{0}\in L^{p}(\Omega \times \Delta (A))$ if the following property is verified: Given $\eta >0$ and $v\in L^{p}(\Omega ;A)$ with $\Vert u_{0}- \widehat{v}\Vert _{L^{p}(\Omega \times \Delta (A))}\leq \frac{\eta }{2}$, there is some $\alpha >0$ such that $\Vert u_{\varepsilon }-v^{\varepsilon }\Vert _{L^{p}(\Omega )}\leq \eta \quad \text{provided }E\ni \varepsilon \leq \alpha .$ \end{definition} \begin{remark} \label{rem3.1} \rm The existence of such $v$'s as in (ii) results from the density of $L^{p}(\Omega ;\mathcal{C}(\Delta (A)))$ in $L^{p}(\Omega ;L^{p}(\Delta (A)))$. \end{remark} We will use the same notation as in Section 2 to briefly express weak and strong $\Sigma$-convergence. Theorem \ref{thm2.1} (together with its proof) carries over to the present setting. Instead of Theorem \ref{thm2.2}, we have here the following notion. \begin{definition} \label{def3.2} \rm The $H$-algebra $A$ is said to be $H^{1}$-\textit{ proper} (or simply proper when there is no risk of confusion) if the following conditions are fulfilled. \begin{itemize} \item[(PR1)] $\mathcal{D}(\Delta (A))$ is dense in $H^{1}(\Delta (A))$. \item[(PR2)] Given a fundamental sequence $E$, and a sequence $(u_{\varepsilon })_{\varepsilon \in E}$ which is bounded in $H^{1}(\Omega )$, one can extract a subsequence $E'$ from $E$ such that as $E'\ni \varepsilon \to 0$, we have $u_{\varepsilon }\to u_{0}$ in $H^{1}(\Omega )$-weak and $\frac{\partial u_{\varepsilon }}{\partial x_{j}} \to \frac{\partial u_{0}}{\partial x_{j}}+\partial _{j}u_{1}$ in $L^{2}(\Omega )$-weak $\Sigma$ $(1\leq j\leq N)$, where $u_{0}\in H^{1}(\Omega )$, $u_{1}\in L^{p}(\Omega ;H_{\#}^{1}(\Delta (A)))$. \end{itemize} \end{definition} The $H$-algebra $A=\mathcal{C}_{\mathrm{per}}(Y)$ (see Section 2) is $H^{1}$-proper. Other examples of $H^{1}$-proper $H$-algebras can be found in \cite {bib9} and \cite{bib10}. Having made the above preliminaries, let us turn now to the statement of a general deterministic homogenization problem for \eqref{eq1.3}-\eqref{eq1.5}. For this purpose, let $\Xi ^{2}$ be the space of functions $u\in L_{ \mathrm{loc}}^{2}(\mathbb{R}_{y}^{N})$ such that $\| u\| _{\Xi ^{2}}=\sup_{0<\varepsilon \leq 1} \Big(\int_{B_{N}}| u(\frac{x}{ \varepsilon }) | ^{2}dx\Big)^{1/2}<\infty ,$ where $B_{N}$ denotes the open unit ball in $\mathbb{R}_{y}^{N}$. $\Xi ^{2}$ is a complex vector space, and the mapping $u\to \|u\| _{\Xi ^{2}}$, denoted by $\| .