\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 221, pp. 1--11.\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-2017/221\hfil Shape differentiation in the non smooth case] {Shape differentiation of steady-state reaction-diffusion problems arising in chemical engineering with non-smooth kinetics\\ with dead core} \author[D. G\'omez-Castro \hfil EJDE-2017/221\hfilneg] {David G\'omez-Castro} \address{David G\'omez-Castro \newline Instituto de Matem\'atica Interdisciplinar, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040 Madrid, Spain} \email{dgcastro@ucm.es} \dedicatory{Communicated by Jes\'us Ildefonso D\'iaz} \thanks{Submitted July 20, 2017. Published September 16, 2017.} \subjclass[2010]{35J61, 46G05, 35B30} \keywords{Shape differentiation; reaction-diffusion; chemical engineering; \hfill\break\indent dead core} \begin{abstract} In this paper we consider an extension of the results in shape differentiation of semilinear equations with smooth nonlinearity presented by D\'iaz and G\'omez-Castro \cite{Diaz+Gomez-Castro:2015shapediff}, to the case in which the nonlinearities might be less smooth. Namely we show that Gateaux shape derivatives exists when the nonlinearity is only Lipschitz continuous, and we will give a definition of the derivative when the nonlinearity has a blow up. In this direction, we study the case of root-type nonlinearities. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this article we consider the shape differentiation of a family of diffusion-reaction problems introduced by Aris in the context of optimization of chemical reactors depending on the spatial domain (see \cite{Aris+Strieder:1973}). It was later shown that the model can be rigorously deduced as a limit of different nonhomogeneous microscopic models (see \cite{Conca+Diaz+Linan+Timofte:2004,diaz+gomez-castro+podolskii+shaposhnikova2017jmaa}). In particular we are interested in the solutions of the problem \begin{equation} \begin{gathered} -\Delta w+\beta (w)={f}, \quad \text{in }\Omega , \\ w=1, \quad \text{on }\partial \Omega , \end{gathered} \label{eq:model problem w} \end{equation}% and their behaviour as we deform the domain $\Omega$. It will be sometimes useful to consider the change in variable $u = 1 -w$, $g(u) = \beta(1) - \beta (1 - u) $ and $\widehat f = \beta (1) - f$, so that we have $u = 0$ on the boundary. After this change in variable we have that $u$ is the solution of \begin{equation} \begin{gathered} -\Delta u+g(u)=\widehat f, \quad \text{in }\Omega , \\ u=0, \quad \text{on }\partial \Omega . \end{gathered}\label{eq:model problem u} \end{equation} These functions will be sometimes denoted $u_\Omega, w_\Omega$ when different domains are considered. In \cite{Diaz+Gomez-Castro:2015shapediff} (see also \cite{Simon:1980differentiation,Henrot+Pierre:2005,Pironneau:1984}) the authors showed that, if $\beta \in W^{2,\infty} (\mathbb R)$ and $f \in L^2(\Omega)$, then the maps \begin{align*} W^{1, \infty} (\mathbb R^n , \mathbb R^n ) & \to H_0^1 (\Omega) \\ \theta & \mapsto u_{(I + \theta) \Omega} \circ ( I + \theta ) \\ W^{1, \infty} (\mathbb R^n , \mathbb R^n ) & \to L^2 (\mathbb R^n) \\ \theta & \mapsto u_{(I + \theta) \Omega}, \end{align*} where the extension by $0$ is considered in $\mathbb R^n \setminus (1+\theta)\Omega$, are Fr\'echet differentiable at $0$. Fixing $\theta \in W^{1,\infty} (\mathbb R^n, \mathbb R^n)$ it was shown in \cite{Diaz+Gomez-Castro:2015shapediff} that the directional derivative (the derivative of $u_\tau = u_{(I + \tau \theta) \Omega}$ with respect to $\tau$, $\frac{d u_\tau}{d\tau} = \frac{d u_\tau}{d\tau}|_{\tau = 0}$) is the solution of the problem \begin{equation} \begin{gathered} -\Delta \frac{d u_\tau} {d\tau}+ g^{\prime }(u_\Omega)\frac{d u_\tau} {d\tau}=0, \quad \text{in }\Omega, \\ \frac{d u_\tau} {d\tau}=-\nabla u_\Omega \cdot \theta , \quad \text{on } \partial \Omega. \end{gathered} \label{eq:u' in terms of theta} \end{equation} Notice that, since $u = 1 -w$, we have that $\frac{d u_\tau} {d\tau} = -\frac{d w_\tau} {d\tau}$. Hence, taking into account that $g'(u) = -\beta' (w)$, we have \begin{equation} \begin{gathered} -\Delta \frac{d w_\tau} {d\tau}+\beta^{\prime }(w_{\Omega})\frac{d w_\tau} {d\tau}=0, \quad \text{in }\Omega, \\ \frac{d w_\tau} {d\tau}=-\nabla w_\Omega \cdot \theta , \quad \text{on } \partial \Omega. \end{gathered} \label{eq:w' in terms of theta} \end{equation} The aim of this paper is to extend this results to the case when $\beta \notin W^{2,\infty}$. First, we will show that, when $\beta \in W^{1,\infty}$, the Gateaux shape derivative exists. However, if $\beta$ is not locally Lipschitz continuous, the solution of \eqref{eq:model problem w} might develop a region of positive measure \begin{equation} \label{eq:dead core} N_\Omega = \{ x \in \Omega: w_\Omega (x) = 0 \}. \end{equation} This region, known as \emph{dead core}, was studied at length in \cite{Diaz:1985,bandle+sperb+stakgold1984}. It is a necessary condition for the existence of this region that $\beta' (w_\Omega) = +\infty$. Hence, equation \eqref{eq:w' in terms of theta} cannot be understood immediately in a standard way. In this setting, we will show that there exists a limit of the previous theory. \section{Statement of results} For the rest of the paper $\Omega \subset \mathbb R^n$ will be a fixed domain, of class $\mathcal C^2$, and $n \ge 2$. \subsection{Existence and estimates of shape derivatives} \subsubsection*{Existence of Gateaux derivative when $\beta \in W^{1,\infty}$} In \cite{Diaz+Gomez-Castro:2015shapediff} the authors prove the existence of a shape derivative in the Fr\'echet sense when $\beta \in W^{2,\infty}(\mathbb R)$. Nonetheless, as is it usually the case, the equation for the derivative is well defined in a straightforward way when $\beta \in W^{1, \infty}(\mathbb R)$. In fact, the following result shows that, if $\beta \in W^{1,\infty}(\mathbb R)$ rather than $W^{2,\infty}(\mathbb R)$, then the shape derivative exists only in the Gateaux sense, which is weaker than the Fr\'echet sense. \begin{theorem} \label{thm:existence of Gateuax derivate} Let $\theta \in W^{1, \infty} (\mathbb R^n, \mathbb R^n)$, $\beta \in W^{1,\infty} (\mathbb R)$ be nondecreasing such that $\beta (0) = 0$ and $f \in H^1 (\mathbb R^n)$. Then, the applications \begin{align*} \mathbb R &\to L^2 (\Omega) \\ \tau &\mapsto u_{(I + \tau \theta) \Omega} \circ (I + \tau \theta), \end{align*} and \begin{align*} \mathbb R &\to L^2 (\mathbb R^n) \\ \tau &\mapsto u_{(I + \tau \theta) \Omega} \end{align*} are differentiable at $0$. Furthermore, $\frac{d u_\tau}{d\tau}|_{\tau = 0}$ is the unique solution of \eqref{eq:u' in terms of theta}. \end{theorem} \begin{remark} \rm In most cases, the process of homogenization mentioned in the introduction gives an homogeneous equation \eqref{eq:model problem w} in which $\beta$ is the same as in the microscopic limit, and thus it is natural that $\beta$ be singular. However, it sometimes happens that the limit kinetic is different. In the homogenization of problems with particles of critical size (see \cite{diaz+gomez-castro+podolskii+shaposhnikova2017generalmmg}) it turns out that the resulting kinetic in the macroscopic homogeneous equation \eqref{eq:model problem w} satisfies $\beta \in W^{1,\infty}$, even when the original kinetic of