\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 40, pp. 1--22.\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-2004/40\hfil Homogenization in chemical reactive flows] {Homogenization in chemical reactive flows} \author[C. Conca, J. I. D\'{\i}az, A. Li\~{n}\'{a}n, C. Timofte \hfil EJDE-2004/40\hfilneg] {Carlos Conca, Jesus Ildefonso D\'{\i}az,\\ Amable Li\~{n}\'{a}n, Claudia Timofte} % in alphabetical order \address{Carlos Conca \hfill\break Departamento de Ingenier\'{\i}a Matem\'{a}tica \\ and Centro de Modelamiento Matem\'{a}tico, UMR 2071 CNRS-U Chile \\ Facultad de Ciencias F\'{\i}sicas y Matem\'{a}ticas \\ Universidad de Chile\\ Casilla 170/3, Santiago, Chile} \email{cconca@dim.uchile.cl} \address{Jesus Ildefonso D\'{\i}az \hfill\break Departamento de Matem\'{a}tica Aplicada \\ Facultad de Matem\'{a}ticas \\ Universidad Complutense \\ 28040 Madrid, Spain} \email{ildefonso\_diaz@mat.ucm.es} \address{Amable Li\~{n}\'{a}n \hfill\break Escuela T. S. de Ingenieros Aeron\'{a}uticos \\ Universidad Polit\'{e}cnica de Madrid \\ Madrid, Spain} \email{linan@tupi.dmt.upm.es} \address{Claudia Timofte \hfill\break Department of Mathematics\\ Faculty of Physics \\ University of Bucharest\\ P.O. Box MG-11, Bucharest-Magurele, Romania} \email{claudiatimofte@hotmail.com} \date{} \thanks{Submitted April 3, 2003. Published March 22, 2004.} \subjclass[2000]{47A15, 46A32, 47D20} \keywords{Homogenization, reactive flows, variational inequality, \hfil\break\indent monotone graph} \begin{abstract} This paper concerns the homogenization of two nonlinear models for chemical reactive flows through the exterior of a domain containing periodically distributed reactive solid grains (or reactive obstacles). In the first model, the chemical reactions take place on the walls of the grains, while in the second one the fluid penetrates the grains and the reactions take place therein. The effective behavior of these reactive flows is described by a new elliptic boundary-value problem containing an extra zero-order term which captures the effect of the chemical reactions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} The general question which will be subject of this paper is the homogenization of chemical reactive flows through the exterior of a domain containing periodically distributed reactive solid grains (or reactive obstacles). We will focus our attention on two nonlinear problems which describe the motion of a reactive fluid having different chemical features. For a nice presentation of the chemical aspects involved in our first model (and also for some mathematical and historical backgrounds) we refer to Antontsev et al.~\cite{Antontsev}, Bear \cite{Bear}, D\'{\i}az \cite{Diaz1,Diaz2,Diaz3} and Norman \cite{Norman}. For the second model, the interested reader can consult the books by Hornung \cite{Hornung} and Norman \cite{Norman} and the references therein. Let $\Omega $ be an open bounded set in $\mathbb{R}^n$ and let us introduce a set of periodically distributed reactive obstacles. As a result, we obtain an open set $\Omega ^{\varepsilon }$ which will be referred to as being the \textit{exterior domain}; $\varepsilon $ represents a small parameter related to the characteristic size of the reactive obstacles. The first nonlinear problem studied in this paper concerns the stationary reactive flow of a fluid confined in $\Omega ^{\varepsilon }$, of concentration $u^{\varepsilon}$, reacting on the boundary of the obstacles. A simplified version of this problem can be written as follows: \begin{equation} \begin{gathered} -D_{f}\Delta u^{\varepsilon }=f\quad \text{in }\Omega ^{\varepsilon }, \\ -D_{f} {\frac{\partial u^{\varepsilon }}{\partial \nu }} =a\varepsilon g(u^{\varepsilon })\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=0\quad \text{on }\partial \Omega . \end{gathered} \end{equation} Here, $\nu $ is the exterior unit normal to $\Omega ^{\varepsilon }$, $a>0$, $f\in L^{2}(\Omega )$ and $S^{\varepsilon}$ is the boundary of our exterior medium $\Omega \setminus \overline{\Omega^{\varepsilon}}$. Moreover, the fluid is assumed to be homogeneous and isotropic, with a constant diffusion coefficient $D_{f}>0$. The semilinear boundary condition on $S^{\varepsilon}$ in problem (1.1) describes the chemical reactions which take place locally at the interface between the reactive fluid and the grains. From strictly chemical point of view, this situation represents, equivalently, the effective reaction on the walls of the chemical reactor between the fluid filling $\Omega^{\varepsilon} $ and a chemical reactant located in the rigid solid grains. The function $g$ in (1.1) is assumed to be given. Two model situations will be considered; the case in which $g$ is a monotone smooth function satisfying the condition $g(0)=0$ and the case of a maximal monotone graph with $g(0)=0$, i.e. the case in which $g$ is the subdifferential of a convex lower semicontinuous function $G$. These two general situations are well illustrated by the following important practical examples \begin{itemize} \item[(a)] $g(v)=\frac{\alpha v}{1+\beta v}$, $\alpha, \beta>0$ (Langmuir kinetics) \item[(b)] $g(v)=|v|^{p-1}v$, $00$ in $\Omega^{\varepsilon}$, although $u^{\varepsilon}$ is not uniformly positive, except in the case in which $g$ is a monotone smooth function satisfying the condition $g(0)=0$, as, for instance, in example (a). The existence and uniqueness of a weak solution of (1.1) can be settled by using the classical theory of semilinear monotone problems (see, for instance, \cite{Brezis}, \cite{Diaz1} and \cite{Lions}). As a result, we know that there exists a unique weak solution $u^{\varepsilon}\in V^{\varepsilon}\bigcap H^{2}(\Omega ^{\varepsilon})$, where \[ V^{\varepsilon}=\{v\in H^{1}(\Omega^{\varepsilon}) : v=0\text{ on } \partial \Omega\}. \] Moreover, if in the second model situation, which is in fact the most general case we treat here, with $\Omega ^{\varepsilon}$ we associate the following nonempty convex subset of $V^{\varepsilon}$: \begin{equation} K^{\varepsilon }=\{ v\in V^{\varepsilon }: G(v)\big|_{S^{\varepsilon }} \in L^{1}(S^{\varepsilon })\}, \end{equation} then $u^{\varepsilon}$ is also known to be characterized as being the unique solution of the following variational problem: \begin{quote} Find $u^{\varepsilon }\in K^{\varepsilon }$ such that \begin{equation} D_{f} \int_{\Omega ^{\varepsilon }}Du^{\varepsilon }D(v^{\varepsilon }-u^{\varepsilon })dx- \int_{\Omega ^{\varepsilon }}f(v^{\varepsilon }-u^{\varepsilon })dx+a\langle \mu ^{\varepsilon },G(v^{\varepsilon })-G(u^{\varepsilon })\rangle \geq 0 \end{equation} for all $v^{\varepsilon }\in K^{\varepsilon }$, where $\mu^{\varepsilon }$ is the linear form on $W_{0}^{1,1}(\Omega )$ defined by \[ \langle \mu ^{\varepsilon },\varphi \rangle =\varepsilon \int_{S^{\varepsilon }}\varphi d\sigma \quad \forall \varphi \in W_{0}^{1,1}(\Omega ). \] \end{quote} From a geometrical point of view, we shall just consider periodic structures obtained by removing periodically from $\Omega$, with period $\varepsilon Y$ (where $Y$ is a given hyper-rectangle in $\mathbb{R^n}$), an elementary reactive obstacle $T$ which has been appropriated rescaled and which is strictly included in $Y$, i.e. $\overline{T}\subset Y$. As usual in homogenization, we shall be interested in obtaining a suitable description of the asymptotic behavior, as $\varepsilon$ tends to zero, of the solution $u^{\varepsilon }$ in such domains. We will wonder, for example, whether the solution $u^{\varepsilon }$ converges to a limit $u$ as $\varepsilon \rightarrow 0$. And if this limit exists, can it be characterized? In the second model situation (in absence of any additional regularity on $g$), the