\| _{\Xi ^{2}}$, is a norm on $\Xi ^{2}$ which makes it a Banach space (this is a simple exercise left to the reader). We define $\mathfrak{X} ^{2}$ to be the closure of $A$ in $\Xi ^{2}$. We provide $\mathfrak{X}^{2}$ with the $\Xi ^{2}$-norm, which makes it a Banach space. Our main goal in the present section is to discuss the homogenization of \eqref{eq1.3}-\eqref{eq1.5} under the assumption $$a_{ij}\in \mathfrak{X}^{2}\quad (1\leq i,j\leq N). \label{eq3.1}$$ As is pointed out in \cite{bib9}, \cite{bib10} and \cite{bib12}, assumption \eqref{eq3.1} covers a great variety of concrete behaviors. In particular, \eqref{eq3.1} generalizes the usual periodicity hypothesis (see Section 2). Indeed, for $A=\mathcal{C}_{\mathrm{per}}(Y)$, we have $\mathfrak{X}^{2}=L_{ \mathrm{per}}^{2}(Y)$ (use Lemma 1 of \cite{bib8}). The approach we follow here is analogous to that which was presented in Section 2. Throughout the rest of the section, it is assumed that \eqref{eq3.1} is satisfied, and $A$ is $H^{1}$-proper. \subsection{A global homogenization theorem} We need a few preliminaries. To begin, we set $\mathcal{G}(\mathbf{\psi })=(\mathcal{G}(\psi ^{i}))_{1\leq i\leq N}$ for any $\mathbf{\psi =}(\psi ^{i})$ with $\psi ^{i}\in A$ $(1\leq i\leq N)$. We have $\mathcal{G}(\mathbf{\psi } )\in \mathcal{C}(\Delta (A))^{N}$, and the transformation $\mathbf{\psi }\to \mathcal{G}(\mathbf{\psi })$ of $A^{N}$ into $\mathcal{C}(\Delta (A))^{N}$ maps in particular $(A_{\mathbb{R} }^{\infty })^{N}$ isomorphically onto $\mathcal{D}(\Delta (A);\mathbb{R} )^{N}$, where we denote $A_{\mathbb{R}}^{\infty }=A^{\infty }\cap \mathcal{C}(\mathbb{R}^{N}; \mathbb{ R}).$ Likewise, letting $\mathbf{J}(\mathbf{u})=(J(u^{i}))_{1\leq i\leq N}$ for $\mathbf{u=}(u^{i})$ with $u^{i}\in H^{1}(\Delta (A))/\mathbb{C}$ $(1\leq i\leq N)$, we have $\mathbf{J}(\mathbf{u})\in H_{\#}^{1}(\Delta (A))^{N}$ and the transformation $\mathbf{u}\to \mathbf{J}(\mathbf{u})$ of $[ H^{1}(\Delta (A))/\mathbb{C}] ^{N}$ into $H_{\#}^{1}(\Delta (A) )^{N}$ maps in particular $[ H^{1}(\Delta (A); \mathbb{R})/ \mathbb{C}] ^{N}$ isometrically into $H_{\#}^{1}(\Delta (A);\mathbb{R})^{N}$, where we denote $H_{\#}^{1}(\Delta (A);\mathbb{R})=\{ u\in H_{\#}^{1}(\Delta (A)):\partial _{i}u\in L^{2}(\Delta (A);\mathbb{R})\quad (1\leq i\leq N)\} .$ We will set \begin{gather*} \mathbb{E}_{0}^{1}=H_{0}^{1}(\Omega ;\mathbb{R})^{N}\times L^{2}(\Omega ;H_{\#}^{1}(\Delta (A);\mathbb{R})^{N}), \\ \mathcal{E}_{0}^{\infty }=\mathcal{D}(\Omega ;\mathbb{R}) ^{N}\times \big( \mathcal{D}(\Omega ;\mathbb{R})\otimes \mathbf{J}[ \mathcal{D}(\Delta (A); \mathbb{R}) / \mathbb{C}] ^{N}\Big), \end{gather*} where $\mathcal{D}(\Delta (A);\mathbb{R})/\mathbb{C} =\mathcal{D}(\Delta (A); \mathbb{R})\cap [ H^{1}(\Delta (A))/ \mathbb{C}]$. $\mathbb{E}_{0}^{1}$ is topologized in an obvious way and $\mathcal{E}_{0}^{\infty }$ is considered without topology. It is clear that $\mathcal{E}_{0}^{\infty }$ is dense in $\mathbb{E}_{0}^{1}$. At the present time, let $\widehat{a}_{\Omega }(\mathbf{u},\mathbf{v}) =\sum_{i,j,k=1}^{N}\iint_{\Omega \times \Delta (A)}\widehat{ a}_{ij}\Big( \frac{\partial