\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 82, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/82\hfil Homogenization of parabolic monotone operators]
{Deterministic  homogenization of parabolic monotone operators with
time dependent coefficients}

\author[G. Nguetseng \& J. Woukeng \hfil EJDE-2004/82\hfilneg]
{Gabriel Nguetseng, Jean Louis Woukeng} % in alphabetical order

\address{Gabriel Nguetseng \hfill\break
University of Yaounde I, Department of Mathematics, P.O. Box 812
Yaounde, Cameroon}
\email{gnguets@uycdc.uninet.cm}

\address{Jean-Louis Woukeng \hfill\break
University of Yaounde I, Department of Mathematics, P.O. Box 812 Yaounde,
Cameroon}
\email{jwoukeng@uycdc.uninet.cm}

\date{}
\thanks{Submitted March 20, 2004. Published June 8, 2004.}
\subjclass[2000]{46J10, 35B40}
\keywords{Deterministic homogenization, homogenization structures, \hfill\break\indent
parabolic equations, monotone operators}

\begin{abstract}
 We study, beyond the classical periodic setting, the homogenization of
 linear and nonlinear parabolic differential equations associated with
 monotone operators. The usual periodicity hypothesis is here substituted by
 an abstract deterministic assumption characterized by a great relaxation of
 the time behaviour. Our main tool is the recent theory of homogenization
 structures by the first author, and our homogenization approach falls under
 the two-scale convergence method. Various concrete examples are worked out
 with a view to pointing out the wide scope of our approach and bringing the
 role of homogenization structures to light.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]

\section{Introduction}

Let $2\leq p<\infty $. Let $(y,\tau ,\lambda )\to a(y,\tau ,\lambda) $ be a
function from $\mathbb{R}^{N}\times \mathbb{R\times R}^{N}$ to
$\mathbb{R}^{N}$ $(N\geq 1)$ with the properties:
\begin{align}
&\parbox{10cm}{\noindent For each fixed $\lambda \in \mathbb{R}^{N}$, the function
$(y,\tau)\to a(y,\tau ,\lambda )$ (denoted by $a(\cdot ,\cdot ,\lambda )$)
from $\mathbb{R}^{N}\times \mathbb{R}$ to $\mathbb{R}^{N}$ is measurable}
\label{1.1} \\[4pt]
&\parbox{10cm}{\noindent $a(y,\tau ,\omega )=\omega$ almost everywhere (a.e.) in
$(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}$,
where $\omega$  denotes the origin in $\mathbb{R}^{N}$}
\label{1.2} \\[4pt]
&\parbox{10cm}{There are two constants $\alpha _{0},\alpha _{1}>0$ such that,
a.e. in $(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}$: \\
(i)\; $(a(y,\tau ,\lambda )-a(y,\tau ,\mu ))\cdot (\lambda -\mu )\geq
\alpha _{0}\mid \lambda -\mu \mid ^{p}$ \\
(ii)\; $| a(y,\tau ,\lambda )-a(y,\tau ,\mu )| \leq \alpha
_{1}(| \lambda | +| \mu | )^{p-2}| \lambda -\mu |$
for all $\lambda ,\mu \in \mathbb{R}^{N}$, where the dot denotes the
usual Euclidean  inner product in $\mathbb{R}^{N}$, and $| \cdot |$
the associated norm.}
\label{1.3}
\end{align}
Let $T$ be a positive real number, $\Omega $ a smooth bounded open set in
$\mathbb{R}_{x}^{N}$ (the space $\mathbb{R}^{N}$ of variables
$x=(x_{1},\dots ,x_{N})$), and
$f\in L^{p'}(0,T;W^{-1,p'}(\Omega ;\mathbb{R}))$ with
$p'=\frac{p}{p-1}$. For each given $\varepsilon >0$, we consider the
initial-boundary value problem
\begin{equation}
\begin{gathered}
\frac{\partial u_{\varepsilon }}{\partial t}
-\mathop{\rm div}a\big( \frac{x}{\varepsilon },\frac{t}{\varepsilon },
Du_{\varepsilon }\big) =f \quad \text{in }Q=\Omega \times (0,T) \\
u_{\varepsilon }=0\quad \text{on }\partial \Omega \times (0,T) \\
u_{\varepsilon }(x,0)=0\quad\text{in }\Omega
\end{gathered} \label{1.4}
\end{equation}
where $D$ denotes the usual gradient, i.e., $D=\left( D_{x_{i}}\right)
_{1\leq i\leq N}$ with $D_{x_{i}}=\frac{\partial }{\partial x_{i}}$, and
$\mathop{\rm div}$ the divergence with respect to the variable $x$.

Provided the diffusion term of the differential operator in (\ref{1.4}) is
rigorously defined (see \cite[Subsection 4.1]{18}) and further an existence
and uniqueness result for (\ref{1.4}) is pointed out (all that will be
accomplished in Section 2), our goal in this paper is to investigate the
limiting behaviour, as $\varepsilon \to 0$, of $u_{\varepsilon }$
(the solution of (\ref{1.4})). In all probability such an undertaking is
hopeless without any further suitable assumption termed a \textit{structure
hypothesis} \cite[20]{18}, which specifies the behaviour of the function
$(y,\tau )\to a(y,\tau ,\lambda )$ (for fixed $\lambda $).

The common structure hypothesis is the so-called periodicity hypothesis. The
latter states that there exist two networks $\mathcal{R}\subset \mathbb{R}
_{y}^{N}$ and $\mathcal{S}\subset \mathbb{R}_{\tau }$, e.g., $\mathcal{R}=\mathbb{Z
}^{N}$ and $\mathcal{S}=\mathbb{Z}$, such that for any given $k\in \mathcal{R}$
and $l\in \mathcal{S}$, we have $a(y+k,\tau +l,\lambda )=a(y,\tau ,\lambda )$
\ a.e. in $(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}$, where $\lambda $ is
arbitrarily fixed. Under the periodicity hypothesis, homogenization results
for problem (\ref{1.4}) are available; see, e.g., \cite[22, 24]{17} (see
also \cite{24} for specific corrector results). It should be mentioned in
passing that the homogenization of linear parabolic operators in the
periodic setting is now a classical theory (see, e.g., \cite[3, 8, 12, 13]{2}
) with, further, an extension to the almost periodic setting (see \cite{26}).

However, much yet remains to be done in this area. To a large extent,
nonstochastic homogenization theory seems to confine itself to the periodic
setting, and that in spite of the gap to be filled between periodic and
stochastic homogenization \cite{par}. No doubt, to arrive~$-$via
homogenization$-$~at a thorough understanding of physical problems we need
to be released from the classical periodicity hypothesis, especially with
regard to the behaviour in the time variable.

Specifically, we study here the homogenization of problem (\ref{1.4}) in a
very general setting characterized by an abstract assumption on $a(y,\tau
,\lambda )$ (for fixed $\lambda $) covering a wide range of behaviours,
especially with respect to the time variable $\tau =\frac{t}{\varepsilon }$. Broadly speaking, this abstract assumption is \textit{proper} \cite{22} with
respect to the space variable $y=\frac{x}{\varepsilon }$ and hence covers a
great variety of concrete behaviours in $y$ (see \cite[Section 5]{22})
whereas, surprisingly enough, with respect to $\tau =\frac{t}{\varepsilon }$
it sets no further significant restriction on $a(y,\tau ,\lambda )$ (fixed
$\lambda $), which we express by referring to the \textit{quasi-properness}
introduced in Definition 3.1. This is a true advance in the homogenization
of parabolic partial differential equations, and a great step towards a
better understanding of evolution phenomena.

Our main tool is the recent theory of homogenization structures earlier
developed in \cite[21]{18} and our homogenization approach falls under the
two-scale convergence method. For an obvious reason (see the diffusion term
of the differential operator in (\ref{1.4})) the present study greatly leans
on the elliptic case \cite{22} of which it is a natural continuation.

The rest of the paper is organized as follows. In Section 2 we rigorously
define the diffusion term of the differential operator in (\ref{1.4}) and we
point out those of its basic properties that ensure an existence and
uniqueness result for the initial-boundary value problem under
consideration. The homogenization of problem (\ref{1.4}) proper begins with
Section 3. Under an abstract deterministic hypothesis on $a(\cdot ,\cdot
,\lambda )$ (for fixed $\lambda $) we achieve fundamental homogenization
results that prove quite similar to those obtained in the periodic setting.
Finally, to illustrate the preceding abstract setting and point out its wide
scope, we consider in Section 4 a few concrete homogenization problems for 
(\ref{1.4}). In particular it is shown how such concrete problems reduce in a
natural way to the abstract setting of Section 3.

In order that we may make use of basic tools provided by the classical
Banach algebras theory, the vector spaces throughout are generally
considered over $\mathbb{C}$ and the scalar functions are assumed to take
complex values. If $X$ and $F$ denote a locally compact space and a Banach
space, respectively, then we write $\mathcal{C}(X;F),\mathcal{B}(X;F)$ and
$\mathcal{K}(X;F)$ for continuous mappings of $X$ into $F$, bounded
continuous mappings of $X$ into $F$, and those mappings in $\mathcal{C}(X;F)$
having compact supports, respectively. We shall always assume that $\mathcal{
B}(X;F)$ is equipped with the supremum norm $\left\| u\right\| _{\infty
}=\sup_{x\in X}\left\| u(x)\right\| $ \ ($\left\| \cdot \right\| $ denotes
the norm in $F$). For shortness we will write
$\mathcal{C}(X)=\mathcal{C}(X;\mathbb{C})$,
$\mathcal{B}(X)=\mathcal{B}(X;\mathbb{C})$ and
$\mathcal{K}(X)=\mathcal{K}(X;\mathbb{C})$. Likewise the usual spaces
$L^{p}(X;F)$ and $L_{\rm loc}^{p}(X;F)$
($X$ provided with a positive Radon measure) will be denoted by $L^{p}(X)$
and $L_{\rm loc}^{p}(X)$, respectively, in the case when $F=\mathbb{C}$. We refer
to \cite[7, 9]{6} for integration theory. On the other hand, for convenience
we will most of the time put
$\mathcal{C}_{\mathbb{R}}(X)=\mathcal{C}(X;\mathbb{R})$,
$\mathcal{B}_{\mathbb{R}}(X)=\mathcal{B}(X;\mathbb{R})$,
$\mathcal{K}_{\mathbb{R}}(X)=\mathcal{K}(X;\mathbb{R})$ and
$L_{\mathbb{R}}^{p}(X)=L^{p}(X;\mathbb{R})$. Finally, the numerical space
 $\mathbb{R}^{d}$ $(d\geq 1)$ and its open
sets are each provided with Lebesgue measure denoted by $dx=dx_{1}\dots dx_{d}$.

\section{Preliminaries}

Let $1<p<\infty $. Let $G$ be a function from $\mathbb{R}^{N}\times \mathbb{
R\times R}^{N}$ to $\mathbb{R}$ with the following properties:
\begin{align}
&\parbox{10cm}{\noindent
For each $\lambda \in \mathbb{R}^{N}$, the function
$(y,\tau)\to G(y,\tau ,\lambda )$  from
$\mathbb{R}^{N}\times \mathbb{R}$  to $\mathbb{R}$, denoted by
$G(\cdot,\cdot ,\lambda )$, is measurable}
\label{2.1} \\[4pt]
&\parbox{10cm}{\noindent
$G(y,\tau ,\omega )=0$ a.e. in $(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}$}
 \label{2.2}\\[4pt]
&\parbox{10cm}{\noindent
There exists a positive constant $\alpha _{1}$  such that
$| G(y,\tau ,\lambda )-G(y,\tau ,\mu )| \leq \alpha _{1}(|\lambda| +| \mu | )^{p-2}
| \lambda -\mu |$ for all $\lambda ,\mu \in \mathbb{R}^{N}$ and for almost all
$(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}$.}
\label{2.3}
\end{align}
With a view to giving a meaning to the diffusion term in (\ref{1.4}), we
wish to define, for each $\mathbf{u}$ in
$L_{\mathbb{R}}^{p}(Q)^{N}=L_{\mathbb{R}}^{p}(Q)\times \dots
\times L_{\mathbb{R}}^{p}(Q)$ ($N$ times), the
function $(x,t)\to G(\frac{x}{\varepsilon },\frac{t}{\varepsilon },
\mathbf{u}(x,t))$  from $Q=\Omega \times (0,T)$ to $\mathbb{R}$, where
$\varepsilon >0$ is freely fixed. As was pointed out in \cite[Subsection 4.1]{18},
it is worth emphasizing that this is a delicate matter because the set
$\mathcal{Q}_{\varepsilon }=\{(x,t,y,\tau ):y=\frac{x}{\varepsilon }$ and
$\tau =\frac{t}{\varepsilon }$ for $(x,t)\in Q\}$ is negligible in
$\mathbb{R}^{N}\times \mathbb{R\times R}^{N}\times \mathbb{R}$.

For $u\in L_{\rm loc}^{1}(Q\times \mathbb{R}_{y}^{N}\times \mathbb{R}_{\tau })$,
we set
\begin{equation}
u^{\varepsilon }(x,t)=u\big( x,t,\frac{x}{\varepsilon },\frac{t}{
\varepsilon }\big) \quad \left( x\in \Omega ,\; 0<t<T\right)
\label{2.4}
\end{equation}
whenever the right-hand side has meaning (see \cite{18}). We will need the
following two basic lemmas. For the proofs we refer to
\cite[Lemmas 2.1 and 2.2]{22}.

\begin{lemma}\label{lm2.1}
The transformation $u\to u^{\varepsilon }$ ($u^{\varepsilon }$ given by
\eqref{2.4}) considered as a mapping of
$\mathcal{C}(\overline{Q})\otimes L^{\infty }(\mathbb{R}^{N+1})$ into
$L^{\infty }(Q)$ extends by continuity to a linear mapping, still denoted
by $u\to u^{\varepsilon }$, of $\mathcal{C}(\overline{Q};L^{\infty }(\mathbb{R}
^{N+1}))$ into $L^{\infty }(Q)$ with $\| u^{\varepsilon }\|
_{L^{\infty }(Q)}\leq \sup_{(x,t)\in \overline{Q}}\| u(x,t)\|
_{L^{\infty }(\mathbb{R}^{N+1})}$ for $u$ in
$\mathcal{C}(\overline{Q};L^{\infty }(\mathbb{R}^{N+1}))$.
\end{lemma}

\begin{lemma} \label{lm2.2}
Let $u\in \mathcal{C}(\overline{Q};L^{\infty }(\mathbb{R}^{N+1}))$.
Suppose for each $(x,t)\in \overline{Q}$ we have
$u(x,t,y,\tau )\geq 0$ a.e. in $(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}$. Then
$u^{\varepsilon }(x,t)\geq 0$  a.e. in $Q$, where $u^{\varepsilon }$ is
considered in the meaning of Lemma \ref{lm2.1}.
\end{lemma}

Now, given $\Phi \in \mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}=\mathcal{C}_{
\mathbb{R}}(\overline{Q})\times \dots \times \mathcal{C}_{\mathbb{R}
}(\overline{Q})$ ($N$ times), it is immediate by (\ref{2.1})-(\ref{2.3})
that the function $(x,t,y,\tau )\to u(x,t,y,\tau )=G(y,\tau ,\Phi
(x,t))$ of $\overline{Q}\times \mathbb{R}^{N}\times \mathbb{R}$ into $\mathbb{R}$
lies in $\mathcal{C}(\overline{Q};L^{\infty }(\mathbb{R}^{N+1}))$. Hence,
the real function $(x,t)\to G(\frac{x}{\varepsilon },\frac{t}{
\varepsilon },\Phi (x,t))$ on $Q$, denoted below by $G^{\varepsilon }(\cdot
,\cdot ,\Phi )$, is defined in the sense of Lemma \ref{lm2.1} as a function
in $L_{\mathbb{R}}^{\infty }(Q)$.

