% ultima correzione: dal maso-ebobisse 18-6-02 \documentclass[a4paper, 11pt]{amsart} \usepackage{amsfonts} %\oddsidemargin 20pt \evensidemargin 20pt % \textheight 550pt \textwidth 420pt % \columnsep 10pt \columnseprule 0pt \oddsidemargin 5pt \evensidemargin 5pt \textwidth16.cm \textheight21.5cm \flushbottom %\usepackage{showkeys} \mathchardef\emptyset="001F \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{definition} \theoremstyle{remark} \numberwithin{equation}{section} \newcommand{\var}{\varphi} \newcommand{\e}{\varepsilon} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\On}{{\mathcal O}(n)} \newcommand{\weakst}{\stackrel{\ast}{\rightharpoonup}} \newcommand{\weak}{\rightharpoonup} \newcommand{\ci}{\stackrel{i}{\to}} \newcommand{\m}{{\bf m}} \newcommand{\dx}{\,dx} \newcommand{\ho}{{\rm hom}} \newcommand{\mo}{{\bf m_0}} \newcommand{\cinfty}{C^{\infty}} \newcommand{\cinftyo}{C^{\infty}_{0}(Q,\R^{N})} \newcommand{\cper}{C^{\infty}_{\rm per}(Q,\R^{N})} \newcommand{\lone}{L^{1}(\Om,\R^{N})} \newcommand{\wone}{W^{1,1}(\Om,\R^{N})} \newcommand{\wonev}{W^{1,1}(V,\R^{N})} \newcommand{\woneq}{W^{1,1}(Q,\R^{N})} \newcommand{\wup}{{\rm W}^{1,p}} \newcommand{\orn}{(\Omega;\R^N)} \newcommand{\woneper}{W^{1,1}_{\rm per}(Q,\R^{N})} \newcommand{\wonepnu}{W^{1,1}_{\rm per}(Q_{\nu},\R^{N})} \newcommand{\M}{{\mathcal M}} \newcommand{\intcauchy}{\mskip 3mu -\mskip -19mu \int} \newcommand{\mpn}{{\rm M}^{p\times N}} \newcommand{\msim}{{\rm M}^{N\times N}_{\rm sym}} \newcommand{\lm}{{\mathcal L}^{N}} \newcommand{\hn}{{\mathcal H}^{N-1}} \newcommand{\no}{\noindent} \newcommand{\F}{{\mathcal F}} \newcommand{\E}{{\mathcal E}} \newcommand{\R}{{\mathbb R}} \newcommand{\Op}{{\mathcal O}} \newcommand{\tr}{{\rm tr}\,} \newcommand{\supp}{{\rm supp}\,} \newcommand{\salt}{\noalign{\vskip .2truecm}} \newcommand{\parent}[3]{\left #1 {#3} \right #2} % racchiude il testo tra il % primo e secondo simbolo \newcommand{\marco}{\hskip 0.6truecm\sqcap\hskip-7pt\sqcup} \newcommand{\graffe}[1]{\parent \{ \}{#1}} % racchiude il testo tra graffe adeg. \newcommand{\res}{\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\dist}{{\rm dist}} \newcommand{\sbd}{{\rm SBD}(\Om)} \newcommand{\bord}{{\partial\Om}} \newcommand{\bordl}{{\partial _L\Om}} \newcommand{\ld}{{\rm LD}(\Om)} \newcommand{\bom}{{\mathcal B}(\Om)} \newcommand{\opom}{{\mathcal O}(\Om)} \newcommand{\omreg}{{\mathcal O}_{\infty}(\Om)} \newcommand{\B}{{\mathcal B}} \newcommand{\diam}{{\rm diam}\,} \newcommand{\dpr}{{\mathcal D}'} \newcommand{\fke}{{\varphi_{k,\e}}} \newcommand{\norm}[1]{\left\Vert {#1} \right\Vert} % norma \title[A stability result for nonlinear Neumann problems under boundary variations]{ A stability result for nonlinear Neumann problems\\ under boundary variations} \author[Gianni Dal Maso]{Gianni Dal Maso} \address[Gianni Dal Maso]{S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy} \email{dalmaso@sissa.it} \author[Fran\c{c}ois Ebobisse]{Fran\c{c}ois Ebobisse} \address[Fran\c{c}ois Ebobisse]{S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy} \email{ebobisse@sissa.it} \author[Marcello Ponsiglione]{Marcello Ponsiglione} \address[Marcello Ponsiglione]{S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy} \email{ponsigli@sissa.it} \begin{document} \baselineskip3.15ex \vskip .2truecm \begin{abstract} \small{In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology. More precisely, for every bounded open subset $\Om$ of $\R ^2$, we consider the problem $$ \begin{cases} -\,{\rm div}\,a(x,\nabla u_\Om)\,+\,b(x, u_\Om)=0 & \mbox{ in }\quad\Om,\\ \salt \quad a(x,\nabla u_\Om)\cdot\nu=0 & \mbox{ on }\quad\partial\Om, \end{cases} $$ where $a\colon \R^2\times\R^2\to\R^2$ and $b\colon \R^2\times\R\to\R$ are two Carath\'eodory functions which satisfy the standard monotonicity and growth conditions of order $p$, with $1<\,p\,\leq 2$. Let $\Om_n$ be a uniformly bounded sequence of open sets in $\R^2$, whose complements $\Om_n^c$ have a uniformly bounded number of connected components. We prove that, if $\Om_n^c\to\Om^c$ in the Hausdorff metric and $|\Om_n|\to|\Om|$, then $u_{\Om_n}\to u_\Om$ and $\nabla u_{\Om_n}\to \nabla u_\Om$ strongly in $L^p$. The proof is obtained by showing the Mosco convergence of the Sobolev spaces $W^{1,p}(\Om_n)$ to the Sobolev space $W^{1,p}(\Om)$. \vskip .7truecm \noindent {\bf Key words:} Boundary value problems, nonlinear elliptic equations, capacity, Hausdorff metric, monotone operators, Mosco convergence \vskip.3truecm \noindent {\bf 2000 Mathematics Subject Classification:} 35J65, 31A15, 47H05, 49Q10, 49J45. } \end{abstract} \maketitle %\vskip .2truecm \section{Introduction}\label{Intro} In this paper we consider nonlinear elliptic equations with Neumann boundary conditions of the form \begin{equation}\label{eqgen} \begin{cases} -\,{\rm div}\,a(x,\nabla u)\,+\,b(x, u)=0 & \mbox{ in }\quad\Om,\\ \salt a(x,\nabla u)\cdot\nu=0 & \mbox{ on }\quad\partial\Om, \end{cases} \end{equation} where $\Om$ is a bounded open subset of $\R ^2$ and $a\colon \R^2\times\R^2\to\R^2$ and $b\colon \R^2\times\R\to\R$ are two Carath\'eodory functions which satisfy suitable monotonicity, coerciveness, and growth conditions (see (\ref{a1})-(\ref{a3}) below). Our purpose is to study the continuity of the mapping $\Om\mapsto u_\Om$ which associates to every $\Om$ the corresponding solution $u_\Om$. The notion of convergence we consider on the sets $\Om$ is given by the Hausdorff complementary topology, which is induced by the Hausdorff distance of the complements of the sets $\Om$ (see Section~\ref{haus}). Many examples (see \cite{daml}, \cite{mur}, \cite{delv}, \cite{con}, \cite{cor}) show that, if we consider a uniformly bounded sequence $\Om_h$ of open subsets of $\R^2$ which converges to an open set $\Om$ in the Hausdorff complementary topology, it may happen that $u_{\Om_h}$ does not converge to $u_{\Om}$. Therefore, in this paper we investigate additional conditions on $\Om_h$ which imply that $u_{\Om_h}$ converges to $u_{\Om}$ for any choice of the functions $a$ and $b$. These stability results are useful in the proof of the existence of solutions of some shape optimization problems. More recently, similar stability results have been applied in \cite{dmro} to study some models in fracture mechanics. In the special case of the linear problems \begin{equation}\label{neumlin} \begin{cases} -\,\Delta u\,+\,u=f & \mbox{ in }\quad\Om,\\ \salt \displaystyle\frac{\partial u}{\partial\nu}=0 & \mbox{ on }\quad\partial\Om, \end{cases} \end{equation} the stability with respect to $\Om$ was first studied by Chenais \cite{chen} under the assumption that the domains $\Om_h$ satisfy a uniform cone condition, which allows to use extension operators with uniformly bounded norms. This condition excludes a large class of domains, like for instance domains with cracks, for which there is an increasing interest in view of the applications to fracture mechanics. The stability of (\ref{neumlin}) in nonsmooth domains is studied in \cite{buzo1}, \cite{buzo2}, \cite{buzo3} under variuos assumptions on $\Om_h$. In \cite{chdo} the problem is studied under the hypothesis that $\Om_h$ converges to $\Om$ in the Hausdorff complementary topology, assuming also a uniform bound for the lenghts of the boundaries ${\mathcal H}^1(\partial\Om_h)$ and for the number of connected components of $\partial\Om_h$. This result has been recently improved in \cite{bv1}, where the bound on ${\mathcal H}^1(\partial\Om_h)$ is replaced by the weaker assumption of convergence of the two-dimensional measures of the domains, i.e., $|\Om_h|\to |\Om|$, which is also necessary for the stability of (\ref{neumlin}). In the present paper we study the stability of the nonlinear problems (\ref{eqgen}) with respect to the Hausdorff complementary topology, assuming that $|\Om_h|\to |\Om|$ and that the number of connected components of the complements $\Om_h^c$ is uniformly bounded. To obtain this result we reduce the problem to the convergence in the sense of Mosco of the Sobolev spaces $W^{1,p}(\Om_h)$ to the Sobolev space $W^{1,p}(\Om)$, where the exponent $p$ is related in the usual way to the growth condition of the functions $a$ and $b$ (see Section~\ref{MC}). The proof of this property for $12$ the stability result for (\ref{eqgen}) and (\ref{eqgenzero}) is not true under our hypotheses, as shown in Remarks~\ref{pgreat2} and \ref{Sobpgreat2}. The general form of the limit problem will be studied in \cite{ebpo}. In the last part of the paper we consider the case of unbounded open sets and the case of mixed boundary value problems, with a Dirichlet condition on a fixed part of the boundary. {\small \tableofcontents} \section{Notation and preliminaries}\label{Notprel} Throughout the paper $p$ and $q$ are real numbers, with $12$, then $C_r(E)>0$ for every nonempty set $E$. On the contrary, if $r=2$ there are nonempty sets $E$ with $C_r(E)=0$ (for instance, $C_r(\{x\})=0$ for every $x\in\R^2$). We say that a property $\mathcal P(x)$ holds $C_r$-{\it quasi everywhere} (abbreviated as {\it $C_r$-q.e.}) in a set $E$ if it holds for all $x\in E$ except a subset $N$ of $E$ with $C_r(N)=0$. We recall that the expression {\it almost everywhere} (abbreviated as {\it a.e.}) refers, as usual, to the Lebesgue measure. A function $u\colon E\to\overline\R$ is said to be {\it quasi-continuous} if for every $\e$ there exists $A_\e\subset\R^2$, with $C_r(A_\e)<\e$, such that the restriction of $u$ to $E\setminus A_\e$ is continuous. If $r>2$ every quasi-continuous function is continuous, while for $r=2$ there are quasi-continuous functions that are not continuous. It is well known that, for any open subset $\Om$ of $\R^2$, every function $u\in W^{1,r}(\Om)$ has a {\it quasi-continuous representative} $\overline u\colon \Om\to\R$, which satisfies $$ \lim_{\rho\to 0^+}\frac{1}{|B(x,\rho)|}\int_{B(x,\rho)}|u(y)-\overline u(x)|\,dy=0\quad\mbox{for q.e.