\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 178, pp. 1--25. \newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/178\hfil Change of homogenized absorption term] {Change of homogenized absorption term in diffusion processes with reaction on the boundary of periodically distributed \\ asymmetric particles of critical size} \author[J. I. D\'iaz, D. G\'omez-Castro, T. A. Shaposhnikova, M. N. Zubova \hfil EJDE-2017/178\hfilneg] {Jes\'us Ildefonso D\'iaz, David G\'omez-Castro, \\ Tatiana A. Shaposhnikova, Maria N. Zubova} \address{Jes\'us Ildefonso D\'iaz \newline Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain} \email{ildefonso.diaz@mat.ucm.es} \address{David G\'omez-Castro \newline Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain} \email{dgcastro@ucm.es} \address{Tatiana A. Shaposhnikova \newline Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia} \email{shaposh.tan@mail.ru} \address{Maria N. Zubova \newline Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia} \email{zubovnv@mail.ru} \thanks{Submitted June 12, 2017. Published July 13, 2017.} \subjclass[2010]{35B25, 35B40, 35J05, 35J20} \keywords{Homogenization; diffusion processes; periodic asymmetric particles; \hfill\break\indent microscopic non-linear boundary reaction; critical sizes} \begin{abstract} The main objective of this article is to get a complete characterization of the homogenized global absorption term, and to give a rigorous proof of the convergence, in a class of diffusion processes with a reaction on the boundary of periodically ``microscopic'' distributed particles (or holes) given through a nonlinear microscopic reaction (i.e. under nonlinear Robin microscopic boundary conditions). We introduce new techniques to deal with the case of non necessarily symmetric particles (or holes) of critical size which leads to important changes in the qualitative global homogenized reaction (such as it happens in many problems of the Nanotechnology). Here we shall merely assume that the particles (or holes) $ G_{\varepsilon }^{j}$, in the $n$-dimensional space, are diffeomorphic to a ball (of diameter $ a_{\varepsilon }=C_0\varepsilon ^{\gamma }$, $\gamma =\frac{n}{n-2}$ for some $C_0>0)$. To define the corresponding ``new strange term'' we introduce a one-parametric family of auxiliary external problems associated to canonical cellular problem associated to the prescribed asymmetric geometry $G_0$ and the nonlinear microscopic boundary reaction $\sigma (s)$ (which is assumed to be merely a H\"older continuous function). We construct the limit homogenized problem and prove that it is a well-posed global problem, showing also the rigorous convergence of solutions, as $\varepsilon \to 0$, in suitable functional spaces. This improves many previous papers in the literature dealing with symmetric particles of critical size. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks %\tableofcontents \section{Introduction} It is well-known that the asymptotic behaviour of the solution of many relevant diffusion processes with reaction on the boundary of periodically \textquotedblleft microscopic\textquotedblright\ distributed particles (or holes) is described through the solution of a global reaction-diffusion problem in which the global reaction term (usually an absorption term if the microscopic reactions are given by monotone non-decreasing functions) maintains the same structural properties as the microscopic reaction (see, for instance, \cite{Conca+Diaz+Linan+Timofte:2004}) and its many references to previous results in the literature). A certain critical size of the \textquotedblleft microscopic particles\textquotedblright\ may be responsible of a change in the nature of the homogenized global absorption term, with respect to the structural assumptions on the microscopic boundary reaction kinetic. It seems that the first result in that direction was presented in the pioneering paper by V. Marchenko and E. Hruslov \cite{Marchenko1964} dealing with microscopic non-homogeneous Neumann boundary condition (see also the study made by E. Hruslov concerning linear microscopic Robin boundary conditions in \cite{Hruslov1977first+bvp,Hruslov:1979second+bvp}). A --perhaps-- more popular presentation of the appearance of some ``strange term'' was due to D. Cioranescu and F.Murat \cite {Cioranescu+Murat:1997} dealing with microscopic Dirichlet boundary conditions (see also \cite{Kaizu1989}). This change of behavior from the microscopic reaction to the global homogeneized reaction term is one of the characteristics of the nanotechnological effects (see, e.g., \cite{Schimpf:2002supported+gold+nanoparticles}) and it does not appear for particles of bigger size (relative to their repetition) than the critical scale (see, e.g., \cite{diaz+gomez-castro+podolskii+shaposhnikova2017jmaa} and the references therein). The total identification of the new or ``strange'' reaction term is an important task which was considered by many authors under different technical assumptions. In the case of nonlinear microscopic boundary reactions the first result in the literature was the 1997 paper by Goncharenko \cite{Goncharenko:1997} (see also the precedent paper \cite{Kaizu1989}). The identification (and the rigorous proof of the convergence in the homogenization process) requires to assume that the particles (or holes) are symmetric balls of diameter $ a_{\varepsilon }=C_0\varepsilon ^{\gamma }$, $\gamma =\frac{n}{n-2}$, for some $C_0>0$. Many other researches were developed for different problems concerning critical sized balls (see \cite{Shaposhnikova+Zubova:2011homogenization+laplacian+strong+convergence+corrector, Podolskii:2015extension+operator+noncritical+general+p, diaz+gomez-castro+podolskii+shaposhnikova2017signorini+p-Laplace+1-2} and the references therein). Recently, a unifying study concerning the homogenization for particles (or holes) given by symmetric balls of critical order was presented in \cite{diaz+gomez-castro+podolskii+shaposhnikova2017generalmmg}: the treatment was extended to a microscopic reaction given by a general maximal monotone graph which allows to include, as special problems, the cases of Dirichlet or nonlinear Robin microscopic boundary conditions. The case of particles of general shape when $n = 2$ was studied in \cite{Perez2014}, with the limit behaviour being similar to the case of spherical inclusions and $n \ge 2$. The main task of this paper is to get a complete characterization of the homogenized global absorption term in the class of problems given through a nonlinear microscopic reaction (i.e. under nonlinear Robin microscopic boundary conditions) and for non necessarily symmetric particles (or holes). Here we will merely assume that the particles (or holes) $G_{\varepsilon }^{j}$ are a rescaled version of a set $G_0$, diffeomorphic to a ball (where the scaling factor is $a_{\varepsilon }=C_0\varepsilon ^{\gamma }$, $\gamma =\frac{n}{n-2}$ for some $C_0>0$). To define the corresponding new ``strange term'' we introduce a one-parametric family of auxiliary external problems associated to canonic cellular problem, which play the role of a ``nonlinear capacity'' of $G_0$ and the nonlinear microscopic boundary reaction $\sigma (s)$ (which is assumed to be merely a H\"older continuous function). We construct the limit homogenized problem and prove that it is well-posed global problem, showing also the rigorous convergence of solutions, as $\varepsilon \to 0$, in suitable functional spaces. \section{Statement of main results} Let $\Omega $ be a bounded domain in $\mathbb{R}^n$ $n \ge 3$ with a piecewise smooth boundary $\partial\Omega$. The case $n=2$ requires some technical modifications which will not be presented here. Let $G_0$ be a domain in $ Y=(-\frac 1 2, \frac 1 2)^n$, and $\overline{G_0}$ be a compact set diffeomorphic to a ball. Let $C_0, \varepsilon > 0$ and set \begin{equation} a_{\varepsilon}=C_0{\varepsilon}^{\alpha} \quad \text{for } \alpha=\frac{n}{ n-2} . \end{equation} For $\delta > 0$ and $B$ a set let $\delta{B}=\{x \, | \, {\delta}^{-1}x\in B \,\}$. Assume that $\varepsilon$ is small enough so that $a_\varepsilon G_0 \subset \varepsilon Y$. We define $\widetilde{\Omega}_{\varepsilon}=\{x \in \Omega \, | \, \rho(x,\partial\Omega)>2{\varepsilon} \,\}.