\begin{proposition} \label{prop2.1}
The transformation $\Phi \to G^{\varepsilon }(\cdot ,\cdot ,\Phi) $
of $\mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}$
into $L^{\infty }(Q)$ extends by continuity to a mapping, still denoted by
$\Phi \to G^{\varepsilon }(\cdot ,\cdot ,\Phi )$, of
$L_{\mathbb{R}}^{p}(Q)^{N}$ into $L^{p'}(Q)$ $(p'=\frac{p}{p-1})$ with
the property
\begin{equation}
\left\| G^{\varepsilon }(\cdot ,\cdot ,\Phi )-G^{\varepsilon }(\cdot ,\cdot
,\Psi )\right\| _{L^{p'}(Q)}\leq
\alpha _{1}\left\| \left| \Phi \right| +\left| \Psi \right| \right\|
_{L^{p}(Q)}^{p-2}\left\| \Phi -\Psi \right\| _{L^{p}(Q)^{N}}
\label{2.5}
\end{equation}
for $\Phi ,\Psi \in L_{\mathbb{R}}^{p}(Q)^{N}$.
\end{proposition}


\begin{proof}
Let $\Phi ,\Psi \in \mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}$. Thanks to (
\ref{2.3}), we may apply Lemma \ref{lm2.2} with
\begin{align*}
u(x,t,y,\tau )&=\alpha _{1}\big( | \Phi (x,t)| +| \Psi(x,t)| \big) ^{p-2}
| \Phi (x,t)-\Psi (x,t)| \\
&\quad -| G( y,\tau ,\Phi (x,t)) -G( y,\tau ,\Psi (x,t))| .
\end{align*}
This leads immediately to
\begin{equation*}
\left| G^{\varepsilon }(\cdot ,\cdot ,\Phi )-G^{\varepsilon }(\cdot ,\cdot
,\Psi )\right| ^{p'}\leq \alpha _{1}^{p'}(\left| \Phi
\right| +\left| \Psi \right| )^{(p-2)p'}\left| \Phi -\Psi \right|
^{p'}\text{.}
\end{equation*}
Considering the functions $\left| \Phi -\Psi \right| ^{p'}$ and
$(\left| \Phi \right| +\left| \Psi \right| )^{(p-2)p'}$ as belonging
to $L^{q}(Q)$ and $L^{q'}(Q)$, respectively, where $q=\frac{p}{
p'}$ and $q'=\frac{q}{q-1}$, and using H\"{o}lder's
inequality, one easily arrives at (\ref{2.5}) with $\mathcal{C}_{\mathbb{R}}(
\overline{Q})^{N}$ in place of $L_{\mathbb{R}}^{p}(Q)^{N}$.
With this in mind, let\ $B_{r}=\{\mathbf{v}\in L_{\mathbb{R}}^{p}(Q)^{N}:\left
\| \mathbf{v}\right\| _{L_{\mathbb{R}}^{p}(Q)^{N}}\leq \frac{r}{2}\}$ with
$r>0. $ Let $g_{r}$ be the restriction to $B_{r}\cap \mathcal{C}_{\mathbb{R}}(
\overline{Q})^{N}$ of the mapping $\mathbf{v}\to G^{\varepsilon
}\left( \cdot ,\cdot ,\mathbf{v}\right) $ (where $\varepsilon >0$ is fixed,
of course). Clearly
\begin{equation}
\left\| g_{r}(\Phi )-g_{r}(\Psi )\right\| _{L^{p'}(Q)}\leq \alpha
_{1}r^{p-2}\left\| \Phi -\Psi \right\| _{L^{p}(Q)^{N}}
\quad \text{for all }\Phi ,\Psi \in B_{r}\cap \mathcal{C}_{\mathbb{R}}(\overline{Q}
)^{N}.
\label{2.6}
\end{equation}
Since $B_{r}\cap \mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}$ is dense in
$B_{r}$ (the verification is an elementary exercise), it follows that $g_{r}$
extends by continuity to a continuous mapping, still denoted by $g_{r}$, of
$B_{r}$ into $L^{p'}(Q)$ such that (\ref{2.6}) holds with $B_{r}$ in
place of $B_{r}\cap \mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}$. Whence we
deduce a sequence $(g_{n})_{n\geq 1}$ of mappings $g_{n}:B_{n}\to
L^{p'}(Q)$ with $g_{n}(\Phi )=G^{\varepsilon }(\cdot ,\cdot ,\Phi )$
for $\Phi \in B_{n}\cap \mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}$. Noticing that $L_{\mathbb{R}}^{p}(Q)^{N}$ is the union of the balls $B_{n}$
$(n\geq 1)$ and, on the other hand, $g_{n+1}(\Phi )=g_{n}(\Phi )$ for $\Phi
\in B_{n}$ ,we are led to a uniquely defined continuous mapping $g:L_{\mathbb{R}
}^{p}(Q)^{N}\to L^{p'}(Q)$ such that $g(\Phi )=$ $
G^{\varepsilon }\left( \cdot ,\cdot ,\Phi \right) $ for any $\Phi \in
\mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}$. Hence the proposition follows
by the density of $\mathcal{C}_{\mathbb{R}}(\overline{Q})^{N}$ in
$L_{\mathbb{R}}^{p}(Q)^{N}$.
\end{proof}

\begin{corollary} \label{corol 2.1}
We have
\begin{gather}
a^{\varepsilon }(\cdot ,\cdot ,\omega )=\omega \quad \text{ a.e. in } Q, \label{2.7}
\\
\left\| a^{\varepsilon }(\cdot ,\cdot ,Du)-a^{\varepsilon }(\cdot ,\cdot
,Dv)\right\| _{L^{p'}(Q)^{N}}\leq
\alpha _{1}\left\| \left| Du\right| +\left| Dv\right| \right\|
_{L^{p}(Q)}^{p-2}\left\| Du-Dv\right\| _{L^{p}(Q)^{N}}
\label{2.8}
\end{gather}
and
\begin{equation}
\begin{aligned}
\Big[ a\big( \frac{x}{\varepsilon },\frac{t}{\varepsilon },Du(x,t)\big)
-a\big( \frac{x}{\varepsilon },\frac{t}{\varepsilon },Dv(x,t)\big)
\Big] \cdot \left( Du(x,t)-Dv(x,t)\right) \\
\geq \alpha _{0}\left| Du(x,t)-Dv(x,t)\right| ^{p}\quad \text{a.e. in }
(x,t)\in Q
\end{aligned} \label{2.9}
\end{equation}
for all $u,v\in L^{p}\left( 0,T;W^{1,p}\left( \Omega ;\mathbb{R}\right) \right)
$, where $a^{\varepsilon }(\cdot ,\cdot ,Du)=\left\{ a_{i}^{\varepsilon
}(\cdot ,\cdot ,Du)\right\} _{1\leq i\leq N}$.
\end{corollary}

Due to (\ref{1.1})-(\ref{1.3}) and Lemma \ref{lm2.2}, this corollary is a
direct consequence of Proposition \ref{prop2.1} with $G=a_{i}$ (the i$^{\text{th}}$
component of the function $(y,\tau ,\lambda )\to a(y,\tau ,\lambda ))$.

\begin{remark} \label{rm 2.1} \rm
Thanks to Proposition \ref{prop2.1}, the diffusion term in (\ref{1.4}) can now
be rigorously defined. Specifically, let $u\in
L^{p}\left( 0,T;W^{1,p}\left( \Omega ;\mathbb{R}\right) \right) $.
Then $a^{\varepsilon }(\cdot ,\cdot ,Du)\in L^{p'}(Q)^{N}$, as
pointed out above. But we may as well view
$a^{\varepsilon }(\cdot ,\cdot,Du)$ as a function in
$L^{p'}(0,T;L^{p'}(\Omega)^{N})$. Consequently,
$\mathop{\rm div}a^{\varepsilon }(\cdot ,\cdot ,Du)$
turns out to precisely represent the function
$t\to \mathop{\rm div} a^{\varepsilon }\left( \cdot ,t,Du(\cdot ,t)\right)$
 of $(0,T)$ into $W^{-1,p'}\left( \Omega ;\mathbb{R}\right) $,
which lies in $L^{p'}(0,T;W^{-1,p'}(\Omega ;\mathbb{R)})$
 (this is straightforward).
\end{remark}

\begin{corollary} \label{corol 2.2}
Let $2\leq p<\infty $. For each given real $\varepsilon >0$, there exists a
unique $u_{\varepsilon }\in L^{p}(0,T;W_{0}^{1,p}(\Omega ;\mathbb{R}))$
satisfying \eqref{1.4}.
\end{corollary}

The statement of this corollary is guaranteed by (\ref{2.7})-(\ref{2.9}).
For more details we refer to, e.g., \cite[16, 26]{1}.

\begin{remark} \label{rm 2.2} \rm
More precisely, $u_{\varepsilon }$ lies in
\begin{equation*}
V^{p}=\Big\{ v\in L^{p}(0,T;W_{0}^{1,p}(\Omega ;\mathbb{R)}):v'=\frac{
\partial v}{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ;
\mathbb{R))}\Big\} .
\end{equation*}
With the norm  $\| v\| _{V^{p}}=\| v\|_{L^{p}\left( 0,T;W_{0}^{1,p}(\Omega )\right) }
+\| v'\|_{L^{p'}\left( 0,T;W^{-1,p'}(\Omega )\right) }$,
$V^{p}$ is a Banach space. For further needs it is worth noting that,
since $p\geq 2$, the space $W_{0}^{1,p}\left( \Omega ;\mathbb{R}\right)$
is continuously and densely embedded in $L_{\mathbb{R}}^{2}(\Omega )$.
Hence, identifying $L_{\mathbb{R}}^{2}(\Omega )$ with its dual,
it follows
\begin{equation*}
W_{0}^{1,p}\left( \Omega ;\mathbb{R}\right) \subset L_{\mathbb{R}}^{2}(\Omega
)\subset W^{-1,p'}\left( \Omega ;\mathbb{R}\right)
\end{equation*}
with continuous embeddings. This has two important consequences:\\
1) We will use the same symbol, to denote both the inner product
in $L_{\mathbb{R}}^{2}(\Omega )$ and the duality between the spaces
$W^{-1,p'}(\Omega ;\mathbb{R})$ and $W_{0}^{1,p}(\Omega ;\mathbb{R})$.\\
2) The space $V^{p}$ is continuously embedded in
$\mathcal{C}([0,T];L_{\mathbb{R}}^{2}(\Omega ))$ (this is a classical
result). Thus, we may define $v(t)$ for $v\in V^{p}$ and
$0\leq t\leq T$, and further the mapping $v\to v(t)$
sends continuously $V^{p}$ into $L_{\mathbb{R}}^{2}(\Omega )$.
Hence, we may consider the space $V_{0}^{p}=\{v\in V^{p}:v(0)=0\}$,
a Banach space with the $V^{p}$-norm, which turns out to contain the
solution $u_{\varepsilon }$ of (\ref{1.4}).
\end{remark}

\section{The Abstract Homogenization problem}

For any notation, notion and result concerning homogenization structures and
homogenization algebras we refer the reader to \cite[21]{18}.
The letter $E$ throughout
will denote exclusively a family of positive real numbers admitting 0 as an
accumulation point. In the particular case where $E=(\varepsilon
_{n})_{n\geq 0}$ with $0<\varepsilon _{n}\leq 1$ and $\varepsilon
_{n}\to 0$ as $n\to \infty $, we will refer to $E$ as a
\textit{fundamental sequence}.

\subsection{Fundamentals of homogenization structures}

To the benefit of the reader we summarize below a few basic notions and
results about the homogenization structures. We refer to \cite[21]{18} for
further details.

We start with one underlying concept. We say that a set $\Gamma \subset
\mathcal{B}(\mathbb{R}_{y}^{N})$ is a \textit{structural representation} on
$\mathbb{R}^{N}$ if

(1) $\Gamma $ is a group under multiplication in $\mathcal{B}(\mathbb{R}_{y}^{N})$

(2) $\Gamma $ is countable

(3) $\gamma \in \Gamma $ implies $\overline{\gamma }\in \Gamma $ ($\overline{
\gamma }$ the complex conjugate of $\gamma $)

(4) $\Gamma \subset \Pi ^{\infty }$.

Here, $\Pi ^{\infty }$ denotes the space of functions
$u\in \mathcal{B}(\mathbb{R}_{y}^{N})$ such that $u^{\varepsilon }\to M(u)$
in $L^{\infty }(\mathbb{R}_{x}^{N})$-weak $*$ as
$\varepsilon \to 0$ ($\varepsilon >0$), where
$u^{\varepsilon }(x)=u\left( \frac{x}{\varepsilon }\right) $
($x\in \mathbb{R}^{N}$) and $M(u)\in \mathbb{C}$.

We recall in passing that the complex mapping $u\to M(u)$ on $\Pi^{\infty }$ is
a positive continuous linear form with $M(1)=1$ and $M(\tau_{h}u)=M(u)$
(for $u\in \Pi ^{\infty }$ and $h\in \mathbb{R}^{N}$) where
$\tau_{h}u(y)=u(y-h)$ ($y\in \mathbb{R}^{N}$). Thus, $M$ is a mean value
(see \cite[19]{18} for further details).

Now, in the collection of all structural representations on $\mathbb{R}^{N}$ we
consider the equivalence relation $\sim $ defined as:
$\Gamma \sim \Gamma'$ if and only if $CLS(\Gamma )=CLS(\Gamma ')$, where
$CLS(\Gamma )$ denotes the closed vector subspace of
$\mathcal{B}(\mathbb{R}_{y}^{N})$ spanned by $\Gamma $.
By an $H$\textit{-structure} on $\mathbb{R}_{y}^{N}$ ($H$ stands for
\textit{homogenization}) is understood any
equivalence class modulo $\sim $.
An $H$-structure is fully determined by its image. Specifically, let $\Sigma
$ be an $H$-structure on $\mathbb{R}^{N}$. Put $A=CLS(\Gamma )$ where $\Gamma $
is any equivalence class representative of $\Sigma $ (such a $\Gamma $ is
termed a \textit{representation} of $\Sigma $). The space $A$ 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:

(5) $A$ with the supremum norm is separable

(6) $A$ contains the constants

(7) If $u\in A$ then $\overline{u}\in A$

(8) $A\subset \Pi ^{\infty }$.

Furthermore, $A$ depends only on $\Sigma $ and not on the chosen
representation $\Gamma $ of $\Sigma $. Thus, we may set
$A=\mathcal{J}(\Sigma )$ (the \textit{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 \cite[Theorem 3.1]{18}).

Let $A$ be an $H$-algebra on $\mathbb{R}_{y}^{N}$. Clearly $A$ (with the
supremum norm) is a commutative $\mathcal{C}^{*}$-algebra with identity (the
involution is here the usual one of complex conjugation). We denote by
$\Delta (A)$ the spectrum of $A$ and by $\mathcal{G}$ the Gelfand
transformation on $A$. We recall 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)=\left\langle
s,u\right\rangle $ ($s\in \Delta (A)$), where $\langle ,\rangle $ denotes
the duality between $A'$ (the topological
dual of $A$) and $A$. The topology on $\Delta (A)$ is the relative weak $*$
topology on $A'$. So topologized, $\Delta (A)$ is a metrizable
compact space, and the Gelfand transformation is an isometric isomorphism of
the $\mathcal{C}^{*}$-algebra $A$ onto the $\mathcal{C}^{*}$-algebra
$\mathcal{C}(\Delta (A))$. For further details concerning the Banach algebras
theory we refer to \cite{15}. The basic measure on $\Delta (A)$ is the
so-called $M$-measure for $A$, 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]{18}).