\ }x\in \Om, $$ where $B(x,\rho)$ is the open ball with centre $x$ and radius $\rho$. We recall that if $u_h$ converges strongly to $u$ in $W^{1,r}(\Om)$, then a subsequence of $\overline u_h$ converges to $\overline u$ pointwise $C_r$-q.e.\ on $\Om$. For these and other properties on quasi-continuous representatives the reader is referred to \cite{EG}, \cite{hekima}, \cite{maz1}, \cite{ziem}. To simplify the notation, we always identify each function $u\in W^{1,r}(\Om)$ with its quasi-continuous representative~$\overline u$. \subsection{Convergence of sets}\label{haus} We recall here the notion of convergence in the sense of {\it Kuratowski}. We say that a sequence $(C_h)$ of closed subsets of $\R^2$ converges to a closed set $C$ in the sense of Kuratowski if the following two properties hold: \begin{itemize} \item[($K_1$)] for every $x\in C,$ there exists a sequence $x_h \in C_h$ such that $x_h\to x$; \item[($K_2$)] if $(h_k)$ is a sequence of indices converging to $\infty$, $(x_k)$ is a sequence such that $x_k\in C_{h_k}$ for every $k$, and $x_k$ converges to some $x\in \R^2$, then $x\in C$. \end{itemize} Let us recall also that the {\it Hausdorff distance} between two nonempty closed subsets $C_1$ and $C_2$ of $\R^2$ is defined by $$ d_{H}(C_1,C_2):=\max\graffe{\displaystyle\sup_{x\in C_1}{\rm dist}\bigl(x,C_2\bigr)\,,\,\displaystyle\sup_{x\in C_2}{\rm dist}\bigl(x,C_1\bigr)}.$$ We say that a sequence $(C_h)$ of nonempty closed subsets of $\R^2$ converges to a nonempty closed subset $C$ in the {\it Hausdorff metric\/} if $d_H(C_h\,,C)$ converges to $0$. A sequence of subsets of $\R^2$ is said to be {\it uniformly bounded\/} if there exists a bounded subset of $\R^2$ which contains all sets of the sequence. The convergence in the Hausdorff metric implies the convergence in the sense of Kuratowski, while in general the converse is false. However, if $(C_h)$ is a uniformly bounded sequence of nonempty closed sets in $\R^2,$ then $(C_h)$ converges to a closed set $C$ in the Hausdorff metric if and only if $(C_h)$ converges to $C$ in the sense of Kuratowski. We say that a sequence $(\Om_h)$ of open subsets of $\R^2$ converges to an open set $\Om$ in the {\it Hausdorff complementary topology\/}, if $d_{H}(\Om_h^c\,,\Om^c)$ converges to $0$, where $\Om_h^c$ and $\Om^c$ are the complements of $\Om_h$ and $\Om$ in $\R^2$, It is well-known (see, e.g., \cite[Blaschke's Selection Theorem]{falc}) that every uniformly bounded sequence of nonempty closed sets is compact with respect to the Hausdorff convergence. This implies that every uniformly bounded sequence of open sets is compact with respect to the Hausdorff complementary topology. Moreover, a uniformly bounded sequence of open sets $(\Om_h)$ converges to an open set $\Om$ in the Hausdorff complementary topology, if and only if the sequence $(\Om_h^c)$ converges to $\Om^c$ in the sense of Kuratowski. \subsection{The Neumann problems} Let $a\colon \R^2\times\R^2\to\R^2$ and $b\colon \R^2\times\R\to\R$ be two Carath\'eodory functions that satisfy the following assumptions: there exist $00;&\label{a1}\\ \salt &|a(x,\xi)|\leq \alpha(x)+c_2|\xi|^{p-1};&\label{a2}\\ \salt &a(x,\xi)\cdot\xi\geq -\beta(x) +c_1|\xi|^p.&\label{a3} \end{eqnarray} We assume that $b$ satisfies the same inequalities for every $\xi,\,\xi_1,\xi_2\in\R$. For every open set $\Om\subset \R^2$, we consider the following nonlinear Neumann boundary value problems, where $\nu$ denotes the outward unit normal to $\partial\Om$: \begin{equation}\label{eq1} \begin{cases} -\,{\rm div}\,a(x,\nabla u)+ b(x,u)=0 & \mbox{ in }\quad\Om,\\ \salt \mbox{ }a(x,\nabla u)\cdot\nu=0 & \mbox{ on }\quad\partial\Om,\end{cases} \end{equation} and \begin{equation}\label{eqaux1} \begin{cases} -\,{\rm div}\,a(x,\nabla v)=0 & \mbox{ in }\quad\Om\,,\\ \salt \mbox{ }a(x,\nabla v)\cdot\nu =0 & \mbox{ on }\quad\partial\Om \,.\end{cases} \end{equation} A function $u$ is a solution of (\ref{eq1}) if \begin{equation}\label{eq2} \begin{cases} u\in W^{1,p}(\Om),\\ \salt \displaystyle\int_{\Om}\bigl[a(x,\nabla u)\cdot\nabla z +b(x,u)z\bigr]dx=0 & \forall z\in W^{1,p}(\Om). \end{cases} \end{equation} while $v$ is a solution of (\ref{eqaux1}) if \begin{equation}\label{eqaux2} \begin{cases} v \in L^{1,p}(\Om), \\ \salt \displaystyle\int_{\Om}a(x,\nabla v)\cdot\nabla z\,\dx=0 & \forall z\in L^{1,p}(\Om). \end{cases} \end{equation} By well-known existence results for nonlinear elliptic equations with strictly monotone operators (see, e.g., Lions \cite{lions}), one can easily see that (\ref{eq2}) has a unique solution in $W^{1,p}(\Om)$. Similarly one can prove that (\ref{eqaux2}) has a solution, and that if $v_1$ and $v_2$ are solutions of (\ref{eqaux2}), then $\nabla v_1=\nabla v_2$ a.e.\ in $\Om$. Note that problem (\ref{eqaux2}) can be formulated in the quotient space $L^{1,p}(\Om)/_\sim$, where a uniqueness result holds. \subsection{Stability of Neumann problems}\label{snp} In order to study the stability of (\ref{eq1}) and (\ref{eqaux1}) with respect to the variations of the open set $\Om$, we should be able to compare two solutions defined in two different domains. For any subset $E$ of $\R^2$, the characteristic function $1_E$ of $E$ is defined by $1_E(x):=1$ for $x\in E$ and $1_E(x):=0$ for $x\in E^c$. For every $u\in L^{1,p}(\Om)$, the functions $u1_\Om$ and $\nabla u1_\Om$ are the extensions of the functions $u$ and $\nabla u$ which vanish in $\Om^c$. By means of these extensions, $W^{1,p}(\Om)$ will be identified with the closed linear subspace $X_\Om$ of $L^p(\R^2)\times L^p(\R^2,\R^2)$ defined by \begin{equation}\label{XOm} X_\Om:=\{(u1_\Om,\nabla u1_\Om): u\in W^{1,p}(\Om)\}, \end{equation} while the quotient space $L^{1,p}(\Om)/_\sim$ will be identified with the closed linear subspace $Y_\Om$ of $ L^p(\R^2,\R^2)$ defined by \begin{equation}\label{YOm} Y_\Om:=\{\nabla u1_\Om: u\in L^{1,p}(\Om)\}. \end{equation} Let $\Om$ be an open subset of $\R^2$ and let $(\Om_h)$ be a sequence of open subsets of $\R^2$. Given a pair of Carath\'eodory functions $a\colon \R^2\times\R^2\to\R^2$ and $b\colon \R^2\times\R\to\R$ satisfying (\ref{a1})--(\ref{a3}), let $u$ be the solution to problem (\ref{eq1}) in $\Om$ and, for every $h$, let $u_h$ be the solution to problem (\ref{eq1}) in~$\Om_h$. \begin{definition}\label{stable} We say that $\Om$ is stable for the Neumann problems (\ref{eq1}) along the sequence $(\Om_h)$ if for every pair of functions $a$, $b$ satisfying (\ref{a1})--(\ref{a3}) the sequence $(u_h1_{\Om_h})$ converges to $u1_\Om$ strongly in $L^p(\R^2)$ and the sequence $(\nabla u_h1_{\Om_h})$ converges to $\nabla u1_\Om$ strongly in $L^p(\R^2,\R^2)$. \end{definition} Similarly, let $v$ be a solution to problem (\ref{eqaux1}) in $\Om$ and, for every $h$, let $v_h$ be a solution to problem (\ref{eqaux1}) in~$\Om_h$. \begin{definition}\label{stableaux} We say that $\Om$ is stable for the Neumann problems (\ref{eqaux1}) along the sequence $(\Om_h)$ if for every function $a$ satisfying (\ref{a1})--(\ref{a3}) the sequence $(\nabla v_h1_{\Om_h})$ converges to $\nabla v1_\Om$ strongly in $L^p(\R^2,\R^2)$. \end{definition} \subsection{Mosco convergence}\label{MC} We shall prove that the notion of stability introduced in the previous definitions is equivalent to a notion of convergence for subspaces of a Banach space introduced by Mosco in \cite{mosco}. Let $\Om_h$ and $\Om$ be open subsets of $\R^2$, and let $X_{\Om_h}$ and $X_\Om$ be the corresponding subspaces defined by (\ref{XOm}). We recall that $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco (see \cite[Definition 1.1]{mosco}) if the following two properties hold: \begin{itemize} \item[($M_1$)] for every $u\in W^{1,p}(\Om)$, there exists a sequence $u_h\in W^{1,p}(\Om_h)$ such that $u_h1_{\Om_h}$ converges to $u1_\Om$ strongly in $L^p(\R^2)$ and $\nabla u_h1_{\Om_h}$ converges to $\nabla u1_\Om$ strongly in $L^p(\R^2,\R^2)$; \item[($M_2$)] if $(h_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k\in W^{1,p}(\Om_{h_k})$ for every $k$, and $u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2)$ to a function $\phi$, while $\nabla u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2,\R^2)$ to a function $\psi$, then there exists $u\in W^{1,p}(\Om)$ such that $\phi=u1_\Om$ and $\psi=\nabla u1_\Om$ a.e.\ in $\R^2$. \end{itemize} \vskip .3truecm Analogously, the convergence in the sense of Mosco of the spaces $Y_{\Om_h}$ to $Y_\Om$ defined by (\ref{YOm}) is obtained by using only the convergence of the extensions of gradients, that is: \begin{itemize} \item[($M_1^\prime$)] for every $u\in L^{1,p}(\Om)$, there exists a sequence $u_h\in L^{1,p}(\Om_h)$ such that $\nabla u_h1_{\Om_h}$ converges strongly to $\nabla u1_\Om$ in $L^p(\R^2,\R^2)$; \item[($M_2^\prime$)] if $(h_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k\in L^{1,p}(\Om_{h_k})$ for every $k$, and $\nabla u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2,\R^2)$ to a function $\psi$, then there exists $u\in L^{1,p}(\Om)$ such that $\psi=\nabla u1_\Om$ a.e.