$ For $j \in \mathbb{Z}^n$ we define \begin{gather*} P_\varepsilon^j = \varepsilon j, \quad Y^j_\varepsilon = P_\varepsilon^j + \varepsilon Y, \quad G^j_\varepsilon = P_\varepsilon^j + a_\varepsilon G_0. \end{gather*} We define the set of admissible indexes: \begin{gather*} \Upsilon_{\varepsilon}=\big\{j\in \mathbb{Z}^n: G^j_{\varepsilon}\cap{ \overline{\widetilde{\Omega}}_{\varepsilon}}\ne \emptyset \big\}. \end{gather*} Notice that $|\Upsilon_{\varepsilon}| \cong d{\varepsilon}^{-n}$ where $d>0$ is a constant. Our problem will be set in the following domain: \begin{gather*} G_{\varepsilon}=\bigcup_{j\in\Upsilon_{\varepsilon}}G^{j}_{ \varepsilon}, \quad \Omega_\varepsilon=\Omega\setminus \overline{G} _\varepsilon \end{gather*} Finally, let \begin{equation*} \partial\Omega_\varepsilon=S_\varepsilon \cup \partial\Omega ,\quad S_\varepsilon=\partial G_\varepsilon . \end{equation*} We consider the following boundary value problem in the domain $ \Omega_{\varepsilon}$ \begin{equation} \begin{gathered} -\Delta{u_{\varepsilon}}=f, \quad x\in \Omega_{\varepsilon},\\ \partial_{\nu}u_{\varepsilon}+\varepsilon^{-\gamma}\sigma(u_{\varepsilon})=0, \quad x\in S_{\varepsilon},\\ u_{\varepsilon}=0, \quad x\in\partial\Omega, \end{gathered} \label{eq:1} \end{equation} where $\gamma=\alpha = \frac{n}{n-2}$, $f\in L^2(\Omega)$, $\nu$ is the unit outward normal vector to the boundary $S_{\varepsilon}$, $\partial_{\nu}u$ is the normal derivative of $u.$ Furthermore, we suppose that the function $ \sigma: \mathbb{R} \to \mathbb{R}$, describing the microscopic nonlinear Neumann boundary condition, is nondecreasing, $\sigma(0)=0$, and there exist constants $k_1, k_{2}$ such that \begin{equation} |\sigma(s) - \sigma(t)|\le k_1 |s-t| ^{\alpha} + k_{2}|s-t| \quad \forall s,t\in \mathbb{R}, \quad \text{for some } 0 < \alpha \le 1. \label{eq:2} \end{equation} \begin{remark} \rm Condition \eqref{eq:2} means that $\sigma$ is locally H\"older continuous, but it is only sublinear towards infinity. This condition is weaker than $u \in \mathcal{C}^{0,\alpha} (\mathbb{R})$ or $\sigma$ Lipschitz, that correspond, respectively, to $k_2 = 0$ and $k_1 = 0$. \end{remark} \begin{remark} \rm Condition \eqref{eq:2} on $\sigma$ is a purely technical requirement. This kind of regularity can probably be improved. In particular, as shown in \cite {Diaz+Gomez-Castro+Podolski+Shaposhnikova:2016homogenization+Heaviside,diaz+gomez-castro+podolskii+shaposhnikova2017generalmmg} the kind of homogenization techniques and result that will be presented later can be expect for any maximal monotone graph $\sigma$. \end{remark} For any prescribed set ${G_0}$, as before, and for any given $u\in \mathbb{ R}$, we define $\widehat{w}(y;{G_0,}u)$, for $y\in $ $\mathbb{R} ^{n}\setminus G_0$, as the solution of the following one-parametric family of auxiliary external problems associated to the prescribed asymmetric geometry $G_0$ and the nonlinear microscopic boundary reaction $\sigma (s)$: \begin{equation} \begin{gathered} - \Delta_y{\widehat{w}}=0 \quad \text{if } y\in \mathbb R^n\setminus\overline{G_0},\\ \partial_{\nu_y}\widehat{w}-C_0\sigma(u-\widehat{w})=0, \quad \text{if } \, y\in \partial G_0,\\ \widehat{w}\to 0 \quad \text{as } | y | \to \infty. \end{gathered} \label{eq:12} \end{equation} We will prove in Section 4 that the above auxiliary external problems are well defined and, in particular, there exists a unique solution $\widehat{w}(y;{G_0,}u)\in H^{1}(\mathbb{R} ^{n}\setminus \overline{G_0})$, for any $u\in \mathbb{R}$. Concerning the corresponding \textquotedblleft new strange term\textquotedblright , for any prescribed asymmetric set ${G_0}$, as before, and for any given $u\in \mathbb{R}$ we introduce the following definition. \begin{definition} \rm Given ${G_0}$ we define $H_{G_0}:\mathbb{R\to }\mathbb{R}$ by means of the identity \begin{equation} \label{eq:15} \begin{aligned} H_{G_0}(u)& :=\int_{\partial {G_0}}\partial _{\nu _{y}} \widehat{w}(y;{G_0,}u) \,\mathrm{d}S_y \\ & =C_0\int_{\partial {G_0}}\sigma (u-\widehat{w}(y;{G_0,}u)) \,\mathrm{d}S_{y}, \quad\text{for any }u\in \mathbb{R}. \end{aligned} \end{equation} \end{definition} \begin{remark} \label{rem:H ball} \rm Let $G_0=B_{1}(0):=\{x\in \mathbb{R}^{n}:|x|<1\}$ be the unit ball in $\mathbb{R}^{n}$. We can find the solution of problem \eqref{eq:12} in the form $\widehat{w}(y;{G_0,}u)=\frac{\mathcal{H}(u)}{|y|^{n-2}}$, where, in this case, $\mathcal{H}(u)$ is proportional to $H_{B_{1}(0)}(u)$. We can compute that \begin{align*} H_{G_0}(u) &=\int_{\partial G_0}\partial_\nu \widehat{w}(u,y) \, \mathrm{d}S_y \\ &=\int_{\partial G_0}(n-2)H_{G_0} (u) \, \mathrm{d}S_y \\ &=(n-2)\mathcal{H}(u)\omega(n), \end{align*} where $\omega(n)$ is the area of the unit sphere. Hence, due to \eqref{eq:15}, $\mathcal H (u)$ is the unique solution of the following functional equation \begin{equation} (n-2)\mathcal{H}(u)=C_0\sigma (u-\mathcal{H}(u)). \end{equation} In this case, it is easy to prove that $H$ is nonexpansive (Lipschitz continuous with constant 1). This equation has been considered in many papers (see \cite{diaz+gomez-castro+podolskii+shaposhnikova2017generalmmg} and the references therein). \end{remark} We shall prove several results on the regularity and monotonicity of the homogenized reaction $H_{G_0}(u)$ in the next section. Concerning the convergence as $\varepsilon \to 0$ the following statement collects some of the more relevant aspects of this process: \begin{theorem} \label{thm:nonlinear case} Let $n\geq 3$, $a_{\varepsilon }=C_0{\varepsilon }^{-\gamma }$, $\gamma =\frac{n}{n-2}$, $\sigma $ a nondecreasing function such that $\sigma (0)=0$ and that satisfies \eqref{eq:2}. Let $u_{\varepsilon }$ be the weak solution of \eqref{eq:1}. Then there exists an extension to $H_0^1(\Omega)$, still denoted by $u_\varepsilon$, such that $u_{\varepsilon }\rightharpoonup u_0$ in $H^{1}(\Omega )$ as $\varepsilon \to 0$, where $u_0\in H_0^{1}(\Omega )$ is the unique weak solution of \begin{equation} \begin{gathered} -\Delta{u_0}+C^{n-2}_0 H_{G_0}(u_0)=f \quad \Omega,\\ u_0=0 \quad \partial\Omega. \end{gathered} \label{eq:27} \end{equation} \end{theorem} \begin{remark} \rm Since $|H_{G_0} (u)| \le C ( 1 + |u|)$ it is clear that $H_{G_0}(u_0) \in L^2 (\Omega)$. \end{remark} \section{On the $\varepsilon $-global problem} Some comments on the well-posednes and some a priori estimates concerning the $\varepsilon $-global problem \eqref{eq:1}, when the nondecreasing function $\sigma \in \mathcal{C}(\mathbb{R})$, $\sigma (0)=0$ satisfies \eqref{eq:2}, are collected in this section. We start by introducing some notations: \begin{definition} \rm Let $U$ be an open set and $\Gamma \subset \partial \Omega $. We define the functional space \begin{equation*} H^{1}(U,\Gamma )=\overline{\{f\in \mathcal{C}^{\infty }(U):f|_{\Gamma }=0\}} ^{H^{1}(U)}. \end{equation*} \end{definition} Thanks to well-known results (see, e.g. the references given in \cite{diaz+gomez-castro+podolskii+shaposhnikova2017generalmmg}) there exists a unique weak solution of problem \eqref{eq:1}: i.e. $u_{\varepsilon }\in H^{1}(\Omega _{\varepsilon },\partial \Omega )$ is the unique function such that \begin{equation} \int_{\Omega _{\varepsilon }}\nabla {u_{\varepsilon }}\nabla \varphi \,\mathrm{d}x+\varepsilon ^{-\gamma }\int_{S_{\varepsilon }}\sigma (u_{\varepsilon })\varphi \,\mathrm{d}S=\int_{\Omega _{\varepsilon }}f\varphi \,\mathrm{d}x, \label{eq:3} \\ \end{equation} for every $\varphi \in H^{1}(\Omega _{\varepsilon },\partial \Omega )$. As a matter of fact, in order to get a proof of the convergence of $u_{\varepsilon }$ as $\varepsilon \to 0$, under the general assumption \eqref{eq:2}, it is useful to recall that, thanks to the monotonicity of $\sigma (u)$, we can write the weak formulation of \eqref{eq:1} in the following equivalent way (for details see \cite{diaz+gomez-castro+podolskii+shaposhnikova2017jmaa}): \begin{equation} \int_{\Omega _{\varepsilon }}\nabla \varphi \cdot \nabla (\varphi -u_{\varepsilon })\,\mathrm{d}x+\varepsilon ^{-\gamma }\int_{S_{\varepsilon }}\sigma (\varphi )(\varphi -u_{\varepsilon })ds\geq \int_{\Omega _{\varepsilon }}f(\varphi -u_{\varepsilon })\, \mathrm{d}x, \label{eq:8} \end{equation} for every $\varphi \in H_0^{1}(\Omega )$. Concerning some initial apriori estimates, we recall that we can work with $ \widetilde{u}_{\varepsilon }\in H_0^{1}(\Omega )$ given as an extension of $u_{\varepsilon }$ to $\Omega $ such that \begin{equation} \| {\widetilde{u}_{\varepsilon }}\| _{H^{1}(\Omega )} \leq K\| {u_{\varepsilon }}\| _{H^{1}(\Omega _{\varepsilon })},\quad \| {\nabla {\widetilde{u}}_{\varepsilon }}\| _{L^{2}(\Omega )} \leq K\| {\nabla {u_{\varepsilon }}}\| _{L^{2}(\Omega _{\varepsilon })}, \label{eq:4} \end{equation} where $K$ does not depend on ${\varepsilon }$. The construction of such an extension is given, e.g., in \cite{Shaposhnikova+Oleinik:1996Lincei} (the $W^{1,p}$ equivalent, for $p \ne 2$, can be found in \cite{Podolskii:2015extension+operator+noncritical+general+p}). Now, considering in the weak formulation \eqref{eq:3} the test function $\varphi =u_{\varepsilon }$, and using the monotonicity of $\sigma $, we obtain \begin{equation} \| {\nabla {u_{\varepsilon }}}\| _{L_{2}(\Omega _{\varepsilon})}^{2}\leq K. \label{eq:5} \\ \end{equation} where $K$ does not depend on $\varepsilon $. From \eqref{eq:5} we derive that there are a subsequence of $\widetilde{u}_{\varepsilon }$ (still denote by ${\widetilde{u}_{\varepsilon }}$) and $u_0\in H_0^{1}(\Omega )$ such that, as ${\varepsilon }\to 0$, we have \begin{align} \widetilde{u}_{\varepsilon }\rightharpoonup u_0\quad \text{weakly in } H_0^{1}(\Omega ), \label{eq:6} \\ \widetilde{u}_{\varepsilon }\to u_0\quad \text{strongly in } L^{2}(\Omega ). \label{eq:7} \end{align} In Section 4 we characterize the limit function $u_0\in H_0^{1}(\Omega )$. \section{On the regularity of the strange term} \subsection{Auxiliary function $\widehat w$} The existence and regularity of solution in domains \begin{equation} \mathcal{O }= \mathbb{R}^n \setminus \overline {G_0} \end{equation} which are commonly known as exterior domains, has been extensively studied (see, e.g., \cite{Galdi2011} and the references therein). Based on the rate of convergence to $0$ as $|y| \to + \infty$ we consider the space \begin{equation} \mathbb{X }= \big\{ w \in L^1_{loc} (\mathcal{O}) : \nabla w \in L^2 ( \mathcal{O}),\; w|_{\partial G_0} \in L^2 (\partial G_0), \; |w| \le \frac{K}{ |y|^{n-2}} \big\} \end{equation} It is a standard result, known as Weyl's lemma, that any harmonic function is smooth (of class $\mathcal C^\infty$) in the interior of the domain. It was first proved for the whole space by Hermann Weyl \cite{Weyl1940}, and later extended by others to any open set in $\mathbb R^n$ (see, e.g., \cite{Ladyzhenskaya1968linear+quasilinear}). \begin{remark} \rm Notice that $\widetilde w (y;G_0,u) = - \widehat w(y;G_0, - u)$ is a solution of \eqref{eq:12} that corresponds to $\widetilde \sigma (s) = - \sigma( - s)$. Hence, any comparison we prove for $u \ge 0$ we automatically prove for $u \le 0$. \end{remark} \subsubsection{A priori estimates} \begin{lemma}[Weak maximum principle in exterior domains] Assume that $w \in \mathbb X$ is such that \begin{gather*} -\Delta w \le 0 \quad \mathcal D'(\mathcal O ), \\ w \le 0 \quad \partial G_0. \end{gather*} Then $w \le 0$ in $\overline{ \mathcal O}$. \end{lemma} \begin{proof} Let $R > 0$. Consider $\mathcal O_R = \mathcal O \cap B_R$. Since $w \in \mathbb X$ then $w \le \frac K {|y|^{ n-2 } }$. Using the hypothesis $w \le 0$ on $\partial G_0$ and this fact, $\frac{K}{R^{n-2}}$ on $\partial \mathcal O_R$. We can apply the standard weak maximum principle for weak solutions in $\mathcal O_R$ to show that $w \le \frac{K}{R^{n-2}}$ on $\overline {\mathcal O}_R$. As $R \to +\infty$ we prove the result. \end{proof} Analogously, we have the strong maximum principle. \begin{lemma} \label{lem:w less u} Let $\sigma$ nondecreasing, $u \in \mathbb{R}$, $\widehat w \in \mathbb{X}$ be a weak solution of \eqref{eq:12}. Then \begin{equation} \min\{0,u\} \le \widehat w \le \max \{0,u\} \end{equation} \end{lemma} \begin{proof} For $u = 0$, $w = 0$ follows from a monotonicity argument. Assume $u > 0$. Let $\psi \in W^{1, \infty} (\mathbb R)$ non-increasing such that \begin{equation*} \psi (s) = \begin{cases} 1 & s < \frac 1 2 \\ 0 & s > 1 \end{cases} \end{equation*} and consider the test function $\varphi = (w - u)_+ \psi \big( \frac{d (\cdot, \partial G_0)}{R} \big)$. Then \begin{align*} &\int _{\mathcal O } | \nabla (w - u)_+|^2 \psi \Big( \frac{d (x, \partial G_0)}{R} \Big) \,\mathrm{d}x + \int_{\mathcal O} (w - u)_+ \frac{\psi'\big(\frac{d (x, \partial G_0)}{R} \big)}{R} \nabla w \cdot \nabla d \,\mathrm{d}x \\ &=C_0 \int_{ \partial G_0 } \sigma(u - w) (w - u)_+ \,\mathrm{d}S \le 0 \end{align*} and \begin{align*} &\Big| \int_{\mathcal O} (w - u)_+ \frac{\psi' \big( \frac{d (x, \partial G_0)}{R} \big)}{R} \nabla w \cdot \nabla d \,\mathrm{d}x \Big| \,\mathrm{d}x \\ &\le C\int_{\{\frac R 2 < d < R\}} \frac{ w } {R } |\nabla w|\,\mathrm{d}x \\ &\le C \Big( \int_{\{\frac R 2 < d < R\}} \frac{|w|^2}{R^2 } \,\mathrm{d}x \Big) ^{1/2} \Big( \int_{\mathcal O} |\nabla w|^2 \,\mathrm{d}x \Big) ^{1/2} \\ &\le \frac{C}{R^{\frac{ n-2 } 2}} \Big( \int_{\mathcal O} |\nabla w|^2 \,\mathrm{d}x \Big) ^{1/2} \to 0, \end{align*} as $R\to \infty$. Therefore, \begin{align*} 0 &\le \int _{0 \le d < \frac R 2} | \nabla (w - u)_+|^2 \,\mathrm{d}x \le \int _{\mathcal O } | \nabla (w - u)_+|^2 \psi \Big( \frac{d (x, \partial G_0)}{R} \Big) \,\mathrm{d}x \\ & \le - \int_{\mathcal O} (w - u)_+ \frac{\psi' \Big( \frac{d (x, \partial G_0)}{R} \Big)}{R} \nabla w \cdot \nabla d \,\mathrm{d}x . \end{align*} As $R \to +\infty$ we obtain that \begin{equation} \int _{\mathcal O} | \nabla (w - u)_+|^2 \,\mathrm{d}x = 0. \end{equation} In particular $(w-u)_+ \ge 0$ is a constant. Since, as $|y| \to +\infty$ we show that the constant must be $(-u)_+ = 0$ we deduce that $w - u \le 0$. If $u < 0$ we apply the previous argument with $\tilde \sigma(s) = - \sigma(-s)$. \end{proof} \begin{lemma} \label{lem:subharmonic w} Let $u \in \mathbb{R}$, $w \in \mathbb X$ such that $w \le u$ in $\partial G_0$ and $-\Delta w \le 0$ and \begin{equation*} K_0 = \max _{z \in \partial G_0} |z|^{n-2}. \end{equation*} Then \[ w\le \frac {K_0u}{|y|^{n-2}} \quad \forall y \in \overline {\mathcal{O}}. \] \end{lemma} \begin{proof} Notice that \begin{equation*} \max _{z \in \partial G_0} w(z) |z|^{n-2} \le u \max _{z \in \partial G_0} |z|^{n-2} = K_0u . \end{equation*} Then \begin{equation*} w \le \frac {K_0u}{|y|^{n-2}} \quad y \in \partial G_0. \end{equation*} Since $w- \frac{K_0 u}{|y|^{n-2}}$ is subharmonic and tends to $0$ as $|y| \to +\infty$, we can apply the weak maximum principle to deduce that \begin{equation*} w(y) \le \frac {K_0 u}{|y|^{n-2}} \quad y \in \mathbb R^n \setminus { G_0}. \end{equation*} This proves the result. \end{proof} By the same argument it is easy to prove that any classical solution $\widehat w \in \mathcal C^2 (\mathcal O) \cap \mathcal C(\overline{ \mathcal O})$ is, in fact, in $\mathbb X$. Furthermore, we have an explicit expression of the $K$ in the definition of $\mathbb X$ for the solutions of \eqref{eq:12}: \begin{lemma} \label{lem:bound hat w} Let $\widehat w \in \mathbb X$ be a solution of \eqref{eq:12}. Then \begin{equation} \label{eq:bound hat w} |\widehat{w}(y;G_0,u)| \le \frac {K_0|u| }{|y|^{n-2}} \quad \forall y \in \overline {\mathcal{O}}. \end{equation} \end{lemma} \begin{lemma} \label{lem:bounds gradient hat w} Let $R_0 = \max_{\partial G_0} |y|$, $ \widehat w \in \mathbb{X}$ be a weak solution of \eqref{eq:12}. Then \begin{equation} \max_{|y| = R} |\nabla \widehat{w}(y;G_0,u) | \le \frac{K|u|}{(R-R_0)^{n-1}} \quad \forall R > R_0 \end{equation} where $K$ does not depend on $u$ or $R$. \end{lemma} \begin{proof} Let $|y_0|=R$. Let $B$ be a ball centered at $y_0$ of radius $\frac{R-R_0}{2} $. In $B$ we have $|y|\ge \frac {R-R_0}2$. Since $\frac{\partial \widehat w}{\partial x_i}$ is a harmonic function, and applying Lemma \ref{lem:bound hat w}, we have \begin{gather*} \frac{\partial \widehat w}{\partial x_i}(y_0) = \frac{1}{|B|} \int _ B \frac{\partial \widehat w}{\partial x_i} \,\mathrm{d}y = \frac{1}{|B|} \int _{\partial B} \widehat w \nu_i\,\mathrm{d}S,\\ \big| \frac{\partial \widehat w}{\partial x_i} (y_0) \big| \le \frac{|\partial B|}{|B|} \frac{K|u|}{(R-R_0)^{n-2}} \le \frac{ K|u|}{(R-R_0)^{n-1}}. \end{gather*} This completes the proof. \end{proof} \subsubsection{Uniqueness, comparison and approximation of solutions} \begin{lemma} \label{lem:comparison w different sigma} Let $u \in \mathbb R$, $\sigma_1, \sigma_2$ be two nondecreasing functions such that $\sigma_1 \le \sigma_2$ in $[0,+\infty)$ and let $w_1, w_2 \in \mathbb X$ satisfy \eqref{eq:12}. Then $w_1 \le w_2$. \end{lemma} \begin{proof} We subtract the two weak formulations, and consider $\varphi = (w_1 - w_2)_+$ as a test function. We obtain that \begin{equation*} \int _{\mathbb R^n \setminus G_0 } |\nabla (w_1 - w_2)_+ |^2 \,\mathrm{d}x = \int_{\partial G_0 } (\sigma_1(u - w_1 ) - \sigma_2 (u - w_2) ) (w_1 - w_2)_+ \,\mathrm{d}S \end{equation*} Thus, in the set $\{w_2 \le w_1\}$ we have that $u - w_2 \ge u - w_1$ and, hence, \begin{equation*} \sigma_2 (u - w_2) \ge \sigma_2 (u - w_1) \ge \sigma_1 (u - w_1), \end{equation*} so \begin{equation*} \sigma_1(u - w_1 ) - \sigma_2 (u - w_2) \le 0. \end{equation*} Thus, since $(w_1 - w_2)_+ \ge 0$ a.e. in $\partial G_0$, we have that \begin{equation} \int _{\mathbb R^n \setminus G_0 } |\nabla (w_1 - w_2)_+ |^2 \,\mathrm{d}x \le 0. \end{equation} Hence $(w_1 - w_2)_+ = c$ constant. Since $(w_1-w_2)_+ \to 0$ as $|y| \to +\infty$, we have that $c = 0$ and thus $w_1 \le w_2$. \end{proof} \begin{corollary} There exists, at most, one solution $w \in \mathbb{X}$ of \eqref{eq:12}. \end{corollary} \begin{lemma} \label{lem:continuity hat w respect sigma} Let $\sigma_1, \sigma_2 \in \mathcal{C}(\mathbb{R})$ be two nondecreasing function. Let $\widehat w_i (\cdot; G_0, u) \in \mathbb{X}$ be a solution of \begin{equation} \begin{gathered} - \Delta_y{\widehat{w}_i }=0 \quad \text{if } y\in \mathbb R^n\setminus\overline{G_0},\\ \partial_{\nu_y}\widehat{w}_i-C_0\sigma_i (u-\widehat{w}_i)=0, \quad \text{if } y\in \partial G_0,\\ \widehat{w_i}\to 0 \quad \text{as } | y | \to \infty. \end{gathered} \end{equation} Then \begin{equation} \|\nabla (\widehat w_1 - \widehat w_2) \|_{L^2 (\mathcal{O})}^2 \le C |u| \| \sigma_1 - \sigma_2 \|_{L^\infty (I)} , \end{equation} where \begin{equation} I = \{ u - \widehat w_1 (y; G_0, u) : y \in \mathbb{R}^n \setminus \overline G_0\} \subset \mathbb{R }, \end{equation} and $C$ is independent of $u$. \end{lemma} \begin{proof} By taking as test function $\varphi = \widehat w_1- \widehat w_2$ in the weak formulation of these equations we have that \begin{align*} \|\nabla (\widehat w_1- \widehat w_2) \|_{L^2 (\mathcal O)}^2 &\le \int _{\mathcal O} | \nabla (\widehat w_1- \widehat w_2) |^2 \,\mathrm{d}x \\ &\quad + \int _{\partial G_0} (\sigma_2(u - \widehat w_2) - \sigma_2 (u - \widehat w_1) ) ( \widehat w_1- \widehat w_2)\,\mathrm{d}S \\ &= \int _{\partial G_0} (\sigma_1(u - \widehat w_1) - \sigma_2 (u - \widehat w_1) ) ( \widehat w_1- \widehat w_2)\,\mathrm{d}S \\ &\le \| \sigma_1- \sigma_2 \|_{\infty} \int _{\partial G_0} | \widehat w_1- \widehat w_2| \,\mathrm{d}S \\ &\le C |u| \| \sigma_1- \sigma_2 \|_{\infty} . \end{align*} This completes the proof. \end{proof} \subsubsection{Existence and regularity} \begin{lemma}\label{lem:existence hat w} Let $u \in \mathbb{R}$ and $\sigma$ uniformly Lipschitz. Then, there exists $\widehat w \in \mathbb{X}$ a weak solution of \eqref{eq:12}. Furthermore, $\widehat w$ satisfies \eqref{eq:bound hat w}. \end{lemma} \begin{proof} Let us assume that $u > 0$. Let $\lambda > 0$, and consider $\mu > 0$ such that \begin{equation*} F: z \mapsto C_0 \sigma(u - z) + \mu z \end{equation*} is nondecreasing. Let $w_0 = 0$. We define the sequence $w_k \in H^1 (\mathcal O)$ as the solutions of \begin{gather*} -\Delta w_{k+1} + \lambda w_{k+1} = \lambda w_k \quad \mathcal O, \\ \partial_ \nu w_{k+1} + \mu w_{k+1} = F(w_k) \quad \partial G_0, \\ w_{k+1} \to 0 \quad |y | \to +\infty. \end{gather*} This sequence is well defined, since $\lambda > 0$ applying the Lax-Milgram theorem. Indeed, if $w_k \in H^1 (\mathcal O)$ then $F(w_k) \in H^{1/2} (\partial G_0)$ so that $w_{k+1} \in H^1 (\mathcal O)$. Let us show that $0 \le w_k \le w_{k+1} \le u$ a.e. in $\mathcal O$ and $\partial G_0$ for every $n \ge 1$. We start by showing that $0 \le w_1$. This is immediate because $F(0) = C_0 \sigma (u) \ge 0$. Let us now show that, if $w_{k-1} \le w_k$ then $w_k \le w_{k+1}$. Considering the weak formulations: \begin{equation} \begin{aligned} & \int _\mathcal O \nabla w_{k+1} \nabla v \,\mathrm{d}x+ \lambda \int _\mathcal O w_{k+1} v \,\mathrm{d}x + \mu \int_ {\partial G_0} w_{k} v\,\mathrm{d}S \\ &= \lambda \int_\mathcal O w_{k+1} v \,\mathrm{d}x + \int_{ \partial G_0 } F(w_k ) v\,\mathrm{d}x \end{aligned} \end{equation} we have that \begin{align*} &\int _\mathcal O \nabla (w_k - w_{k+1}) \nabla v\,\mathrm{d}x + \lambda \int _\mathcal O (w_k - w_{k+1}) \varphi\,\mathrm{d}x + \mu \int_ {\partial G_0} (w_k - w_{k+1}) v \,\mathrm{d}S\\ &=\lambda \int_\mathcal O (w_{k-1} - w_{k}) v \,\mathrm{d}x + \int_{ \partial G_0 } ( F(w_{k-1}) - F(w_k ) ) v \,\mathrm{d}S \end{align*} Consider $v = (w_k - w_{k+1})_+ \ge 0$. We have that $w_{k-1} \le w_k $ therefore $w_{k-1} - w_k, F(w_{k-1}) - F(w_k) \le 0$. Hence \begin{align*} &\int _\mathcal O |\nabla (w_k - w_{k+1})_+|^2 \,\mathrm{d}x + \lambda \int _\mathcal O |(w_k - w_{k+1})_+|^2 \,\mathrm{d}x + \mu \int_ {\partial G_0} |(w_k - w_{k+1})_+|^2 \,\mathrm{d}S \\ &=\lambda \int_\mathcal O (w_k - w_{k+1})v \,\mathrm{d}x + \int_{ \partial G_0 } ( F(w_{k-1}) - F(w_k ) ) v\,\mathrm{d}S \le 0 \end{align*} so that $(w_k - w_{k+1})_+ = 0$. Hence $w_k \le w_{k+1}$ a.e. in $\mathcal O$ and in $\partial G_0$. With an argument similar to the one in Lemma \ref{lem:w less u}, one proves that $w_{k+1} \le u$ a.e. in $\mathcal O$ and $\partial G_0$. The sequence $w_k$ is pointwise increasing a.e. in $\mathcal O$. Therefore, there exists a function $w$ such that \begin{equation} w_k (y) \nearrow w(y) \quad \text{a.e. } \mathcal O. \end{equation} Taking traces, the same happens in $\partial G_0$. Hence \begin{equation} w_k (y) \nearrow w(y) \quad \text{a.e. } \partial G_0. \end{equation} Thus $F(w_k) \nearrow F(w)$ a.e. in $L^2 (\partial G_0)$. Since $F(w) \le F(u)$ and $\partial G_0$ has bounded measure, we have that \begin{equation} \label{eq:existence convergence of boundary term} F(w_k) \to F(w) \quad \text{in } L^2 (\partial G_0). \end{equation} We have that \begin{equation*} -\Delta w_{k+1} = \lambda (w_k - w_{k+1}) \le 0 \quad \mathcal O. \end{equation*} Hence, $w_{k}$ are all subharmonic. Then, since $w_{k} \to 0$ as $|y| \to 0$ and $w_{k} \in \mathbb X$ and $w_{k} \le u$ on $\partial G_0$, by Lemma \ref{lem:subharmonic w} we have that \begin{equation*} 0 \le w_{k} \le \frac{K_0 u}{|y|^{n-2}}. \end{equation*} in particular $w_{n} \in \mathbb X$. Passing to the limit we deduce that \begin{equation*} 0 \le w \le \frac{K_0 u}{|y|^{n-2}} \quad y \in \mathcal O \end{equation*} Hence $ w \to 0$ as $|y| \to +\infty$. Applying an equivalent argument to the one in Lemma \ref{lem:continuity hat w respect sigma} we have that $\nabla w_k$ is a Cauchy sequence in $L^2 (\mathcal O)^n$. In particular, there exists $\xi \in L^2 (\mathcal O)^n$ such that \begin{equation*} \nabla w_k \to \xi \quad \text{in }L^2 (\mathcal O)^n. \end{equation*} Consider $\mathcal O' \subset \mathcal O$ open and bounded. Then we have that \begin{equation*} \int _{\mathcal O'} |\nabla w_k |^2 \,\mathrm{d}y \le \int _{\mathcal O} |\nabla w_n |^2 \,\mathrm{d}y \end{equation*} is bounded, and \begin{equation*} \int _{\mathcal O'} |w_k|^2 \,\mathrm{d}y \le |u|^2 |\mathcal O'| . \end{equation*} Hence, there is convergent subsequence in $H^1 (\mathcal O')$. Any convergent subsequence must have the same limit, so $ w_k \rightharpoonup w$ $ H^1 (\mathcal O')$. In particular \begin{equation*} \xi = \nabla w \quad \text{a.e. } \mathcal O'. \end{equation*} Since this works for every $\mathcal O'$ bounded we have that $\nabla w \in L^2 (\mathcal O)^n$, hence $w \in \mathbb X$, and \begin{equation*} \nabla w_n \to \nabla w \quad \text{in } L^2 (\mathcal O)^n. \end{equation*} Using this fact and \eqref{eq:existence convergence of boundary term}, we can pass to the limit in the weak formulation to deduce that \begin{gather*} -\Delta w = 0 \quad \mathcal O,\\ \frac{\partial w}{\partial n} = C_0 \sigma (u - w) \quad \partial G_0. \end{gather*} In particular, a solution of \eqref{eq:12}. The same reasoning applies to case $u < 0$. \end{proof} \begin{corollary} Let $\sigma \in \mathcal{C}(\mathbb{R})$ be nondecreasing be such that \begin{equation} |\sigma (u) | \le C (1 + |u|). \end{equation} Then, there exists a unique solution of \eqref{eq:12}. \end{corollary} \begin{proof} Let us assume first that $u > 0$. Let $ \sigma_m \in C^1 ([0,|u|])$ be a pointwise increasing sequence that approximates $\sigma$ uniformly in $[0,|u|]$. Since $\sigma_m$ is Lipschitz, then $\widehat w_m$ exists by the previous part. Because of Lemma \ref{lem:comparison w different sigma}, the sequence $\widehat w_m$ of solutions of \eqref{eq:12} is pointwise increasing. Since we know that, we have that $\widehat w_m \le u$ then, for a.e. $y \in \mathcal O$, $\widehat w_m (y)$ is a bounded and increasing sequence \begin{equation*} \widehat w_m (y) \nearrow w (y). \end{equation*} for some $w(y)$. In particular \begin{equation*} 0 \le w(y) \le \frac{K_0 u}{|y|^{n-2} } \quad y \in \mathcal O. \end{equation*} Applying as in the proof of Lemma~\ref{lem:existence hat w} we deduce that $w \in \mathbb X$ and it is a solution of \eqref{eq:12}. The proof for $u < 0$ follows in the same way, by taking a pointwise decreasing sequence $\sigma_m$. \end{proof} With the same techniques we can prove the following (applying that $u - w \ge 0$ for $u \ge 0$ and $u - w \le 0$ for $u \le 0$): \begin{lemma} \label{lem:convergence hat w} Let $u \in \mathbb{R}$, $\mathcal{O}'\subset \mathcal{O}$ bounded, $\sigma, \sigma_m$ be nondecreasing continuous functions such that $\sigma(0) = \sigma_m (0) = 0 $ and $|\sigma_m|\le|\sigma|$ and $\sigma_m \to \sigma$ in $\mathcal{C}([-2|u|,2|u|])$. Then: \begin{equation} \widehat w_m (\cdot; G_0, u )\to \widehat w (\cdot; G_0, u)\quad \text{strongly in } H^1 (\mathcal{O}'). \end{equation} Furthermore, \begin{enumerate} \item If $u \ge 0$ then $\widehat w_m \nearrow \widehat w$ a.e. $y \in \mathcal O$ and $y \in \partial G_0$. \item If $u \le 0$ then $\widehat w_m \searrow \widehat w$ a.e. $y \in \mathcal O$ and $y \in \partial G_0$. \end{enumerate} \end{lemma} \subsubsection{Lipschitz continuity with respect to $u$} \begin{lemma} For every $y \in \mathbb{R}^n \setminus G_0$, $\widehat{w}(y;{G_0,}u)$ is a nondecreasing Lipschitz-continuous function with respect to $u$. In fact, \begin{equation} |\widehat{w}(u_1;G_0,y) -\widehat{w}(y;G_0,u_2)|\le | u_1-u_2|\quad \forall u_1,u_2 \in \mathbb{R}, \; \forall y \in \mathbb{R}^n \setminus G_0. \label{eq:14} \end{equation} Furthermore, for every $y \in \partial G_0$, $\partial_\nu \widehat{w}(y;G_0,u)$ is also nondecreasing in $u$. \end{lemma} \begin{proof} Let us consider first that $\sigma \in \mathcal C^1 (\mathbb R)$. We have that $\widehat{w}(\cdot;G_0,u) \in \mathcal C(\overline{ \mathcal O})\cap \mathcal C^2 (\mathcal O)$ for every $u \in \mathbb R$ and the equation is satisfied pointwise (see \cite{Ladyzhenskaya1968linear+quasilinear}). Let us first consider $u_1 > u_2$. We want to prove the following \begin{gather} \label{eq:hat w u1 less u2} 0 \le \widehat{w}(u_1;G_0,y) - \widehat{w}(y;G_0,u_2) \le u_1 - u_2\\ \label{eq:monotonicity normal derivative of hat w} \partial_ \nu \widehat{w}(u_1;G_0,y) \ge \partial_\nu \widehat{w}(y;G_0,u_2). \end{gather} That \begin{equation} \label{eq: w u1 greater w u2} \widehat{w}(u_1;G_0,y) \ge \widehat{w}(y;G_0,u_2). \end{equation} follows from the comparison principle. Indeed, let us plug $\widehat{w}(u_1;G_0,y)$ in the equation for $\widehat{w}(y;G_0,u_2)$: \begin{equation} \begin{gathered} -\Delta \widehat{w}(u_1;G_0,y) = 0 \quad \mathbb R^n \setminus G_0, \\ \begin{aligned} &\partial_{{\nu_y}} \widehat{w}(u_1;G_0,y) - C_0 \sigma(u_2 - \widehat{w}(u_1;G_0,y)) \\ & = C_0 \left(\sigma(u_1 - \widehat{w}(u_1;G_0,y))-\sigma(u_2 - \widehat{w}(y;G_0,u_2)) \right) \ge 0 \quad \partial G_0, \end{aligned}\\ \widehat{w}(u_1;G_0,y) \to 0 \quad |y| \to +\infty. \end{gathered} \end{equation} Therefore, $\widehat{w}(u_1;G_0,y)$ is a supersolution of the problem for $\widehat{w}(y;G_0,u_2)$. Applying the comparison principle we deduce \eqref{eq: w u1 greater w u2}. We define \begin{equation} g(u_1,u_2,y)=\widehat{w}(u_1;G_0,y)-\widehat{w}(y;G_0,u_2) \ge 0. \end{equation} The function $g$ is the solution of the following elliptic problem: \begin{equation} \begin{gathered} \Delta_y{g}=0 \quad \text{if } y\in \mathbb R^n\setminus\overline{G_0}, \\ \partial_{\nu_y}{g} -C_0\Big(\sigma(u_1-\widehat{w}(u_1;G_0,y)) -\sigma(u_2-\widehat{w}(y;G_0,u_2))\Big)=0 \quad \text{if } y\in \partial G_0, \\ g\to 0 \quad \text{as } | y | \to \infty . \end{gathered} \label{eq:13} \end{equation} Let us consider the boundary condition for $y \in \partial G_0$: \[ \partial_{\nu_y}{g} (y)=C_0(\sigma(u_1-\widehat{w}(u_1;G_0,y))-\sigma(u_2-\widehat{w}(y;G_0,u_2))) \] multiplying by $u_1-u_2-g(u_1,u_2,y)$, and applying the monotonicity of $\sigma$, we have \begin{equation} \label{eq:monotonicity g} (\partial_{\nu}{g}(y))(u_1-u_2-g(y))\ge 0\quad \forall y \in \partial G_0. \end{equation} Let $g(y_0) = \max_{\partial G_0} g$ for some $y_0 \in \partial G_0$. By the strong maximum principle $g(y_0) = \max_{\mathbb R^n \setminus {G_0}} g$. Hence $g(y) \le g(y_0)$ for $y \in \mathbb R^n \setminus G_0$ and we have \begin{equation*} \partial_{{\nu_y}} g (y_0) \ge 0. \end{equation*} Assume, first, that $\sigma$ is strictly increasing. We study two cases. If $\partial_{{\nu_y}} g(y_0) > 0$ then, by \eqref{eq:monotonicity g}, \begin{equation*} u_1 - u_2 \ge g(y_0) \ge g(y) \quad \forall y \in \mathbb R^n \setminus G_0. \end{equation*} If $\partial_\nu g (y_0) = 0$ then, by \eqref{eq:13}, \begin{gather*} \sigma(u_1 -\widehat{w}(y_0;G_0,u_1)) = \sigma(u_2 -\widehat{w}(y_0;G_0,u_2) ) \\ u_1 -\widehat{w}(y_0;G_0,u_1) = u_2 -\widehat{w}(y_0;G_0,u_2) \\ u_1 - u_2 = g(y_0) \ge g(y) \quad \forall y \in \mathbb R^n \setminus G_0. \end{gather*} Either way, we deduce that \eqref{eq:hat w u1 less u2} holds. Hence, \begin{equation*} \sigma(u_1-\widehat{w}(u_1;G_0,y)) \ge \sigma(u_2-\widehat{w}(y;G_0,u_2)) \quad \forall y \in \partial G_0 \end{equation*} so \eqref{eq:monotonicity normal derivative of hat w} holds. This concludes the proof when $\sigma$ is strictly increasing. Let $\sigma$ be a nondecreasing function and $U= \max\{|u_1|,|u_2|\}$. We consider an approximation sequence $\sigma_m$ of $\sigma$ in $[-2U,2U]$ by strictly increasing smooth functions such that $|\sigma_m| \le |\sigma|$. Consider $\widehat w_m$ as defined in Lemma \ref{lem:approximation of hat w}. We have that \begin{equation*} u_i - \widehat{w}(y;G_0,u_i) \in [-2U,2U] \quad \forall i =1,2, \forall y \in \mathbb R^n \setminus \overline G_0. \end{equation*} By the previous part $\widehat w_m$ satisfies \eqref{eq:hat w u1 less u2} and \eqref{eq:monotonicity normal derivative of hat w}. Applying Lemma \ref{lem:convergence hat w} we have a.e.-pointwise convergence $\widehat w_m (u_i, y) \to \widehat w (u_i,y) $ for $i = 1,2$, up to a subsequence, as $m \to +\infty$. Therefore \eqref{eq:hat w u1 less u2} and \eqref{eq:monotonicity normal derivative of hat w} hold almost everywhere in $y$. Since $\widehat w$ is continuous, \eqref{eq:hat w u1 less u2} and \eqref{eq:monotonicity normal derivative of hat w} hold everywhere. This concludes the proof in the case $u_1 > u_2$. If $u_1 < u_2$ we can exchange the roles of $u_1$ and $u_2$ in \eqref{eq:hat w u1 less u2} to deduce \eqref{eq:14}. This concludes the proof. \end{proof} \subsubsection{Auxiliary function $\widehat w_\varepsilon^j$} We conclude this section by introducing the following function: \begin{definition} \rm Let $u \in \mathbb{R}$, $j \in \Upsilon_\varepsilon$ and $\varepsilon > 0$. We define \begin{equation} \widehat w _\varepsilon ^j(x; G_0, u) = \widehat{w}\Big(\frac{x-P^j_\varepsilon}{a_\varepsilon};G_0,u\Big) . \end{equation} \end{definition} It is clear that this function is the solution of the problem \begin{equation} \begin{gathered} -\Delta \widehat w_\varepsilon^j = 0 \quad\mathbb R^n \setminus G_\varepsilon^j, \\ \partial_n \widehat w_\varepsilon^j - \varepsilon^{-\gamma} \sigma (u - \widehat w_\varepsilon^j) = 0 \quad \partial G_\varepsilon^j \\ \widehat w_\varepsilon^j \to 0 \quad |x| \to + \infty . \end{gathered} \end{equation} We have the following estimates: \begin{lemma}\label{lem:bounds hat w eps} Let $\varepsilon,r > 0$ and $x \in \partial T_{r\varepsilon} ^j $. Then \begin{equation} |\widehat{w}_\varepsilon^j (x; G_0, u) |\le \frac{K|u|}{\left|\frac{x - P_\varepsilon^j}{a_\varepsilon} \right|^{n-2}}\le\frac{K|u| a_\varepsilon^{n-2}}{r^{n-2} \varepsilon^{n-2}} \le \frac{ K|u| } {r^{n-2}} \varepsilon^2 \end{equation} where $K$ does not depend on $r, |u|$ or $\varepsilon$. \end{lemma} \begin{lemma} \label{lem:bounds gradient hat w eps} For $\varepsilon,r > 0$ be such that $a_\varepsilon < \frac{ r \varepsilon }{2R_0}$. Let $x \in \partial T_{r\varepsilon} ^j$. Then \begin{align} |\nabla \widehat{w}_\varepsilon^j (x; G_0, u)|\le \frac{ K| u |} {r^{n-1} } \varepsilon , \end{align} where $K$ does not depend on $r, \varepsilon$ or $j$. \end{lemma} \begin{proof} By the definition of $\widehat w _\varepsilon^j$ we have \begin{equation*} \nabla \widehat w_\varepsilon^j (x; G_0,u) = a_\varepsilon^{-1} (\nabla \widehat w ) \Big( \frac{x - P_\varepsilon^j}{a_\varepsilon}; G_0, u \Big) \end{equation*} Therefore, for $x \in \partial T_{r \varepsilon }^j $, \begin{align*} |\nabla \widehat w_\varepsilon^j | &= a_\varepsilon^{ -1} \big| \nabla \widehat w \Big( \frac{x - P_\varepsilon^j}{a_\varepsilon} \Big) \big| \le \frac{K|u| a_\varepsilon^{-1}}{\left( | \frac{x - P_\varepsilon^j}{a_\varepsilon} | - R_0 \right)^{n-1}} \\ & \le \frac{K|u|a_\varepsilon^{n-2}}{ \left( r \varepsilon - a_\varepsilon R_0 \right)^{n-1} } \le \frac{K |u| a_\varepsilon^{n-2} }{\left( \frac {r\varepsilon}{2} \right) ^{n-1} } \\ & \le \frac{ K| u |} {r^{n-1} } \varepsilon. \end{align*} This completes the proof. \end{proof} \subsection{Properties of $H_{G_0}$} \begin{lemma} \label{lem:regularity of H} $H_{G_0}$ is a nondecreasing function. Furthermore: \begin{enumerate} \item If $\sigma$ satisfies \eqref{eq:2}, then so does $H_{G_0}$. \item If $\sigma \in \mathcal{C}^{0,\alpha} (\mathbb{R})$, then so is $H_{G_0}$. \item If $\sigma \in \mathcal{C}^1 (\mathbb{R})$, then $H_{G_0}$ is locally Lipschitz continuous. \item If $\sigma \in W^{1, \infty} (\mathbb{R})$, then so is $H_{G_0}$. \end{enumerate} \end{lemma} \begin{proof} Let us prove the monotonicity of $H_{G_0} (u)$ given by \eqref{eq:15}. Let $u_1 > u_2$. By applying \eqref{eq:monotonicity normal derivative of hat w} we deduce that $H_{G_0} (u_1 ) \ge H_{G_0} (u_2)$. Assume \eqref{eq:2}. Indeed, taking into account \eqref{eq:14} we deduce \begin{equation} \begin{aligned} &|H_{G_0} (u)-H_{G_0} (v)| \\ &\le C_0\int_{\partial{G_0}}|\sigma(u-\widehat{w}(y;{G_0,}u)) -\sigma(v-\widehat{w}(y;G_0,v))|\,\mathrm{d}S _{y} \\ &\le C_0k_1\int_{\partial{G_0}}\Bigl(|u-v|+|\widehat{w}(y;{G_0,}u) -\widehat{w}(y;G_0,v)|\Bigr)^\alpha\,\mathrm{d}S_{y} \\ &\quad + C_0k_2\int_{\partial{G_0}}\Bigl(|u-v|+|\widehat{w}(y;{G_0,}u) -\widehat{w}(y;G_0,v)|\Bigr)\,\mathrm{d}S_{y} \\ &\le K_1 |u-v|^\alpha + K_2 |u-v| \end{aligned}\label{eq:equicontinuity of H} \end{equation} In particular, if $u \in \mathcal C^{0,\alpha} (\mathbb R)$, then $k_2 = 0$ and $K_2 = 0$. Assume now that $\sigma \in \mathcal C^1 (\mathbb R)$. Let $u_1, u_2 \in \mathbb R$. We have that, for $y \in \partial G_0$ \begin{align*} &|\partial _{\nu_y} \widehat{w}(u_1;G_0,y) - \partial _{\nu _ y} \widehat w(u_2,y ) |\\ &=C_0 |\sigma (u_1 - \widehat{w}(u_1;G_0,y)) - \sigma(u_2 - \widehat{w}(y;G_0,u_2)) | \\ &\le C|\sigma' (\xi) | \Big( |u_1 - u_2|+ |\widehat{w}(u_1;G_0,y) - \widehat{w}(y;G_0,u_2)|\Big) \\ &\le C |\sigma'(\xi)| |u_1 - u_2|. \end{align*} for some $\xi$ between $u_1 - \widehat{w}(y;G_0,u_1)$ and $u_2 - \widehat{w}(y;G_0,u_2)$. Since $|\widehat w(u,y)|\le |u|$, for every $K \subset \mathbb R$ compact there exists a constant $C_K$ such that \begin{equation*} |\partial _{\nu_y} \widehat{w}(y;G_0,u_1) - \partial _{\nu _ y} \widehat{w}(y;G_0,u_2) | \le C_K |u_1 - u_2| \quad \forall u_1, u_2 \in K. \end{equation*} Therefore, \begin{equation*} |H_{G_0} (u) - H_{G_0} (v)| \le \widetilde C_K |u_1-u_2| \quad \forall u_1, u_2 \in K. \end{equation*} Let $\sigma \in W^{1,\infty}(\mathbb R)$. By approximation by nondecreasing functions $\sigma_n \in W^{1,\infty} \cap \mathcal C^1$, we obtain that \begin{equation} |\partial _{\nu_y} \widehat{w}(y;G_0,u_1) - \partial _{\nu _ y} \widehat{w}(y;G_0,u_2) | \le 2\| \sigma' \|_\infty |u_1 - u_2|. \end{equation} Therefore, \begin{equation} |H_{G_0} (u) - H_{G_0} (v)| \le 2 \|\sigma'\|_{\infty} |\partial G_0||u_1 - u_2| \quad \forall u_1, u_2 \in \mathbb R. \end{equation} This completes the proof. \end{proof} \begin{lemma} \label{lem:convergence of H uniformly over compacts} Let $u \in \mathbb{R}$. Let $\sigma, \sigma_m$ be nondecreasing continuous functions such that $\sigma(0) = \sigma_m (0) = 0$ satisfy \eqref{eq:2} with the same constants $k_1, k_2$ and $\alpha$, $|\sigma_m| \le |\sigma|$ and $\sigma_m \to \sigma$ in $\mathcal{C}([ -2U,2U])$ for some $U > 0$. Then \begin{equation} H_{G_0,m} \to H_{G_0} \quad \text{in }\mathcal{C}([-U,U]). \end{equation} \end{lemma} \begin{proof} Let $u \in [0,U]$. By Lemma \ref{lem:convergence hat w} we know that \begin{equation*} u- \widehat w_m(y; G_0, u) \searrow u - \widehat w(y; G_0, u) \text{ for a.e. } y \in \partial G_0. \end{equation*} In particular, due to the dominated convergence theorem, $H_{G_0,m} (u) \to H_{G_0} (u)$. An equivalent argument applies to $u \in [-U,0]$. Hence \begin{equation*} H_{G_0,m} \to H_{G_0} \quad \text{pointwise in } [-U,U]. \end{equation*} Since all $\sigma_m$ satisfy \eqref{eq:2} with the same $k_1, k_2, \alpha$, we know that $H_m$ satisfies \eqref{eq:equicontinuity of H} with the same $K_1, K_2$ and $\alpha$. Hence, $H_{G_0,m}$ is an equicontinuous sequence. Applying the Ascoli-Arzela theorem we know that the sequence is relatively compact in $\mathcal C ([-U,U])$ with the supremum norm. It has, at least, a uniformly convergent subsequence. Since every convergent subsequence has to converge to $H_{G_0}$, we know that the the whole sequence $H_{G_0,m}$ converges to $H_{G_0}$ uniformly in $[-U,U]$. \end{proof} \begin{remark} \rm When $\partial G_0$ is assumed $C^{2}$ it is possible to develop other type of techniques (which we shall not present in detail here) showing the existence and uniqueness of solution $\widehat{w}(y;{G_0,}u)$. Indeed, the existence of $\widehat{w}(y;{G_0,}u)$ can be built through passing to the limit after a truncation of the domain process (with the artificial boundary condition $\widehat{w}(y;{G_0,}u)=0$ on the new truncated boundary). The maximum principle for classical solutions (see, e.g. \cite{Ladyzhenskaya1968linear+quasilinear}, \cite[p.206]{friedman2008partial} or \cite{Alikakos:1981regularityparabolic}) allows to get universal a priori estimates which justify the weak convergence and thanks to the monotonicity of the nonlinear term in the interior boundary condition the passing to the limit can well-justified. In addition, it can be proved (see the indicated references) that the limit is also a classical solution on the whole exterior domain. Moreover, the same technique (i.e. the maximum principle for classical solutions) implies the comparison, uniqueness and continuous dependence of the solution $\widehat{w}(y;G_0,u)$. \end{remark} \section{Proof in the smooth case $\sigma \in \mathcal{C}^1 (\mathbb{R})$} \subsection{Auxiliary function $w_\protect\varepsilon^j$} To pass to the limit as ${\varepsilon}\to 0$ in \eqref{eq:8} we need some auxiliary functions. \begin{definition} \rm Let $u \in \mathbb{R}$, $\varepsilon > 0$ and $j \in \Upsilon_\varepsilon$. We define the function $w^j_{\varepsilon}(\cdot; G_0,u)$ as the solution of the problem \begin{equation} \begin{gathered} \Delta w^j_{\varepsilon}=0 \quad \text{if } x\in T^j_{\varepsilon/4}\setminus\overline{G^j_\varepsilon},\\ \partial_{\nu_x}w^j_{\varepsilon}-\varepsilon^{-\gamma}\sigma(u-w^j_\varepsilon)=0 \quad \text{if } x\in \partial G^j_{\varepsilon},\\ w^j_{\varepsilon}=0 \quad \text{if } x\in \partial T^j_{\varepsilon/4}, \end{gathered} \label{eq:9} \end{equation} where \begin{equation} T^j_r =\{ x\in \mathbb{R}^n : | x-P^j_\varepsilon|\le r \}, \end{equation} $P^j_\varepsilon$ is the center of $Y^j_{\varepsilon}$. Finally, we define \begin{equation} W_{\varepsilon}(x; G_0, u) = \begin{cases} w^j_{\varepsilon}(x; G_0, u) & \text{if } x\in T^j_{\varepsilon/4}\setminus {\overline{G^{j}_{\varepsilon}}} , j\in\Upsilon_\varepsilon , \\ 0 & \text{if } x\in \mathbb{R}^n\setminus \cup_{j\in\Upsilon_\varepsilon}\overline{T^j_{\varepsilon/4}}. \end{cases} \label{eq:10} \end{equation} \end{definition} Applying the comparison principle we obtain the following result. \begin{lemma} \label{lem:comparison ws} Let $u \ge 0$. Then $0 \le w_\varepsilon^j (\cdot; G_0, u) \le \widehat w_\varepsilon^j (\cdot ; G_0, u) $. If $u \le 0$ then $\widehat w_\varepsilon^j(\cdot; G_0, u) \le w_\varepsilon^j (\cdot; G_0, u) \le 0$. \end{lemma} \begin{remark}\label{rem:monotonicity} \rm Note that, if $u = 0$, then $w_\varepsilon^j (\cdot; G_0, 0) \equiv 0$. \end{remark} Let us prove some properties of $W_{\varepsilon}(x; G_0, u)$. First, we introduce the following lemma. \begin{lemma}[Uniform trace constant] \label{eq:trace constant} There exists a constant $C_T > 0$ such that, for all $\varepsilon > 0$ \begin{equation} \varepsilon^{-\gamma } \int _{ \partial G_ \varepsilon ^j } |f|^2 \,\mathrm{d}S \le C_T \int _ {T_{\varepsilon/4}^j \setminus G_\varepsilon^j } |\nabla f|^2 \,\mathrm{d}x \quad \forall f \in H^1 \big(T_{\varepsilon/4}^j \setminus \overline {G_\varepsilon^j}, \partial T_{\frac \varepsilon 4} ^j \big) \end{equation} \end{lemma} \begin{proof} First, we extend $f$ to $H_0^1(Y_\varepsilon^j)$ where $Y_\varepsilon^j = \varepsilon j + \varepsilon Y $. In \cite{Shaposhnikova+Oleinik:1996Lincei} we find that \begin{equation} \varepsilon^{-\gamma} \int_{\partial G_\varepsilon^j } |f|^2 \le C \Big( \int_ {Y_\varepsilon^j} |f|^2 + \int _{Y_\varepsilon^j } |\nabla f|^2 \Big). \end{equation} Since $f = 0$ on $\partial Y_\varepsilon^j$, taking $\tilde f(y) = f(P_\varepsilon^j + \varepsilon y)$, we have \begin{equation} \int _{Y} |\tilde f|^2 \,\mathrm{d}y \le C \int_{Y} |\nabla \tilde f|^2 \,\mathrm{d}y. \end{equation} Since $\nabla_x f = \varepsilon \nabla_y \tilde f $ we have \begin{equation} \int_ {Y_\varepsilon^j} |f|^2 \le \varepsilon^2 \int _{Y_\varepsilon^j } |\nabla f|^2 . \end{equation} Hence, the result is proved. \end{proof} We have some precise estimates on the norm of $W_\varepsilon$: \begin{lemma}\label{lem:1} For