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 $A^{1}=\{\psi \in \mathcal{C}
^{1}(\mathbb{R}^{N}):$ $\psi ,D_{y_{i}}\psi \in A$ ($1\leq i\leq N$)$\}$.
Higher order derivatives are defined analogously. At the present time, let
$A^{\infty }$ be the space of $\psi \in \mathcal{C}^{\infty }(\mathbb{R}
_{y}^{N}) $ such that $D_{y}^{\alpha }\psi =\frac{\partial ^{\left| \alpha
\right| }\psi }{\partial y_{1}^{\alpha _{1}}\dots \partial
y_{N}^{\alpha _{N}}}\in A$ 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 \cite
{18}), $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
distribution on $\Delta (A)$. 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. If we assume that $A^{\infty }$ is dense in $A$ (this
condition is always fulfilled 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{18} for more details). Hence
we may define
\begin{equation*}
W^{1,p}(\Delta (A))=\{u\in L^{p}(\Delta (A)):\text{ }\partial _{i}u\in
L^{p}(\Delta (A))\text{ (}1\leq i\leq N\text{)}\}
\end{equation*}
where the derivative $\partial _{i}u$ is taken in the distribution sense on
$\Delta (A)$ (exactly as the Schwartz derivative is taken in the classical
case). 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))}\text{ \thinspace }
\left( u\in W^{1,p}(\Delta (A))\right) ,
\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
\left( u\in W^{1,p}(\Delta (A))/\mathbb{C}\right) .
\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{6} and page 29 of \cite{9}). $W_{\#}^{1,p}(\Delta (A))$
is a Banach space and $W_{\#}^{1,2}(\Delta (A))$ is a Hilbert space.
Furthermore, as pointed out in \cite{18}, 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))}\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 $) for which we refer to \cite{18}.

\subsection{The abstract structure hypothesis}

Let $\Sigma _{y}$ and $\Sigma _{\tau }$ be two $H$-structures of class
$\mathcal{C}^{\infty }$ on $\mathbb{R}_{y}^{N}$ and $\mathbb{R}_{\tau }$,
respectively, and let $\Sigma =\Sigma _{y}\times \Sigma _{\tau }$ be their
product, which is an $H$-structure of class $\mathcal{C}^{\infty }$ on $\mathbb{
R}^{N}\times \mathbb{R}$. We introduce their respective images (i.e., the
associated $H$-algebras) : $A_{y}=\mathcal{J}(\Sigma _{y})$, $A_{\tau }=
\mathcal{J}(\Sigma _{\tau })$ and $A=\mathcal{J}(\Sigma )$. The same letter,
$\mathcal{G}$, will denote the Gelfand transformation on $A_{y}$, $A_{\tau
}, $ and $A$, as well. Points in $\triangle (A_{y})$ (resp. $\triangle
(A_{\tau })$) are denoted by $s$ (resp. $s_{0}$). The compact space
$\triangle (A_{y}) $ (resp. $\triangle (A_{\tau })$) is equipped with the $M$
-measure, $\beta _{y}$ (resp. $\beta _{\tau }$), for $A_{y}$ (resp. $A_{\tau
}$). We have $\triangle (A)=$ $\triangle (A_{y})\times \triangle (A_{\tau })$
(Cartesian product) and the $M$-measure for $A$, with which $\triangle (A)$
is equipped, is precisely the product $\beta =\beta _{y}\otimes \beta _{\tau}$.

Now, let $1\leq p<\infty $. Let $\Xi ^{p}$ denote the space of all $u\in
L_{\rm loc}^{p}\left( \mathbb{R}_{y}^{N}\times \mathbb{R}_{\tau }\right) $ for which
the sequence $\left( u^{\varepsilon }\right) _{0<\varepsilon \leq 1}$ [where
$u^{\varepsilon }(x,t)=u\left( \frac{x}{\varepsilon },\frac{t}{\varepsilon }
\right) $ \ $\left( x\in \mathbb{R}^{N},t\in \mathbb{R}\right) $] is bounded in
$L_{\rm loc}^{p}\left( \mathbb{R}_{x}^{N}\times \mathbb{R}_{t}\right) $. This is a
Banach space with norm
\begin{equation*}
\left\| u\right\| _{\Xi ^{p}}=\sup_{0<\varepsilon \leq 1}
\Big(\int_{B_{N+1}}\Big| u \Big(\frac{x}{\varepsilon },\frac{t}{\varepsilon }
\Big)\Big| ^{p}dx\,dt\Big) ^{1/p}
\end{equation*}
where $B_{N+1}$ is the open unit ball in $\mathbb{R}^{N+1}$. Next, we define
$\mathfrak{X}_{\Sigma }^{p}$ to be the closure of $A$ in $\Xi ^{p}$. We equip
$\mathfrak{X}_{\Sigma }^{p}$ with the $\Xi ^{p}$-norm, which makes it a Banach
space. It is worth recalling that the Gelfand transformation $\mathcal{G}
:A\to \mathcal{C}\left( \triangle (A)\right) $ extends by continuity
to a continuous linear mapping, still denoted by $\mathcal{G}$, of $\mathfrak{X}
_{\Sigma }^{p}$ into $L^{p}\left( \triangle (A)\right) $. This is referred
to as the canonical mapping of $\mathfrak{X}_{\Sigma }^{p}$ into $L^{p}\left(
\triangle (A)\right) $.
We are now in a position to state the so-called abstract homogenization
problem for (\ref{1.4}). Let
\begin{equation*}
A_{\mathbb{R}}=A\cap \mathcal{C}_{\mathbb{R}}\left( \mathbb{R}^{N}\times \mathbb{R}
\right) .
\end{equation*}

The main purpose of the present section is to investigate the limiting
behaviour, as $\varepsilon \to 0$, of $u_{\varepsilon }$ (the
solution of (\ref{1.4})) under the \textit{abstract structure hypothesis}
\begin{equation}
a_{i}\left( \cdot ,\cdot ,\Psi \right) \in \mathfrak{X}_{\Sigma }^{p'}
\quad \text{for all }\Psi \in \left( A_{\mathbb{R}}\right) ^{N}\;
(1\leq i\leq N)  \label{3.1}
\end{equation}
with $2\leq p<\infty $ and $p'=\frac{p}{p-1}$, where $a_{i}(\cdot
,\cdot ,\Psi )$ denotes the function $(y,\tau )\to a_{i}(y,\tau
,\Psi (y,\tau ))$ from $\mathbb{R}^{N}\times \mathbb{R}$ to $\mathbb{R}$, which
belongs to $L_{\mathbb{R}}^{\infty }\left( \mathbb{R}^{N}\times \mathbb{R}\right) $
(see point (4.1) of \cite{22}). The problem thus stated is precisely what is
referred to as the \textit{abstract homogenization problem} for (\ref{1.4})
in a deterministic setting.

However, as will be seen later, one further assumption on $\Sigma $, the
\textit{quasi-properness} hypothesis, will be necessary to the resolution of
the preceding abstract homogenization problem. Meanwhile, let us prove a few
basic results we will need. In the sequel we assume that (\ref{3.1}) holds.
Thus, if $\Psi \in \left( A_{\mathbb{R}}\right) ^{N}$, then $a_{i}\left( \cdot
,\cdot ,\Psi \right) $ lies in $\mathfrak{X}_{\Sigma }^{p',\infty }=
\mathfrak{X}_{\Sigma }^{p'}\cap $ $L_{\mathbb{R}}^{\infty }\left( \mathbb{R}
^{N}\times \mathbb{R}\right) $. Consequently, $\mathcal{G}\left( a_{i}\left(
\cdot ,\cdot ,\Psi \right) \right) \in L^{\infty }\left( \triangle
(A)\right) $ \cite[corollary 2.2]{18}. With this in mind, let the index
$1\leq i\leq N$ be arbitrarily fixed. For $\mathbf{\varphi }=\left( \varphi
_{j}\right) _{1\leq j\leq N}$ in $\mathcal{C}_{\mathbb{R}}\left( \triangle
(A)\right) ^{N}$, let
\begin{equation*}
b_{i}\left( \mathbf{\varphi }\right) =\mathcal{G}\left( a_{i}\left( \cdot
,\cdot ,\mathcal{G}^{-1}\mathbf{\varphi }\right) \right)
\end{equation*}
where $\mathcal{G}^{-1}\mathbf{\varphi }=\left( \mathcal{G}^{-1}\varphi
_{j}\right) _{1\leq j\leq N}$. This defines a transformation $b_{i}$ of
$\mathcal{C}_{\mathbb{R}}\left( \triangle (A)\right) ^{N}$ into $L^{\infty
}\left( \triangle (A)\right) $.

\begin{proposition} \label{prop3.1}
Let $2\leq p<\infty $. Suppose \eqref{3.1} holds. For $\Psi
=\left( \psi _{j}\right) _{1\leq j\leq N}$ in $\mathcal{C}(\overline{Q}
;(A_{\mathbb{R}})^{N})$, let $b_{i}(\widehat{\Psi }(x,t))=\mathcal{G}
\left( a_{i}\left( \cdot ,\cdot ,\Psi \left( x,t\right) \right) \right) $
for $(x,t)\in \overline{Q}$, where
$\widehat{\Psi }=(\widehat{\psi }_{j})_{1\leq j\leq N}$ with
$\widehat{\psi }_{j}=\mathcal{G}\circ \psi _{j}$. This defines a
mapping $(x,t)\to b_{i}(\widehat{\Psi }(x,t))$, still
denoted by $b_{i}(\widehat{\Psi })$, of $\overline{Q}$ into
$L^{\infty}(\triangle (A))$. The following assertions are true:
\begin{itemize}
\item[(i)]  We have $b_{i}(\widehat{\Psi })\in \mathcal{C(}\overline{Q}
;L^{\infty }(\triangle (A)))$  and
\begin{equation}
a_{i}^{\varepsilon }\left( \cdot ,\cdot ,\Psi ^{\varepsilon }\right)
\to b_{i}(\widehat{\Psi })\quad \text{in }L^{p'}(Q)\text{-weak }
\Sigma \text{ as }\varepsilon \to 0,  \label{3.3}
\end{equation}
where $\Psi ^{\varepsilon }=(\psi _{j}^{\varepsilon })_{1\leq j\leq N}$,
$\psi _{j}^{\varepsilon }$ defined as in (\ref{2.4}).

\item[(ii)] The mapping
$\Phi \to b(\Phi )=(b_{i}(\Phi ))_{1\leq i\leq N}$ of
$\mathcal{C}(\overline{Q};\mathcal{C}_{\mathbb{R}}(\triangle (A))^{N})$ into
$L^{p'}(Q\times \triangle (A))^{N}$ extends by continuity to a
mapping, still denoted by
$b$, of the space $L^{p}(Q;L_{\mathbb{R}}^{p}(\triangle (A))^{N})$
into $L^{p'}(Q\times \triangle (A))^{N}$  such that
\begin{equation}
\left\| b\left( \mathbf{u}\right) -b\left( \mathbf{v}\right) \right\|
_{L^{p'}(Q\times \triangle (A))^{N}}
\leq \alpha _{1}\left\| \left| \mathbf{u}\right| +\left| \mathbf{v}\right|
\right\| _{L^{p}(Q\times \triangle (A))}^{p-2}\left\| \mathbf{u}-\mathbf{v}
\right\| _{L^{p}(Q;L_{\mathbb{R}}^{p}(\triangle (A))^{N})}
\label{3.4}
\end{equation}
and
\begin{equation}
\left( b\left( \mathbf{u}\right) -b\left( \mathbf{v}\right) \right) \cdot
\left( \mathbf{u}-\mathbf{v}\right) \geq \alpha _{0}\left| \mathbf{u}-
\mathbf{v}\right| ^{p}\text{ a.e. in }Q\times \triangle (A)  \label{3.5}
\end{equation}
for all $\mathbf{u},\mathbf{v}\in L^{p}(Q;L_{\mathbb{R}}^{p}(\triangle(A))^{N})$.
\end{itemize}
\end{proposition}

The proof of \cite[Proposition 4.1]{22} carries over mutatis mutandis to the
present setting.

\begin{remark} \label{rm 3.1} \rm
We have in particular \begin{itemize}
\item[(1)] $b(\omega )=\omega $
\item[(2)] $| b(\lambda )-b(\mu )| \leq \alpha _{1}(\left| \lambda
\right| +\left| \mu \right| )^{p-2}\left| \lambda -\mu \right|$
 ($\lambda,\mu \in \mathbb{R}^{N}$)
\item[(3)] $(b(\lambda )-b(\mu ))\cdot (\lambda -\mu )\geq \alpha _{0}\left|
\lambda -\mu \right| ^{p}$ ($\lambda ,\mu \in \mathbb{R}^{N}$).
\end{itemize}
\end{remark}

As a consequence of Proposition \ref{3.1}, there is the following important
corollary.

\begin{corollary} \label{corol 3.1}
Let
\begin{equation}
\Phi _{\varepsilon }=\psi _{0}+\varepsilon \psi _{1}^{\varepsilon }, \label{3.6}
\end{equation}
i.e.,  $\Phi _{\varepsilon }(x,t)=\psi _{0}( x,t) +\varepsilon
\psi _{1}( x,t,\frac{x}{\varepsilon },\frac{t}{\varepsilon }) $
for $(x,t)\in Q$, where $\psi _{0}\in \mathcal{D}_{\mathbb{R}}(Q)=\mathcal{K}_{
\mathbb{R}}(Q)\cap \mathcal{C}^{\infty }(Q)$ and
$\psi _{1}\in \mathcal{D}_{\mathbb{R}}(Q)\otimes A_{\mathbb{R}}^{\infty }$
with $A_{\mathbb{R}}^{\infty}=A^{\infty }\cap A_{\mathbb{R}}$. Then, as
$\varepsilon \to 0$,
\begin{equation*}
a_{i}^{\varepsilon }\left( \cdot ,\cdot ,D\Phi _{\varepsilon }\right)
\to b_{i}(D\psi _{0}+\partial \widehat{\psi }_{1})\text{ in }
L^{p'}(Q)\text{-weak }\Sigma \;\;\left( 1\leq i\leq N\right)
\end{equation*}
where $\partial $ stands for the gradient operator on $\Delta (A_{y})$
[specifically, we have here $\partial \widehat{\psi }_{1}=(\partial _{j}
\widehat{\psi }_{1})_{1\leq j\leq N}$ with
$\partial _{j}\widehat{\psi }_{1}=\partial _{j}\circ \widehat{\psi }_{1}$
viewed as a function of $\overline{Q}\times \Delta (A_{\tau })$ into
$\mathcal{D}(\Delta (A_{y}))$, where $\partial _{j}$ is the partial
derivative of index $j$ on $\Delta (A_{y})$]. Furthermore, if
$\left( v_{\varepsilon }\right) _{\varepsilon \in E}$ is a sequence in
$L^{p}(Q)$ such that $v_{\varepsilon }\to v_{0}$
in $L^{p}(Q)$-weak $\Sigma $ as $E\ni \varepsilon \to 0$, then, as
$E\ni \varepsilon \to 0$,
\begin{equation*}
\int_{Q}a_{i}^{\varepsilon }\left( \cdot ,\cdot ,D\Phi _{\varepsilon
}\right) v_{\varepsilon }dxdt\to \int \int_{Q\times \Delta
(A)}b_{i}(D\psi _{0}+\partial \widehat{\psi }_{1})v_{0}\text{\thinspace }
\,dx\,dt\,d\beta \quad \left( 1\leq i\leq N\right) .
\end{equation*}
\end{corollary}

The proof of this corollary is a simple adaptation of the proof of
\cite[Corollary 4.1]{22}.