\ in $\R^2$. \end{itemize} \begin{theorem}\label{mosccont} Let $\Om_h$ and $\Om$ be open subsets of $\R^2$, and let $X_{\Om_h}$ and $X_\Om$ be the corresponding subspaces defined by (\ref{XOm}). Then $\Om$ is stable for the Neumann problems (\ref{eq1}) along the sequence $(\Om_h)$ if and only if $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco. \end{theorem} \begin{proof} Assume that $\Om$ is stable for the Neumann problems (\ref{eq1}) along the sequence $(\Om_h)$. We want to prove that $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco by using only the stability of the solutions corresponding to functions $a$ and $b$ of the special form \begin{equation}\label{gdm20} a(x,\xi):=a_0(x)+a_1(\xi),\qquad\qquad b(x,t):=b_0(x)+b_1(t), \end{equation} with \begin{equation}\label{gdm21} a_1(\xi):=|\xi|^{p-2}\xi, \qquad b_1(t):=|t|^{p-2}t, \qquad a_0\in L^q(\R^2,\R^2), \qquad b_0\in L^q(\R^2). \end{equation} Let us prove ($M_1$). Given $u\in W^{1,p}(\Om)$, let $a_0:=-|\nabla u|^{p-2}\nabla u1_\Om$ and $b_0:=-|u|^{p-2}u1_\Om$. Then $u$ is the solution of (\ref{eq2}) in $\Om$ with $a$ and $b$ given by (\ref{gdm20}). Let $u_h\in W^{1,p}(\Om_h)$ be the solution of (\ref{eq2}) in $\Om_h$ with the same $a$ and $b$. By Definition \ref{stable} the sequence $(u_h1_{\Om_h})$ converges to $u1_\Om$ strongly in $L^p(\R^2)$ and $(\nabla u_h1_{\Om_h})$ converges to $\nabla u1_\Om$ strongly in $L^p(\R^2,\R^2)$. This proves ($M_1$). Let us prove ($M_2$). Let $(h_k)$ be a sequence of indices converging to $\infty$ and let $(u_k)$ be a sequence, with $u_k\in W^{1,p}(\Om_{h_k})$ for every $k$, such that $(u_k1_{\Om_{h_k}})$ converges weakly in $L^p(\R^2)$ to a function $\phi$, while $(\nabla u_k1_{\Om_{h_k}})$ converges weakly in $L^p(\R^2,\R^2)$ to a function $\psi$. Let $a_0:=-a_1(\psi)=-|\psi|^{p-2}\psi$ and $b_0:=-b_1(\phi)=-|\phi|^{p-2}\phi$, let $a$ and $b$ be defined by (\ref{gdm20}), and let $u^*$ and $u^*_{h_k}$ be the solutions of problems (\ref{eq2}) in $\Om$ and $\Om_{h_k}$ respectively. By the stability assumption the sequence $(u^*_{h_k}1_{\Om_{h_k}})$ converges to $u^*1_\Om$ strongly in $L^p(\R^2)$ and $(\nabla u^*_{h_k}1_{\Om_{h_k}})$ converges to $\nabla u^*1_\Om$ strongly in $L^p(\R^2,\R^2)$. This implies that $a(x, \nabla u^*_{h_k}1_{\Om_{h_k}})$ converges to $a(x, \nabla u^*1_{\Om})$ strongly in $L^q(\R^2,\R^2)$ and $b(x, u^*_{h_k}1_{\Om_{h_k}})$ converges to $b(x, u^*1_{\Om})$ strongly in $L^q(\R^2)$. Therefore \begin{eqnarray}\nonumber &\displaystyle \lim_{k\to\infty}\int_{\Om_{h_k}} \bigl[a(x,\nabla u^*_{h_k})\cdot(\nabla u_{h_k}-\nabla u^*_{h_k}) +b(x,u^*_{h_k})(u_{h_k}- u^*_{h_k})\bigr]dx= \\ &\displaystyle =\int_{\R^2}\bigl[a(x,\nabla u^*1_\Om)\cdot(\psi-\nabla u^*1_\Om) +b(x,u^*1_\Om)(\phi- u^*1_\Om)\bigr]dx.\label{gdm22} \end{eqnarray} By (\ref{eq2}) the left hand side of (\ref{gdm22}) is zero. Therefore, using (\ref{gdm20}) and (\ref{gdm21}) we obtain $$ \int_{\R^2}\bigl[(a_1(\nabla u^*1_\Om)-a_1(\psi))\cdot(\psi-\nabla u^*1_\Om) +(b_1(u^*1_\Om)-b_1(\phi))(\phi- u^*1_\Om)\bigr]dx=0. $$ Using the strict monotonicity of $a_1$ and $b_1$ we obtain that $\psi=\nabla u^*1_\Om$ and $\phi=u^*1_\Om$ a.e.\ in~$\R^2$. Conversely, assume now that $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco and let us prove the stability. Let $a$ and $b$ be two Carath\'eodory functions satisfying (\ref{a1})--(\ref{a3}) and let $u_h$ and $u$ be the solutions to problems (\ref{eq1}) in $\Om_h$ and $\Om$. The weak convergence in $L^p(\R^2)\times L^p(\R^2,\R^2)$ of $(u_h1_{\Om_h}, \nabla u_h1_{\Om_h})$ to $(u1_{\Om},\nabla u1_{\Om})$ is a particular case of \cite[Theorem A]{mosco}. For the reader's convenience, we give here the simple proof. Using $z:=u_h$ as test function in (\ref{eq2}) for $\Om_h$, from (\ref{a3}) we obtain that $\|u_h\|_{W^{1,p}(\Om_h)}$ is bounded. Passing to a subsequence, we have that $u_h1_{\Om_h}$ converges weakly in $L^p(\R^2)$ to a function $\phi$, while $\nabla u_h1_{\Om_h}$ converges weakly in $L^p(\R^2,\R^2)$ to a function $\psi$. By ($M_2$) there exists $u^*\in W^{1,p}(\Om)$ such that $\phi=u^*1_\Om$ and $\psi=\nabla u^*1_\Om$ a.e.\ in $\R^2$. By monotonicity for every $v\in W^{1,p}(\Om)$ we have \begin{eqnarray} &\displaystyle \int_{\R^2} \bigl[a(x,\nabla v1_{\Om}) \cdot(\nabla v1_\Om-\nabla u_h1_{\Om_h}) +b(x,v1_{\Om})(v1_\Om-u_h1_{\Om_h}) \bigr]dx\ge \label{gdm10} \\ &\displaystyle \ge \int_{\R^2} \bigl[a(x,\nabla u_h1_{\Om_h}) \cdot(\nabla v1_\Om-\nabla u_h1_{\Om_h}) +b(x,u_h1_{\Om_h})(v1_\Om-u_h1_{\Om_h}) \bigr]dx. \nonumber \end{eqnarray} By ($M_1$) there exists a sequence $v_h\in W^{1,p}(\Om_h)$ such that $v_h1_{\Om_h}$ converges to $v1_\Om$ strongly in $L^p(\R^2)$ and $\nabla v_h1_{\Om_h}$ converges to $\nabla v1_\Om$ strongly in $L^p(\R^2,\R^2)$. As $v_h-u_h\in W^{1,p}(\Om_h)$, by (\ref{eq2}) we have \begin{eqnarray} &\displaystyle \int_{\R^2} \bigl[a(x,\nabla u_h1_{\Om_h}) \cdot(\nabla v1_\Om-\nabla u_h1_{\Om_h}) +b(x,u_h1_{\Om_h})(v1_\Om-u_h1_{\Om_h}) \bigr]dx= \label{gdm11} \\ &\displaystyle = \int_{\R^2} \bigl[a(x,\nabla u_h1_{\Om_h}) \cdot(\nabla v1_\Om-\nabla v_h1_{\Om_h}) +b(x,u_h1_{\Om_h})(v1_\Om-v_h1_{\Om_h}) \bigr]dx. \nonumber \end{eqnarray} Since $a(x,\nabla u_h1_{\Om_h})$ is bounded in $L^q(\R^2,\R^2)$ and $b(x,u_h1_{\Om_h})$ is bounded in $L^q(\R^2)$, passing to the limit in (\ref{gdm10}) and (\ref{gdm11}) we obtain \begin{eqnarray}\label{gdm12} &\displaystyle \int_{\Om} \bigl[a(x,\nabla v) \cdot(\nabla v-\nabla u^*) +b(x,v)(v-u^*) \bigr]dx\ge \\ &\displaystyle \ge \lim_{h\to\infty} \int_{\R^2} \bigl[a(x,\nabla u_h1_{\Om_h}) \cdot(\nabla v1_\Om-\nabla v_h1_{\Om_h}) +b(x,u_h1_{\Om_h})(v1_\Om-v_h1_{\Om_h}) \bigr]dx = 0. \nonumber \end{eqnarray} Then we take $v=u^*\pm\e z$ in (\ref{gdm12}), with $z\in W^{1,p}(\Om)$ and $\e>0$. Dividing by $\e$, and passing to the limit as $\e$ tends to $0$, we obtain that $u^*$ satisfies (\ref{eq2}) in $\Om$. This proves that $u^*=u$. Therefore $u_h1_{\Om_h}$ converges to $u1_\Om$ weakly in $L^p(\R^2)$ and $\nabla u_h1_{\Om_h}$ converges to $\nabla u1_\Om$ weakly in $L^p(\R^2,\R^2)$. Taking $v:=u$ in (\ref{gdm12}) we obtain that \begin{eqnarray*}%\label{gdm13} &\displaystyle %\lim_{h\to\infty} \int_{\R^2} (a(x,\nabla u_h1_{\Om_h})-a(x,\nabla u1_{\Om})) \cdot(\nabla u_h1_{\Om_h} - \nabla u1_\Om)dx + %= 0, \\ %\label{gdm14} &\displaystyle %\lim_{h\to\infty} + \int_{\R^2} (b(x,u_h1_{\Om_h})- b(x,u1_{\Om})) (u_h1_{\Om_h}-u1_\Om)dx %= 0. \end{eqnarray*} tends to $0$ as $h\to \infty$. Using the monotonicity of $a$ and $b$ we conclude that each integral tends to~$0$. The strong convergence of $(u_h1_{\Om_h}, \nabla u_h1_{\Om_h})$ is now a consequence of the following lemma. \end{proof} \begin{lemma}\label{lemmastrong} Let $(\psi_h)$ be a sequence in $L^p(\R^2,\R^2)$ converging weakly in $L^p(\R^2,\R^2)$ to a function $\psi$. Assume that \begin{equation}\label{gdm15} \lim_{h\to\infty} \int_{\R^2} (a(x,\psi_h)-a(x,\psi)) \cdot(\psi_h-\psi)dx = 0. \end{equation} Then $\psi_h$ converges to $\psi$ strongly in $L^p(\R^2,\R^2)$. \end{lemma} \begin{proof} Various forms of this lemma have been used in the study of Leray-Lions operators (see, e.g., \cite[Lemma 5]{BO-Mu-Pu}). For the sake of completeness, we give here the short proof of the present version. Let $g_h:=(a(x,\psi_h)-a(x,\psi))\cdot(\psi_h-\psi)$. By monotonicity we have $g_h\ge 0$ a.e.\ in $\R^2$, thus (\ref{gdm15}) implies that $g_h$ converges to $0$ strongly in $L^1(\R^2)$. Passing to a subsequence, we may assume that $g_h$ converges to $0$ a.e.\ in $\R^2$. Using the Cauchy inequality, from (\ref{a2}) and (\ref{a3}) we obtain for every $\e>0$ \begin{eqnarray*} &c_1|\psi_h|^p\le \beta + a(x,\psi_h)\cdot\psi_h= \beta + g_h + a(x,\psi_h)\cdot\psi + a(x,\psi)\cdot(\psi_h-\psi)\le \\ &\displaystyle \le g_h + \beta + \alpha|\psi| + |a(x,\psi)||\psi| + c_2 \big(\frac{\e^q}{q} + \frac{\e^p}{p}\big)|\psi_h|^p + c_2\big( \frac{|\psi|^p}{p\e^p} + \frac{|a(x,\psi)|^q}{q\e^q}\big). \end{eqnarray*} Choosing $\e$ small enough, we obtain that there exist a constant $c_3>0$ and a function $\gamma\in L^1(\R^2)$ such that \begin{equation}\label{gdm16} c_3|\psi_h|^p\le g_h + \gamma. \end{equation} Let us fix a point $x\in \R^2$ where $\gamma(x)<+\infty$ and where $g_h(x)$ tends to $0$. By (\ref{gdm16}) the sequence $\psi_h(x)$ is bounded in $\R^2$, thus a subsequence (depending on $x$) converges to a vector $\xi\in \R^2$. By the definition of $g_h(x)$ and by the continuity of $a(x,\cdot)$ we get $(a(x,\xi)-a(x,\psi(x)))\cdot(\xi-\psi(x))=0$, which implies $\xi=\psi(x)$ by (\ref{a1}). Therefore the whole sequence $\psi_h(x)$ converges to $\psi(x)$. Since this is true for a.e.\ $x\in \R^2$, the strong convergence in $L^p(\R^2,\R^2)$ follows from (\ref{gdm16}) by the dominated convergence theorem. \end{proof} \begin{remark}\label{MS->mis} {\rm Let us observe that, if $(\Om_h)$ is a uniformly bounded sequence of open subsets of $\R^2$ such that $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco, then $|\Om_h\triangle\Om|$ converges to $0$. Let $\Sigma\subset\R^2$ be a bounded closed set such that $\Om_h\subset\Sigma$ for every $h$. {}From property ($M_1$) it follows that $\Om\subset\Sigma$. Indeed, if $\Om\setminus\Sigma\neq\emptyset$, let $B\subset\Om\setminus\Sigma$ be an open ball and let $\varphi\in C^\infty_c(B)$ with $\varphi\neq 0$. By property ($M_1$), there exists $u_h\in W^{1,p}(\Om_h)$ such that $u_h1_{\Om_h}\to \varphi$ strongly in $L^p(\R^2)$. So, $$0<\int_B|\varphi|^p\dx=\lim_{h\to\infty}\int_B|u_h1_{\Om_h}|^p\dx=0,$$ which is absurd. Now, let $u:=1_\Om$. Since $u\in W^{1,p}(\Om)$, by property ($M_1$) there exists $u_h\in W^{1,p}(\Om_h)$ such that $u_h1_{\Om_h}\to 1_\Om$ strongly in $L^p(\R^2)$. As $|u_h1_{\Om_h}- 1_\Om|^p=1$ a.e.