u_{0}^{k}}{\partial x_{j}}+\partial _{j}u_{1}^{k}\Big)\Big( \frac{\partial v_{0}^{k}}{\partial x_{i}} +\partial _{i}v_{1}^{k}\Big) dx\,d\beta$ for $\mathbf{u}=(\mathbf{u}_{0},\mathbf{u}_{1})$ and $\mathbf{v} =(\mathbf{v} _{0},\mathbf{v}_{1})$ in $\mathbb{E}_{0}^{1}$ with, of course, $\mathbf{u} _{0}=(u_{0}^{k})$, $\mathbf{u}_{1}=(u_{1}^{k})$ (and analogous expressions for $\mathbf{v}_{0}$ and $\mathbf{v}_{1}$), where $\widehat{a}_{ij}=\mathcal{ G}(a_{ij})$. This gives a bilinear form $\widehat{a}_{\Omega }$ on $\mathbb{E }_{0}^{1}\times \mathbb{E}_{0}^{1}$, which is symmetric, continuous, and coercive (see \cite{bib9}). We also define $b_{\Omega }$ and $L$ as in Subsection 2.2 but with $\mathbb{E}_{0}^{1}$ in place of $\mathbb{F}_{0}^{1}$. Now, let $V_{A}=\{ \mathbf{u=}(u^{i})\in H_{\#}^{1}(\Delta (A);\mathbb{R})^{N}: \widehat{\mathop{\rm div}}\mathbf{u=}0\},$ where $\widehat{\mathop{\rm div}}\mathbf{u}=\sum_{i=1}^{N}\partial _{i}u^{i}.$ Equipped with the $H_{\#}^{1}(\Delta (A))^{N}$-norm, $V_{A}$ is a Hilbert space. We next put $\mathbb{F}_{0}^{1}=V\times L^{2}(\Omega ;V_{A})$ provided with an obvious norm. It is an easy exercise to check that Lemma \ref{lem2.1} together with its proof can be carried over mutatis mutandis to the present setting. This leads us to the analogue of Theorem \ref{thm2.3}. \begin{theorem} \label{thm3.1} Suppose \eqref{eq3.1} holds and further $A$ is $H^{1}$-proper. On the other hand, let \eqref{eq2.6} be satisfied. For each real $0<\varepsilon <1$, let $\mathbf{u}_{\varepsilon }=(u_{\varepsilon}^{k})\in H_{0}^{1} (\Omega ;\mathbb{R})^{N}$ be defined by \eqref{eq1.3}-\eqref{eq1.5} (or equivalently by \eqref{eq2.7}). Then, as $\varepsilon \to 0$, \begin{gather} \mathbf{u}_{\varepsilon }\to \mathbf{u}_{0}\quad \text{ in } H_{0}^{1}(\Omega )^{N}\text{-weak}, \label{eq3.2} \\ \frac{\partial u_{\varepsilon }^{k}}{\partial x_{j}}\to \frac{ \partial u_{0}^{k}}{\partial x_{j}}+\partial _{j}u_{1}^{k}\quad \text{in } L^{2}(\Omega )\text{-weak }\Sigma \; (1\leq j,k\leq N), \label{eq3.3} \end{gather} where $\mathbf{u}=(\mathbf{u}_{0},\mathbf{u}_{1})$ (with $\mathbf{u} _{0}=(u_{0}^{k})$ and $\mathbf{u}_{1}=( u_{1}^{k})$) is the unique solution of \eqref{eq2.9}. \end{theorem} \begin{proof} This is an adaptation of the proof of Theorem \ref{thm2.3} and we will not go too deeply into details. Starting from (\ref{eq2.12}), we see that the generalized sequences $(\mathbf{u}_{\varepsilon }) _{0<\varepsilon <1}$ and $(p_{\varepsilon })_{0<\varepsilon <1}$ are bounded in $H_{0}^{1}(\Omega ; \mathbb{R})^{N}$ and $L^{2}(\Omega ;\mathbb{R})/ \mathbb{R}$, respectively. Hence, from any given fundamental sequence $E$ one can extract a subsequence $E'$ such that as $E'\ni \varepsilon \to 0$, we have (\ref{eq2.13}), \eqref{eq3.2} and \eqref{eq3.3}, where $p$ lies in $L^{2}(\Omega ;L^{2}(\Delta (A);\mathbb{R}))$ and $\mathbf{u}=(\mathbf{u}_{0},\mathbf{u} _{1})$ lies in $\mathbb{F}_{0}^{1}$. Now, for each real $0<\varepsilon <1$, let $$\mathbf{\Phi }_{\varepsilon }=\mathbf{\psi }_{0}+\varepsilon \mathbf{\psi } _{1}^{\varepsilon }\quad \text{ with }\mathbf{\psi }_{0}\in \mathcal{D} (\Omega ;\mathbb{R})^{N},\mathbf{\psi }_{1}\in \mathcal{D}(\Omega ; \mathbb{R})\otimes (A_{\mathbb{R}}^{\infty }/\mathbb{C})^{N} \label{eq3.4}$$ and $\mathbf{\Phi }=\big(\mathbf{\psi }_{0},\mathbf{J}(\widehat{\mathbf{\psi }} _{1})\big),$ where: $A_{\mathbb{R}}^{\infty }/\mathbb{C=}\{\psi \in A_{\mathbb{R} }^{\infty }:M(\psi )=0\}$, $\widehat{\mathbf{\psi }}_{1}$ stands for the function $x\rightarrow \mathcal{G}(\mathbf{\psi }_{1}(x,.))$ of $\Omega$ into $[\mathcal{D}(\Delta (A);\mathbb{R})/\mathbb{C}]^{N}$ ($\mathbf{\psi } _{1}$ being viewed as a function say in $\mathcal{C}(\Omega ;A^{N})$), $\mathbf{J}(\widehat{\mathbf{\psi }}_{1})$ stands for the function $x\rightarrow \mathbf{J}(\widehat{\mathbf{\psi }}_{1}(x,.))$ of $\Omega$ into $H_{\#}^{1}(\Delta (A);\mathbb{R})^{N}$. It is clear that $\mathbf{\Phi }\in \mathcal{E}_{0}^{\infty }$. With this in mind, we can pass to the limit in (\ref{eq2.15}) (with $\mathbf{\Phi }_{\varepsilon }$ given by (\ref{eq3.4} )) as $E'\ni \varepsilon \rightarrow 0$, and we obtain $\widehat{a}_{\Omega }(\mathbf{u},\mathbf{\Phi })+b_{\Omega }(\mathbf{u}, \mathbf{u},\mathbf{\Phi })-\iint_{\Omega \times \Delta (A)}p(\mathop{\rm div} \mathbf{\psi }_{0}+\widehat{\mathop{\rm div}}\widehat{\mathbf{\psi }} _{1})dxd\beta =(\mathbf{f},\mathbf{\psi }_{0}).$ Therefore, thanks to the density of $\mathcal{E}_{0}^{\infty }$ in $\mathbb{E }_{0}^{1}$, $$\widehat{a}_{\Omega }(\mathbf{u},\mathbf{v})+b_{\Omega }(\mathbf{u},\mathbf{u },\mathbf{v})-\iint_{\Omega \times \Delta (A)}p(\mathop{\rm div}\mathbf{v} _{0}+\widehat{\mathop{\rm div}}\mathbf{v}_{1})dxd\beta =(\mathbf{f},\mathbf{v }_{0}), \label{eq3.5}$$ and that for all $\mathbf{v=}(\mathbf{v}_{0},\mathbf{v}_{1})\in \mathbb{E} _{0}^{1}$. Taking in particular $\mathbf{v}\in \mathbb{F}_{0}^{1}$ leads us immediately to \eqref{eq2.9}. Hence the theorem follows by the same argument as used in the proof of Theorem \ref{thm2.3}. \end{proof} As pointed out in Section 2, it is of interest to give a suitable representation of $\mathbf{u}_{1}$ (in Theorem \ref{thm3.1}). To this end, let $\widehat{a}(\mathbf{v},\mathbf{w})=\sum_{i,j,k=1}^{N}\int_{\Delta (A)} \widehat{a}_{ij}\partial _{j}v^{k}\partial _{i}w^{k}d\beta$ for $\mathbf{v=}(v^{k})$ and $\mathbf{w=}(w^{k})$ in $H_{\#}^{1}(\Delta (A); \mathbb{R})^{N}$. This defines a bilinear form $\widehat{a}$ on $H_{\#}^{1}(\Delta (A);\mathbb{R})^{N}\times H_{\#}^{1}(\Delta (A);\mathbb{R} )^{N}$, which