\subsection{Quasi-proper H-structures}

The basic notation and hypotheses are as in the preceding subsection. Now,
for $1\leq p<\infty $, we put
\begin{equation*}
\mathcal{H}=L^{p}(\Delta (A_{\tau });W_{\#}^{1,p}(\Delta (A_{y});\mathbb{R})),
\end{equation*}
a Banach space with an obvious norm. The canonical mapping of
$W^{1,p}(\Delta (A_{y}))/\mathbb{C}$ into its separated completion,
$W_{\#}^{1,p}(\Delta (A_{y}))$, will be denoted by $J_{y}$.

\begin{definition} \label{def 3.1} \rm
The $H$-structure $\Sigma =\Sigma _{y}\times \Sigma _{\tau }$
is said to be quasi-proper for some real $p>1$ if the following two
conditions are fulfilled:
\begin{itemize}
\item[(QP1)] $\Sigma _{y}$ is total for $p$, i.e.,
$\mathcal{D}(\Delta (A_{y}))$ is dense in $W^{1,p}(\Delta (A_{y}))$

\item[(QP2)] Given a bounded sequence
$(u_{\varepsilon})_{\varepsilon \in E}$ in $V^{p}$ (see Remark 2.2),
where $E$ is a fundamental sequence, there exist a subsequence $E'$
from $E$ and some
$\mathbf{u}=(u_{0},u_{1})\in V^{p}\times L^{p}(Q;\mathcal{H})$
such that, as $E'\ni \varepsilon \to 0$,
\begin{gather}
u_{\varepsilon }\to u_{0}\quad \text{in  }V^{p}\text{-weak} \label{3.7} \\
\frac{\partial u_{\varepsilon }}{\partial x_{j}}\to \frac{\partial
u_{0}}{\partial x_{j}}+\partial _{j}u_{1}\;\;in\;L^{p}(Q)\text{-weak }
\Sigma \;\;(1\leq j\leq N).  \label{3.8}
\end{gather}
\end{itemize}
\end{definition}

\begin{remark} \label{rm 3.2} \rm
The partial derivative $\partial _{j}u_{1}$ in (\ref{3.8}) needs an explanation.
First, let us once for all keep in mind
that for $1\leq j\leq N$, the symbol $\partial _{j}$ denotes
the partial derivative of index $j$ on $\Delta (A_{y})$
whereas $\partial _{0}$ denotes the derivative on $\Delta (A_{\tau})$.
Now, let $1\leq j\leq N$. It is to be noted that
$\partial _{j}$ yields a transformation, still denoted by
$\partial_{j}$, that maps continuously and linearly
$W_{\#}^{1,p}\left( \Delta(A_{y})\right) $ into $L^{p}\left( \Delta (A_{y})\right) $
and in particular $W_{\#}^{1,p}\left( \Delta (A_{y});\mathbb{R}\right) $
into $L_{\mathbb{R}}^{p}\left( \Delta (A_{y})\right) $ (see \cite{22}).
With this in mind, if $\Phi \in \mathcal{H}$, then
$\partial_{j}\Phi $ is understood as $\partial _{j}\circ \Phi $ (usual
composition). We have
$\partial _{j}\Phi \in L_{\mathbb{R}}^{p}\left( \Delta(A)\right) $, and
the transformation $\Phi \to \partial_{j}\Phi $ maps continuously and linearly
$\mathcal{H}$ into $L_{\mathbb{R}}^{p}\left( \Delta (A)\right) $.
Accordingly if $u_{1}\in L^{p}\left( Q;\mathcal{H}\right)$, then
$\partial _{j}u_{1}$ is naturally defined as being the function
$(x,t)\to \partial_{j}\left( u_{1}(x,t)\right) $ from $Q$ to
$L_{\mathbb{R}}^{p}\left( \Delta (A)\right) $. We have
$\partial _{j}u_{1}\in L_{\mathbb{R}}^{p}\left( Q\times \Delta (A)\right)$,
 and the transformation $u_{1}\to \partial _{j}u_{1}$ maps
continuously and linearly $L^{p}\left( Q;\mathcal{H}\right) $ into
$L_{\mathbb{R}}^{p}\left( Q\times \Delta (A)\right) $.
\end{remark}

\begin{remark} \label{rm 3.3} \rm
Let $E\ni \varepsilon \to 0$. In order
that (\ref{3.7}) hold, it is necessary and sufficient that we have
$u_{\varepsilon }\to u_{0}$ in $L^{p}(0,T;W_{0}^{1,p}(\Omega))$-weak  and
$\frac{\partial u_{\varepsilon }}{\partial t}\to
\frac{\partial u_{0}}{\partial t}$ in
$L^{p'}(0,T;W^{-1,p'}(\Omega ))$-weak.
\end{remark}

\subsection{Homogenization results}

Throughout this subsection we assume that $2\leq p<\infty $ and the
$H$-structure $\Sigma =\Sigma _{y}\times \Sigma _{\tau }$ is quasi-proper for
$p $. In the sequel, the space
$\mathbb{H}=L^{p}(0,T;W_{0}^{1,p}(\Omega ;\mathbb{R}))\times L^{p}(Q;\mathcal{H})$
is equipped with the norm $\left\| \mathbf{v}
\right\| _{\mathbb{H}}=\left\| v_{0}\right\| _{L^{p}(0,T;W_{0}^{1,p}(\Omega ;
\mathbb{R}))}+\left\| v_{1}\right\| _{L^{p}(Q;\mathcal{H})}$,
$\mathbf{v=}(v_{0},v_{1})\in \mathbb{H}$, which makes it a Banach space. We will
need the following lemma.

\begin{lemma} \label{lm 3.1}
$\mathfrak{F}_{0}^{\infty }=\mathcal{D}_{\mathbb{R}}(Q)\times (
\mathcal{D}_{\mathbb{R}}(Q)\otimes [\mathcal{D}_{\mathbb{R}}(\Delta (A_{\tau
}))\otimes J_{y}(\mathcal{D}_{\mathbb{R}}(\Delta (A_{y}))/\mathbb{C})])$ is dense
in $L^{p}(0,T;W_{0}^{1,p}(\Omega ;\mathbb{R}))\times L^{p}(Q;\mathcal{H})$.
\end{lemma}

\begin{proof}
In view of (QP1) (Definition \ref{3.1}), the space
$\mathcal{D}_{\mathbb{R}}(\Delta (A_{\tau }))\otimes J_{y}(\mathcal{D}_{\mathbb{R}
}(\Delta (A_{y}))/\mathbb{C})$ is dense in $\mathcal{H}$
(use \cite[Remark 3.5]{22} and the fact that $\Sigma _{\tau }$ is of class
$\mathcal{C}^{\infty }$). We deduce immediately
that $\mathcal{D}_{\mathbb{R}}(Q)\otimes [\mathcal{
D}_{\mathbb{R}}(\Delta (A_{\tau }))\otimes J_{y}(\mathcal{D}_{\mathbb{R}}(\Delta
(A_{y}))/\mathbb{C})]$ is dense in $L^{p}(Q;\mathcal{H})$. Hence, the lemma
follows by the density of $\mathcal{D}_{\mathbb{R}}(Q)$ in
$L^{p}(0,T;W_{0}^{1,p}(\Omega ;\mathbb{R}))$.
\end{proof}

\begin{remark} \label{rm 3.4} \rm
We have
\begin{equation*}
\mathcal{D}_{\mathbb{R}}\left( \Delta (A_{\tau })\right) \otimes \left[
\mathcal{D}_{\mathbb{R}}\left( \Delta (A_{y})\right) /\mathbb{C}\right] =\mathcal{G
}\left( _{\mathbb{R}}A_{\tau }^{\infty }\otimes \left[ _{\mathbb{R}}A_{y}^{\infty
}/\mathbb{C}\right] \right)
\end{equation*}
where $\mathcal{G}$ is here the Gelfand transformation on $A$, and
where $_{\mathbb{R}}A_{\tau }^{\infty }=A_{\tau }^{\infty }\cap
\mathcal{C}_{\mathbb{R}}(\mathbb{R})$ and
$_{\mathbb{R}}A_{y}^{\infty }/\mathbb{C}=\left\{ \psi \in A_{y}^{\infty }
\cap \mathcal{C}_{\mathbb{R}}(\mathbb{R}^{N}):M(\psi )=0\right\} $
($M$ denotes the mean value on $\mathbb{R}^{N}$ in the sense of
\cite[Subsection 2.1]{18}).
\end{remark}

\begin{lemma} \label{lm 3.2}
The variational problem
\begin{equation}
\begin{gathered}
\mathbf{u}=(u_{0},u_{1})\in \mathbb{F}_{0}^{1,p}=V_{0}^{p}\times L^{p}(Q;
\mathcal{H})\quad \text{:} \\
\int_{0}^{T}(u_{0}'(t),v_{0}(t))dt+\int \int_{Q\times \Delta
(A)}b(Du_{0}+\partial u_{1})\cdot (Dv_{0}+\partial v_{1})\,dx\,dt\,d\beta \\
=\int_{0}^{T}(f(t),v_{0}(t))dt,
\end{gathered} \label{3.10}
\end{equation}
for all $\mathbf{v} =(v_{0},v_{1})\in \mathbb{F}_{0}^{1,p}$,
has at most one solution.
\end{lemma}

The proof of this lemma follows in a quite classical way
(use in particular (\ref{3.5}) and $b(\omega )=\omega $).
We are now in a position to state and prove the main result in the present
section.

\begin{theorem} \label{th 3.1}
Let $2\leq p<\infty $. Suppose \eqref{3.1} holds and
$\Sigma=\Sigma _{y}\times \Sigma _{\tau }$ is quasi-proper for $p$. For each fixed
real number $\varepsilon >0$, let $u_{\varepsilon }$ be the solution of the
initial-boundary value problem \eqref{1.4}. As $\varepsilon \to 0$,
we have
\begin{gather}
u_{\varepsilon }\to u_{0}\quad\mbox{in }L^{p}(0,T;W_{0}^{1,p}(\Omega ))
\text{-weak} \label{3.11} \\
\frac{\partial u_{\varepsilon }}{\partial t}\to \frac{\partial u_{0}
}{\partial t}\quad\mbox{in } L^{p'}(0,T;W^{-1,p'}(\Omega ))\text{-weak}
\label{3.12} \\
\frac{\partial u_{\varepsilon }}{\partial x_{j}}\to \frac{\partial
u_{0}}{\partial x_{j}}+\partial _{j}u_{1}\quad\mbox{in }
L^{p}(Q)\text{-weak }\Sigma \quad (1\leq j\leq N),  \label{3.13}
\end{gather}
where $\mathbf{u}=\left( u_{0},u_{1}\right) $ is the unique solution of
\eqref{3.10}.
\end{theorem}


\begin{proof}
The first point is to check that the sequence
$(u_{\varepsilon})_{\varepsilon >0}$ is bounded in $V^{p}$. To this end,
observe that $u_{\varepsilon }\in V_{0}^{p}$ (Remark \ref{2.2}) and
\begin{equation}
\int_{0}^{T}(u_{\varepsilon }'(t),v(t))dt+\int_{Q}a^{\varepsilon
}(x,t,Du_{\varepsilon }(x,t))\cdot Dv(x,t)dx\,dt
=\int_{0}^{T}(f(t),v(t))dt
 \label{3.14}
\end{equation}
for all $v\in V_{0}^{p}$,
where $\varepsilon >0$ is arbitrarily fixed. Taking in particular
$v=u_{\varepsilon }$ and using
\begin{equation}
\int_{0}^{T}(u_{\varepsilon }'(t),u_{\varepsilon }(t))dt=\frac{1}{2}
\left\| u_{\varepsilon }(T)\right\| _{L^{2}(\Omega )}^{2}\geq 0  \label{3.15}
\end{equation}
and (\ref{2.7})-(\ref{2.9}), we obtain by mere routine
\begin{equation}
\underset{\varepsilon >0}{\sup }\left\| u_{\varepsilon }\right\|
_{L^{p}(0,T;W_{0}^{1,p}(\Omega ))}<\infty .  \label{3.16}
\end{equation}
Using (\ref{2.7})-(\ref{2.8}), once again, it follows
\begin{equation*}
\underset{\varepsilon >0}{\sup }\left\| a^{\varepsilon }(\cdot ,\cdot
,Du_{\varepsilon })\right\| _{L^{p'}(Q)^{N}}<\infty ,
\end{equation*}
hence  $\underset{\varepsilon >0}{\sup }\left\| \mathop{\rm div}a^{\varepsilon
}(\cdot ,\cdot ,Du_{\varepsilon })\right\| _{L^{p'}(0,T;W^{-1,p'}(\Omega ))}<\infty $. We deduce by (\ref{1.4}) that
\begin{equation*}
\underset{\varepsilon >0}{\sup }\Big\| \frac{\partial u_{\varepsilon }}{
\partial t}\Big\| _{L^{p'}(0,T;W^{-1,p'}(\Omega ))}<\infty ,
\end{equation*}
which combines with (\ref{3.16}) to show that the sequence $(u_{\varepsilon
})_{\varepsilon >0}$ is bounded in $V^{p}$, hence also in $V_{0}^{p}$.

Thus, given an arbitrary fundamental sequence $E$, the quasi-properness of
$\Sigma $ (see especially (QP2)) guarantees the existence of a
subsequence $E'$ from $E$ and of some $\mathbf{u}=(u_{0},u_{1})\in
\mathbb{F}_{0}^{1,p}=V_{0}^{p}\times L^{p}(Q;\mathcal{H})$ such that as
$E'\ni \varepsilon \to 0$, (\ref{3.11})-(\ref{3.13}) hold
true (see Remark \ref{rm 3.3}). Therefore, thanks to Lemma \ref{lm 3.2}, the
theorem is proved once we have established that the vector function $\mathbf{
u}=(u_{0},u_{1})$ satisfies the variational equation in (\ref{3.10}) (the
conclusive argument is classical; see, e.g., the proof of
\cite[Theorem 4.1]{22}).

To do this, let $\Phi \in \mathfrak{F}_{0}^{\infty }$ (see Lemma \ref{lm 3.1}),
i.e., $\Phi =(\psi _{0},J_{y}(\widehat{\psi }_{1}))$ with $\psi _{0}\in
\mathcal{D}_{\mathbb{R}}(Q),\;\psi _{1}\in \mathcal{D}_{\mathbb{R}}(Q)\otimes [_{
\mathbb{R}}A_{\tau }^{\infty }\otimes (_{\mathbb{R}}A_{y}^{\infty }/\mathbb{C})]$
(see Remark \ref{rm 3.4}), $\widehat{\psi }_{1}=\mathcal{G}\circ \psi _{1}$
and $J_{y}(\widehat{\psi }_{1})=\;J_{y}\circ \widehat{\psi }_{1}$
($\widehat{\psi }_{1}$ viewed as a function of $\overline{Q}\times \Delta (A_{\tau
}) $ into $\mathcal{D}(\Delta (A_{y}))/\mathbb{C})$. Define $\Phi
_{\varepsilon }$ as in (\ref{3.6}). Clearly $\Phi _{\varepsilon }\in
\mathcal{D}_{\mathbb{R}}(Q). $ In (\ref{3.14}), take $v=\Phi _{\varepsilon }$
and then use (\ref{2.9}) to get
\begin{equation*}
0\leq \int_{0}^{T}(f(t)-u_{\varepsilon }'(t),u_{\varepsilon
}(t)-\Phi _{\varepsilon }(t))dt-\int_{Q}a^{\varepsilon }(\cdot ,\cdot ,D\Phi
_{\varepsilon })\cdot (Du_{\varepsilon }-D\Phi _{\varepsilon })dxdt
\end{equation*}
or, according to (\ref{3.15}),
\begin{equation}
\begin{aligned}
\frac{1}{2}\left\| u_{\varepsilon }(T)\right\| _{L^{2}(\Omega )}^{2}
&\leq \int_{0}^{T}(f(t),u_{\varepsilon }(t)-\Phi _{\varepsilon
}(t))dt+\int_{0}^{T}(u_{\varepsilon }'(t),\Phi _{\varepsilon }(t))dt\\
&\quad -\int_{Q}a^{\varepsilon }(\cdot ,\cdot ,D\Phi _{\varepsilon })\cdot
(Du_{\varepsilon }-D\Phi _{\varepsilon })dxdt
\end{aligned} \label{3.17}
\end{equation}
and that for any $\varepsilon >0$. Our goal now is to pass to the limit when
$E'\ni \varepsilon \to 0$.