the microscopic problem was a general maximal monotone graph. \end{remark} \subsubsection*{From $W^{2,\infty}$ to $W^{1,\infty} \cap \mathcal C^1$} Let us show that the shape derivative is continuously dependent on the nonlinearity, and thus that we can make a smooth transition from the Fr\'echet scenario presented in \cite{Diaz+Gomez-Castro:2015shapediff} to our current case. For the rest of the paper we will use the notation: \begin{equation} v = \frac{d w_\tau}{d\tau}\Big|_{\tau = 0} \end{equation} \begin{lemma} \label{thm:approximation} Let $f \in L^2 (\mathbb R^n)$, $\beta \in W^{1,\infty} (\mathbb R)$ be nondecreasing functions such that $\beta(0) = 0$ and let $\beta_n \in W^{2,\infty} (\mathbb R)$ nondecreasing such that $\beta_n (0) = 0$. Let $w_n$ be the unique solution of \begin{equation} \label{eq:model wn} \begin{gathered} -\Delta w_n + \beta_n (w_n) = f \quad\text{in } \Omega, \\ w_n = 1 \quad\text{on } \partial \Omega. \end{gathered} \end{equation} Then \begin{gather} \|w_n - w \|_{H^1 (\Omega)} \le C \| \beta_n - \beta \|_{L^\infty} \label{eq:H1 norm wm minus w}\\ \| w_n - w\|_{H^2 (\Omega)} \le C (1 + \| \beta' \|_{L^\infty} ) \| \beta_n - \beta \|_{L^\infty}. \label{eq:H2 norm wm minus w} \end{gather} Furthermore, let $\beta \in C^1 (\mathbb R) \cap W^{1,\infty} (\mathbb R)$ and $v_n$ be the unique solution of \begin{equation} \label{eq:model vn} \begin{gathered} -\Delta v_n + \beta_n' (w_n) v_n = 0 \quad \text{in } \Omega, \\ v_n + \nabla w_n \cdot \theta = 0 \quad \text{on } \partial \Omega. \end{gathered} \end{equation} Then \begin{equation} v_n \rightharpoonup v \quad \text{ in } H^1 (\Omega). \end{equation} \end{lemma} \begin{remark} \rm In \eqref{eq:H1 norm wm minus w} the notation \begin{equation*} \| \beta_n - \beta \|_{L^\infty} = \sup_{x \in \mathbb R} |\beta_n (x) - \beta(x)| \end{equation*} does not mean that either $\beta_n$ or $\beta$ are $L^\infty (\mathbb R)$ functions themselves, but rather that their difference is pointwise bounded, and, in fact, this bound is destined to go $0$ as $n\to +\infty$. We will use this notation throughout the paper. \end{remark} \subsubsection*{Shape derivative with a dead core} We can prove that the shape derivative in the smooth case has, under some assumptions, a natural limit when $\beta$ not smooth. In some cases in the applications (see \cite{Diaz:1985}) we can take $\beta$ so that $\beta' (w_\Omega)$ has a blow up. It is common, specially in Chemical Engineering, that $\beta' (0) = +\infty$ and $N_\Omega$ exists (see \cite{Diaz:1985}). In this case $\beta' (w_\Omega) = +\infty$ in $N_\Omega$. Because of this fact, the natural behaviour of the weak solutions of \eqref{eq:w' in terms of theta} is $v = 0$ in $N_\Omega$. We have the following result \begin{theorem} \label{thm:existence of derivative} Let $\beta$ be nondecreasing, $\beta(0) = 0$, $\beta' (0) = + \infty$, $$ \beta \in \mathcal C(\mathbb R) \cap \mathcal C^1 (\mathbb R \setminus \{0\}), $$ and assume that $|N_\Omega|> 0$, $\theta \in W^{1,\infty} (\mathbb R^n, \mathbb R^n)$ and $ 0 \le f \le \beta(1)$. Then, there exists $v$ a solution of \begin{equation} \label{eq:model v} \begin{gathered} -\Delta v + \beta'(w_\Omega) v = 0 \quad \Omega \setminus N_\Omega, \\ v = 0 \quad \partial N_\Omega, \\ v = - \nabla w_\Omega \cdot \theta \quad \partial \Omega, \end{gathered} \end{equation} in the sense that $v \in H^1 (\Omega)$, $v = 0$ in $N_\Omega$, $v = - \nabla w_\Omega \cdot \theta$ in $L^2 (\partial \Omega)$, $\beta' (w_\Omega) v^2 \in L^1 (\Omega)$ and \begin{equation} \int _{\Omega \setminus N_\Omega} \nabla v \nabla \varphi + \int _{\Omega\setminus N_\Omega} \beta'(w) v \varphi = 0 \end{equation} for every $\varphi \in W^{1, \infty}_c (\Omega \setminus N_\Omega) $. Furthermore, for $m \in \mathbb N$, consider $\beta_m$ defined by $$ \beta_m'(s) = \min \{ m , \beta' (s) \}, \quad \beta_m(0) = \beta(0) = 0, $$ and let $w_m, v_m$ be the unique solutions of \eqref{eq:model wn} and \eqref{eq:model vn}. Then, \begin{equation} v_m \rightharpoonup v, \quad \text{in }H^1 (\Omega), \end{equation} where $v$ is a solution of \eqref{eq:model v}. \end{theorem} The uniqueness of solutions of \eqref{eq:model v} when $\beta'(w_\Omega)$ blows up is by no means trivial. Problem \eqref{eq:model v} can be written in the following way: \begin{equation} \label{eq:linear problem with potential} -\Delta v + V v = f \end{equation} where $V = \beta' (w_\Omega)$ may blow up as a power of the distance to a piece of the boundary. This kind of problems are common in Quantum Physics, although their mathematical treatment is not always rigorous (cf. \cite{Diaz2015ambiguousSchrodinger,Diaz2017ambiguousSchrodinger}). In the next section we will show estimates on $\beta' (w_\Omega)$. Let us state here some uniqueness results depending on the different blow-up rates. When the blow-up is subquadratic (i.e. not too rapid), by applying Hardy's inequality and the Lax-Migram theorem, we have the following result (see \cite{Diaz2015ambiguousSchrodinger,Diaz2017ambiguousSchrodinger}). \begin{corollary} Let $N_\Omega$ have positive measure and $\beta'(w(x)) \le C d(x, N_\Omega)^{-2}$ for a.e. $x \in \Omega \setminus N_\Omega$. Then the solution $v$ is unique. \end{corollary} The study of solutions of problem \eqref{eq:linear problem with potential} in $\Omega$ when $V \in L^1_{\rm loc} (\Omega)$ by many authors (see \cite{Diaz+Rakotoson:2010,Diaz+Gomez-Castro:2015veryweak} and the references therein). Existence and uniqueness of this problem in the case $V(x) \ge C d (x, \partial \Omega)^{-r}$ with $r > 2$ was proved in \cite{Diaz+Gomez-Castro:2015veryweak}. Applying these techniques one can show that \begin{corollary} Let $N_\Omega$ have positive measure and $\beta'(w(x)) \ge C d(x, N_\Omega)^{-r}, r > 2$ for a.e. $x \in \Omega\setminus N_\Omega$. Then the solution $v$ is unique. \end{corollary} Similar techniques can be applied to the case $\beta'(w(x)) \ge C d(x, N_\Omega)^{-2}$. This will be the subject of a further paper. \subsection{Estimates of $w_\Omega$ close to $N_\Omega$} Let us study the solution $w_\Omega$ on the proximity of the dead core and the blow up behaviour of $\beta' (w_\Omega)$. First, we present a known example \begin{example}\rm Explicit radial solutions with dead core are known when $\beta (w) = |w|^{q-1}w$ ($0 < q < 1$), $\Omega$ is a ball of large enough radius and $f$ is radially symmetric. In this case it is known that $N_\Omega$ exists, has positive measure and $$ \frac{1}{C} d(x, N_\Omega)^{-2}\le \beta' (w_\Omega) \le C d(x, N_\Omega)^{-2}. $$ For the details see \cite{Diaz:1985}. \end{example} In fact, we present here a more general result to study the behaviour in the proximity of the dead core, based on estimates from \cite{Diaz:1985}. \begin{proposition} \label{prop:general nonlinearity} Let $f = 0$, $\beta$ be continuous, monotone increasing such that $\beta(0) = 0$, $w$ be a solution of \eqref{eq:model problem w} that develops a dead core $N_\Omega$ of positive measure and $\partial N_\Omega \in \mathcal C^1$. Assume that \begin{equation} G(t) = \sqrt 2 \Big( \int_0^t \beta (\tau) d \tau + \alpha t \Big)^{1/2}, \quad \text{where } \alpha = \max \big\{ 0 , \min_{x\in\partial \Omega} H(x) \frac{\partial w}{\partial n}(x) \big\} , \end{equation} is such that $\frac 1 G \in