\ on $\Om\setminus\Om_h$, we have \begin{equation}\label{limmis1} \lim_{h\to\infty}|\Om\setminus\Om_h| \,\leq\, \lim_{h\to\infty}\int_{\R^2}|u_h1_{\Om_h}- 1_\Om|^p\dx\, =\,0. \end{equation} On the other hand, up to a subsequence, $1_{\Om_h}$ converges weakly in $L^p(\R^2,\R^2)$ to some $\phi$. Hence from property ($M_2$), there exists $v\in W^{1,p}(\Om)$ such that $\phi=v1_\Om$ a.e.\ in $\R^2$. So we have that $$\lim_{h\to\infty}|\Om_h\setminus\Om|= \lim_{h\to\infty}\int_{\R^2} 1_{\Om_h}1_{\Sigma\setminus\Om}\dx =\int_{\R^2}v1_\Om1_{\Sigma\setminus\Om}\dx=0,$$ which together with (\ref{limmis1}) gives $|\Om_h\triangle\Om|\to 0$. Note that, if the open sets $\Om_h$ are not uniformly bounded, it is possible that $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco while $|\Om_h\triangle\Om|$ does not converge to $0$. Consider, for example, the sequence of open sets $\Om_h:=B(0,1)\cup ((B(0,h+h^{-1})\setminus\overline{B(0,h)})$ and $\Om:=B(0,1)$. We have that $|\Om_h\triangle\Om|=2\pi+h^{-2}\pi\to 2\pi$. Let us verify that $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco. For every $u\in W^{1,p}(\Om)$, property ($M_1$) is satisfied by the sequence $u_h:=u1_\Om$. For property ($M_2$), let $(h_k)$ be a sequence of indices converging to $\infty$; let $(u_k)$ be a sequence, with $u_k\in W^{1,p}(\Om_{h_k})$ for every $k$, such that $u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2)$ to a function $\phi$, while $\nabla u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2,\R^2)$ to a function $\psi$. We set $u:=\phi|_\Om$. As $u_k\rightharpoonup u$ weakly in $L^p(\Om)$ and $\nabla u_k\rightharpoonup \psi|_\Om$ weakly in $L^p(\Om,\R^2)$, we have $u\in W^{1,p}(\Om)$ and $\nabla u=\psi|_\Om$ a.e.\ on $\Om$. Now, for every ball $D\subset \Om^c$ we have $D\subset \Om^c_h$ for $h$ large enough, hence $$\int_D\phi\dx\,=\, \lim_{k\to\infty}\int_Du_k1_{\Om_{h_k}}dx\,=\,0.$$ So, $\phi =0$ a.e.\ in $\Om^c$ and similarly also $\psi =0$ a.e.\ in $\Om^c$. Hence, $\phi =u1_\Om$ and $\psi=\nabla u1_\Om$ a.e.\ in~$\R^2$. Note that, in this case, $\Om_h$ converges to $\Om$ in the Hausdorff complementary topology, since $d_H(\Om_h^c,\Om^c)=h^{-1}\to0$. By adding a small strip whose width tends to zero one can obtain an example with connected sets. } \end{remark} The following theorem can be proved as Theorem~\ref{mosccont}. \begin{theorem}\label{mosccontaux} Let $\Om_h$ and $\Om$ be open subsets of $\R^2$, and let $Y_{\Om_h}$ and $Y_\Om$ be the corresponding subspaces defined by (\ref{YOm}). Then $\Om$ is stable for the Neumann problems (\ref{eqaux1}) along the sequence $(\Om_h)$ if and only if $Y_{\Om_h}$ converges to $Y_\Om$ in the sense of Mosco. \end{theorem} \begin{remark}\label{MDLmis} {\rm If $(\Om_h)$ is a uniformly bounded sequence of open subsets of $\R^2$ such that $Y_{\Om_h}$ converges to $Y_\Om$ in the sense of Mosco, then $|\Om_h\triangle\Om|$ converges to $0$. Let $\Sigma\subset\R^2$ be a bounded set such that $\Om_h\subset\Sigma$ for every $h$. Arguing as in Remark \ref{MS->mis}, we get that $\Om\subset\Sigma$. Now, let $u(x):=\xi\cdot x$ with $\xi\in \R^2$ and $|\xi|=1$. Since $u\in L^{1,p}(\Om)$, by property ($M_1'$) there exists $u_h\in L^{1,p}(\Om_h)$ such that $\nabla u_h1_{\Om_h}\to \nabla u1_\Om$ strongly in $L^p(\R^2,\R^2)$. As $|\nabla u_h1_{\Om_h}-\nabla u1_\Om|^p=1$ a.e.\ on $\Om\setminus\Om_h$, we have \begin{equation}\label{limmis2} \lim_{h\to\infty}|\Om\setminus\Om_h| \,\leq\, \lim_{h\to\infty}\int_{\R^2}|\nabla u_h1_{\Om_h}-\nabla u1_\Om|^p\dx\,=\,0. \end{equation} On the other hand, we consider the functions $v_h\in L^{1,p}(\Om_h)$ defined by $v_h(x):=\xi\cdot x$. Up to a subsequence, $\nabla v_h1_{\Om_h}=\xi1_{\Om_h}$ converges weakly in $L^p(\R^2,\R^2)$ to some function $\psi$. By property ($M_2'$), there exists a function $v\in L^{1,p}(\Om)$ such that $\psi =\nabla v1_\Om$ a.e.\ in $\R^2$. So, it follows that $$\xi\lim_{h\to\infty}|\Om_h\setminus\Om| =\lim_{h\to\infty}\int_{\R^2}\nabla v_h 1_{\Om_h} 1_{\Sigma\setminus\Om}\dx =\int_{\R^2}\nabla v 1_\Om 1_{\Sigma\setminus\Om}\dx\,=\,0,$$ which together with (\ref{limmis2}) gives $|\Om\triangle\Om_h|\to0$. } \end{remark} \begin{remark}\label{stabiauxmono} {\rm Theorems~\ref{mosccont} and \ref{mosccontaux} can be applied in the following easy case. If $(\Om_h)$ is increasing and $\Om$ is the union of the sequence, then it is easy to see that $X_{\Om_h}$ converges to $X_\Om$ and $Y_{\Om_h}$ converges to $Y_\Om$ in the sense of Mosco. Therefore every open set is stable for the Neumann problems (\ref{eq1}) and (\ref{eqaux1}) along any increasing sequence converging to it.} \end{remark} \section{Mosco convergence of Deny-Lions spaces}\label{MDL} In this section we study the Mosco convergence of the subspaces $Y_\Om$ introduced in~(\ref{YOm}) and corresponding to the Deny-Lions spaces $L^{1,p}(\Omega)$. By Theorem \ref{mosccontaux}, this is equivalent to the stability for the Neumann problems~(\ref{eqaux1}). \begin{theorem}\label{moscograd} Let $(\Om_h)$ be a uniformly bounded sequence of open subsets of $\R^2$ that converges to an open set $\Om$ in the Hausdorff complementary topology. Assume that $|\Om_h |$ converges to $|\Om|$ and that $\Om_h^c$ has a uniformly bounded number of connected components. Then $\Om$ is stable for the Neumann problems (\ref{eqaux1}) along the sequence $(\Om_h)$. \end{theorem} To prove Theorem~\ref{moscograd} we use the following lemmas. \begin{lemma}\label{convcharac} Let $(\Om_h)$ be a uniformly bounded sequence of open subsets of $\R^2$ which converges to an open set $\Om$ in the Hausdorff complementary topology. Assume that $|\Om_h | \to |\Om|$. Then $1_{\Om_h}\to 1_\Om$ in measure, i.e., $|\Om_h\triangle\Om|\to 0$. Moreover, if $\var_h\rightharpoonup \var$ weakly in $L^r(\R^2)$ for some $12$ the result follows from the Sobolev embedding theorem, which yields the continuity of $v$. \end{proof} \begin{lemma}\label{moscostante1} Let $(\Om_h)$ be a uniformly bounded sequence of open subsets of $\R^2$ which converges to an open set $\Om$ in the Hausdorff complementary topology, and let $(v_h)$ be a sequence in $W^{1,q}(\R^2)$, which converges to a function $v$ weakly in $W^{1,q}(\R^2)$. Assume that $\Om_h^c$ has a uniformly bounded number of connected components and that every function $v_h$ is constant $C_q$-q.e.\ in each connected component of $\Om_h^c$. Then $v$ is constant $C_q$-q.e.\ in each connected component of $\Om^c$. \end{lemma} \begin{proof} Let $C_h^1,\ldots,C_h^{n_h}$ be the connected components of $\Om_h^c$. Passing to a subsequence, we can assume that $n_h$ does not depend on~$h$, and that the sets $C_h^i$ converge in the Hausdorff metric to some connected sets $C^i$ as $h\to \infty$. Let us prove that $v$ is constant q.e.\ in each~$C^i$. This is trivial if $C^i$ contains only a single point. If $C^i$ has more than one point, there exists $r>0$ such that ${\rm diam}(C_h^i)>2r$ for $h$ large enough. Let us prove that the constant values $c_h^i$ taken by $v_h$ on $C_h^i$ are bounded uniformly with respect to~$h$. To this aim let us consider a point $x_h\in C^i_h$. Since ${\rm diam}(C_h^i)>2r$, we have $C^i_h\setminus B(x_h,r)\neq\emptyset$, and by connectedness \begin{equation}\label{polar} C^i_h\cap\partial B(x_h,\rho)\neq\emptyset\qquad \hbox{ for every } 0<\rho0$ let $C_\e:=\{x\in\R^2: {\rm dist}(x,C)<\e\}$, and let $u_\e$ be a solution of problem (\ref{eqaux1}) in $\Om_\e:=\Om\setminus \overline C_\e$. Let $v_\e$ be the unique function in $W^{1,q}(\R^2)$ such that $\nabla v_\e=Ra(x,\nabla u_\e)1_{\Om_\e}$ a.e.\ in $\R^2$. By Remark \ref{stabiauxmono}, $\nabla u_\e$ converges to $\nabla u$ strongly in $L^p(\R^2,\R^2)$ and so $v_\e$ converges to $v$ strongly in $W^{1,q}(\R^2)$. By construction $\nabla v_\e=0$ in $C_\e$. As $C_\e$ is a connected open set containing $C$, we have that $v_\e$ is constant $C_q$-q.e.\ on $C$. Since a subsequence of $v_\e$ converges to $v$ $C_q$-q.e.\ on $\R^2$, we conclude that $v$ is constant $C_q$-q.e.\ on $C$. \end{proof} \begin{proof}[Proof of Theorem~\ref{moscograd}.] Let $a\colon \R^2\times\R^2\to\R^2$ be a Carath\'eodory function satisfying (\ref{a1})--(\ref{a3}) and let $u_h$ and $u$ be solutions to problems (\ref{eqaux1}) in $\Om_h$ and $\Om$. Taking $u_h$ as test function in (\ref{eqaux2}) in $\Om_h$ and using (\ref{a3}) we obtain that $\nabla u_h1_{\Om_h}$ is bounded in $L^p(\R^2,\R^2)$. By (\ref{a2}) we obtain also that $a(x,\nabla u_h)1_{\Om_h}$ is bounded in $L^q(\R^2)$. Passing to a subsequence, we may assume that $\nabla u_h1_{\Om_h}\rightharpoonup\Psi$ weakly in $L^p(\R^2,\R^2)$ and $a(x,\nabla u_h)1_{\Om_h}\rightharpoonup\Phi$ weakly in $L^q(\R^2,\R^2)$. By (\ref{eqaux2}) we have ${\rm div}( a(x,\nabla u_h)1_{\Om_h})=0$ in ${\mathcal D}'(\R^2)$, hence ${\rm div}\,\Phi=0$ in ${\mathcal D}'(\R^2)$. If $\Om'\subset\subset\Om$, by the Hausdorff complementary convergence we have $\Om'\subset\subset\Om_h$ for $h$ large enough. Since the set of gradients of functions of $L^{1,p}(\Om')$ is closed in $L^p(\Om',\R^2)$, the vectorfield $\Psi$ is the gradient of a function of $L^{1,p}(\Om')$. As $\Om'$ is arbitrary, we can construct $u^* \in L^{1,p}(\Om)$ such that $\Psi=\nabla u^*$ a.e.\ in $\Om$. On the other hand, by Lemma \ref{convcharac} we have $\Psi=0$ a.e.\ in $\Om^c$, hence $\Psi=\nabla u^*1_\Om$ a.e.\ in $\R^2$. Let us prove that $\Phi=a(x,\nabla u^*)1_\Om$ a.e.