First, as $\varepsilon \to 0$, we have
\begin{gather}
\frac{\partial \Phi _{\varepsilon }}{\partial x_{j}}\to \frac{
\partial \psi _{0}}{\partial x_{j}}+\partial _{j}\widehat{\psi }
_{1}\quad\mbox{in }L^{q}(Q)\text{-weak }\Sigma \quad (1\leq j\leq N)  \label{3.18}\\
\frac{\partial \Phi _{\varepsilon }}{\partial t}\to \frac{\partial
\psi _{0}}{\partial t}+\partial _{0}\widehat{\psi }_{1}\quad\mbox{in }L^{q}(Q)
\text{-weak }\Sigma ,  \label{3.19}
\end{gather}
and that for any given $1\leq q<\infty $. Choosing in particular $q=p$ and
using \cite[Propositions 2.5 and 4.4]{18}, it follows that $\Phi
_{\varepsilon }\to \psi _{0}$ in $W_{0}^{1,p}(Q)$-weak. Hence
$\Phi _{\varepsilon }\to \psi _{0}$ in $L^{p}(0,T;W_{0}^{1,p}(\Omega
))$-weak as $\varepsilon \to 0$, since $W_{0}^{1,p}(Q)$ is
continuously embedded in $L^{p}(0,T;W_{0}^{1,p}(\Omega ))$. Recalling
(\ref{3.11}) (when $E'\ni \varepsilon \to 0\;$), we finally
arrive at  $\int_{0}^{T}(f(t),u_{\varepsilon }(t)-\Phi _{\varepsilon
}(t))dt\to \int_{0}^{T}(f(t),u_{0}(t)-\psi _{0}(t))dt$  when
$E'\ni \varepsilon \to 0$.
Next, observe that
\begin{equation*}
\int_{0}^{T}(u_{\varepsilon }'(t),\Phi _{\varepsilon
}(t))dt=-\int_{Q}u_{\varepsilon }\frac{\partial \Phi _{\varepsilon }}{
\partial t}dxdt.
\end{equation*}
Thanks to the fact that $V^{p}\;$(for $2\leq p<\infty $) is compactly
embedded in the space $L^{p}(0,T;L^{2}(\Omega ))$ (this is a classical
property; use, e.g., \cite[p.58, Theorem 5.1]{16}) and that the latter is
continuously embedded in $L^{2}(Q)$, we have (from (\ref{3.11})-(\ref{3.12}
))\ \ $u_{\varepsilon }\to u_{0}$ in $L^{2}(Q)$ as $E'\ni
\varepsilon \to 0$. Combining this with (\ref{3.19}) (for $q=2$), it
follows that
\begin{equation*}
\int_{0}^{T}(u_{\varepsilon }'(t),\Phi _{\varepsilon
}(t))dt\to \int_{0}^{T}(u_{0}'(t),\psi _{0}(t))dt\quad
\text{as  }E'\ni \varepsilon \to 0.
\end{equation*}

Now, based on (\ref{3.13}) (when $E'\ni \varepsilon \to 0$,
of course) and (\ref{3.18}) (with $q=p$), a quick application of Corollary
3.1 yields
\begin{equation*}
\int_{Q}a^{\varepsilon }(\cdot ,\cdot ,D\Phi _{\varepsilon })\cdot
(Du_{\varepsilon }-D\Phi _{\varepsilon })dxdt\to \;\int
\int_{Q\times \Delta (A)}b(\mathbb{D}\Phi )\cdot \mathbb{D}(\mathbf{u}-\Phi
)\,dx\,dt\,d\beta
\end{equation*}
as $E'\ni \varepsilon \to 0$, where, for $\mathbf{v}
=(v_{0},v_{1})\in L^{p}(0,T;\;W_{0}^{1,p}(\Omega ))\;\times L^{p}(Q;\mathcal{
H})$, we denote $\mathbb{D}\mathbf{v}=Dv_{0}+\partial v_{1}$ with
$D=(D_{x_{i}})_{1\leq i\leq N}$ and $\partial =(\partial _{i})_{1\leq i\leq
N}$.
Finally, as pointed out in Remark \ref{rm 2.2}, the transformation
$v\to \left\| v(T)\right\| _{L^{2}(\Omega )}^{2}$ is continuous on $V_{0}^{p}$. On
the other hand, according to (\ref{3.11})-(\ref{3.12}), we have
$u_{\varepsilon }\to u_{0}$ in $V_{0}^{p}$-$weak$ as $E'\ni
\varepsilon \to 0$. Hence, by a classical argument it follows that
\begin{equation*}
\left\| u_{0}(T)\right\| _{L^{2}(\Omega )}^{2}\leq \underset{E'\ni
\varepsilon \to 0}{\lim \inf }\left\| u_{\varepsilon }(T)\right\|
_{L^{2}(\Omega )}^{2}.
\end{equation*}
Therefore, taking the $\lim \inf_{E'\ni \varepsilon \to 0}$
of both sides of (\ref{3.17}) and using
\begin{equation*}
\frac{1}{2}\left\| u_{0}(T)\right\| _{L^{2}(\Omega
)}^{2}=\int_{0}^{T}(u_{0}'(t),u_{0}(t))dt,
\end{equation*}
one arrives at
\begin{equation*}
0\leq \int_{0}^{T}(f(t)-u_{0}'(t),u_{0}(t)-\psi _{0}(t))dt-\int
\int_{Q\times \Delta (A)}b(\mathbb{D}\Phi )\cdot \mathbb{D}(\mathbf{u}-\Phi
)dx\,dt\,d\beta
\end{equation*}
and that for any $\Phi \in \mathfrak{F}_{0}^{\infty }$. Thanks to Lemma 3.1,
this still holds true for $\Phi \in L^{p}(0,T;W_{0}^{1,p}(\Omega ;\mathbb{R}
))\times L^{p}(Q;\mathcal{H})$, hence for $\Phi \in \mathbb{F}_{0}^{1,p}$.
Therefore the theorem follows by a classical line of reasoning (proceed as
in the proof of \cite[Theorem 4.1]{22}).
\end{proof}

The variational problem (\ref{3.10}) is called the global homogenized
problem for (\ref{1.4}) under the abstract structure hypothesis (\ref{3.1})
with $\Sigma $ quasi-proper (for the given real $p\geq 2$). The term \textit{
global} is used here to lay emphasis on the fact that (\ref{3.10}) includes
both the local (or microscopic) equation for $u_{1}(x,t)$ (where $(x,t)$ is
fixed in $Q$) and the macroscopic homogenized equation for $u_{0}$.
Specifically, by choosing in (\ref{3.10}) the test function $\mathbf{v}
=(v_{0},v_{1})$ such that $v_{0}=0$ and $v_{1}(x,t)=\varphi (x,t)w$ with
$\varphi \in \mathcal{D}_{\mathbb{R}}(Q)$ and $w\in \mathcal{H}$, we obtain the
so-called local equation at (fixed) $(x,t)\in Q$ :
\begin{equation}
\int_{\Delta (A)}b(Du_{0}(x,t)+\partial u_{1}(x,t))\cdot \partial w\text{
\thinspace }d\beta =0\text{ \ for all }w\in \mathcal{H}.  \label{3.20}
\end{equation}

As regards the derivation of the macroscopic homogenized equation, let $r\in
\mathbb{R}^{N}$ be freely fixed. Consider the so-called cell problem
\begin{gather*}
\pi (r)\in \mathcal{H}: \\
\int_{\Delta (A)}b(r+\partial \pi (r))\cdot \partial w\text{\thinspace }
d\beta =0\quad \text{for all }w\in \mathcal{H}
\end{gather*}
which uniquely determines $\pi (r)$, thanks to Remark \ref{rm 3.1} (see
\cite[Chap.3]{14}). Then, taking in particular $r=Du_{0}(x,t)$ with $(x,t)$
arbitrarily fixed in $Q$, and comparing with (\ref{3.20}), it follows at
once
\begin{equation}
u_{1}=\pi (Du_{0})  \label{3.21}
\end{equation}
where the right-hand side stands for the function $(x,t)\to \pi
(Du_{0}(x,t))$ from $Q$ to $\mathcal{H}$. Hence, substituting (\ref{3.21})
in (\ref{3.10}) and choosing there the test functions $\mathbf{v}
=(v_{0},v_{1})$ such that $v_{1}=0$, we are led to the so-called macroscopic
homogenized problem for (\ref{1.4}), viz.
\begin{equation}
\begin{gathered}
\frac{\partial u_{0}}{\partial t}-\mathop{\rm div}q(Du_{0})=f\quad \text{in }Q \\
u_{0}=0\quad \text{on }\partial \Omega \times (0,T) \\
u_{0}(x,0)=0\quad \text{in }\Omega,
\end{gathered}  \label{3.22}
\end{equation}
where
$q(r)=\int_{\Delta (A)}b(r+\partial \pi (r))d\beta$ ($r\in \mathbb{R}^{N}$).


\begin{remark} \label{rm 3.5} \rm
A vector function $\mathbf{u}=(u_{0},u_{1})$
satisfies (\ref{3.10}) if and only if the macroscopic component $u_{0}$
 solves (\ref{3.22}) and the microscopic component, $u_{1}(x,t)$,
at a given point $(x,t)\in Q$ solves (\ref{3.20}). Thanks to
Lemma \ref{lm 3.2}, this guarantees the uniqueness in (\ref{3.22}).
\end{remark}

\begin{remark} \label{rm 3.6} \rm
We have $q(\omega )=0$ and further it can be
shown that the function $r\to q(r)$ satisfies inequalities
of the same type \textit{mutatis mutandis} as in Remark \ref{rm 3.1}.
\end{remark}

\subsection{Study of a concrete case. Harmonic H-structures}

We start with the following definition.

\begin{definition} \label{def 3.2} \rm
The $H$-structure (of class $\mathcal{C}^{\infty }$) $\Sigma_{y}$ on
$\mathbb{R}^{N}$ is termed $p$-harmonic (for some given $1<p<\infty $)
if the following conditions are satisfied:
\begin{itemize}
\item[(H1)] $\Sigma _{y}$ is total for $p$
\item[(H2)]  To any
$\mathbf{f}=(f_{j})_{1\leq j\leq N}\in L^{p}(\Delta(A_{y})^{N}$ satisfying
\begin{equation}
\int_{\Delta (A_{y})}\mathbf{f\cdot }\widehat{\Psi }d\beta _{y}\equiv
\sum_{j=1}^{N}\int_{\Delta (A_{y})}f_{j}\widehat{\psi }_{j}d\beta _{y}=0\;\;
\text{(with }\widehat{\psi }_{j}=\mathcal{G}(\psi _{j})\text{)}  \label{3.23}
\end{equation}
for all $\Psi =(\psi _{j})\in \mathcal{V}_{div}=\{\mathbf{u}\in
(A_{y}^{\infty })^{N}:\mathop{\rm div}_{y}\mathbf{u}=0\}$,  there
is attached a unique $\chi \in W_{\#}^{1,p}(\Delta (A_{y}))$ such
that $f_{j}=\partial _{j}\chi$ ($1\leq j\leq N$).
\end{itemize}
\end{definition}

We turn now to the proof of the following statement.

\begin{proposition} \label{prop 3.2}
Suppose $\Sigma _{y}$ is $p$-harmonic (for some given real $p>1$).
Then $\Sigma =\Sigma _{y}\times \Sigma _{\tau }$ is quasi-proper for $p$.
\end{proposition}

\begin{proof}
We need verify only (QP2). So let $(u_{\varepsilon })_{\varepsilon \in E}$
be a bounded sequence in $V^{p}$, $E$ being fundamental. Based on the
reflexivity of $V^{p}$ and on the $\Sigma $-reflexivity of $L^{p}(Q)$
\cite[Theorem 4.1]{18}, we can find a subsequence $E'$ from $E$, a
function $u_{0}\in V^{p}$ and a family $(z_{j})_{1\leq j\leq N}\subset L_{
\mathbb{R}}^{p}(Q\times \Delta (A))$ such that as $E'\ni \varepsilon
\to 0$, we have $u_{\varepsilon }\to u_{0}$ in $V^{p}$-$weak$
and $\frac{\partial u_{\varepsilon }}{\partial x_{j}}\to z_{j}$ in
$L^{p}(Q)$-$weak\;\Sigma \;\;(1\leq j\leq N)$. Thus, the proposition is
proved if we can establish that there is some function $u_{1}\in L^{p}(Q;
\mathcal{H})$ such that
\begin{equation}
z_{j}=\frac{\partial u_{0}}{\partial x_{j}}+\partial
_{j}u_{1}\quad (1\leq j\leq N).  \label{3.24}
\end{equation}
To do this, let $\Phi =(\phi _{j})_{1\leq j\leq N},$\ $\phi _{j}\in
L^{p'}(Q;A),$ with
\begin{equation*}
\phi _{j}(x,t,y,\tau )=\varphi (x,t)\psi _{j}(y)w(\tau )
\quad ((x,t)\in Q,\; y\in \mathbb{R}^{N},\; \tau \in \mathbb{R}),
\end{equation*}
where $\varphi \in \mathcal{D}(Q)$,
$\Psi =(\psi _{j})\in \mathcal{V}_{div}$ and $w\in A_{\tau }^{\infty }$.
 Clearly
\begin{equation*}
\sum_{j=1}^{N}\int_{Q}\frac{\partial u_{\varepsilon }}{\partial x_{j}}\psi
_{j}^{\varepsilon }w^{\varepsilon }\varphi
dx\,dt=-\sum_{j=1}^{N}\int_{Q}u_{\varepsilon }\psi _{j}^{\varepsilon
}w^{\varepsilon }\frac{\partial \varphi }{\partial x_{j}}dx\,dt.
\end{equation*}
Passing to the limit (as $E'\ni \varepsilon \to 0$) on both
sides gives
\begin{equation*}
\sum_{j=1}^{N}\int \int_{Q\times \Delta (A)}z_{j}\widehat{\psi }_{j}\widehat{
w}\varphi \,dx\,dt\,d\beta =\sum_{j=1}^{N}\int \int_{Q\times \Delta (A)}\frac{
\partial u_{0}}{\partial x_{j}}\widehat{\psi }_{j}\widehat{w}\varphi
\,dx\,dt\,d\beta
\end{equation*}
where, regarding the right-hand side, we have used the facts that
$u_{\varepsilon }\to u_{0}$ in $L^{2}(Q)$ as $E'\ni
\varepsilon \to 0$ (see the proof of Theorem \ref{3.1}) and
$\psi_{j}^{\varepsilon }w^{\varepsilon }\to \int_{\Delta (A)}\widehat{
\psi }_{j}\widehat{w}d\beta $ in $L^{2}(Q)$-weak as
$\varepsilon \to 0$.
Using first the arbitrariness of $\varphi $ and then that of $w$, we quickly
arrive at (\ref{3.23}) for all $\Psi \in \mathcal{V}_{div}$, where
\begin{equation*}
f_{j}(s)=z_{j}(x,t,s,s_{0})-\frac{\partial u_{0}}{\partial x_{j}}
(x,t)\quad (s\in \Delta (A_{y})),
\end{equation*}
$(x,t)\in Q$ and $s_{0}\in \Delta (A_{\tau })$ being fixed. Thanks to the
$p$-harmonicity of $\Sigma _{y}$, this yields a function
$u_{1}\in L^{p}(Q; \mathcal{H})$ such that (\ref{3.24}) holds
(to show this is an easy matter),
as claimed. \end{proof}

This is worth illustrating the results above.

\begin{example} \label{exple 3.1} \rm
Suppose $\Sigma _{y}$ is an almost periodic
$H$-structure on $\mathbb{R}^{N}$ \cite[Example 3.3]{18}. Then
$\Sigma =\Sigma _{y}\times \Sigma _{\tau }$ (where $\Sigma _{\tau }$
 is any $H$-structure of class $\mathcal{C}^{\infty }$
on $\mathbb{R}$) is quasi-proper for $p=2$. Indeed, $\Sigma_{y}$
is $2$-harmonic (this is established in a preprint by
the first author) and so the claimed property follows by Proposition
\ref{prop 3.2}.
\end{example}

\begin{example} \label{exple 3.2} \rm
Suppose $\Sigma _{y}$ is the periodic $H$-structure on $\mathbb{R}^{N}$
 represented by a network $\mathcal{R}\subset \mathbb{R}^{N}$, say
$\mathcal{R}=\mathbb{Z}^{N}$ (see \cite[Example 3.2]{18}).
Then $\Sigma _{y}$\emph{\ is }$p$-harmonic for any real
$p>1$ (see \cite[Subsection 3.3]{22}).
Consequently, according to Proposition \ref{prop 3.2}, the $H$-structure
$\Sigma =\Sigma _{y}\times \Sigma _{\tau }$ (where
$\Sigma _{\tau }$ is an arbitrary $H$-structure of class
$\mathcal{C}^{\infty }$ on $\mathbb{R}$) is quasi-proper for any
real $p>1$.
\end{example}

\section{Concrete homogenization problems for \eqref{1.4}}

This section provides concrete examples of homogenization problems
for (\ref{1.4}). More precisely, we study here the limiting behaviour, as
$\varepsilon \to 0$, of $u_{\varepsilon }$ (the solution of (\ref{1.4}))
under various \textit{concrete }structure hypotheses. It should be noted
that in practice the statement of a homogenization problem makes no mention
of the concept of a homogenization structure, still less of that of a
quasi-proper $H$-structure. The term \textit{concrete} used above is
precisely intended to stress this fact, as opposed to the abstract nature of
(\ref{3.1}).