solution $u^{\varepsilon}$, properly extended to the whole of $\Omega$, converges to the unique solution of the variational inequality: $u\in H^{1}_{0}(\Omega)$, \begin{equation} \int_{\Omega }QDuD(v-u)dx\geq \int_{\Omega }f(v-u)dx-a\frac{| \partial T|}{| Y\setminus T | } \int_{\Omega}(G(v)-G(u))dx, \end{equation} for all $v \in H^{1}_{0}(\Omega)$. Here, $Q=((q_{ij}))$ is the classical homogenized matrix, whose entries are \begin{equation} q_{ij}=D_{f}\Big( \delta _{ij}+\frac{1}{|Y\setminus T|} \int_{Y\setminus T}\frac{\partial \chi _{j}}{\partial y_{i}}dy\Big) \end{equation} in terms of the functions $\chi _{i}$, $i=1,\dots ,n$, solutions of the so-called cell problems \begin{equation} \begin{gathered} -\Delta \chi _{i}=0 \quad \text{in } Y\setminus T, \\ \frac{\partial (\chi _{i}+y_{i})}{\partial \nu }=0 \quad \text{on }\partial T, \\ \chi _{i}\quad\text{is $Y$-periodic.} \end{gathered} \end{equation} We remark that if $g$ is smooth, then $g$ is the classical derivative of $G$. The chemical situation behind the second nonlinear problem that we will treat in this paper is slightly different from the previous one; it also involves a chemical reactor containing reactive grains, but we assume that now there is an internal reaction inside the grains, instead just on their boundaries. In fact, it is therefore a transmission problem with an unknown flux on the boundary of each grain. To simplify matters, we shall just focus on the case of a function $g$ which is continuous, monotone increasing and such that $g(0)=0$; examples (a) and (b) are both covered by this class of functions $g's$ and, of course, both are still our main practical examples. A simplified setting of this kind of models is as follows: \begin{equation} \begin{gathered} -D_{f}\Delta u^{\varepsilon }=f\quad \text{in }\Omega ^{\varepsilon }, \\ -D_{p}\Delta v^{\varepsilon }+ag(v^{\varepsilon })=0,\quad \text{in }\Omega \setminus \overline{\Omega ^{\varepsilon }} \\ -D_{f}{\frac{\partial u^{\varepsilon }}{\partial \nu }}=D_{p} {\frac{\partial v^{\varepsilon }}{\partial \nu }}\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=v^{\varepsilon }\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=0\quad \text{on }\partial \Omega, \end{gathered} \end{equation} where $D_p$ is a second diffusion coefficient characterizing the granular material filling the reactive obstacles. As in the previous case, the classical semilinear theory guarantees the well-posedness of this problem. When we define $\theta^{\varepsilon}$ as \[ \theta ^{\varepsilon }(x)=\begin{cases} u^{\varepsilon }(x)& x\in \Omega ^{\varepsilon }, \\ v^{\varepsilon }(x)& x\in \Omega \setminus \overline{\Omega^{\varepsilon }}, \end{cases} \] and we introduce \[ A=\begin{cases} D_{f}Id & \text{in }Y\setminus T \\ D_{p}Id & \text{in }T,% \end{cases} \] then our main result of convergence for this model shows that $\theta^{\varepsilon}$ converges weakly in $H^{1}_{0}(\Omega)$ to the unique solution of the homogenized problem \begin{equation} \begin{gathered} -{\sum_{i,j=1}^{n}a_{ij}^{0}}{\frac{\partial ^{2}u}{\partial x_{i} \partial x_{j}}}+a{\frac{|T|}{|Y\setminus T |}}g(u)=f\quad \text{in }\Omega, \\ u=0\quad \text{on }\partial \Omega. \end{gathered} \end{equation} Here, $A^{0}=((a_{ij}^{0}))$ is the homogenized matrix, whose entries are \begin{equation} a_{ij}^{0}=\frac{1}{|Y|}\int_{Y}\big( a_{ij}+a_{ik} \frac{\partial \chi_{j}}{\partial y_{k}}\big)dy, \end{equation} in terms of the functions $\chi _{j}$, $j=1,\dots ,n$, solutions of the so-called cell problems \begin{equation} \begin{gathered} -\mathop{\rm div}(AD(y_{j}+\chi _{j}))=0\quad \text{in }Y, \\ \chi_{j} \quad\mbox{is $Y$-periodic}. \end{gathered} \end{equation} Note that the two reactive flows studied in this paper, namely (1.1) and (1.7), lead to completely different effective behavior. The macroscopic problem (1.4) arises