\ in $\R^2$. By Lemma \ref{convcharac} it is enough to prove the equality in every open ball $B\subset\subset\Om$. Note that by the Hausdorff complementary convergence we have $B\subset\subset\Om_h$ for $h$ large enough. By adding suitable constants, we may assume that the mean values of $u_h$ and $u^*$ on $B$ are zero. Thus the Poincar\'e inequality and the Rellich theorem imply that $u_h$ converges to $u^*$ strongly in $L^p(B)$. Let $z\in W^{1,p}(B)$ and let $\var\in C^\infty_c(B)$ with $\var\ge 0$. For $h$ large enough we have $B\subset\subset\Om_h$, thus by monotonicity we have \begin{equation}\label{gdm30} \int_B (a(x,\nabla z)-a(x,\nabla u_h))\cdot(\nabla z-\nabla u_h)\var dx\ge0. \end{equation} By (\ref{eqaux2}) we have also $$ \int_B a(x,\nabla u_h)\cdot\nabla (( z- u_h)\,\var) dx =0, $$ which, together with (\ref{gdm30}), gives \begin{equation}\label{gdm31} \int_B a(x,\nabla z)\cdot\nabla (( z- u_h)\var) dx -\int_B (a(x,\nabla z)-a(x,\nabla u_h))\cdot\nabla \var \,(z-u_h)dx\ge 0. \end{equation} We can pass to the limit in each term of (\ref{gdm31}) and we get \begin{equation}\label{gdm32} \int_B a(x,\nabla z)\cdot\nabla (( z- u^*)\var) dx -\int_B (a(x,\nabla z)-\Phi)\cdot\nabla \var \,(z-u^*)dx\ge0. \end{equation} As ${\rm div}\,\Phi=0$ in ${\mathcal D}'(B)$, we have \begin{equation}\label{gdm33} \int_B \Phi\cdot\nabla (( z- u^*)\var) dx =0. \end{equation} {}{}From (\ref{gdm32}) and (\ref{gdm33}) we obtain $$ \int_B (a(x,\nabla z)-\Phi)\cdot(\nabla z-\nabla u^*)\,\var dx\ge0. $$ As $\var$ is arbitrary, we get $(a(x,\nabla z)-\Phi)\cdot(\nabla z-\nabla u^*)\ge0$ a.e.\ in $B$. In particular, taking $z(x):= u^*(x)\pm\e \xi\cdot x$, with $\xi\in\R^2$ and $\e>0$, we obtain $\pm (a(x,\nabla u^*\pm\e \xi)-\Phi)\cdot\xi\ge0$ a.e.\ in $B$. As $\e$ tends to zero we get $(a(x,\nabla u^*)-\Phi)\cdot\xi=0$ a.e.\ in $B$, which implies that $a(x,\nabla u^*)=\Phi$ a.e.\ in $B$ by the arbitrariness of $\xi$. Let us prove now that \begin{equation}\label{eqlimit} \int_\Om a(x,\nabla u^*)\cdot\nabla z\dx=0\qquad\forall z\in L^{1,p}(\Om). \end{equation} By Lemma \ref{conj} for every $h$ there exists $v_h\in W^{1,q}(\R^2)$ such that $\nabla v_h= Ra(x, \nabla u_h)1_{\Om_h}$ a.e.\ in $\R^2$. Moreover $v_h$ is constant $C_q$-q.e.\ on each connected component of $\Om_h^c$. As $a(x, \nabla u_h)1_{\Om_h}$ converges to $a(x, \nabla u^*)1_{\Om}$ weakly in $L^q(\R^2,\R^2)$, there exists a function $v\in W^{1,q}(\R^2)$ such that $ v_h \rightharpoonup v$ weakly in $W^{1,q}(\R^2)$ and $\nabla v=R a(x, \nabla u^*)1_\Om$ a.e.\ in $\R^2$. So, we have to prove that \begin{equation}\nonumber \int_\Om R\nabla v\cdot \nabla z\dx=0\qquad\forall z\in L^{1,p}(\Om). \end{equation} {}From the Lemma \ref{moscostante1} it follows that $v$ is constant $C_q$-q.e.\ on the connected components of $ \Om^c$. By \cite[Theorem 4.5]{hekima} we can approximate $v$ strongly in $W^{1,q}(\R^2)$ by a sequence of functions $v_h \in C^\infty_c(\R^2)$ that are constant in suitable neighborhoods $U^i_h$ of each connected component $C^i$ of $\Om^c$. Let $z\in L^{1,p}(\Om)$ and $z_h \in W_0^{1,p}(\Om)$ such that $ z_h=z$ in $\Om \setminus \bigcup_{i} U^i_h$. Then, we have \begin{equation}\label{lasteqn} \int_\Om R\nabla v_h \cdot \nabla z\dx = \int_\Om R \nabla v_h \cdot \nabla z_h \dx = 0, \end{equation} where the last equality follows from the fact that the vector field $R\nabla v_h$ is divergence free. Then, passing to the limit in (\ref{lasteqn}) for $h\to\infty$, we get \[ \int_\Om a(x,\nabla u^*)\cdot \nabla z\dx= -\int_\Om R\nabla v \cdot \nabla z dx=0.\] So $u^*$ is a solution of (\ref{eqaux2}) in $\Om$, hence $\nabla u^*=\nabla u$ a.e.\ in $\Om$ by uniqueness of the gradients. This implies that $\nabla u_h1_{\Om_h}$ converges to $\nabla u1_{\Om}$ weakly in $L^p(\R^2,\R^2)$ and $a(x,\nabla u_h)1_{\Om_h}$ converges to $a(x,\nabla u)1_{\Om}$ weakly in $L^q(\R^2,\R^2)$. Since $|\Om_h \triangle \Om|$ tends to $0$ by Lemma \ref{convcharac}, from the identity $a(x,\nabla u_h1_{\Om_h})=a(x,\nabla u_h)1_{\Om_h}+ a(x,0)1_{\Om_h^c}$ we conclude also that $a(x,\nabla u_h1_{\Om_h})$ converges to $a(x,\nabla u1_{\Om})$ weakly in $L^q(\R^2,\R^2)$. To prove the strong convergence, we consider the integral $$ I_h:=\int_{\R^2} (a(x,\nabla u_h1_{\Om_h})-a(x,\nabla u1_{\Om})) \cdot(\nabla u_h1_{\Om_h} - \nabla u1_\Om)dx. $$ Since by (\ref{eqaux2}) $$ \int_{\Om_h} a(x,\nabla u_h)\cdot\nabla u_h \dx=0 \qquad\mbox{and}\qquad \int_{\Om} a(x,\nabla u)\cdot\nabla u \dx=0, $$ we have $$ I_h=-\int_{\Om} a(x,\nabla u_h1_{\Om_h}) \cdot\nabla u\,dx - \int_{\Om_h}a(x,\nabla u1_{\Om})\cdot\nabla u_h1_{\Om_h} \dx. $$ Therefore \begin{equation}\label{gdm40} \lim_{h\to\infty} I_h=-2\int_{\Om} a(x,\nabla u) \cdot\nabla u\dx = 0, \end{equation} where the last equality can be deduced from (\ref{eqaux2}). The strong convergence in $L^p(\R^2,\R^2)$ of $\nabla u_h1_{\Om_h}$ to $\nabla u1_{\Om}$ follows now from (\ref{gdm40}) and from Lemma~\ref{lemmastrong}. \end{proof} \begin{remark}\label{pgreat2} {\rm As we said in the introduction, in the case $p>2$ the stability result for problem (\ref{eqaux1}) is not true under our hypotheses. Indeed, let us consider $$S:=[1,3]\times\{0\},\quad S_h:=([1,2-1/h]\cup [2 + 1/h,3])\times\{0\},$$ $$\Om:= B(0,3)\setminus(\overline{B(0,1)}\cup S) \quad\mbox{ and }\quad\Om_h:= B(0,3)\setminus(\overline{B(0,1)}\cup S_h).$$ Let $\var\in C^\infty_c(0,\infty)$ be such that $\var(\rho)=\rho^{-p}$ for $1\leq \rho\leq 3$. We set $$ a(x,\xi):=|\xi|^{p-2}\xi - \var(|x|)Rx, $$ where $R$ is the rotation by $\pi/2$ defined by $Rx:=(-x_2,x_1)$. Let $u_h$ and $u$ be solutions of problems (\ref{eqaux1}) in $\Om$ and $\Om_h$, with $\int_{\Om_h} u_h\dx=\int_\Om u\dx=0$. For every $x\in\Om$, let $0<\theta(x)<2\pi$ be the angle between $x$ and the positive $x_1$-axis. As $\nabla \theta(x) =Rx/|x|^2$, we have that $u=\theta -\pi$ in $\Om$. If the open set $\Om$ were stable for problem (\ref{eqaux1}) along the sequence $(\Om_h)$, then $\nabla u_h$ would converge strongly to $\nabla u$ in $L^p(\Om,\R^2)$. By the Poincar\'e inequality we would have that $u_h$ converges strongly to $u$ in $W^{1,p}(\Om)$. For every $v\in W^{1,p}(\Om)$, let $v^+$ and $v^-$ be the upper and lower traces of $v$ on $S$, defined by \begin{equation}\label{traces} v^+(x):=\lim_{\substack{y\to x\\{y_2}>0}}v(y)\quad\mbox{ and }\quad v^-(x):=\lim_{\substack{y\to x\\{y_2}<0}}v(y). \end{equation} From the convergence of $u_h$ to $u$ in $W^{1,p}(\Om)$, we obtain that $u_h^+\to u^+$ and $u_h^-\to u^-$ uniformly on $S$ (recall that $p>2$ here). Since $u_h^+(2,0)=u_h^-(2,0)$ by the continuity of $u_h$, we obtain $u^+(2,0)=u^-(2,0)$, which contradicts the fact that $u^+(2,0)=-\pi$ and $u^-(2,0)=\pi$, being $u={\theta -\pi}$. } \end{remark} \section{Mosco convergence of Sobolev spaces} In this section we study the convergence in the sense of Mosco of the subspaces $X_{\Om}$ introduced in (\ref{XOm}) and corresponding to the Sobolev spaces $W^{1,p}(\Om)$. The convergence of $X_{\Om_h}$ to $X_{\Om}$ will be obtained from the convergence of $Y_{\Om_h}$ to $Y_{\Om}$ and from the following approximation theorem for functions which are locally constant on the limit open set $\Om$. \begin{theorem}\label{apcostanti} Let $(\Om_h)$ be a uniformly bounded sequence of open subsets of $\R^2$ which converges to an open set $\Om$ in the Hausdorff complementary topology. Assume that $|\Om_h|$ converges to $|\Om|$ and that $\Om_h^c$ is connected for every $h$. Then for every $u\in W^{1,p}(\Om)$ with $\nabla u=0$ a.e.\ in $\Om$, there exists a sequence $u_h\in W^{1,p}(\Om_h)$ such that $u_h1_{\Om_h}$ converges to $u1_{\Om}$ strongly in $L^p(\R^2)$ and $\nabla u_h1_{\Om_h}$ converges to $0$ strongly in $L^p(\R^2,\R^2)$. \end{theorem} The proof of this theorem is postponed. We are now in a position to state the main result of the paper. \begin{theorem}\label{moscosobolev} Let $(\Om_h)$ be a uniformly bounded sequence of open subsets of $\R^2$ which converges to an open set $\Om$ in the Hausdorff complementary topology, with $|\Om_h |$ converging to $|\Om|$. Assume that $\Om_h^c$ has a uniformly bounded number of connected components. Then $X_{\Om_h}$ converges to $X_\Om$ in the sense of Mosco. \end{theorem} To prove Theorem \ref{moscosobolev} we need the following localization lemma. \begin{lemma}\label{Mloc->Mglob} Let $(\Om_h)$ be a uniformly bounded sequence of open subsets of $\R^2$, and let $\Om$ be a bounded open subset of $\R^2$. Assume that for every $x\in \R^2$ there exists $\e>0$ such that the sequence $X_{B(x,\e)\cap\Om_h}$ converges to $X_{B_\e(x)\cap\Om}$ in the sense of Mosco. Then $X_{\Om_h}$ converges to $X_{\Om}$ in the sense of Mosco. \end{lemma} \begin{proof} Condition ($M_2$) is easy, and condition ($M_1$) can be obtained by using a partition of unity. \end{proof} \begin{proof}[Proof of Theorem \ref{moscosobolev}.] {\it Step 1.} Assume that $\Om^c_h$ is connected for every $h$. Let us prove ($M_2$). Let $(h_k)$ be a sequence of indices converging to $\infty$, $(u_k)$ be a sequence, with $u_k\in W^{1,p}(\Om_{h_k})$ for every $k$, such that $u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2)$ to a function $\phi$, while $\nabla u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2,\R^2)$ to a function $\psi$. {}From Lemma \ref{convcharac} it follows that $\phi$ and $\psi$ vanish a.e.\ in $\Om^c$. Let $\Om'\subset\subset\Om$ be an open set. By the Hausdorff complementary convergence we have $\Om'\subset\subset\Om_h$ for $h$ large enough. So, $u_k|_{\Om'}$ converges weakly to $\phi|_{\Om'}$ in $L^p(\Om')$ and $\nabla u_k|_{\Om'}$ converges weakly to $\psi|_{\Om'}$ in $L^p(\Om',\R^2)$. Hence $\phi|_{\Om'}\in W^{1,p}(\Om')$ and $\psi|_{\Om'}=\nabla\phi|_{\Om'}$ in $\Om'$. {}From the arbitrariness of $\Om'$, it follows that the function $u:=\phi|_\Om$ belongs to $ W^{1,p}(\Om)$, $\phi= u1_\Om$ and $\psi =\nabla u1_\Om$ a.e.