In fact, in view of the fundamental results achieved in the preceding
section, our only concern in each example under consideration below will be
to show that the concrete structure hypothesis supplementing (\ref{1.4}) (so
as to yield a solvable homogenization problem) can be reduced to (\ref{3.1})
for a suitable quasi-proper $H$-structure $\Sigma $. This is the general
point of view. We will see that the particular case where the diffusion term
in (\ref{1.4}) is linear entails considerable simplifications with regard to
practice.

\subsection{General case}

Just as in the preceding subsections, it is not specified here whether the
diffusion term in (\ref{1.4}) is linear or nonlinear.

\subsubsection*{Problem I} (Periodic setting)
As we mentioned in Section 1, the homogenization of (\ref{1.4}) under the
periodicity hypothesis has been sufficiently investigated. We will only draw
attention to the fact that the present study includes the periodic setting.
Indeed, suppose for each fixed $\lambda \in \mathbb{R}^{N}$, the function
$(y,\tau )\to a(y,\tau ,\lambda )$ is $Y$-periodic in $y\in \mathbb{R}
^{N}$ and $Z$-periodic in $\tau \in \mathbb{R}$ with, e.g., $Y=(0,1)^{N}$ and
$Z=(0,1)$. It amounts to saying that for any $k\in \mathcal{R}=\mathbb{Z}^{N}$
and any $l\in \mathcal{S}=\mathbb{Z}$, we have $a(y+k,\tau +l,\lambda
)=a(y,\tau ,\lambda )$ \ a.e. in $(y,\tau )\in \mathbb{R}^{N}\mathbb{\times R}$. Immediately we see that the appropriate homogenization structures are the
periodic $H$-structures $\Sigma _{y}=\Sigma _{\mathcal{R}}$ and $\Sigma
_{\tau }=\Sigma _{\mathcal{S}}$ represented by $\mathcal{R}=\mathbb{Z}^{N}$ and
$\mathcal{S}=\mathbb{Z}$, respectively (see \cite[Example 3.2]{18}). In other
words, in the present case we have $A_{y}=\mathcal{C}_{\rm per}(Y),\;A_{\tau }=
\mathcal{C}_{\rm per}(Z)$, and hence $A=\mathcal{C}_{\rm per}(Y\times Z)$. Then, as
pointed out in Example \ref{exple 3.2}, the product homogenization structure
$\Sigma =\Sigma _{y}\times \Sigma _{\tau }$ is quasi-proper for any
$1<p<\infty $. Hence, the results of Subsection 3.4 (and especially the
conclusion of Theorem \ref{th 3.1}) are valid, as claimed. Finally, for the
sake of exactness, we should change there $\Delta (A_{y})\;$(resp. $\Delta
(A_{\tau })$, $\Delta (A)$) in $Y$, (resp. $Z,\;Y\times Z$), $\partial _{j}$
in $D_{y_{j}}\;(1\leq j\leq N)$, $\partial _{0}$ in $\frac{d}{d\tau }$, and
$\beta _{y}\;$(resp. $\beta _{\tau },\;\beta $) in $dy\;$(resp. $d\tau,dyd\tau $).

\subsubsection*{Problem II} (Almost periodic setting)
For the benefit of the reader we begin by recalling the notion of an almost
periodic function \cite[11, 15]{4} we will be dealing with. By an almost
periodic continuous complex function on $\mathbb{R}^{d}$ ($d\geq 1$) is meant a
function $u\in \mathcal{B}(\mathbb{R}^{d})$ whose translates $\{\tau
_{h}u\}_{h\in \mathbb{R}^{d}}$ (with $\tau _{h}u(y)=u(y-h)$, $y\in \mathbb{R}^{d}$
) form a relatively compact set in $\mathcal{B}(\mathbb{R}^{d})$. Such
functions form a closed subalgebra of $\mathcal{B}(\mathbb{R}^{d})$ denoted by
$AP(\mathbb{R}^{d})$. This fundamental notion, which is due to Bohr \cite{5},
has been generalized to $L_{\rm loc}^{p}$ spaces. A function $u\in L_{\rm loc}^{p}(
\mathbb{R}^{d})$ ($1\leq p<\infty $) is said to be almost periodic in Stepanoff
sense if $u$ lies in the amalgam space $(L^{p},l^{\infty })(\mathbb{R}^{d})$
\cite[20]{10} and further the translates $\{\tau _{h}u\}_{h\in \mathbb{R}^{d}}$
form a relatively compact set in $(L^{p},l^{\infty })(\mathbb{R}^{d})$. Such
functions form a closed vector subspace of $(L^{p},l^{\infty })(\mathbb{R}^{d})$
denoted by $L_{AP}^{p}(\mathbb{R}^{d})$. It seems useful to recall that
$(L^{p},l^{\infty })(\mathbb{R}^{d})$ is the space of functions $u\in
L_{\rm loc}^{p}(\mathbb{R}^{d})$ such that
\begin{equation*}
\left\| u\right\| _{p,\infty }=\sup_{k\in \mathbb{Z}^{d}}
\Big(\int_{k+(0,1)^{d}}\left| u(y)\right| ^{p}dy\Big) ^{1/p}<\infty .
\end{equation*}
Equipped with the norm $\left\| \cdot \right\| _{p,\infty }$,
$(L^{p},l^{\infty })(\mathbb{R}^{d})$ is a Banach space. The appropriate norm on
$L_{AP}^{p}(\mathbb{R}^{d})$ is the $(L^{p},l^{\infty })(\mathbb{R}^{d})$-norm. It
is also worth noting that $AP(\mathbb{R}^{d})$ is a dense vector subspace of
$L_{AP}^{p}(\mathbb{R}^{d})$.

Now, let $M$ denote the mean value on $\mathbb{R}^{d}$ as stated in Subsection
3.1. Considered as defined on $AP(\mathbb{R}^{d})$, the mapping $M$ extends to
a continuous linear form on $L_{AP}^{p}(\mathbb{R}^{d})$ still denoted by $M$.
This follows immediately by the inequality
\begin{equation}
\Big( \int_{B}\left| u\left( \frac{x}{\varepsilon }\right) \right|
^{p}dx\Big) ^{1/p}\leq c\left\| u\right\| _{p,\infty}\quad
(0<\varepsilon \leq 1)  \label{4.1}
\end{equation}
for all $u\in (L^{p},l^{\infty })(\mathbb{R}^{d})$, where $B$ is a fixed
bounded open set in $\mathbb{R}^{d}$ and $c$ is a positive constant independent
of both $\varepsilon $ and $u$.

Now, given a countable subgroup $\mathcal{R}$ of $\mathbb{R}^{d}$, we will put
\begin{gather*}
AP_{\mathcal{R}}(\mathbb{R}^{d})=\{u\in AP(\mathbb{R}^{d}):Sp(u)\subset \mathcal{R}
\} \\
L_{AP,\mathcal{R}}^{p}(\mathbb{R}^{d})=\{u\in L_{AP}^{p}(\mathbb{R}
^{d}):Sp(u)\subset \mathcal{R}\}
\end{gather*}
where $Sp(u)$ stands for the spectrum of $u$, i.e.,
$Sp(u)=\{k\in \mathbb{R}^{d}:M(u\overline{\gamma }_{k})\neq 0\}$ with
$\gamma _{k}(y)=\exp (2i\pi k\cdot y)\;\;(y\in \mathbb{R}^{d})$.
Equipped with the $(L^{p},l^{\infty })$-norm,
$L_{AP,\mathcal{R}}^{p}(\mathbb{R}^{d})$ is a Banach space. As regards
$AP_{\mathcal{R}}(\mathbb{R}^{d})$, this is a homogenization algebra on
$\mathbb{R}^{d}$, the associated $H$-structure being the so-called almost
periodic $H$-structure represented by $\mathcal{R}$
\cite[Examples 2.2 and 3.3]{18}. It
is worth knowing that $AP_{\mathcal{R}}(\mathbb{R}^{d})$ is dense in
$L_{AP,\mathcal{R}}^{p}(\mathbb{R}^{d})$. Indeed, given
$u\in L_{AP,\mathcal{R}}^{p}(\mathbb{R}^{d})\;(1\leq p<\infty )$,
the same procedure as followed in the
proof of \cite[Proposition 3.2]{22} leads to
\begin{equation*}
\left\| u*\theta _{n}-u\right\| _{p,\infty }^{p}\leq \int \theta
_{n}(x)\left\| u-\tau _{x}u\right\| _{p,\infty }^{p}dx\quad
(\text{integers }n\geq 1),
\end{equation*}
where $\theta _{n}\in \mathcal{D}(\mathbb{R}^{d})$ is a mollifier. By using the
fact that $\tau _{x}u\to u$ in $(L^{p},l^{\infty })(\mathbb{R}^{d})$ as
$\left| x\right| \to 0$ (there is no serious difficulty in verifying
this), we deduce that  $u*\theta _{n}\to u$  in
$(L^{p},l^{\infty })(\mathbb{R}^{d})$ as $n\to \infty $. But
$u*\theta_{n}\in AP(\mathbb{R}^{d})$ and
$M([u*\theta _{n}]\overline{\gamma }_{k})
=M(u\overline{\gamma }_{k})\mathcal{F}\theta _{n}(-k)$, where $\mathcal{F}$
denotes the Fourier transformation on $\mathbb{R}^{d}$. Hence $u*\theta _{n}$
$\in AP_{\mathcal{R}}(\mathbb{R}^{d})$ and so the claimed result follows.

Having made this point, let us turn now to one fundamental result.

\begin{proposition} \label{prop 4.1}
Let $(f_{i})_{i\in I}$ be a countable family in $L_{AP}^{p}(
\mathbb{R}^{d})$. There exists a countable subgroup $\mathcal{R}$ of $\mathbb{R}
^{d}$such that $f_{i}\in L_{AP,\mathcal{R}}^{p}(\mathbb{R}^{d})$ for all $i\in
I$. \end{proposition}

\begin{proof}
Indeed, the set $\mathcal{U}=\cup _{i\in I}Sp(f_{i})$ is countable, since $I$
and $Sp(f_{i})$ (for each fixed $i\in I$) are countable. Therefore, the set
$\mathcal{R}$ of finite combinations $\sum_{\text{finite}}t_{i}k_{i}$\ \
$(t_{i}\in \mathbb{Z},$\ $k_{i}\in \mathcal{U})$ is countable. But $\mathcal{R}$
is a subgroup of $\mathbb{R}^{d}$ and $Sp(f_{i})\subset \mathcal{R}$ for all
$i\in I$. Hence the proposition follows.
\end{proof}

As a consequence of this, we have the following result.

\begin{corollary} \label{corol 4.1}
Let $X$ be a separable metric space and let $(\varphi
_{i})_{i\in I}\subset \mathcal{C}(X;L_{AP}^{p}(\mathbb{R}^{d}))$, where the
index set $I$ is countable. Then, there is some countable subgroup $\mathcal{
R}$ of $\mathbb{R}^{d}$ such that $\varphi _{i}\in \mathcal{C}(X;L_{AP,\mathcal{
R}}^{p}(\mathbb{R}^{d}))$ for every $i\in I$.
\end{corollary}

\begin{proof}
Let $D$ be a dense countable set in $X$. Thanks to Proposition \ref{prop 4.1},
there is a countable subgroup $\mathcal{R}$ of $\mathbb{R}^{d}$ such that
$\varphi _{i}(\zeta )\in L_{AP,\mathcal{R}}^{p}(\mathbb{R}^{d})$ for all $i\in I$
and all $\zeta \in D$. Now let $i\in I$ be fixed. Fix also some $x\in X$.
Let $\eta >0$. By the continuity of $\varphi $ and the density of $D$ in $X$,
we may consider some $\zeta \in D$ such that
$\left\| \varphi_{i}(x)-\varphi _{i}(\zeta )\right\| _{p,\infty }
\leq \frac{\eta }{c}$ where $c$ is a positive constant such that
$\left| M(u)\right| \leq c\left\|u\right\| _{p,\infty }$
($u\in L_{AP}^{p}(\mathbb{R}^{d})$). It follows that
$\left| M(\varphi _{i}(x)\overline{\gamma }_{k})-M(\varphi _{i}(\zeta )
\overline{\gamma }_{k})\right| \leq \eta $ for all $k\in \mathbb{R}^{d}$.
 But $M(\varphi _{i}(\zeta )\overline{\gamma }_{k})=0$ for all
$k\in \mathbb{R}^{d}\backslash \mathcal{R}$. By the arbitrariness of $\eta $
we deduce that $M(\varphi _{i}(x)\overline{\gamma }_{k})=0$ for all
$k\in \mathbb{R}^{d}\backslash \mathcal{R}$. Hence
$\varphi _{i}(x)\in L_{AP,\mathcal{R}}^{p}(\mathbb{R}^{d})$.
This completes the proof.
\end{proof}

We are now in a position to study the almost periodic homogenization of
(\ref{1.4}).

\begin{example} \label{exple 4.1} \rm
Our goal here is to investigate the limiting behaviour, as
$\varepsilon \to 0$, of $u_{\varepsilon }$, the solution of (\ref{1.4}) for
$p=2$, under the structure hypothesis
\begin{equation}
a_{i}(\cdot ,\cdot ,\lambda )\in L_{AP}^{2}(\mathbb{R}^{N+1})\quad
\text{ for fixed }\lambda \in \mathbb{R}^{N}\quad (1\leq i\leq N).
\label{4.2}
\end{equation}
\end{example}

According to Theorem \ref{th 3.1}, this homogenization problem is quite
solvable and the results are available in Subsection 3.4 if we can find a
suitable quasi-proper $H$-structure
$\Sigma =\Sigma _{y}\times \Sigma _{\tau}$ for $p=2$ such that (\ref{3.1})
holds for $p=2$.
To achieve this, we shall require the following property:
For $\Psi \in AP(\mathbb{R}^{N+1};\mathbb{R})^{N}$, we have
\begin{equation}
\sup_{k\in \mathbb{Z}^{N+1}} \int_{k+Z}\left| a(y-r,\tau -\sigma
,\Psi (y,\tau ))-a(y,\tau ,\Psi (y,\tau ))\right| ^{2}dyd\tau \to 0
\label{4.3}
\end{equation}
as $\left| r\right| \to 0$ and $\sigma \to 0$, where $Z=(0,1)^{N+1}$.


\begin{remark} \label{rm 4.1} \rm
Condition (\ref{4.3}) is satisfied if the following condition holds:
For each bounded set $\Lambda \subset \mathbb{R}^{N}$
and each real $\eta >0$, there exists a real $\rho >0$ such that
\begin{equation}
\left| a(y-r,\tau -\sigma ,\lambda )-a(y,\tau ,\lambda )\right| \leq
\eta \label{4.4}
\end{equation}
for all $\lambda \in \Lambda$ and for almost all $(y,\tau )\in \mathbb{R}^{N+1}$
 provided $\left| r\right| +\left| \sigma \right| \leq \rho$.