from the homogenization of a boundary-value problem in the exterior of some periodically distributed obstacles and the zero-order term occurring in (1.4) has its origin in this particular structure of the model. The influence of the chemical reactions taking place on the boundaries of the reactive obstacles is reflected in the appearance of this zero-order extra-term. On the other hand, the second model is again a boundary-value problem, but this time in the whole domain $\Omega$, with discontinuous coefficients. Its macroscopic behavior (see (1.8)) also involves a zero-order term, but of a completely different nature; it is originated in the chemical reactions occurring inside the grains. The approach we used is the so-called energy method introduced by Tartar \cite{Tartar1}, \cite{Tartar2} for studying homogenization problems. It consists of constructing suitable test functions that are used in our variational problems. However, it is worth mentioning that the $\Gamma $-convergence of integral functionals involving oscillating obstacles could be a successful alternative. Extensive references on this topic can be found in the monographs of Dal Maso \cite{DalMaso} and of Braides and Defranceschi \cite{Braides-Defranceschi}. For example, our main result in Chapter 2 (cf. Theorem \ref{thm2.6}) can also be interpreted as a $\Gamma$-convergence-type result for the functionals \[ v\mapsto \frac {1}{2} D_{f}\int_{\Omega ^{\varepsilon }}DvDvdx+a\langle \mu ^{\varepsilon },G(v)\rangle-\int_{\Omega ^{\varepsilon }}fvdx +I_{K^{\varepsilon }}(v) \] (where $I_{K^{\varepsilon }}$ is the indicator function of the set $% K^{\varepsilon }$, i.e. $I_{K^{\varepsilon }}$ is equal to zero if v belongs to $K^{\varepsilon }$ and $+\infty $ otherwise) to the limit functional \[ v\mapsto \frac{1}{2}\int_{\Omega}QDvDvdx+a \frac{|\partial T|}{|Y\setminus T |}% \int_{\Omega }G(v)dx-\int_{\Omega}fvdx, \] which is the energy functional associated to (1.3). Also, let us mention that another possible way to get the limit problem (1.8) could be to use the two-scale convergence technique, coupled with periodic modulation, as in \cite{Bourgeat-Luckhaus-Mikelic}. Regarding our second problem, i.e. chemical reactive flows through periodic array of cells, a related work was completed by {Hornung et al.} \cite{Hornung-Jager-Mikelic} using nonlinearities which are essentially different from the ones we consider in the present paper. The proof of these authors is also different, since it is mainly based on the technique of two-scale convergence, which, as already mentioned, proves to be a successful alternative for this kind of problems. However, we have decided to use the energy method, coupled with monotonicity methods and results from the theory of semilinear problems, because it offered us the possibility to cover the nonlinear cases of practical importance mentioned above. The structure of our paper is as follows: first, let us mention that we shall just focus on the case $n\geq 3$, which will be treated explicitly. The case $n=2$ is much more simpler and we shall omit to treat it. In Section 2 we start by analyzing the first nonlinear problem, namely (1.1). We begin with the case of a monotone smooth function $g$ and we prove the convergence result using the energy method. Next, we treat the case of a maximal monotone graph, by writing our microscopic problem in the form of a variational inequality. The case of a reactive flow penetrating a periodical structure of grains is addressed in Section 3. Finally, notice that throughout the paper, by $C$ we shall denote a generic fixed strictly positive constant, whose value can change from line to line. \section{Chemical reactions on the walls of a chemical reactor} In this section, we will be concerned with the stationary reactive flow of a fluid confined in the exterior of some