\ in $\R^2$. Now let us prove ($M_1$). Let $u\in W^{1,p}(\Om)$. We write $\Om:=\bigcup_{i=1}^m\Om_i$, where $1\leq m\leq\infty$ and $(\Om_i)$ is the family of connected components of $\Om$. Since the set of functions $u$ satisfying ($M_1$) is a closed linear subspace of $W^{1,p}(\Om)$, by a density argument it is enough to prove ($M_1$) when $u$ belongs to $L^\infty(\Om)$ and vanishes on all connected components of $\Om$ except one. By renumbering the sequence $(\Om_i)$, we may assume that $u$ vanishes on $\Om_i$ for every $i\geq 2$. {}From Theorem \ref{moscograd} on the convergence of $Y_{\Om_h}$ to $Y_\Om$ in the sense of Mosco, there exists a sequence $z_h \in L^{1,p}(\Om_h)$ such that $\nabla z_h1_{\Om_h}$ converges strongly to $\nabla u1_\Om $ in $L^p(\R^2,\R^2).$ Let us fix a nonempty open set $A_0\subset\subset\Om_1$. We can assume that $\int_{A_0}z_hdx=\int_{A_0} udx$. For every smooth connected open set $A$, with $A_0\subset\subset A\subset\subset\Om_1$, by the Poincar\'e inequality we have that $z_h|_A\to u|_A $ strongly in $W^{1,p}(A)$. We consider now $w_h:=(-\|u\|_\infty)\vee z_h\wedge \|u\|_\infty$. We have that $w_h|_A\to u|_A $ strongly in $W^{1,p}(A)$ for every open set $A\subset\subset\Om_1$. Moreover, for every open set $E\subset\subset\Om$ the function $w_h|_E$ belongs to $W^{1,p}(E)$ for $h$ large enough. As $\|w_h\|_\infty\leq\|u\|_\infty$ and $|\nabla w_h|_E|\leq |\nabla z_h|_E|$, the sequence $(w_h|_E)$ is bounded in $W^{1,p}(E)$. By the Rellich theorem, there exists $w\in W^{1,p}(\Om)$ such that $w_h|_E$ converges to $w|_E$ strongly in $L^p(E)$ for every open set $E\subset\subset\Om$ with smooth boundary. As $w_h|_A\to u|_A $ strongly in $W^{1,p}(A)$ for every open set $A\subset\subset\Om_1$, we have that $w=u$ a.e. in $\Om_1$. For every open set $E\subset\subset\Om\setminus\Om_1$, since $|\nabla w_h|\leq |\nabla z_h|$ and $\nabla z_h|_E\to\nabla u|_E=0$ strongly in $L^p(E,\R^2)$, we have $\nabla w_h|_E\to 0$ strongly in $L^p(E,\R^2)$. Therefore we get $\nabla w=0=\nabla u$ a.e. in $\Om\setminus\Om_1$, which together with the result obtained in $\Om_1$ implies that $\nabla w_h|_E$ converges to $\nabla u|_E$ strongly in $L^p(E,\R^2)$ for every $E\subset\subset\Om$. In particular, we obtain that the function $u-w$ is locally constant in $\Om$. We claim that $\nabla w_h1_{\Om_h}$ converges to $\nabla w1_\Om$ strongly in $L^p(\R^2,\R^2)$. Indeed for every $E\subset\subset\Om$ we have $ \|\nabla w_h1_{\Om_h}-\nabla u1_\Om\|_{L^p(\R^2,\R^2)} \leq \|\nabla w_h1_{E}-\nabla u1_{E}\|_{L^p(E,\R^2)} +\, \|\nabla z_h1_{\Om_h}1_{E^c}\|_{L^p(\R^2,\R^2)} +\, \|\nabla u1_{\Om\setminus E}\|_{L^p(\R^2,\R^2)}$. Hence $$\limsup_{h\to\infty}\|\nabla w_h1_{\Om_h}- \nabla u1_\Om\|_{L^p(\R^2,\R^2)} \le 2\|\nabla u1_{\Om\setminus E}\|_{L^p(\R^2,\R^2)}, $$ and by letting $E\nearrow\Om$ we prove the claim. In a similar way, we obtain also that $ w_h1_{\Om_h}$ converges to $w1_\Om$ strongly in $L^p(\R^2)$. Since $u-w$ is locally constant in $\Om$, from Theorem \ref{apcostanti} there exists $v_h\in W^{1,p}(\Om_h)$ such that $v_h1_{\Om_h}\to (u-w)1_\Om$ strongly in $L^p(\R^2)$ and $\nabla v_h1_{\Om_h}\to 0$ strongly in $L^p(\R^2,\R^2)$. Therefore $w_h+v_h\in W^{1,p}(\Om_h)$, $(w_h+v_h)1_{\Om_h}\to u1_\Om$ strongly in $L^p(\R^2)$, and $\nabla (w_h+v_h)1_{\Om_h}\to \nabla u1_\Om$ strongly in $L^p(\R^2,\R^2)$, which give property ($M_1$). \vskip .2truecm \noindent {\it Step 2.} We now remove the hypothesis that $\Om^c_h$ is connected. Let $C_h^1,\ldots,C_h^{n_h}$ be the connected components of $\Om_h^c$. Passing to a subsequence we can assume that $n_h$ does not depend on $h$ and that the sets $C_h^i$ converge in the Hausdorff metric to some connected sets $C^i$ as $h\to\infty$. Let $C^{i_1},\ldots,C^{i_d}$ be those $C^i$ having at least two points. We set $$\Om^*:=\Bigl(\bigcup_{j=1}^dC^{i_j}\Bigr)^c\quad\mbox{ and } \quad \Om^*_h:=\Bigl(\bigcup_{j=1}^dC^{i_j}_h\Bigr)^c.$$ We have that $\Om\subset\Om^*$, $\,\Om_h\subset\Om^*_h$, and, by construction, $\Om^*_h$ converges in the Hausdorff complementary topology to $\Om^*$ and $|\Om^*_h|\to|\Om^*|$ (because $|C_h^i|\to 0$ if $i\neq i_1,\ldots,i_d$). There exists some $\eta>0$ such that ${\rm diam}\,C^{i_j}>\eta$ for every $j$, hence ${\rm diam}\,C^{i_j}_h>\eta$ for $h$ large enough. Let us observe that, for every $x\in\R^2$, the sequence $B(x,\eta /2)\cap\Om^*_h$ converges to $B(x,\eta /2)\cap\Om^*$ in the Hausdorff complementary topology and also $|B(x,\eta /2)\cap\Om^*_h|\to |B(x,\eta /2)\cap\Om^*|$ (by Lemma \ref{convcharac}). As ${\rm diam}\,C^{i_j}_h>\eta$, it is easy to see that $(B(x,\eta /2)\cap\Om^*_h)^c$ is connected for $h$ large enough. So, from Step 1 we obtain the Mosco convergence of $X_{B(x,\eta /2)\cap\Om^*_h}$ to $ X_{B(x,\eta /2)\cap\Om^*}$. Now, using Lemma \ref{Mloc->Mglob} we get the Mosco convergence of $X_{\Om^*_h}$ to $X_{\Om^*}$. As $\Om_h^*\setminus\Om_h$ is the union of a uniformly bounded number of sets with diameter tending to $0$, using the fact that $10$. Let $U$ be an open subset of $\R^2$ such that $U^c$ is connected and $d_H(U^c,\Om^c)<\delta$. Then there exists $x\in\Om^c$ such that the closed square $Q(x,9\delta)$, with centre $x$ and side length $9\delta$, intersects any curve contained in $U$ and joining the points $a$ and $b$. \end{lemma} \begin{proof}[Proof of Theorem \ref{apcostanti}.] By a density argument, it is sufficient to prove that for every connected component $\Om^0$ of $\Om$ there exists a sequence $u_h\in W^{1,p}(\Om_h)$ such that $u_h1_{\Om_h}$ converges strongly to $ 1_{\Om^0}$ in $L^p(\R^2)$ and $\nabla u_h1_{\Om_h}$ converges strongly to $ 0$ in $L^p(\R^2, \R^2)$. Let $a^0\in \Om^0$ and let $\Om^0_h$ be the connected component of $\Om_h$ which contains $a^0$ (which is defined for $h$ large enough). Up to a subsequence, $\Om^0_h$ converges in the Hausdorff complementary topology to some open set $E$. {}From Lemma \ref{compopalla2} it follows that $$E=\bigcup_{i=0}^m\Om^i,$$ where $0\leq m\leq\infty$ and $(\Om^i)$ is a family of connected components of $\Om$ (including $\Om^0$), and \begin{equation}\label{stabimise} |\Om^0_h|\to |E|. \end{equation} Let $0<\e<|\Om^0|$ be fixed. There exist a finite integer $n_\e\geq 1$ and an open set $\Om_\e$ such that \begin{equation}\label{omegaeps} E=\bigcup_{i=0}^{n_\e}\Om^i\cup\Om_\e, \end{equation} where $|\Om_\e|\leq\e$ and $\Om^i\cap\Om_\e=\emptyset$ for every $i\leq n_\e$. We fix now a point $a^i$ in each set $\Om^i$. Let $\delta>0$ be such that ${\rm dist}(a^i,\Om^c)>10\delta$ for every $i\leq n_\e$. {}From Lemma \ref{squaresep}, for $h$ big enough there exist some points $(x^{\delta,i}_h)_{i=1}^{n_\e}$, uniformly bounded in $\Om^c$, such that for every $i\leq n_\e$ the square $Q(x^{\delta,i}_h,9\delta)$ intersects any curve contained in $\Om^0_h$ and joining the points $a^0$ and $a^i$. Up to a subsequence, we have that $x^{\delta,i}_h\to x^{\delta,i}$ as $h\to\infty$, for some $x^{\delta,i}\in\Om^c$. Once again up to a subsequence, we have that $x^{\delta,i}\to x^{i}$ as $\delta\to 0$, for some $x^{i}\in\Om^c$. Let $$K^{\delta,\e}:=\bigcup_{i=1}^{n_\e}\overline{B(x^i,10\delta)}$$ and, for $i=0,\ldots,n_\e$, let $\Om_h^{\delta,\e,i}$ be the connected component of $\Om^0_h\setminus K^{\delta,\e}$ containing $a^i$. As $K^{\delta,\e}\supset Q(x^{\delta,i}_h,9\delta)$ for $\delta$ small enough and $h$ large enough, we have \begin{equation}\label{ut1} \Om_h^{\delta,\e,0}\neq\Om_h^{\delta,\e,i}\quad\mbox{ for }i\neq 0. \end{equation} Let $\var^{\delta,\e}$ be the $C_p$-capacitary potential of $K^{\delta,\e}$, i.e., the solution of the minimum problem $$\min\graffe{\int_{\R^2}[|\nabla\var|^p +|\var|^p]\dx :\var\in W^{1,p}(\R^2) \mbox{, }\var=1\mbox{ }C_p\mbox{-a.e.\ on } K^{\delta,\e}}.$$ We set \begin{equation}\label{approssimante} u_h^{\e,\delta}:=\begin{cases} 1 & \mbox{ in } \Om_h^{\delta,\e,0},\\ \salt \var^{\delta,\e} & \mbox{ in } \Om_h\setminus \Om_h^{\delta,\e,0}. \end{cases} \end{equation} As $\Om_h\cap\partial\Om_h^{\delta,\e,0}\subset K^{\delta,\e}$, we have that $u_h^{\e,\delta}\in W^{1,p}(\Om_h)$. We observe that $$\|u_h^{\delta,\e}1_{\Om_h}-1_{\Om_h^{\delta,\e,0}}\|_{L^p(\R^2)} + \|\nabla u_h^{\delta,\e}1_{\Om_h}\|_{L^p(\R^2,\R^2)} \,\leq\, 2C_p(K^{\delta,\e})^{\frac{1}{p}}.