Indeed, if (\ref{4.4}) holds and if $\Psi $ is given in
$AP(\mathbb{R}^{N+1};\mathbb{R})^{N}$, then by choosing
$\Lambda =\Psi (\mathbb{R}^{N+1})$ (range of $\Psi $) we get at once (\ref{4.3}).
\end{remark}

This being so, let $(\theta _{n})_{n\geq 1}$ be a sequence with $\theta
_{n}\in \mathcal{D}_{\mathbb{R}}(\mathbb{R}^{N+1}),\theta _{n}\geq
0$, $\mathop{\rm Supp}\theta _{n}\subset \frac{1}{n}\overline{B}_{N+1}$
($B_{N+1}$ the open unit ball of $\mathbb{R}^{N+1}$,
$\overline{B}_{N+1}$ its closure) and
$\int \theta _{n}(y,\tau )dyd\tau =1$. Let
\begin{equation*}
\zeta _{n}^{i}(y,\tau ,\lambda )=\int \theta _{n}(r,\sigma )a_{i}(y-r,\tau
-\sigma ,\lambda )drd\sigma \quad (1\leq i\leq N)
\end{equation*}
for $\lambda ,y\in \mathbb{R}^{N}$ and $\tau \in \mathbb{R}$, which defines a
function $(y,\tau ,\lambda )\to \zeta _{n}^{i}(y,\tau ,\lambda )$ of
$\mathbb{R}^{N}\times \mathbb{R}\times \mathbb{R}^{N}$ into $\mathbb{R}$.
 Clearly $\zeta _{n}^{i}(\cdot ,\cdot ,\lambda )\in AP(\mathbb{R}^{N+1})$ for each
$\lambda \in \mathbb{R}^{N}$, and further $\left| \mathbf{\zeta }_{n}(y,\tau
,\lambda )-\mathbf{\zeta }_{n}(y,\tau ,\mu )\right| \leq \alpha _{1}\left|
\lambda -\mu \right| $ for all $\lambda ,\mu ,y\in \mathbb{R}^{N}$ and all
$\tau \in \mathbb{R}$, where $\mathbf{\zeta }_{n}=(\zeta _{n}^{i})_{1\leq i\leq N}$.
Now, thanks to Corollary \ref{corol 4.1}, there exists a countable subgroup
$R$ of $\mathbb{R}^{N+1}$ such that $\zeta _{n}^{i}(\cdot ,\cdot ,\lambda )\in
AP_{R}(\mathbb{R}^{N+1})\;(1\leq i\leq N)$ for all $\lambda \in \mathbb{R}^{N}$
and all integers $n\geq 1$. Let $\mathcal{R}_{y}=pr_{y}(R)$ and $\mathcal{R}
_{\tau }=pr_{\tau }(R)$, where $pr_{y}$ (resp. $pr_{\tau }$) stands for the
natural projection of $\mathbb{R}^{N+1}=\mathbb{R}_{y}^{N}\times \mathbb{R}_{\tau }$
onto $\mathbb{R}_{y}^{N}$ (resp. $\mathbb{R}_{\tau }$). The set $\mathcal{R}_{y}$
(resp. $\mathcal{R}_{\tau }$) is a countable subgroup of $\mathbb{R}^{N}$
(resp. $\mathbb{R}$). Therefore $\mathcal{R=R}_{y}\times \mathcal{R}_{\tau }$
is a countable subgroup of $\mathbb{R}^{N+1}$ with moreover
$R\subset \mathcal{R}$. Hence
\begin{equation}
\zeta _{n}^{i}(\cdot ,\cdot ,\lambda )\in A=AP_{\mathcal{R}}(\mathbb{R}
^{N+1})\quad (\lambda \in \mathbb{R}^{N},\; n\in \mathbb{N}^{*},\;1\leq i\leq
N)  \label{4.5}
\end{equation}
and
\begin{equation}
\Sigma _{\mathcal{R}}=\Sigma _{\mathcal{R}_{y}}\times \Sigma _{\mathcal{R}
_{\tau }}\quad \text{(see \cite[Example 3.6]{18})}  \label{4.6}
\end{equation}
where $\Sigma _{\mathcal{R}}$ (resp. $\Sigma _{\mathcal{R}_{y}},\Sigma _{
\mathcal{R}_{\tau }}$) is the almost periodic $H$-structure on $\mathbb{R}
^{N+1} $ (resp. $\mathbb{R}^{N},\;\mathbb{R}$) represented by $\mathcal{R}$ (resp.
$\mathcal{R}_{y},\;\mathcal{R}_{\tau }$). Recalling that $\Sigma _{\mathcal{R
}}$ is quasi-proper for $p=2$ (see Example \ref{exple 3.1}), we see that the
problem under consideration is completely solved if we show that (\ref{3.1})
holds with $\Sigma =\Sigma _{\mathcal{R}}$ and $p=2$. To this end, starting
from (\ref{4.5}) and following the same line of reasoning as in
\cite[Subsection 5.5]{22} leads to $\zeta _{n}^{i}(\cdot ,\cdot ,\Psi )\in A$
for all $\Psi \in (A_{\mathbb{R}})^{N}\;(n\in \mathbb{N}^{*},\;1\leq i\leq N)$. On
the other hand, by an obvious adaptation of the procedure in
\cite[Subsection 5.6]{22} one quickly arrives at the following result :
\begin{itemize}
\item[] Given $\Psi \in (A_{\mathbb{R}})^{N}$  and $1\leq i\leq N$, to
each $\eta >0$ there is assigned some integer
$\nu \geq 1$  such that $\left\| \zeta _{n}^{i}(\cdot ,\cdot ,\Psi
)-a_{i}(\cdot ,\cdot ,\Psi )\right\| _{2,\infty }\leq \eta$  for
all $n\geq \nu$.
\end{itemize}
Since $(L^{2},l^{\infty })(\mathbb{R}^{N+1})$ is continuously embedded in $\Xi
^{2}(\mathbb{R}^{N+1})$ (this follows immediately by (\ref{4.1})), the desired
result follows from all that.

\begin{remark} \label{rm 4.2}\rm
If instead of (\ref{4.2}) we consider the structure
hypothesis:
\begin{equation*}
a_{i}(\cdot ,\cdot ,\lambda )\in AP(\mathbb{R}^{N+1})\quad \text{for fixed }
\lambda \in \mathbb{R}^{N}\quad (1\leq i\leq N),
\end{equation*}
then (\ref{4.3}) may be disregarded. Indeed, proceeding directly as in
\cite[Subsection 5.5]{22} we arrive at $a_{i}(\cdot ,\cdot ,\Psi )\in A$
 for all $\Psi \in (A_{\mathbb{R}})^{N}\;\;(1\leq i\leq N)$,
which leads at once to (\ref{3.1}) with $\Sigma _{\mathcal{R}}$ as
in (\ref{4.6}), and with $p=2$, of course.
\end{remark}

\subsubsection*{Problem III}
The present problem deals with two closely connected examples.

\begin{example} \label{exple 4.2} \rm
We assume here that the family
$\{a(\cdot ,\cdot ,\lambda )\}_{\lambda \in \mathbb{R}^{N}}$ satisfies the
condition
\begin{itemize}
\item[(BUE)] For each bounded set $\Lambda \subset \mathbb{R}^{N}$
 and each real $\eta >0$, there exists a real $\rho >0$ such that
$\left|a(y-r,\tau -\sigma ,\lambda )-a(y,\tau ,\lambda )\right| \leq \eta$
for all $\lambda \in \Lambda$  and all
$(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}$  provided
$\left|r\right| +\left| \sigma \right| \leq \rho$.
\end{itemize}
\end{example}

\begin{remark} \label{rm 4.3} \rm
Condition (BUE) is more practical than its analog (UE) in \cite[Subsection 5.4]{22}.
In fact, \cite[Proposition 5.2]{22} and its proof remain unchanged if the sole
points $\lambda $ considered in (UE) are those lying in an arbitrarily fixed
bounded set $\Lambda \subset \mathbb{R}^{N}$. This remark carries over
\textit{mutatis mutandis} to \cite[Subsection 5.7]{22}.
\end{remark}

Assuming (BUE), we want to study the homogenization of (\ref{1.4}) (for any
given real $p\geq 2$) under the structure hypothesis
\begin{equation}
a_{i}(\cdot ,\cdot ,\lambda )\in \mathcal{B}_{\infty }(\mathbb{R};\mathcal{C}
_{\rm per}(Y))\text{ for any }\lambda \in \mathbb{R}^{N}\quad (1\leq i\leq N)
\label{4.7}
\end{equation}
where $Y=(0,1)^{N}$. We recall that $\mathcal{C}_{\rm per}(Y)$ denotes the space
of continuous complex functions on $\mathbb{R}^{N}$ that are $Y$-periodic
(i.e., that satisfy \ $f(y+k)=f(y)$ \ for all $y\in \mathbb{R}^{N}$and all
$k\in \mathbb{Z}^{N}$),\ and $\mathcal{B}_{\infty }(\mathbb{R};\mathcal{C}
_{\rm per}(Y)) $ denotes the space of those $f\in \mathcal{C}(\mathbb{R};\mathcal{C}
_{\rm per}(Y))$ such that $f(\tau )$ has a limit in $\mathcal{B}(\mathbb{R}^{N})$
when $\left| \tau \right| \to \infty $.
Now, let $\Sigma _{\mathbb{Z}^{N}}$ be the periodic $H$-structure on $\mathbb{R}
^{N}$ represented by the network $\mathbb{Z}^{N}$, and $\Sigma _{\mathbb{\infty }}$
be the $H$-structure on $\mathbb{R}$\ of which $\mathcal{B}_{\infty }(\mathbb{R})$
is the image (see \cite[Example 3.4]{18}). The product $H$-structure $\Sigma
=\Sigma _{\mathbb{Z}^{N}}\times \Sigma _{\mathbb{\infty }}$ on $\mathbb{R}^{N}\times
\mathbb{R}$ is quasi-proper for any $1<p<\infty $ (see Example \ref{exple 3.2})
and its image is precisely $A=\mathcal{B}_{\infty }(\mathbb{R};\mathcal{C}
_{\rm per}(Y)$) (see \cite[Proposition 3.3]{18}). Thus, the present study falls
under the framework of Section 3 provided it is shown that (\ref{3.1}) holds
true for the above $H$-structure. Clearly it suffices to check that
$a_{i}(\cdot ,\cdot ,\Psi )\in A$ for all $\Psi \in (A_{\mathbb{R}
})^{N}\;\;(1\leq i\leq N)$. But this follows by proceeding exactly as in the
proof of \cite[Proposition 5.2]{22}.

\begin{example} \label{exple 4.3}\rm
Assuming here that (\ref{4.4}) holds true, let us
consider the homogenization problem for (\ref{1.4})
(for any real $p\geq 2$) under the structure hypothesis
\begin{equation}
a_{i}(\cdot ,\cdot ,\lambda )\in \mathcal{B}_{\infty }(\mathbb{R}
;L_{\rm per}^{\infty }(Y))\text{\emph{\ for any} }\lambda \in \mathbb{R}
^{N}\quad (1\leq i\leq N).  \label{4.8}
\end{equation}
\end{example}

By a simple adaptation of \cite[Subsection 5.7]{22} one is led to (\ref{3.1}
) with $\Sigma =\Sigma _{\mathbb{Z}^{N}}\times \Sigma _{\infty }$ as above.
Hence, thanks to Theorem \ref{th 3.1}, the same conclusion as above follows.

\subsection{The linear case}

In this subsection we assume that the function $\lambda \to a(\cdot
,\cdot ,\lambda )$ of $\mathbb{R}^{N}$ into itself is linear. Then, there is a
family $\{a_{ij}\}_{1\leq i,j\leq N},\;a_{ij}\in L_{\mathbb{R}}^{\infty }(\mathbb{R
}^{N}\times \mathbb{R})$ (thanks to (\ref{1.1})-(\ref{1.2}) and part $(ii)$ of (
\ref{1.3})), such that
\begin{equation*}
a_{i}(y,\tau ,\lambda )=\sum_{j=1}^{N}a_{ij}(y,\tau )\lambda _{j}\text{ \
for all }\lambda \in \mathbb{R}^{N}\quad (1\leq i\leq N).
\end{equation*}
In the sequel we suppose $p=2$. Now, it is clear that the results obtained
in Subsection 4.1 remain valid in the present case. In addition, by turning
the linearity to good account we can get round technical difficulties and
thus, with the help of further concrete examples, point out the wide scope
of Theorem \ref{th 3.1}.

This being so, it is immediate that (\ref{3.1}) holds true if and only if
\begin{equation}
a_{ij}\in \mathfrak{X}_{\Sigma }^{2}\quad (1\leq i,j\leq N).  \label{4.9}
\end{equation}
Thus, in the sequel, the aim will be to reduce to (\ref{4.9}) the concrete
examples under consideration.

\subsubsection*{Problem IV}
Our purpose here is to study the homogenization of (\ref{1.4}) under the
structure hypothesis
\begin{equation}
a_{ij}\in F_{\infty ,AP}\quad (1\leq i,j\leq N)  \label{4.10}
\end{equation}
where $F_{\infty ,AP}$ denotes the closure of
$\mathcal{B}_{\infty }(\mathbb{R};AP(\mathbb{R}^{N}))$ in
$(L^{2},l^{\infty })(\mathbb{R}^{N+1})$. To this end, let
us consider
\begin{equation*}
\zeta _{nij}\in \mathcal{B}_{\infty }(\mathbb{R};AP(\mathbb{R}^{N}))\quad
(n\in \mathbb{N},\;1\leq i,j\leq N)
\end{equation*}
such that as $n\to \infty ,\;\zeta _{nij}\to a_{ij}$ in
$(L^{2},l^{\infty })(\mathbb{R}^{N+1})$ for $1\leq i,j\leq N$. According to
Corollary \ref{corol 4.1}, there exists a countable subgroup $\mathcal{R}$
of $\mathbb{R}^{N}$ such that $\zeta _{nij}\in \mathcal{B}_{\infty }(\mathbb{R}
;AP_{\mathcal{R}}(\mathbb{R}^{N}))$ for all integers $n\in \mathbb{N}$ and all
indices $1\leq i,j\leq N$. Let $\Sigma =\Sigma _{\mathcal{R}}\times \Sigma _{
\mathbb{\infty }}$, where $\Sigma _{\mathcal{R}}$ is the almost periodic $H$
-structure on $\mathbb{R}^{N}$ represented by $\mathcal{R}$ and $\Sigma _{\mathbb{
\infty }}$ is the $H$-structure on $\mathbb{R}$ introduced in Example
\ref{exple 4.2}. The $H$-structure $\Sigma $ on $\mathbb{R}^{N}\times \mathbb{R}\;$is
quasi-proper for $p=2$ (Example \ref{exple 3.1}) and its image is precisely
the $H$-algebra $A=\mathcal{B}_{\infty }(\mathbb{R};AP_{\mathcal{R}}(\mathbb{R}
^{N}))\;$\cite[Proposition 3.3]{18}. Therefore, the homogenization problem
before us is solved through Theorem \ref{th 3.1} if we can check that (\ref
{4.9}) holds for the preceding $H$-structure. But this follows immediately
by the fact ( already pointed out before) that $(L^{2},l^{\infty })(\mathbb{R}
^{N+1})$ is continuously embedded in $\Xi ^{2}(\mathbb{R}^{N+1})$.
Let us illustrate this.

\begin{example} \label{exple 4.4}\rm
The structure hypothesis (generalizing (\ref{4.7}))
$a_{ij}\in \mathcal{B}_{\infty }(\mathbb{R};AP(\mathbb{R}^{N}))$ $(n\in \mathbb{N}$,
$1\leq i,j\leq N)$ reduces to (\ref{4.10}). The same is true of the
structure hypothesis (generalizing (\ref{4.8}))
$a_{ij}\in \mathcal{B}_{\infty }(\mathbb{R};L_{AP}^{2}(\mathbb{R}^{N}))$
 $(1\leq i,j\leq N)$.
Indeed,the first assertion is evident. Regarding the next one, observe that
$\mathcal{B}_{\infty }(\mathbb{R};AP(\mathbb{R}^{N}))$ is dense in
$\mathcal{B}_{\infty }(\mathbb{R};L_{AP}^{2}(\mathbb{R}^{N}))$ provided with
the $\mathcal{B}(\mathbb{R};(L^{2},l^{\infty })(\mathbb{R}^{N}))$-norm, and
the latter is continuously embedded in $(L^{2},l^{\infty })(\mathbb{R}^{N+1})$.
\end{example}

\begin{example} \label{exple 4.5}\rm
The structure hypothesis $a_{ij}\in L^{2}(\mathbb{R};L_{AP}^{2}(\mathbb{R}^{N}))$
($1\leq i,j\leq N$) reduces to (\ref{4.10}). Indeed,
$\mathcal{K}(\mathbb{R};AP(\mathbb{R}^{N}))$
(a subspace of $\mathcal{B}_{\infty }(\mathbb{R};AP(\mathbb{R}^{N}))$) is dense in
$L^{2}(\mathbb{R};L_{AP}^{2}(\mathbb{R}^{N}))$ and the latter is
continuously embedded in $(L^{2},l^{\infty })(\mathbb{R}^{N+1})$.
\end{example}

\begin{example} \label{exple 4.6} \rm
Suppose our goal is to study the homogenization of (\ref{1.4}) under the
following structure hypothesis, where the two indices
$1\leq i,j\leq N$ are arbitrarily fixed:
\begin{itemize}
\item[(1)] The function $\tau \to a_{ij}(\cdot ,\tau )$
maps continuously $\mathbb{R}$ into $(L^{2},l^{\infty })(\mathbb{R}^{N})$

\item[(2)] As $\left| \tau \right| \to \infty $,
$a_{ij}(\cdot ,\tau )$ has a limit in $(L^{2},l^{\infty })(\mathbb{R}^{N})$

\item[(3)] For each fixed $\tau \in \mathbb{R}$, the function
$a_{ij}(\cdot ,\tau )$ is $Y_{\tau }$-periodic, where
$Y_{\tau }=(0,c_{\tau })^{N}$ with $c_{\tau }>0$.
\end{itemize}
Then this leads us to Problem IV. Indeed, it is not hard to check that
the preceding structure hypothesis implies that $a_{ij}$
belongs to $\mathcal{B}_{\infty }(\mathbb{R};L_{AP}^{2}(\mathbb{R}^{N}))$
 (Example \ref{exple 4.4}).
\end{example}

Our last problem states as follows.