is symmetric, continuous and coercive. For each pair of indices $1\leq i,k\leq N$, we consider the variational problem $$\begin{gathered} \mathbf{\chi }_{ik}\in V_{A}: \\ \widehat{a}(\mathbf{\chi }_{ik},\mathbf{w}) =\sum_{l=1}^{N}\int_{\Delta (A)}\widehat{a}_{li}\partial _{l}w^{k}d\beta \quad \text{for all }\mathbf{w=}(w^{j})\in V_{A}, \end{gathered} \label{eq3.6}$$ which uniquely determines $\mathbf{\chi }_{ik}$. \begin{lemma} \label{lem3.1} Under the assumptions and notation of Theorem \ref{thm3.1}, we have $$\mathbf{u}_{1}(x,s)=-\sum_{i,k=1}^{N}\frac{\partial u_{0}^{k}}{ \partial x_{i}}(x)\mathbf{\chi }_{ik}(s) \label{eq3.7}$$ almost everywhere in $(x,s)\in \Omega \times \Delta (A)$. \end{lemma} \begin{proof} This is a simple adaptation of the proof of Lemma \ref{lem2.2}; the verification is left to the reader. \end{proof} \subsection{Macroscopic homogenized equations} The aim here is to derive from (\ref{eq3.5}) a well-posed boundary value problem for the pair $(\mathbf{u}_{0},p_{0})$, where $\mathbf{u}_{0}$ is the weak limit in \eqref{eq3.2} and $p_{0}$ is the mean of $p$ (in \eqref{eq3.5}), i.e., $p_{0}(x)=\int_{\Delta (A)}p(x,s)d\beta (s)$ for $x\in \Omega$. We will proceed exactly as in Subsection 2.3. First, for $1\leq i,j,k,h\leq N$, let $q_{ijkh}=\delta _{kh}\int_{\Delta (A)}\widehat{a}_{ij}( s)d\beta (s)-\sum_{l=1}^{N}\int_{\Delta (A)} \widehat{a}_{il}(s)\partial _{l}\mathcal{ \chi } _{jh}^{k}(s)d\beta (s),$ where $\mathbf{\chi }_{jh}=(\mathcal{\chi }_{jh}^{k})$ is defined as in (\ref{eq3.6}). To these coefficients we associate the differential operator $\mathcal{Q}$ on $\Omega$ given by (\ref{eq2.20}). Finally, we consider the boundary value problem \eqref{eq2.21}-\eqref{eq2.23}. \begin{lemma} \label{lem3.2} Under the hypotheses of Theorem \ref{thm3.1}, the boundary value problem \eqref{eq2.21}-\eqref{eq2.23} admits at most one weak solution $(\mathbf{u}_{0},p_{0})$ with $\mathbf{u}_{0}\in H_{0}^{1}(\Omega ;\mathbb{R} )^{N}$, $p_{0}\in L^{2}( \Omega ;\mathbb{R})/ \mathbb{R}$. \end{lemma} \begin{proof} It is an easy exercise to show that if a pair $(\mathbf{u} _{0},p_{0})\in H_{0}^{1}(\Omega ;\mathbb{R})^{N}\times L^{2}(\Omega ;\mathbb{R})$ is a solution of \eqref{eq2.21}-\eqref{eq2.23}, then the pair $\mathbf{u=}( \mathbf{u}_{0},\mathbf{u} _{1})$ (in which $\mathbf{u}_{1}$ is given by (\ref{eq3.7})) satisfies \eqref{eq2.9} and is therefore unique. Hence Lemma \ref{lem3.2} follows at once. \end{proof} We are now in a position to state and prove the next theorem. \begin{theorem} \label{thm3.2} Let the hypotheses of Theorem \ref{thm3.1} be satisfied. For each real $0<\varepsilon <1$, let $(\mathbf{u}_{\varepsilon },p_{\varepsilon })\in H_{0}^{1}(\Omega ;\mathbb{R}) ^{N}\times [ L^{2}(\Omega ;\mathbb{R})/ \mathbb{R }]$ be defined by \eqref{eq1.3}-\eqref{eq1.5}. Then, as $\varepsilon \to 0$, we have $\mathbf{u}_{\varepsilon }\to \mathbf{u}_{0}$ in $H_{0}^{1}(\Omega )^{N}$-weak and $p_{\varepsilon }\to p_{0}$ in $L^{2}(\Omega )$-weak, where the pair $(\mathbf{u}_{0},p_{0})$ lies in $H_{0}^{1}(\Omega ; \mathbb{R})^{N}\times [ L^{2}(\Omega ;\mathbb{R}) / \mathbb{R}]$ and is the unique weak solution of \eqref{eq2.21}- \eqref{eq2.23}. \end{theorem} \begin{proof} As was pointed out above, from any arbitrarily given fundamental sequence $E$ one can extract a subsequence $E'$ such that as $E'\ni \varepsilon \rightarrow 0$, we have \eqref{eq3.2}-\eqref{eq3.3} and \eqref{eq2.13} hence $p_{\varepsilon }\rightarrow p_{0}$ in $L^{2}(\Omega )$-weak, where $p_{0}$ is the mean of $p$ and thus $p_{0}\in L^{2}(\Omega ; \mathbb{R})/\mathbb{R}$, and where $\mathbf{u=}(\mathbf{u}_{0},\mathbf{u} _{1})\in \mathbb{F}_{0}^{1}$. Furthermore, (\ref{eq3.5}) holds for all $\mathbf{v=}(\mathbf{v}_{0},\mathbf{v}_{1})\in \mathbb{E}_{0}^{1}$. Substituting (\ref{eq3.7}) in (\ref{eq3.5}) and then choosing therein the particular test functions $\mathbf{v=}(\mathbf{v}_{0},\mathbf{v}_{1})\in \mathbb{E}_{0}^{1}$ with $\mathbf{v}_{1}=0$ leads to Theorem \ref{thm3.2}, thanks to Lemma \ref{lem3.2}. \end{proof} It is possible to present $q_{ijkh}$ in a suitable form as in Remark \ref {rem2.2}. For this purpose, we introduce the space $\mathcal{M}$ of all $N\times N$ matrix functions with entries in $L^{2}(\Delta ( A);\mathbb{R})$. Specifically, $\mathcal{M}$ denotes the space of $\mathbf{F=}(F^{ij})_{1\leq i,j\leq N}$ with $F^{ij}\in L^{2}(\Delta (A);\mathbb{R})$. Provided with the norm $\| \mathbf{F}\| _{\mathcal{M}}=\Big( \sum_{i,j=1}^{N}\| F^{ij}\| _{L^{2}(\Delta ( A))}^{2}\Big)^{1/2}, \quad \mathbf{F}=(F^{ij})\in \mathcal{M },$ $\mathcal{M}$ is a Hilbert space. Now, let $\mathcal{A}(\mathbf{F},\mathbf{G}) =\sum_{i,j,k=1}^{N}\int_{\Delta (A)} \widehat{a}_{ij}( s)F^{jk}(s)G^{ik}(s)d\beta (s)$ for $\mathbf{F=}(F^{jk})$ and $\mathbf{G=}(G^{ik})$ in $\mathcal{M}$. This gives a bilinear form $\mathcal{A}$ on $\mathcal{M} \times \mathcal{M}$, which is symmetric, continuous and coercive. Furthermore, $\widehat{a}(\mathbf{u},\mathbf{v})=\mathcal{A}\Big(\widehat{ \nabla }\mathbf{ u},\widehat{\nabla }\mathbf{v}\Big), \quad \mathbf{u}, \mathbf{v}\in H_{\#}^{1}(\Delta (A);\mathbb{R})^{N},$ where $\widehat{\nabla }\mathbf{u=}(\partial _{j}u^{k})$ for any $\mathbf{u} =(u^{k})\in H_{\#}^{1}(\Delta (A);\mathbb{R})^{N}$. Now, by the same line of proceeding as followed in \cite{bib2} (see also \cite{bib8}) one can quickly show that $q_{ijkh}=\mathcal{A}(\widehat{\nabla }\mathbf{\chi }_{ik}-\mathbf{\ \theta } _{ik},\widehat{\nabla }\mathbf{\chi }_{jh}-\mathbf{\theta } _{jh}),$ where, for any pair of indices $1\leq i,k\leq N$, $\mathbf{\chi }_{ik}$ is defined by (\ref{eq3.6}), and $\mathbf{\theta }_{ik}=(\theta _{ik}^{lm})\in \mathcal{M}$ with $\theta _{ik}^{lm}=\delta _{il}\delta _{km}$. Having made this point, Remark \ref{rem2.2} can then be carried over to the present setting. \subsection{Some concrete examples} In the present subsection we consider a few examples of homogenization problems for \eqref{eq1.3}-\eqref{eq1.5} in a concrete setting (as opposed to the abstract assumption \eqref{eq3.1}) and we show how their study leads naturally to the abstract setting of Subsection 3.1 and so we may conclude by merely applying Theorems \ref{thm3.1} and \ref{thm3.2}. \begin{example}[Almost periodic setting] \label{ex1} \rm The aim here is to study the homogenization of \eqref{eq1.3}-\eqref{eq1.5} under the almost periodicity hypothesis $$a_{ij}\in L_{AP}^{2}(\mathbb{R}_{y}^{N})\quad (1\leq i,j\leq N), \label{eq3.8}$$ where $L_{AP}^{2}(\mathbb{R}_{y}^{N})$ denotes the space of all functions $w\in L_{loc}^{2}(\mathbb{R}_{y}^{N})$ that are almost periodic in the sense of Stepanoff (see, e.g., \cite[Section 4]{bib14}). According to \cite[ Proposition 4.1]{bib14}, the hypothesis (\ref{eq3.8}) yields a countable subgroup $\mathcal{R}$ of $\mathbb{R}_{y}^{N}$ such that $a_{ij}\in L_{AP, \mathcal{R}}^{2}(\mathbb{R}_{y}^{N})$ $(1\leq i,j\leq N)$, where $L_{AP, \mathcal{R}}^{2}(\mathbb{R}_{y}^{N})=\{u\in L_{AP}^{2}(\mathbb{R} _{y}^{N}):Sp(u)\subset \mathcal{R}\}$, $Sp(u)$ being the spectrum of $u$, i.e., $Sp(u)=\{k\in \mathbb{R}^{N}:M(u\overline{\gamma }_{k})\neq 0\}$ with $\gamma _{k}(y)=\exp (2i\pi k.y)$ $(y\in \mathbb{R}^{N})$. The appropriate $H$-algebra is here $AP_{\mathcal{R}}(\mathbb{R}_{y}^{N})=\{u\in AP(\mathbb{R} _{y}^{N}):Sp(u)\subset \mathcal{R}\}$, where $AP(\mathbb{R}_{y}^{N})$ denotes the space of almost periodic continuous complex functions on $\mathbb{R}_{y}^{N}$ (see, e.g., \cite[Chapter 5]{bib3} and \cite[Chapter 10] {bib4}). The $H$-algebra $A=AP_{\mathcal{R}}(\mathbb{R}_{y}^{N})$ is $H^{1}$-proper (see \cite{bib9}) and further (\ref{eq3.1}) is satisfied, since $L_{AP,\mathcal{R}}^{2}(\mathbb{R}_{y}^{N})\subset \mathfrak{X}^{2}$ (use \cite[Lemma 1]{bib8}). Hence the study of the problem under consideration reduces to the abstract analysis in Subsections 3.2 and 3.3. \end{example} \begin{example} \label{ex2} \rm Let $(L^{2},\ell ^{\infty })$ be the space of all $u\in L_{loc}^{2}(\mathbb{R}_{y}^{N})$ such that $\Vert u\Vert _{2,\infty }=\sup_{k\in \mathbb{Z}^{N}}\Big( \int_{k+Y}|u(y)|^{2}dy\Big) ^{1/2}<\infty ,$ where $Y=(-\frac{1}{2},\frac{1}{2})^{N}$. This is a Banach space under the norm $\Vert \cdot \Vert _{2,\infty }$. We denote by $L_{\infty ,per}^{2}(Y)$ the closure in $(L^{2},\ell ^{\infty })$ of the space of all finite sums $$\sum \varphi _{i}u_{i}\quad (\varphi _{i}\in \mathcal{B}_{\infty }(\mathbb{R} _{y}^{N}),\quad u_{i}\in \mathcal{C}_{per}(Y)), \label{eq3.9}$$ where $\mathcal{C}_{per}(Y)$ is defined in Subsection 2.1, and $\mathcal{B} _{\infty }(\mathbb{R}_{y}^{N})$ is the space of all $u\in \mathcal{C}( \mathbb{R}_{y}^{N})$ such that $\lim_{|y|\rightarrow \infty }u(y)=\xi \in \mathbb{C}$ ($\xi$ depending on $u$, $|y|$ the Euclidean norm of $y$ in $\mathbb{R}^{N}$). The problem to be worked out here states as in Example \ref {ex1} except that (\ref{eq3.8}) is replaced by $$a_{ij}\in L_{\infty ,per}^{2}(Y)\quad (1\leq i,j\leq N). \label{eq3.10}$$ We define $A$ to be the closure in $\mathcal{B}(\mathbb{R}_{y}^{N})$ of the finite sums in (\ref{eq3.9}). This is an $H^{1}$ -proper homogenization algebra on $\mathbb{R}_{y}^{N}$ (see \cite[Example 5.4]{bib9}) and further \eqref{eq3.1} holds because the space $(L^{2},\ell ^{\infty })$ is continuously embedded in $\Xi ^{2}$ (use \cite[Lemma 1]{bib8}). Therefore, we arrive at the same conclusion as above. \end{example} \begin{example} \label{ex3} \rm We assume here that the coefficients $a_{ij}$ are constant on each cell $k+Y$ ($k\in \mathbb{Z}^{N}$, $Y$ as above). More precisely, we assume that there exists a family of functions $r_{ij}:\mathbb{ Z} ^{N}\to \mathbb{R}$ $(1\leq i,j\leq N)$ such that for each $k\in \mathbb{Z }^{N}$, we have $a_{ij}(y)=r_{ij}( k)$ a.e. in $y\in k+Y$, and that for $1\leq i,j\leq N$. We also assume the following behaviour: $r_{ij}\in \mathcal{B}_{\infty }( \mathbb{Z}^{N})$ $(1\leq i,j\leq N)$, i.e., each $r_{ij}(k)$ tends to a finite limit as $| k| \to \infty$. Under these hypotheses in place of (\ref{eq3.10}), we consider the above homogenization problem. As is explained in detail in \cite{bib10}, one can find an $H^{1}$-proper homogenization algebra $A$ on $\mathbb{R}_{y}^{N}$ such that \eqref{eq3.1} holds true, which leads us to the same conclusion as above. \end{example} \subsection*{Acknowledgements} The authors wish to thank the anonymous referees for their useful suggestions. \begin{thebibliography}{10} \bibitem[1]{bib1} G. Allaire; Homogenization and two-scale convergence, \textit{SIAM J. Math. Anal., }\textbf{23} (1992), 1482-1518. \bibitem[2]{bib2} A. Bensoussan, J. L. Lions and G. Papanicolaou; \textit{ Asymptotic Analysis for Periodic Structures,} North-Holland, 1978. \bibitem[3]{bib3} A. Guichardet; \textit{Analyse Harmonique Commutative}, Dunod, Paris, 1968. \bibitem[4]{bib4} R. Larsen; \textit{Banach Algebras}, Dekker, New York, 1973. \bibitem[5]{bib5} J. L. Lions; \textit{Quelques m\'{e}thodes de r\'{e} solution des probl\`{e}mes aux limites non lin\'{e}aires,} Dunod, Paris, 1969. \bibitem[6]{bib6} D. Lukkassen, G. 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