periodically distributed obstacles, reacting on the boundaries of the obstacles. We will treat separately the situation in which the nonlinear function $g$ in (1.1) is a monotone smooth function satisfying the condition $g(0)=0$ and the situation in which $g$ is a maximal monotone graph with $g(0)=0$. Let $\Omega $ be a smooth bounded connected open subset of $\mathbb{R}^{n}$ $% (n\geq 3)$ and let $Y$ $=[0,l_{1}[\times \dots [0,l_{n}[$ be the representative cell in $\mathbb{R}^{n}$. Denote by $T$ an open subset of $Y$ with smooth boundary $\partial T$ such that $\overline{T}\subset Y$. We shall refer to $% T $ as being \textit{the elementary obstacle}. Let $\varepsilon $ be a real parameter taking values in a sequence of positive numbers converging to zero. For each $\varepsilon $ and for any integer vector $k\in \mathbb{Z}^{n}$, set $T_{k}^{\varepsilon }$ the translated image of$\ \varepsilon T$ by the vector $% kl=(k_{1}l_{1},\dots ,k_{n}l_{n}):$% \[ T_{k}^{\varepsilon }=\varepsilon (kl+T). \] The set $T_{k}^{\varepsilon }$ represents the obstacles in $\mathbb{R}^{n}$. Also, let us denote by $T^{\varepsilon }$ the set of all the obstacles contained in $\Omega $, i.e. \[ T^{\varepsilon }=\bigcup \left\{ T_{k}^{\varepsilon } : \overline{T_{k}^{\varepsilon }}\mathbf{\subset }\Omega , k\in \mathbb{% Z}^{n}\right\} . \] Set \[ \Omega ^{\varepsilon }=\Omega \setminus {\overline{T^{\varepsilon }}}. \] Hence, $\Omega ^{\varepsilon }$ is a periodical domain with periodically distributed obstacles of size of the same order as the period. Remark that the obstacles do not intersect the boundary $\partial \Omega $. Let \[ S^{\varepsilon }=\cup \{\partial T_{k}^{\varepsilon }\mid \overline{% T_{k}^{\varepsilon }}\mathbf{\subset }\Omega , k\in \mathbb{Z}^{n}\}. \] So \[ \partial \Omega ^{\varepsilon }=\partial \Omega \cup S^{\varepsilon }. \] We shall also use the following notation: $|\omega |$ is the Lebesgue measure of any measurable subset $\omega$ of $\mathbb{R}^{n}$, $\chi _{\omega }$ is the characteristic function of the set $\omega$, $Y^{*}=Y\setminus \overline{T}$, and \begin{equation} \rho =\frac{|Y^{*}|}{|Y|}. \end{equation} Moreover, for an arbitrary function $\psi \in L^{2}(\Omega ^{\varepsilon })$, we shall denote by $\widetilde{\psi }$ its extension by zero inside the obstacles: \[ \widetilde{\psi }=\begin{cases} \psi & \text{in } \Omega ^{\varepsilon }, \\ 0 & \text{in } \Omega \setminus \overline{\Omega ^{\varepsilon }}. \end{cases} \] Also, for any open subset $D\subset \mathbb{R}^{n}$ and any function $g\in L^{1}(D)$, we set \begin{equation} \mathcal{M}_{D}(g)=\frac{1}{|D|}\int_{D}gdx. \end{equation} In the sequel we reserve the symbol $\#$ to denote periodicity properties. \subsection{Setting of the problem} As already mentioned, we are interested in studying the behavior of the solution, in such a periodical domain, of the problem \begin{equation} \begin{gathered} -D_{f}\Delta u^{\varepsilon }=f\quad \text{in }\Omega ^{\varepsilon }, \\ -D_{f} {\frac{\partial u^{\varepsilon }}{\partial \nu }} =a\varepsilon g(u^{\varepsilon })\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=0\quad \text{on }\partial \Omega . \end{gathered} \end{equation} Here, $\nu $ is the exterior unit normal to $\Omega ^{\varepsilon }$, $a>0$, $f\in L^{2}(\Omega )$ and $g$ is assumed to be given. Two model situations will be considered; the case in which $g$ is a monotone smooth function satisfying the condition $g(0)=0$ and the case of a maximal monotone graph with $g(0)=0$, i.e. the case in which $g$ is the subdifferential of a convex lower semicontinuous function $G$. These two general situations are well illustrated by the following important practical examples: \begin{itemize} \item[(a)] $g(v)=\dfrac{\alpha v}{1+\beta v}$, $\alpha, \beta>0$ (Langmuir kinetics) \item[(b)] $g(v)=|v|^{p-1}v$, $00$ in $\Omega^{\varepsilon}$, although $u^{\varepsilon}$ is not uniformly positive except in the case in which $g$ is a monotone smooth function satisfying the condition $g(0)=0$, as, for instance, in example $a)$. Moreover, since $u$ represents a concentration, it could be natural to assume that $f\leq 1$, and again one can prove that, in this case, $u \leq 1$. Without loss of generality, in what follows we shall assume that $D_{f}=1$. \subsection{First model situation: $g$ smooth} Let $g$ be a continuously differentiable function, monotonously non-decreasing and such that $g(v)=0$ if and only if $v=0$. We shall suppose that there exist a positive constant $C$ and an exponent $q$, with $0\leq q< n/(n-2)$, such that \begin{equation} |\frac{\partial g}{\partial v}|\leq C(1+|v|^{q}). \end{equation} Let us introduce the functional space \[ V^{\varepsilon }=\left\{ v\in H^{1}(\Omega ^{\varepsilon }) : v=0\text{on }\partial \Omega \right\} , \] with $\| v\| _{V^{\varepsilon }}=\| \nabla v\|_{L^{2}(\Omega ^{\varepsilon })}$. The weak formulation of problem (2.3) (written for $D_{f}=1$) is:\\ Find $u^{\varepsilon }\in V^{\varepsilon }$ such that \begin{equation} {\int_{\Omega ^{\varepsilon }}\nabla u^{\varepsilon }\cdot \nabla \varphi dx+a\varepsilon \int_{S^{\varepsilon }}g(u^{\varepsilon })\varphi d\sigma =\int_{\Omega ^{\varepsilon }}f\varphi dx}\quad \forall \varphi \in V^{\varepsilon }. \end{equation} By classical existence results (see \cite{Brezis}), there exists a unique weak solution $u^{\varepsilon }\in V^{\varepsilon }\cap H^{2}(\Omega ^{\varepsilon })$ of problem (2.3). The solution $u^{\varepsilon }$ of problem (2.3) being defined only on $% \Omega ^{\varepsilon }$, we need to extend it to the whole of $\Omega $ to be able to state the convergence result. In order to do that, let us recall the following well-known extension result (see \cite{Cioranescu-Paulin}). \begin{lemma} \label{lm2.1} There exists a linear continuous extension operator $$ P^{\varepsilon }\in \mathcal{L}(L^{2}(\Omega ^{\varepsilon });L^{2}(\Omega )) \cap \mathcal{L} (V^{\varepsilon}; H_{0}^{1}(\Omega )) $$ and a positive constant $C$, independent of $\varepsilon $, such that for any $v\in V^{\varepsilon }$, \begin{gather*} \| P^{\varepsilon }v\| _{L^{2}(\Omega )}\leq C\| v\|_{L^{2} (\Omega ^{\varepsilon })},\\ \| \nabla P^{\varepsilon }v\|_{L^{2}(\Omega )}\leq C\| \nabla v\|_{L^{2}(\Omega ^{\varepsilon })}\,. \end{gather*} \end{lemma} An immediate consequence of the previous lemma is the following Poincar\'{e}'s inequality in $V^{\varepsilon }$. \begin{lemma} \label{lm2.2} There exists a positive constant $C$, independent of $\varepsilon $, such that for any $v\in V^{\varepsilon }$, \[ \| v\|_{L^{2}(\Omega ^{\varepsilon })}\leq C\| \nabla v\|_{L^{2}(\Omega ^{\varepsilon })}\,. \] \end{lemma} The main result of this section is as follows. \begin{theorem} \label{thm2.3} One can construct an extension $P^{\varepsilon }u^{\varepsilon }$ of the solution $u^{\varepsilon }$ of the variational problem (2.5) such that $P^{\varepsilon }u^{\varepsilon }\rightharpoonup u$ weakly in $H_{0}^{1}(\Omega )$, where $u$ is the unique solution of \begin{equation} \begin{gathered} -{\sum_{i,j=1}^{n}q_{ij}}{\frac{\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+a{\frac{|\partial T|}{|Y^{*}|} }g(u)=f\quad \text{in }\Omega, \\ u=0\quad \text{on }\partial \Omega \,. \end{gathered} \end{equation} Here, $Q=((q_{ij}))$ is the classical homogenized matrix, whose entries are \begin{equation} q_{ij}=\delta _{ij}+\frac{1}{|Y^{*}|} \int_{Y^{*}}\frac{\partial \chi _{j}}{\partial y_{i}}dy \end{equation} in terms of the functions $\chi _{i}$, $i=1,\dots ,n$, solutions of the so-called cell problems \begin{equation} \begin{gathered} -\Delta \chi _{i}=0 \quad \text{in } Y^{*}, \\ \frac{\partial (\chi _{i}+y_{i})}{\partial \nu }=0 \quad\text{on } \partial T, \\ \chi _{i}\quad\text{is $Y$-periodic.