\subsubsection*{Problem V}
Let $A_{\tau }$ be an $H$-algebra on $\mathbb{R}$ with the property that
$A_{\tau }^{\infty }$ is dense in $A_{\tau }$. The matter in hand here is to
study the homogenization of (\ref{1.4}) under the hypothesis that
\begin{equation}
a_{ij}\text{ lies in the closure of }AP(\mathbb{R}^{N})\otimes A_{\tau }\text{
in }(L^{2},l^{\infty })(\mathbb{R}^{N+1})\;(1\leq i,j\leq N).  \label{4.11}
\end{equation}
To begin with, let $\zeta _{nij}\in AP(\mathbb{R}^{N})\otimes A_{\tau
}\;\;(n\in \mathbb{N},\;1\leq i,j\leq N)$ be such that $\zeta _{nij}\to
a_{ij}$ in $(L^{2},l^{\infty })(\mathbb{R}^{N+1})\;\;(1\leq i,j\leq N)$ as
$n\to \infty $. By Proposition \ref{prop 4.1} one is easily led to
some countable subgroup $\mathcal{R}$ of $\mathbb{R}^{N}$ such that $\zeta
_{nij}\in AP_{\mathcal{R}}(\mathbb{R}^{N})\otimes A_{\tau }$ for all $n\in \mathbb{
N}$ and all indices $1\leq i,j\leq N$. Let $\Sigma =\Sigma _{\mathcal{R}
}\times \Sigma _{\mathbb{\tau }}$, where $\Sigma _{\mathcal{R}}$ is as in
Problem IV and $\Sigma _{\mathbb{\tau }}$ is the $H$-structure of class
$\mathcal{C}^{\infty }$ on $\mathbb{R}$ of which $A_{\tau }$ is the image. The
$H $-structure $\Sigma $ on $\mathbb{R}^{N}\times \mathbb{R}$ is quasi-proper for
$p=2 $ and its image is the closure, $A$, of $AP_{\mathcal{R}}(\mathbb{R}
^{N})\otimes A_{\tau }$ in $\mathcal{B}(\mathbb{R}^{N}\times \mathbb{R)}$ (see
\cite[Proposition 3.2]{18}). Thus, we will be through if we have shown that (
\ref{4.9}) holds. But this is a direct consequence of the fact that
$(L^{2},l^{\infty })(\mathbb{R}^{N+1})$ is continuously embedded in $\Xi ^{2}(
\mathbb{R}^{N+1})$. Therefore, the homogenization problem under consideration
lies within the scope of Theorem \ref{th 3.1} and so we are led to the
results of Subsection 3.4.

\begin{remark} \label{rm 4.4} \rm
According to (\ref{4.11}), the function $(y,\tau)\to a_{ij}(y,\tau )$
is almost periodic in $y\in \mathbb{R}^{N} $ whereas in the variable
$\tau \in \mathbb{R}$ it admits a great variety of behaviours.
This is illustrated below.
\end{remark}

\begin{example} \label{exple 4.7} \rm
Property (\ref{4.11}) includes (\ref{4.10}) as a
particular case. Indeed, this follows by choosing
$A_{\tau }=\mathcal{B}_{\infty }(\mathbb{R})$ in (\ref{4.11}) and observing
that $AP(\mathbb{R}^{N})\otimes \mathcal{B}_{\infty }(\mathbb{R})$ is a dense
subspace of $\mathcal{B}_{\infty }(\mathbb{R};AP(\mathbb{R}^{N}))$.
\end{example}

\begin{example} \label{exple 4.8} \rm
Our purpose in the present example is to study the homogenization of
(\ref{1.4}) under the following assumptions, where the pair of indices
$1\leq i,j\leq N$ is arbitrarily fixed:
\begin{itemize}
\item[(SH1)]  $a_{ij}(\cdot ,\tau )\in L_{AP}^{2}(\mathbb{R}^{N})$
a.e. in $\tau \in \mathbb{R}$

\item[(SH2)]  The function $\tau \to a_{ij}(\cdot ,\tau) $ from
$\mathbb{R}$ to $L_{AP}^{2}(\mathbb{R}^{N})$ is piecewise
constant in the sense that there exists a mapping
$q_{ij}:\mathbb{Z}\to L_{AP}^{2}(\mathbb{R}^{N})$ such that
\begin{equation}
a_{ij}(\cdot ,\tau )=q_{ij}(k)\quad \text{a.e. in } k\leq \tau
<k+1\;\; (k\in \mathbb{Z}).  \label{4.12}
\end{equation}
\end{itemize}
However, in order to have a ``well posed'' homogenization problem,
we need to be
informed about the behaviour of the coefficient $q_{ij}$. We assume here
that
\begin{itemize}
\item[(SH3)]
$q_{ij}\in \mathcal{B}_{\infty }(\mathbb{Z};L_{AP}^{2}(\mathbb{R}^{N})$
where $\mathcal{B}_{\infty }(\mathbb{Z};L_{AP}^{2}(\mathbb{R}^{N}))$ denotes the
space of mappings $q:\mathbb{Z}\to L_{AP}^{2}(\mathbb{R}^{N})$ that
converge at infinity, i.e., such that $q(k)$ has a limit in $L_{AP}^{2}(\mathbb{
R}^{N})$ when $\left| k\right| \to \infty $.
\end{itemize}
It is well to note in passing that such a $q$ is necessarily bounded.

Let us show that the structure hypothesis made up of (SH1)--(SH3)
reduces to (\ref{4.11}), so that the problem under consideration is quite
solvable.
\end{example}

\begin{proposition} \label{prop 4.2}
Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}
\to \mathbb{C}$ of the form
\begin{equation}
f=\sum_{k\in \mathbb{Z}}r(k)\tau _{k}\varphi \quad
(r\in \mathcal{B}_{\infty }(\mathbb{Z}),\;\varphi \in \mathcal{K}(Z))  \label{4.13}
\end{equation}
with $Z=(0,1)$, where $\mathcal{K}(Z)$ is identified with the space of
functions in $\mathcal{K}(\mathbb{R})$ with supports contained in $Z$. Let
$A_{\tau }$ be the closure in $\mathcal{B}(\mathbb{R})$ of the space of
functions of the form
\begin{equation*}
\psi =c+\sum_{finite}f_{i}\quad (c\in \mathbb{C},\;f_{i}\in \mathcal{F}).
\end{equation*}
Then the following assertions are true.
\begin{itemize}
\item[(i)] $A_{\tau }$  is an $H$-algebra on $\mathbb{R}$ with the
further property that $A_{\tau }^{\infty }$ is dense in $A_{\tau }$.
\item[(ii)] The family $\{a_{ij}\}_{1\leq i,j\leq N}$  satisfies
(\ref{4.11}).
\end{itemize}
\end{proposition}

\begin{proof}
Part (i) is proved in \cite{ngu1}. Thus, we need only show (ii). Let a
pair of indices $1\leq i,j\leq N$ be freely fixed. Based on the density of
$AP(\mathbb{R}^{N})\otimes \mathcal{B}_{\infty }(\mathbb{Z})\;$in $\mathcal{B}
_{\infty }(\mathbb{Z};AP(\mathbb{R}^{N}))$ (this follows by
\cite[page 46, Proposition 5]{6}) and recalling that $AP(\mathbb{R}^{N})$ is a
dense subspace of $L_{AP}^{2}(\mathbb{R}^{N})$, we see immediately that $AP(
\mathbb{R}^{N})\otimes \mathcal{B}_{\infty }(\mathbb{Z})$ is dense in $\mathcal{B}
_{\infty }(\mathbb{Z};L_{AP}^{2}(\mathbb{R}^{N}))$. Hence, in view of (SH3), we may consider a sequence $(q_{nij})_{n\in \mathbb{N}}$ in $AP(\mathbb{R}
^{N})\otimes \mathcal{B}_{\infty }(\mathbb{Z})$ such that $q_{nij}\to
q_{ij}$ in $\mathcal{B}_{\infty }(\mathbb{Z};L_{AP}^{2}(\mathbb{R}^{N}))$ as
$n\to \infty $. It is useful to specify that $q_{nij}$ writes as
\begin{equation}
q_{nij}(y,k)=\sum_{l\in I}\zeta _{nij}^{l}(k)u_{nij}^{l}(y)\quad
(y\in \mathbb{R}^{N},\;k\in \mathbb{Z)}  \label{4.14}
\end{equation}
where $\zeta _{nij}^{l}\in \mathcal{B}_{\infty }(\mathbb{Z}),\;u_{nij}^{l}\in
AP(\mathbb{R}^{N})$, and $I$ is a finite set (depending on $q_{nij}$). On the
other hand, it is worth bearing in mind that (\ref{4.12}) is equivalent to
\begin{equation*}
a_{ij}(y,\tau )=\sum_{k\in \mathbb{Z}}q_{ij}(y,k)\chi _{k+Z}(\tau )\quad
\text{ a.e. in }(y,\tau )\in \mathbb{R}^{N}\times \mathbb{R}
\end{equation*}
where $\chi _{k+Z}$ denotes the characteristic function of $k+Z=(k,k+1)$ in
$\mathbb{R}$, and where the sum on the right is locally finite.
Now, let
\begin{equation}
a_{nij}(y,\tau )=\sum_{k\in \mathbb{Z}}q_{nij}(y,k)\chi _{k+Z}(\tau )\quad
(y\in \mathbb{R}^{N},\;\text{a.e. in }\tau \in \mathbb{R}).  \label{4.15}
\end{equation}
It is clear that $a_{nij}$ lies in $L^{\infty }(\mathbb{R};AP(\mathbb{R}
^{N}))\subset (L^{2},l^{\infty })(\mathbb{R}^{N+1})$ and further
\begin{equation*}
\left\| a_{nij}-a_{ij}\right\| _{2,\infty }\leq
\sup_{k\in \mathbb{Z}}\left\| q_{nij}(k)-q_{ij}(k)\right\| _{2,\infty }.
\end{equation*}
Hence
$a_{nij}\to a_{ij}$ in $(L^{2},l^{\infty })(\mathbb{R}^{N+1})$
 as $n\to \infty$.

On the other hand, substituting (\ref{4.14}) in (\ref{4.15}) yields
\begin{equation*}
a_{nij}(y,\tau )=\sum_{l\in I}u_{nij}^{l}(y)f_{nij}^{l}(\tau
)\quad (y\in \mathbb{R}^{N},\text{ a.e. in }\tau \in \mathbb{R})
\end{equation*}
where $f_{nij}^{l}\in L^{\infty }(\mathbb{R})$ with
\begin{equation*}
f_{nij}^{l}(\tau )=\sum_{k\in \mathbb{Z}}\zeta _{nij}^{l}(k)\chi _{k+Z}(\tau
)\quad (\text{a.e. in }\tau \in \mathbb{R}).
\end{equation*}
But if $\eta >0$ is arbitrarily given and if $\varphi \in \mathcal{K}(Z)$ is
such that $\left\| \chi _{Z}-\varphi \right\| _{L^{2}(\mathbb{R})}=\left\|
1-\varphi \right\| _{L^{2}(Z)}\leq \frac{\eta }{c}$, where $c>0$ with
$\left| \zeta _{nij}^{l}(k)\right| \leq c\;(k\in \mathbb{Z})$, then $\left\|
f_{nij}^{l}-\psi _{nij}^{l}\right\| _{2,\infty }\leq \eta $ \thinspace
\thinspace \thinspace with $\psi _{nij}^{l}=\underset{k\in \mathbb{Z}}{\sum }
\zeta _{nij}^{l}(k)\tau _{k}\varphi $ (see (\ref{4.13})).

Finally, let
\begin{equation*}
\Phi _{nij}(y,\tau )=\sum_{l\in I}u_{nij}^{l}(y)\psi _{nij}^{l}(\tau
)\quad (y\in \mathbb{R}^{N},\;\tau \in \mathbb{R}),
\end{equation*}
which defines a function in $AP(\mathbb{R}^{N})\otimes A_{\tau }$. It is an
elementary exercise to deduce from the preceding development that for any
$\eta >0$, there is some integer $n\in \mathbb{N}$ such that
$\left\|a_{ij}-\Phi _{nij}\right\| _{2,\infty }\leq \eta $.
This completes the proof.
\end{proof}

\begin{example} \label{exple 4.9} \rm
The case to be examined here states as in Example \ref{exple 4.8}
except that in (SH3),
$\mathcal{B}_{\infty }(\mathbb{Z};L_{AP}^{2}(\mathbb{R}^{N}))$
is substituted by the space
$\ell ^{1}(\mathbb{Z};L_{AP}^{2}(\mathbb{R}^{N}))$ of mappings
$q:\mathbb{Z}\to L_{AP}^{2}(\mathbb{R}^{N})$ such that
$\sum_{k\in \mathbb{Z}}\left\| q(k)\right\| _{2,\infty}<\infty $.
Without going too deeply into details let us verify that
the present case leads to the same conclusion as in the preceding example.
First, let $\ell _{0}^{1}(\mathbb{Z})$ denote the closure in
$\ell^{\infty }(\mathbb{Z})$ of the set of functions
$r\in \ell ^{\infty }(\mathbb{Z})$ of the form $r=c+r_{0}$ with
$c\in \mathbb{C}$ and $r_{0}\in \ell ^{1}(\mathbb{Z})$.
We claim that the statement of
Proposition \ref{prop 4.2} is still valid when
$\mathcal{B}_{\infty }(\mathbb{Z})$, in (\ref{4.13}), is replaced by
$\ell _{0}^{1}(\mathbb{Z})$.
Indeed, there is no real difficulty in verifying that the proof of the said
proposition holds when the symbol $\mathcal{B}_{\infty }$ is
replaced by $\ell ^{1}$ (not $\ell _{0}^{1}!$). The details
are left to the reader.
\end{example}

\begin{remark} \rm
The coefficient $q_{ij}$ in Example \ref{exple 4.8} is
$q_{ij}(k)=\int_{k}^{k+1}a_{ij}(\cdot ,\tau )d\tau $
 $(k\in \mathbb{Z})$.
\end{remark}

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\end{document}