$$ As $C_p(K^{\delta,\e})\leq n_\e C_p(B(0,10\delta))$ and $p\leq 2$, we conclude that \begin{equation}\label{approx0} \limsup_{\delta\to 0}\,\limsup_{h\to\infty}\bigl[\|u_h^{\delta,\e,0}1_{\Om_h}-1_{\Om_h^{\delta,\e}}\|_{L^p(\R^2)} + \|\nabla u_h^{\delta,\e}1_{\Om_h}\|_{L^p(\R^2,\R^2)}\bigr]\,=\,0. \end{equation} We claim that \begin{equation}\label{approxi1} \limsup_{\delta\to 0}\,\limsup_{h\to\infty}\|u_h^{\e,\delta,0}1_{\Om_h}-1_{\Om^0}\|^p_{L^p(\R^2)}\leq\e, \end{equation} and \begin{equation}\label{approxi2} \limsup_{\delta\to 0}\,\limsup_{h\to\infty}\|\nabla u_h^{\e,\delta}1_{\Om_h}\|_{L^p(\R^2,\R^2)}=0, \end{equation} from which the proof of the theorem is achieved by the arbitrariness of $\e$. It is easy to see that (\ref{approxi2}) follows from (\ref{approx0}), while (\ref{approxi1}) is a consequence of (\ref{approx0}) and of the following inequality \begin{equation}\label{apmeas} \limsup_{\delta\to 0}\,\limsup_{h\to\infty}|\Om_h^{\delta,\e,0}\triangle\Om^0|\leq\e. \end{equation} So, let us prove (\ref{apmeas}). For every $i=0,\ldots n_\e$, up to a subsequence, $\Om_h^{\delta,\e,i}$ converges in the Hausdorff complementary topology, when $h\to\infty$, to some open set $\Om^{\delta,\e,i}\subset E$. We observe that $\Om_h^0\setminus K^{\delta,\e}$ converges to $E\setminus K^{\delta,\e}$ in the Hausdorff complementary topology when $h\to\infty$. Let $E^{\delta,\e,i}$ be the connected component of $E\setminus K^{\delta,\e}$ which contains $a^i$. It is easy to see that \begin{equation}\label{ut} E^{\delta,\e,i}\subset \Om^{\delta,\e,i}. \end{equation} Note that, as $\delta\searrow 0$, $K^{\delta,\e}$ converges decreasingly to the set $\{x_1,\ldots, x_{n_\e}\}$, $E^{\delta,\e,i}$ converges increasingly to $\Om^i$, and $\Om^{\delta,\e,i}$ converges increasingly to some open set $\Om^{\e,i}\subset E$. {}From (\ref{ut}), it follows that $\Om^i\subset \Om^{\e,i}$. {}From (\ref{stabimise}) and from Lemma \ref{convcharac} it follows that \begin{equation}\label{ut3} |\Om_h^0\setminus K^{\delta,\e}|\to |E\setminus K^{\delta,\e}|. \end{equation} By Lemma \ref{compopalla2} applied to $\Om_h^1:=\Om_h^{\delta,\e,0}$ and $\Om^2_h:=(\Om^0_h\setminus K^{\delta,\e})\setminus \Om_h^{\delta,\e,0}$, we have that $\Om^{\delta,\e,0}\cap\Om^{\delta,\e,i}=\emptyset$ for every $i\neq 0$, from which it follows that $\Om^{\e,0}\cap\Om^{\e,i}=\emptyset$ and hence $\Om^{\e,0}\cap\Om^{i}=\emptyset$ for every $1\leq i\leq n_\e$. Therefore, there exists an open set $\Om_\e'$, contained in the set $ \Om_\e$ introduced in (\ref{omegaeps}), such that $$\Om^{\e,0}=\Om^0\cup\Om_\e'.$$ {}From (\ref{ut3}) and from Lemmas \ref{compopalla2} and \ref{convcharac}, it follows that \begin{equation}\label{ut4} |\Om_h^{\delta,\e,0} \triangle\Om^{\delta,\e,0}|\to 0. \end{equation} As $\Om^{\delta,\e,0}\subset\Om^{\e,0}=\Om^0\cup\Om_\e'$, it follows that \begin{equation}\label{ds1} |\Om^{\delta,\e,0}_h\setminus\Om^0| \,\leq\, |\Om^{\delta,\e,0}_h\setminus\Om^{\delta,\e,0}| +|\Om^{\delta,\e,0}\setminus\Om^0|\,\leq |\Om^{\delta,\e,0}_h\setminus\Om^{\delta,\e,0}| + \,|\Om_\e'|, \end{equation} and \begin{equation}\label{ds2} |\Om^0\setminus\Om^{\delta,\e,0}_h|\,\leq\,|\Om^0 \setminus\Om^{\delta,\e,0}| +|\Om^{\delta,\e,0}\setminus\Om^{\delta,\e,0}_h|\leq\,|\Om^{\e,0} \setminus\Om^{\delta,\e,0}| +|\Om^{\delta,\e,0}\setminus\Om^{\delta,\e,0}_h|. \end{equation} As $\Om^{\delta,\e,0}$ converges increasingly to $\Om^{\e,0}$ as $\delta\to 0^+$, we have $|\Om^{\e,0}\setminus\Om^{\delta,\e,0}|\to 0$ as $\delta\to 0$. By (\ref{ut4}), passing to the limit in (\ref{ds1}) and (\ref{ds2}) first as $h\to\infty$ and then as $\delta\to 0^+$ we obtain $$ \limsup_{\delta\to 0}\,\limsup_{h\to\infty}|\Om^{\delta,\e,0}_h\setminus\Om^0|\leq |\Om_\e| +\limsup_{h\to\infty}|\Om^{\delta,\e,0}_h\setminus\Om^{\delta,\e,0}|\leq\e$$ and $$\limsup_{\delta\to 0}\,\limsup_{h\to\infty}|\Om^0\setminus\Om^{\delta,\e,0}_h|=0,$$ which give inequality (\ref{apmeas}). \end{proof} \begin{remark}\label{Sobpgreat2} {\rm In the case $p>2$ the stability result for problem (\ref{eqgen}) is not true under our hypotheses. Indeed, let us consider $$S:=[-1,1]\times\{0\},\quad S_h:=([-1,-1/h]\cup [1/h,1])\times\{0\},$$ $$\Om:= (-1,1)^2\setminus S, \quad\mbox{ and }\quad\Om_h:= (-1,1)^2\setminus S_h.$$ We set $$ a(x,\xi):=|\xi|^{p-2}\xi\quad\mbox{ and }\quad b(x,\eta):=|\eta|^{p-2}\eta -x_2, $$ where $x=(x_1,x_2)$. Let $u_h$ and $u$ be solutions of problems (\ref{eqgen}) in $\Om_h$ and $\Om$ respectively. By the symmetry of $\Om$, the solution $u$ will depend only on $x_2$. Therefore, for every $x=(x_1,x_2)\in\Om$ $$u(x)=\begin{cases} \mbox{ }w(x_2) &\mbox{ if } x_2\in (0,1),\\ -w(-x_2) &\mbox{ if } x_2\in (-1,0) \end{cases}$$ where $w$ is the solution of the one-dimensional problem \begin{equation}\label{1D} \begin{cases} -(|w'|^{p-2}w')'+|w|^{p-2}w=t & \mbox{ in }\, (0,1),\\ \salt \quad w'(0)= w'(1)=0,& \mbox{ } \end{cases} \end{equation} which turns out to be of class $C^1([0,1])$. For every $v\in W^{1,p}(\Om)$, let $v^+$ and $v^-$ be the upper and lower traces of $v$ on $S$, defined as in (\ref{traces}). If the open set $\Om$ were stable for problem (\ref{eqgen}) along the sequence $(\Om_h)$, then $u_h$ would converge strongly to $u$ in $W^{1,p}(\Om)$. Hence we would have that $u_h^+\to u^+$ and $u_h^-\to u^-$ uniformly on $S$ (recall that $p>2$ here). Since $u_h^+(0,0)=u_h^-(0,0)$ by the continuity of $u_h$, we would obtain $u^+(0,0)=u^-(0,0)$, which implies that $w(0)=0$. Let us prove that this is false. Indeed, by the maximum principle we have that $w(t)\geq 0$ for every $t\in [0,1]$. Since $w'(0)=0$ and $p>2$, we have that $w^{p-1}(t)-t<0$ in a small neighborhood $I$ of $0$ in $[0,1]$. So, from equation (\ref{1D}) the function $|w'|^{p-2}w'$ is decreasing in $I$ and hence $w'(t)<0$ for every $t\in I$. If $w(0)$ were equal to $0$, we would obtain $w(t)<0$ for every $t\in I$, which contradicts the fact that $w(t)\geq 0$ for every $t\in [0,1]$. This proves that $w(0)>0$, and hence $\Om$ is not stable for problem (\ref{eqgen}) along the sequence $(\Om_h)$ for $p>2$. } \end{remark} \section{The case of unbounded domains} We now extend the results of the previous sections to the case of unbounded domains. \begin{theorem}\label{msnlim} Let $(\Om_h)$ be a sequence of open subsets of $\R^2$ such that $(\Om^c_h)$ converges to $\Om^c$ in the sense of Kuratowski for some open subset $\Om$. Assume that, for every $R>0$, $|\Om_h\cap B(0,R)|$ converges to $|\Om\cap B(0,R)|$ and that the number of connected components of $(\Om_h\cap B(0,R))^c$ is uniformly bounded with respect to $h$. Then the sequence of subspaces $X_{\Om_h}$ {\rm (}resp. $Y_{\Om_h}${\rm )} converges to $X_{\Om}$ {\rm (}resp.\ $Y_{\Om}${\rm )}. \end{theorem} \begin{proof} We prove only the Mosco convergence of $X_{\Om_h}$ to $X_{\Om}$, since the convergence of $Y_{\Om_h}$ to $Y_\Om$ can be proved in the same way. First of all note that, from the convergence of $\Om^c_h$ to $\Om^c$ in the sense of Kuratowski, it follows that the sequence $\Om_h\cap B(0,R)$ converges to $\Om\cap B(0,R)$ in the Hausdorff complementary topology. Moreover, by the assumptions of the theorem, we can apply Theorem \ref{moscosobolev} to the sequence $\Om_h\cap B(0,R)$. So, we get that $X_{\Om_h\cap B(0,R)}$ converges to $X_{\Om\cap B(0,R)}$ in the sense of Mosco. Now let us prove ($M_2$) for $X_{\Om_h}$ and $X_\Om$. Let $(h_k)$ be a sequence of indices converging to $\infty$, $(u_k)$ be a sequence, with $u_k\in W^{1,p}(\Om_{h_k})$ for every $k$, such that $u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2)$ to a function $\phi$, while $\nabla u_k1_{\Om_{h_k}}$ converges weakly in $L^p(\R^2,\R^2)$ to a function $\psi$. It follows that $u_k1_{\Om_{h_k}\cap B(0,R)}$ converges to $\phi 1_{B(0,R)}$ weakly in $L^p(\R^2)$, while $\nabla u_k1_{\Om_{h_k}\cap B(0,R)}$ converges to $\psi 1_{B(0,R)}$ weakly in $L^p(\R^2,\R^2)$. So, by property ($M_2$) relative to the Mosco convergence of $X_{\Om_h\cap B(0,R)}$ to $X_{\Om\cap B(0,R)}$, there exists a function $u_R\in W^{1,p}(\Om\cap B(0,R))$ such that $\phi 1_{B(0,R)}=u_R1_{\Om\cap B(0,R)}$ and $\psi 1_{B(0,R)}=\nabla u_R1_{\Om\cap B(0,R)}$ a.e.\ in $\R^2$. Since $R$ is arbitrary, it is easy to construct $u\in W^{1,p}(\Om)$ such that $\phi =u1_\Om$ and $\psi=\nabla u1_\Om$ a.e.\ in $\R^2$. Let us prove property ($M_1$). Let $u\in W^{1,p}(\Om)$ and let $\e>0$. There exists $R_\e>0$ such that $$\int_{\Om\setminus B(0,R_\e)}[|u|^p+|\nabla u|^p]\dx\leq\e.$$ By property ($M_1$) relative to the Mosco convergence of $X_{\Om_h\cap B(0,R_\e+1)}$ to $X_{\Om\cap B(0,R_\e+1)}$, there exists a sequence $w^\e_h\in W^{1,p}(\Om_h\cap B(0,R_\e+1))$ such that $w^\e_h1_{\Om_h\cap B(0,R_\e+1)}$ converges strongly to $u 1_{\Om\cap B(0,R_\e+1)}$ in $ L^p(\R^2)$ and $\nabla w^\e_h1_{\Om_h\cap B(0,R_\e+1)}$ converges strongly to $\nabla u 1_{\Om\cap B(0,R_\e+1)}$ in $L^p(\R^2,\R^2)$. Let $\var_\e\in C^1_c(B(0,R_\e+1))$ such that $0\leq\var_\e\leq 1$,\,\,$\var_\e=1$ in $B(0,R_\e)$, and $\|\nabla\var_\e\|_\infty\leq C$. Now we set $u^\e_h:=\var _\e w^\e_h$. By construction $u_h^\e\in W^{1,p}(\Om_h)$, $u_h^\e1_{\Om_h}\to\var_\e u1_\Om$ strongly in $L^p(\R^2)$, and $\nabla u_h^\e1_{\Om_h}\to\var_\e\nabla u1_\Om +u\nabla\var_\e1_\Om$ strongly in $L^p(\R^2,\R^2)$. On the other hand $$ \limsup_{h\to\infty}\int_{\R^2}|u^\e_h1_{\Om_h}-u1_{\Om}|^p + |\nabla u^\e_h1_{\Om_h}-\nabla u1_{\Om}|^p\dx\,\leq \,2^{p-1}(C^p+1)\e, $$ $\var_\e u1_\Om\to u1_\Om$ strongly in $L^p(\R^2)$, and $\var_\e\nabla u1_\Om +u\nabla\var_\e1_\Om\to \nabla u1_\Om$ strongly in $L^p(\R^2,\R^2)$. Therefore, we can construct a sequence $u_h\in W^{1,p}(\Om_h)$ which satisfies ($M_1$) by a standar argument on double sequences. \end{proof} \section{Problems with Dirichlet boundary conditions} In this section we study the Mosco convergence of Sobolev and Deny-Lions spaces with prescribed Dirichlet conditions on part of the boundary. Let $A\subset\R^2$ be a bounded open set with Lipschitz boundary $\partial A$, and let $\partial_DA$ be a relatively open subset of $\partial A$ with a finite number of connected components. For every compact set $K\subset\overline A$, for every $g\in W^{1,p}(A)$, and for every pair of function $a$ and $b$ satisfying the properties (\ref{a1})--(\ref{a3}), we consider the solutions $u$ and $v$ of the mixed problems \begin{equation}\label{eqmix1} \begin{cases} -\,{\rm div}\,a(x,\nabla u)+ b(x,u)=0 & \mbox{ in }\quad A\setminus K,\\ \salt \mbox{ }u=g & \mbox{ on }\quad \partial_DA\setminus K,\\ \salt \mbox{ }a(x,\nabla u)\cdot\nu=0 & \mbox{ on }\quad \partial (A\setminus K)\setminus (\partial_DA\setminus K), \end{cases} \end{equation} and \begin{equation}\label{eqmix2} \begin{cases} -\,{\rm div}\,a(x,\nabla v)=0 & \mbox{ in }\quad A\setminus K,\\ \salt \mbox{ }v=g & \mbox{ on }\quad \partial_DA\setminus K,\\ \salt \mbox{ }a(x,\nabla v)\cdot\nu=0 & \mbox{ on }\quad \partial (A\setminus K)\setminus (\partial_DA\setminus K). \end{cases} \end{equation} Let $(K_h)$ be a sequence of compact subsets of $\overline A$, let $(g_h)$ be a sequence in $W^{1,p}(A)$, and let $(u_h)$ be the sequence of the solutions of problems (\ref{eqmix1}) corresponding to $K_h$ and $g_h$. \begin{definition}\label{stabledc} We say that the pair $(K,g)$ is stable for the mixed problems (\ref{eqmix1}) along the sequence $(K_h,g_h)$ if for every pair of functions $a$, $b$ satisfying (\ref{a1})--(\ref{a3}) the sequence $(u_h1_{K^c_h})$ converges to $u1_{K^c}$ strongly in $L^p(A)$ and the sequence $(\nabla u_h1_{K^c_h})$ converges to $\nabla u1_{K^c}$ strongly in $L^p(A,\R^2)$. The stability for problems (\ref{eqmix2}) is defined in a similar way by using only the convergence of the gradients {\rm (}as in Definition ~\ref{stableaux}{\rm )}. \end{definition} The stabilty for problems (\ref{eqmix2}) has been recently studied in \cite{dmro} in the case $a(x,\xi)=\xi$. In this section we will study the stability in the general case by using again the notion of Mosco convergence. We set $$W^{1,p}_g(A\setminus K,\partial_DA\setminus K):=\{u\in W^{1,p}(A\setminus K): u=g\mbox{ on } \partial_DA\setminus K\},$$ and $$ L^{1,p}_g(A\setminus K,\partial_DA\setminus K):=\{u\in L^{1,p}(A\setminus K): u=g\mbox{ on } \partial_DA\setminus K\},$$ where the equality $u=g$ on $\partial_DA\setminus K$ is intended in the usual sense of traces. As in Section \ref{snp} the space $W^{1,p}_g(A\setminus K,\partial_DA\setminus K)$ will be identified with the closed linear subspace $X_K^g(A)$ of $L^p(A)\times L^p(A,\R^2)$ defined by \begin{equation}\label{XAg} X_K^g(A):=\{(u1_{K^c},\nabla u1_{K^c}): u\in W^{1,p}_g(A\setminus K,\partial_DA\setminus K)\}. \end{equation} For problem (\ref{eqmix2}), we consider in $L^{1,p}_g(A\setminus K,\partial_DA\setminus K)$ the equivalence relation $\sim$ defined in (\ref{defreleq}). Note that in this case $v_1\sim v_2$ if and only if $v_1=v_2$ a.e.\ in those connected components of $A\setminus K$ whose boundary intersects $\partial_DA\setminus K$ and $\nabla v_1=\nabla v_2$ a.e.\ in the other connected components of $A\setminus K$. The corresponding quotient space, denoted by $L^{1,p}_g(A\setminus K,\partial_DA\setminus K)/_\sim$, will be identified with the closed linear subspace $Y_K^g(A)$ of $ L^p(A,\R^2)$ defined by \begin{equation}\label{YAg} Y_K^g(A):=\{\nabla u1_{K^c}: u\in L^{1,p}_g(A\setminus K,\partial_DA\setminus K)\}. \end{equation} Let $K_h$, $K$ be compact subsets of $\overline A$ and let $g_h$, $g\in W^{1,p}(A)$. Let $X_{K_h}^{g_h}(A)$ and $X_K^g(A)$ be the corresponding subspaces defined by (\ref{XAg}). We recall that $X_{K_h}^{g_h}(A)$ converges to $X_K^g(A)$ in the sense of Mosco if the following two properties hold: \begin{itemize} \item[($M_1''$)] for every $u\in W^{1,p}_g(A\setminus K,\partial_DA\setminus K)$, there exists a sequence $u_h\in W^{1,p}_{g_h}(A\setminus K_h,\partial_DA\setminus K_h)$ such that $u_h1_{K^c_h}$ converges strongly to $u1_{K^c}$ in $L^p(A)$ and $\nabla u_h1_{K^c_h}$ converges strongly to $\nabla u1_{K^c}$ in $L^p(A,\R^2)$; \item[($M_2''$)] if $(h_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence, with $u_k\in W^{1,p}_{g_{h_k}}(A\setminus K_{h_k}, \partial_DA\setminus K_{h_k})$ for every $k$, such that $u_k1_{K^c_{h_k}}$ converges weakly in $L^p(A)$ to a function $\phi$, while $\nabla u_k1_{K^c_{h_k}}$ converges weakly in $L^p(A,\R^2)$ to a function $\psi$, then there exists $u\in W^{1,p}_g(A\setminus K,\partial_DA\setminus K)$ such that $\phi=u1_{K^c}$ and $\psi=\nabla u1_{K^c}$ a.e.\ in $A$. \end{itemize} Analogously, the convergence of $Y_{K_h}^{g_h}(A)$ to $Y_K^g(A)$ in the sense of Mosco can be characterized by using only the convergence of the extensions of the gradients. \begin{remark}\label{rem} {\rm As in Section \ref{MC} we can prove that the Mosco convergence of $X_{K_h}^{g_h}(A)$ to $X_K^g(A)$ (resp.\ of $Y_{K_h}^{g_h}(A)$ to $Y_K^g(A)$) is equivalent to the stability of $(K,g)$ for the mixed problems (\ref{eqmix1}) (resp.\ (\ref{eqmix2}) along the sequence $(K_h,g_h)$).} \end{remark} The following theorem is the main result of this section. \begin{theorem}\label{moscodati} Let $A$ be a bounded open subset of $\R^2$ with Lipschitz boundary $\partial A$ and let $\partial_DA$ be a relatively open subset of $\partial A$ with a finite number of connected components. Let $(g_h)$ be a sequence in $W^{1,p}(A)$ converging strongly to a function $g$ in $W^{1,p}(A)$, and let $(K_h)$ be a sequence of compact subsets of $\overline A$ converging to a set $K$ in the Hausdorff metric. Assume that $|K_h |$ converges to $|K|$ and that the sets $K_h$ have a uniformly bounded number of connected components. Then $X_{K_h}^{g_h}(A)$ converges to $X_K^g(A)$ {\rm (}resp.\ $Y_{K_h}^{g_h}(A)$ converges to $Y_K^g(A)${\rm )} in the sense of Mosco. \end{theorem} \begin{proof} The main idea of this proof is due to Chambolle. Let us first prove the Mosco convergence of $X_{K_h}^{g_h}(A)$ to $X_K^g(A)$. Let $\Sigma$ be an open ball in $\R^2$ such that $\overline A\subset\Sigma$. Let $\tilde g_h$, $\tilde g\,\in W^{1,p}(\Sigma)$ be extensions of $g_h$ and $g$ to $\Sigma$ such that $\tilde g_h$ converges to $\tilde g$ strongly in $W^{1,p}(\Sigma)$. We set $$\Om_h:=\Sigma\setminus(K_h\cup (\partial A\setminus\partial_DA))\quad\mbox{ and } \quad\Om:=\Sigma\setminus(K\cup (\partial A\setminus\partial_DA)).$$ Note that $\Om_h$ and $\Om$ satisfy the assumptions of Theorem \ref{moscosobolev}. Let us prove property $(M2'')$. Let $(h_k)$ be a sequence of indices that tends to $\infty$, and $u_k\in W^{1,p}_{g_{h_k}}(A\setminus K_{h_k},\partial_DA\setminus K_{h_k})$ such that $u_k1_{K^c_{h_k}}$ converges weakly to $\phi$ in $L^p(A)$ and $\nabla u_k1_{K^c_{h_k}}$ converges weakly to $\psi$ in $L^p(A,\R^2)$. Let $\tilde u_k$ be the extension of $u_k$ defined by $$ \tilde u_k:=\begin{cases} u_k1_{K^c_{h_k}} &\mbox{ in }\, A,\\ \tilde g_{h_k} &\mbox{ in } \Sigma\setminus A,\end{cases} $$ and let $\tilde\phi$ and $\tilde\psi$ be defined by $$ \tilde \phi:=\begin{cases} \phi &\mbox{ in }\, A,\\ \tilde g &\mbox{ in } \Sigma\setminus A,\end{cases} \qquad\qquad\mbox{ and }\qquad\qquad \tilde \psi:=\begin{cases} \psi &\mbox{ in }\, A,\\ \nabla\tilde g &\mbox{ in } \Sigma\setminus A.\end{cases}$$ As $u_k=g_{h_k}$ on $\partial_DA\setminus K_{h_k}$, we have $\tilde u_k\in W^{1,p}(\Om_{h_k})$. Since $\tilde u_k1_{\Om_{h_k}}$ converges to $\tilde\phi 1_\Sigma$ weakly in $L^p(\R^2)$ and $\nabla \tilde u_k1_{\Om_{h_k}}$ converges to $\tilde\psi 1_\Sigma$ weakly in $L^p(\R^2,\R^2)$, by Theorem \ref{moscosobolev} we conclude that there exists $\tilde u\in W^{1,p}(\Om)$ such that $\tilde\phi 1_\Sigma =\tilde u 1_\Om$ and $\tilde\psi 1_\Sigma =\nabla\tilde u 1_\Om$. Let $u$ be the restriction of $\tilde u$ to $A\setminus K$. Then $u\in W^{1,p}(A\setminus K)$ and we have that $\phi=u1_{K^c}$ and $\psi=\nabla u1_{K^c}$ a.e.\ in $A$. As $\tilde u\in W^{1,p}(\Om)$, the traces of $\tilde u$ on both sides of $\partial_DA\setminus K$ coincide. Since $\tilde u=\tilde g$ a.e.\ in $\Sigma\setminus A$, we conclude that $u=\tilde g=g$ in the sense of traces on $\partial_DA\setminus K$. Therefore $u\in W^{1,p}_g(A\setminus K,\partial_DA\setminus K)$. Now we prove property ($M_1''$). Let $u\in W^{1,p}_g(A\setminus K,\partial_DA\setminus K)$. Let $\tilde u$ be the extension of $u$ defined by \[ \tilde u= \begin{cases} u1_{K^c} & \text{ a.e.\ in } A,\\ \tilde g & \text{ a.e.\ in } \Sigma\setminus A. \end{cases} \] As $u=g$ on $\partial_DA\setminus K$, we have that $\tilde u\in W^{1,p}(\Om)$. By Theorem \ref{moscosobolev} there exists a sequence $\tilde u_h\in W^{1,p}(\Om_h)$ such that $\tilde u_h1_{\Om_h}$ converges to $\tilde u1_\Om$ strongly in $L^p(\R^2)$ and $\nabla \tilde u_h1_{\Om_h}$ converges to $\nabla\tilde u1_\Om $ strongly in $L^p(\R^2,\R^2).$ We consider the function $$\phi_h:=( \tilde u_h - \tilde g_h)|_{\Sigma\setminus\overline A}.$$ By construction, $\phi_h\to 0$ strongly in $W^{1,p}(\Sigma\setminus\overline A)$. Therefore there exists a sequence $v_h\in W^{1,p}(\Sigma)$, converging to $0$ strongly in $W^{1,p}(\Sigma)$, such that $v_h|_{\Sigma\setminus\overline A}=\phi_h$ a.e.\ in $\Sigma\setminus\overline A$. We set $$u_h: = \bigl( \tilde u_h - v_h \bigr)|_{A\setminus K_h}.$$ By construction, we have that $u_h \in W^{1,p}(A \setminus K_h)$ and $u_h = g_h$ in the sense of traces on $\partial_D A\setminus K_h$. Moreover, we have that $ u_h1_{K^c_h}$ converges to $u1_{K^c}$ strongly in $L^p(A)$ and $\nabla u_h1_{K^c_h}$ converges to $\nabla u1_{K^c}$ strongly in $L^p(A,\R^2)$. \vskip .2truecm Now let us prove that $Y_{K_h}^{g_h}(A)$ converges to $Y_K^g(A)$ in the sense of Mosco. Property ($M_2''$) is obtained arguing as in \cite[Lemma 4.1]{dmro}. So, let us prove ($M_1''$). Let $u\in L^{1,p}_g(A\setminus K,\partial_DA\setminus K)$. We set for every $k\in\N$ $$u^k:=(g-k)\vee u\wedge(g+k).$$ Then $ u^k\in W^{1,p}_g(A\setminus K,\partial_DA\setminus K)$ and $\nabla u^k1_{K^c}\to\nabla u1_{K^c}$ strongly in $L^p(A,\R^2)$. {}From property ($M_1''$) proved above for the Mosco convergence of $X_{K_h}^{g_h}(A)$ to $X_K^g(A)$, for every $k$ there exists $u^k_h\in W^{1,p}_{g_h}(A\setminus K_h,\partial_DA\setminus K_h)$ such that $\nabla u^k_h1_{K_h^c}\to\nabla u^k1_{K^c}$ in $L^p(A,\R^2)$. 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