} \end{gathered} \end{equation} The constant matrix $Q$ is symmetric and positive-definite. \end{theorem} \begin{proof} We divide the proof into four steps. \noindent\textit{First step.} Let $u^{\varepsilon }\in V^{\varepsilon }$ be the solution of the variational problem (2.5) and let $P^{\varepsilon }u^{\varepsilon }$ be the extension of $u^{\varepsilon }$ inside the obstacles given by Lemma \ref{lm2.1}. Taking $\varphi =u^{\varepsilon }$ as a test function in (2.5), using Schwartz and Poincar\'{e}'s inequalities, we easily get \[ \| P^{\varepsilon }u^{\varepsilon }\|_{H_{0}^{1}(\Omega )}\leq C. \] Consequently, by passing to a subsequence, still denoted by $P^{\varepsilon }u^{\varepsilon }$, we can assume that there exists $u\in H_{0}^{1}(\Omega )$ such that \begin{equation} P^{\varepsilon }u^{\varepsilon }\rightharpoonup u\quad \text{weakly in }% H_{0}^{1}(\Omega ). \end{equation} It remains to identify the limit equation satisfied by $u$. \noindent\textit{Second step}. In order to get the limit equation satisfied by $u$ we have to pass to the limit in (2.5). For getting the limit of the second term in the left hand side of (2.5), let us introduce, for any $h\in L^{s'}(\partial T)$, $1\leq s'\leq \infty $, the linear form $\mu_{h}^{\varepsilon }$ on $W_{0}^{1,s}(\Omega )$ defined by \[ \langle \mu _{h}^{\varepsilon },\varphi \rangle =\varepsilon \int_{S^{\varepsilon }}h(\frac{x}{\varepsilon })\varphi d\sigma \quad \forall \varphi \in W_{0}^{1,s}(\Omega ), \] with $1/s+1/s'=1$. It is proved in \cite{Cioranescu-Donato} that \begin{equation} \mu _{h}^{\varepsilon }\rightarrow \mu _{h}\quad \text{strongly in }% (W_{0}^{1,s}(\Omega ))', \end{equation} where $\langle \mu _{h},\varphi \rangle =\mu _{h}\int_{\Omega}\varphi dx$, with \[ \mu _{h}=\frac{1}{|Y|}\int_{\partial T}h(y)d\sigma . \] In the particular case in which $h\in L^{\infty }(\partial T)$ or even when $h$ is constant, we have \[ \mu _{h}^{\varepsilon }\rightarrow \mu _{h}\quad \text{strongly in }% W^{-1,\infty }(\Omega ). \] In what follows, we shall denote by $\mu ^{\varepsilon }$ the above introduced measure in the particular case in which $h=1$. Notice that in this case $\mu _{h}$ becomes $\mu _{1}=|\partial T|/|Y|$. Let us prove now that for any $\varphi \in \mathcal{\ D}(\Omega )$ and for any $v^{\varepsilon }\rightharpoonup v$ weakly in $H_{0}^{1}(\Omega )$, we get \begin{equation} \varphi g(v^{\varepsilon })\rightharpoonup \varphi g(v)\quad \text{weakly in }W_{0}^{1,\overline{q}}(\Omega ), \end{equation} where \[ \overline{q}=\frac{2n}{q(n-2)+n}. \] To prove (2.11), let us first note that \begin{equation} \sup \| \nabla g(v^{\varepsilon })\|_{L^{\overline{q}}(\Omega )}<\infty . \end{equation} Indeed, from the growth condition (2.4) imposed to $g$, we get \begin{align*} \int_{\Omega }\big|\frac{\partial g}{\partial x_{i}}(v^{\varepsilon })\big|^{\overline{q}}dx &\leq C\int_{\Omega }\big( 1+|v^{\varepsilon }| ^{q\overline{q}}\big) |\frac{\partial v^{\varepsilon }}{\partial x_{i}}|^{\overline{q}}dx\\ &\leq C( 1+( \int_{\Omega }|v^{\varepsilon }|^{q\overline{q}\gamma }dx) ^{1/\gamma }) ( \int_{\Omega }|\nabla v^{\varepsilon }|^{\overline{q}\delta }dx) ^{1/\delta }, \end{align*} where we took $\gamma $ and $\delta $ such that $\overline{q}\delta =2$, $% 1/\gamma +1/\delta =1$ and $q\overline{q}\gamma =2n/(n-2)$. Note that from here we get $\overline{q}={\frac{2n}{q(n-2)+n}}$. Also, since $0\leq q< n/(n-2)$, we have $\overline{q}> 1$. Now, since \[ \sup \| v^{\varepsilon }\|_{L^{\frac{2n}{n-2}}(\Omega )}<\infty, \] we get immediately (2.12). Hence, to get (2.11), it remains only to prove that \begin{equation} g(v^{\varepsilon })\rightarrow g(v)\quad \text{strongly in }L^{\overline{q}% }(\Omega ). \end{equation} But this is just a consequence of the following well-known result (see \cite{DalMaso} and \cite{Lions}). \begin{theorem} \label{thm2.4} Let $G:\Omega \times \mathbb{R}\rightarrow \mathbb{R}$ be a Carath\'{e}odory function, i.e. \begin{itemize} \item[(a)] For every $v$ the function $G(\cdot ,v)$ is measurable with respect to $x\in \Omega $. \item[(b)] For every (a.e.) $x\in \Omega $, the function $G(x,\cdot )$ is continuous with respect to $v$. \end{itemize} Moreover, if we assume that there exists a positive constant $C$ such that \[ |G(x,v)|\leq C\big( 1+|v|^{r/t}\big) , \] with $r\geq 1$ and $t<\infty $, then the map $v\in L^{r}(\Omega )\mapsto G(x,v(x))\in L^{t}(\Omega )$ is continuous in the strong topologies. \end{theorem} Indeed, since \[ |g(v)|\leq C(1+|v|^{q+1}), \] applying the above theorem for $G(x,v)=g(v)$, $t=\overline{q}$ and $% r=(2n/(n-2))-r'$, with $r'>0$ such that $q+10$ (see, for instance, \cite{Aris}). The correct mathematical treatment needs the problem to be reformulated by using the maximal monotone graph of $\mathbb{% R^2}$ associated to the Heaviside function $\beta(v)=\{0\}$ if $v<0$, $% \beta(0)=[0,1]$ and $\beta(v)={1}$ if $v>0$. The existence and uniqueness of a solution can be found, for instance, in Br\'{e}zis \cite{Brezis} and D\'{\i}az \cite{Diaz1}. The solution is obtained by passing to the limit in a sequence of problems associated to a monotone sequence of Lipschitz functions approximating $\beta$ and the results of this section remain true. Notice that now the homogenized problem becomes \begin{gather*} -{\sum_{i,j=1}^{n}q_{ij} \frac{\partial^{2}u}{\partial x_{i}\partial x_{j}} +a \frac{|\partial T|}{|Y^{*}|}\beta(u)\ni f}\quad \text{in } \Omega, \\ u=0 \quad \text{on }\partial \Omega. \end{gather*} A curious fact is that this type of problems arises in very different contexts (see, for instance, \cite{Ughi}). \end{remark} \begin{remark} \label{rmk2.9}\rm Under the assumptions of this section, $g$ does not need to be Lipschitz continuous (as, for instance, in the second example or in the multivalued example of the previous remark) and so the solution of the homogenized problem may give rise to a ``dead zone" (where $u(x)=0$) when a suitable balance between the ``size" of some norm of $f$ and the ``size" of the greatest ball included in $\Omega$ holds (see D\'{\i}az \cite{Diaz3}). \end{remark} \begin{remark} \rm The case of a spherically symmetric isolated particle under singular reaction kinetics was considered by Vega and Li\~n\'{a}n \cite{Vega-Liņan}. \end{remark} \section{Chemical reactive flow through grains} As already mentioned in Introduction, the chemical situation behind the second nonlinear problem we will treat here involves a chemical reactor with the grains constituted by solid catalyst particles. We assume that now the chemical reactions take place inside the grains, instead just on their boundaries. In fact, the problem corresponds to a transmission problem between the solutions of two separated equations. A simplified version of this kind of models can be formulated as follows: \begin{equation} \begin{gathered} -D_{f}\Delta u^{\varepsilon }=f\quad \text{in }\Omega ^{\varepsilon }, \\ -D_{p}\Delta v^{\varepsilon }+ag(v^{\varepsilon })=0,\quad \text{in }\Pi^{\varepsilon }, \\ -D_{f}{\frac{\partial u^{\varepsilon }}{\partial \nu }}=D_{p}% {\frac{\partial v^{\varepsilon }}{\partial \nu }}\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=v^{\varepsilon }\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=0\quad \text{on }\partial \Omega . \end{gathered} \end{equation} Here, $\Pi ^{\varepsilon }=\Omega \setminus \overline{\Omega ^{\varepsilon }}$, $\nu $ is the exterior unit normal to $\Omega ^{\varepsilon }$, $a$, $D_{f}$, $D_{p}>0$, $f\in L^{2}(\Omega )$ and $g$ is a continuous function, monotonously non-decreasing and such that $g(v)=0$ if and only if $v=0$. Moreover, we shall suppose that there exist a positive constant $C$ and an exponent $q$, with $0\leq q