all $u \in \mathbb{R}$, we have \begin{gather} \|\nabla W_\varepsilon\|^2_{L^{2}(\Omega_\varepsilon)} \le K( |u| + |u|^2), \label{eq:bound nabla W} \\ \| W_\varepsilon\|^2_{L^{2}(\Omega_\varepsilon)} \le K( |u| + |u|^2) \varepsilon^2 . \label{eq:11} \end{gather} \end{lemma} \begin{proof} Let $u \in \mathbb R$ be fixed. If we take $w^j_\varepsilon$ as a test function in weak formulation of problem \eqref{eq:9} we obtain \begin{equation*} \int_{T^j_{\varepsilon/4}\setminus G^j_{\varepsilon}}| \nabla w^j_\varepsilon|^2\,\mathrm{d}x -\varepsilon^{-\gamma}\int_{\partial G^j_{\varepsilon}}\sigma(u-w^j_\varepsilon) w^j_\varepsilon\,\mathrm{d}S=0. \end{equation*} We rewrite this as follows: \begin{equation*} \int_{T^j_{\varepsilon/4} \setminus G^j_{\varepsilon}}| \nabla w^j_\varepsilon|^2\,\mathrm{d}x +\varepsilon^{-\gamma}\int_{\partial G^j_{\varepsilon}}\sigma(u-w^j_\varepsilon)(u- w^j_\varepsilon)\,\mathrm{d}S =\varepsilon^{-\gamma}\int_{\partial G^j_{\varepsilon}}\sigma(u-w^j_\varepsilon)u\,\mathrm{d}S. \end{equation*} Since $\sigma$ is nondecreasing we have that \begin{align*} \|\nabla w^j_\varepsilon\|^2_{L^{2}(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})} &\le \varepsilon^{-\gamma} | u|\int_{\partial G^j_{\varepsilon}}| \sigma( u-w^j_\varepsilon )|\,\mathrm{d}S. \end{align*} Because of \eqref{eq:2} and that $|s| ^\alpha \le 1 + |s|$ for every $s \in \mathbb R$, we have \begin{align*} \varepsilon^ {-\gamma} \int_{\partial G^j_{\varepsilon}}| \sigma( u-w^j_\varepsilon )|\,\mathrm{d}S &\le k_1 \varepsilon^ {-\gamma}\int _{\partial G_\varepsilon^j } |u - w^j_\varepsilon |^\alpha\,\mathrm{d}S + k_2 \varepsilon^ {-\gamma}\int_{\partial G_\varepsilon^j } |u-w^j_\varepsilon|\,\mathrm{d}S \\ & \le k_1 \varepsilon^ {-\gamma} |\partial G_\varepsilon^j | + (k_1 + k_2)\varepsilon^ {-\gamma}\int _{\partial G_\varepsilon^j } |u - w^j_\varepsilon |\,\mathrm{d}S. \end{align*} Applying Lemma \ref{eq:trace constant} and that, for every $a,b,C \in \mathbb R$ it holds that $ab \le \frac {C^2} 2 a^2 + \frac{1}{2C^2} b^2$, we obtain \begin{align*} (k_1 + k_2) \varepsilon^{-\gamma} \int_{\partial G^j_{\varepsilon}}| u-w^j_\varepsilon|\,\mathrm{d}S & \le \varepsilon^{ -\gamma } C |\partial G_\varepsilon ^j | + \frac{1}{2 C_T |u|} \varepsilon^{ -\gamma } \int_{ \partial G_\varepsilon^j } |u - w_\varepsilon^j | ^2\,\mathrm{d}S \\ & \le C |u| \varepsilon^{ -\gamma } |\partial G^j_{\varepsilon}|+\frac{1}{2C_T| u|} \| u-w^j_\varepsilon\|^2_{L^{2}(\partial G^j_{\varepsilon})}\\ & \le C |u|\varepsilon ^n +\frac{1}{2| u|}\| \nabla (u-w^j_\varepsilon) \|^2_{L^{2}(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})}\\ & = C |u| \varepsilon^n +\frac{1}{2| u|}\| \nabla w^j_\varepsilon\|^2_{L^{2} (T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})}. \end{align*} Therefore, \begin{align*} \|\nabla w^j_\varepsilon\|^2_{L^{2}(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})} &\le K ( |u| + |u|^2) \varepsilon^n +\frac{1}{2}\| \nabla w^j_\varepsilon\|^2_{L^{2} (\partial G^j_{\varepsilon})}. \end{align*} Thus, we have \begin{equation*} \|\nabla w^j_\varepsilon\|^2_{L^{2}(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})} \le K ( |u| + |u|^2) \varepsilon^n. \end{equation*} Adding over $j\in \Upsilon_{\varepsilon}$, and taking into account that $\# \Upsilon_{\varepsilon} \le d \varepsilon^{-n}$, we deduce that \eqref{eq:bound nabla W} holds. Using Friedrich's inequality we obtain \begin{equation*} \| w^j_\varepsilon\|^2_{L^{2}(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})} \le \varepsilon^2 K\|\nabla w^j_\varepsilon\|^2_{L^{2}(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})}, \end{equation*} so \eqref{eq:11} holds. This completes the proof. \end{proof} \subsection{Auxiliary function $v_\protect\varepsilon^j = w_\varepsilon^j - \widehat w_\protect\varepsilon^j$} Let us define: \begin{equation} v_\varepsilon^j = w_\varepsilon^j - \widehat w_\varepsilon^j. \end{equation} This functions is the solution of the problem \begin{equation} \begin{gathered} \Delta v^j_{\varepsilon}=0 \quad \text{if } x\in T^j_{\varepsilon/4}\setminus\overline{ G^j_{\varepsilon}}, \\ \partial_{\nu}v^j_{\varepsilon}-\varepsilon^{-\gamma}(\sigma(u-w^j_\varepsilon)- \sigma(u-\widehat{w}_\varepsilon^j))=0, \quad\text{if } x\in \partial G^j_{\varepsilon},\\ v^j_{\varepsilon}=-\widehat{w}_\varepsilon^j (x; G_0, u), \quad \text{if } x\in \partial T^j_{\varepsilon/4} . \end{gathered} \label{eq:16} \end{equation} \begin{lemma} \label{lem:2} The following estimates hold \begin{gather} \sum_{j\in\Upsilon_\varepsilon}\|\nabla(w^j_\varepsilon(x; G_0, u)- \widehat{w}_\varepsilon^j (x; G_0, u))\|^2_{L^{2}(T^j_{\frac{\varepsilon}{4} }\setminus G^j_\varepsilon)} \le K(|u| + |u|^2)\varepsilon^2, \\ \sum_{j\in\Upsilon_{\varepsilon}}\| w^j_\varepsilon(x; G_0, u)- \widehat{w} _\varepsilon^j (x; G_0, u) \|^2_{L^{2}(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})}\le K(|u| + |u|^2)\varepsilon^4. \end{gather} \end{lemma} \begin{proof} From Lemma \ref{lem:comparison ws} it is clear that \begin{equation} |v^j_{\varepsilon}(x; G_0, u)|\le |\widehat{w}_\varepsilon^j(x; G_0, u)| \quad \forall x \in \overline {T_{\varepsilon/4}^j }\setminus G_\varepsilon^j. \end{equation} Integrating by parts $v^j_{\varepsilon}( \Delta v_\varepsilon^j)$ and using \eqref{eq:16} we deduce that \begin{align*} &\int_{T^j_{\varepsilon/4}\setminus G^j_{\varepsilon}}|\nabla v^j_{\varepsilon}|^2\,\mathrm{d}x -\varepsilon^{-\gamma}\int_{\partial G^j_\varepsilon}(\sigma(u-w^j_\varepsilon) -\sigma(u-\widehat{w})) v^j_{\varepsilon}\,\mathrm{d}S\\ &=-\int_{\partial T^j_{\varepsilon/4}}(\partial_{\nu}v^j_{\varepsilon}) \widehat{w}_\varepsilon^j(x; G_0, u)\,\mathrm{d}S \end{align*} By the monotonicity of $\sigma$ and applying Green's first identity, we have \begin{align*} \|\nabla v^j_{\varepsilon}\|^2_{L_2(T^j_{\varepsilon/4}\setminus G^j_{\varepsilon})} &\le -\int_{\partial T^j_{\varepsilon/4}}(\partial_{\nu}v^j_{\varepsilon}) \widehat{w}_\varepsilon^j(x; G_0, u)\,\mathrm{d}S\\ &= -\int_{T^j_{\varepsilon/4}\setminus T^j_{\varepsilon/8}}\nabla v^j_{\varepsilon} \nabla\widehat{w}_\varepsilon^j\,\mathrm{d}x+\int_{\partial T^j_{\varepsilon/8}} (\partial_{\nu}v^j_{\varepsilon})\widehat{w}_\varepsilon^j\,\mathrm{d}S. \end{align*} Applying Lemmas \ref{lem:bounds hat w eps} and \ref{lem:bounds gradient hat w eps} we have \begin{gather*} | v^j_{\varepsilon}(x; G_0, u)| \le |\widehat{w}_\varepsilon^j (x; G_0, u) |\le K|u|\varepsilon^2. \\ |\nabla \widehat w _\varepsilon^j (x; G_0, u) | \le K|u| \varepsilon \end{gather*} for all $x \in T_{\varepsilon/8}^j $, where $K$ does not depend on $\varepsilon$. Since $v_\varepsilon^j$ is harmonic, denoting $T^x_r = \{ z \in \mathbb R^n : |x-z| 0$ there exists $A_\delta$ measurable such that $|A_\delta| < \delta$ and $H_m(u_m) \to H (\widetilde u)$ uniformly $\Omega \setminus A_\delta$. Since $H_m(u_m) \rightharpoonup \widetilde H$ in $L^2 (\Omega \setminus A_\delta)$ we have that $H (\widetilde u) = \widetilde H$ a.e. in $\Omega \setminus A_\delta$. Hence $H(\widetilde u) = \widetilde H$ in a.e. $\Omega$, so \begin{equation*} H_m (u_m) \rightharpoonup H(\widetilde u) \textrm{ in } L^2 (\Omega). \end{equation*} By passing to the limit in the weak formulation we deduce that $\widetilde u = u$. \end{proof} \begin{thebibliography}{00} \bibitem{Alikakos:1981regularityparabolic} N.~D. 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