L^1(\mathbb R)$. Then \begin{equation} w_\Omega(x) \le \Psi^{-1} (d(x,{N_{\Omega}})), \quad \text{where } \Psi (s) = \int_{0}^s \frac{dt} { G(t) }, \end{equation} in a neighbourhood of $N _\Omega$. \end{proposition} \begin{example}[Root type reactions] \rm Let $f = 0$, $\beta(s) = \lambda |s|^{q-1} s$ with $0 < q < 1$ and $\Omega$ be convex such that $N_\Omega$ exists and $\partial N_\Omega \in \mathcal C^1$. Then \begin{equation} w_\Omega(x) \le C d(x, {N_\Omega} )^{\frac 2 {1-q}}. \end{equation} Furthermore \begin{equation} \beta'(w_\Omega (x)) \ge Cd(x, {N_\Omega}) ^{ -2 }. \end{equation} \end{example} \section{Proof of Theorem \ref{thm:existence of Gateuax derivate}} For the rest of this paper let us denote \begin{equation} u_\tau = u_{(I + \tau \theta) \Omega}. \end{equation} Notice that $u_0 = u_\Omega$. Let us define $U_\tau = u_{(I + \tau \theta)\Omega} \circ (I + \tau \theta) \in H_0^1 (\Omega)$. Again $U_0 = u_0 = u_\Omega$. We have \begin{equation} \int _\Omega A_\tau \nabla U_\tau \nabla \varphi + \int _ \Omega g (U_\tau) \varphi J_\tau = \int _ \Omega f_\tau \varphi J_\tau , \end{equation} where $J_\tau$ is the Jacobian of the transformation. $f_\tau = f \circ (I + \tau \theta)$ and $A_\tau$ is the corresponding diffusion matrix (see \cite{Diaz+Gomez-Castro:2015shapediff} for the explicit expression). Fortunately, $J_\tau \ge 0$ and, for $\tau$ small, we have that $\xi \cdot A_\tau \xi \ge A_0 |\xi|^2$ for some $A_0 > 0$ constant. Considering the difference of the weak formulations of $U_\tau$ and $U_0 = u_\Omega$ we have \begin{align*} & \int_{\Omega} A_\tau \nabla (U_\tau - u_0) \nabla \varphi + \int _ \Omega (g(U_\tau) - g(u_0) ) J_\tau \varphi \\ & = \int _ \Omega (f_\tau J_\tau - f) \varphi + + \int _ \Omega (I - A_\tau) \nabla u_0 \nabla \varphi + \int _ \Omega (J_\tau -1 ) g(u_0) \varphi. \end{align*} Hence, by the monotonicity of $g$, we have \begin{align*} & \| \nabla \big( \frac{U_\tau - u}{\tau } \big) \|_{L^2 } \\ &\le C \Big( \| \frac{f_\tau - f}{\tau} \|_{L^2} + \| \frac{ A_\tau - I}{\tau } \| _{L^\infty} \| \nabla u_0 \|_{L^2} + \| \frac{ J _ \tau - 1}{\tau} \|_{L^\infty} \|g(u_0)\|_{L^2} \Big) \end{align*} Since $f_\tau, A_\tau$ and $J_\tau$ are differentiable at $0$, there is weak $H_0^1 (\Omega)$ limit. Hence, the limit is strong in $L^2 (\Omega)$. Therefore, the function \begin{equation} u_\tau = U_\tau \circ (I + \tau \theta )^{-1} \end{equation} is differentiable with respect to $\tau \in \mathbb R$ with images in $L^2 (\Omega)$ at $\tau = 0$. Also \begin{equation} H_0^1 (\Omega) \ni \frac{d U_\tau}{d \tau} \Big|_{\tau = 0} = \frac{du_\tau}{d\tau} \Big|_{\tau = 0} + \nabla u_0 \cdot \theta. \end{equation} To characterize the derivative, we differentiate on the variational formulation \begin{equation*} \int_{\mathbb{R}^n} f \varphi = \int_{\mathbb{R}^n} \left( -u_\tau \Delta \varphi + g(u_\tau) \varphi \right) \quad \forall \varphi \in \mathcal C_c ^\infty (\Omega). \end{equation*} Considering the difference of the equations for $u_\tau$ and $u_0$ and diving by $\tau$, \begin{align} 0 &= \int _{\mathbb R^n } \Big( -\frac{u_\tau - u_0}{\tau } \Delta \varphi + \frac{g(u_\tau) - g(u_0)}{\tau} \varphi \Big) \\ &= \int _{\mathbb R^n } \frac{u_\tau - u_0}{\tau } \Big( - \Delta \varphi + \frac{g(u_\tau) - g(u_0)}{u_\tau - u_0} \varphi \Big). \end{align} Notice that \begin{equation*} \big| \frac{g(u_\tau) - g(u_0)}{u_\tau - u_0} \big| \le \| g ' \|_{L^\infty}. \end{equation*} Therefore, up to a subsequence, $\frac{g(u_\tau) - g(u_0)}{u_\tau - u_0}$ converges weakly in $L^2 (\Omega)$. On the other hand since $u_\tau \to u_0$ pointwise, again up to a subsequence, so \begin{equation} \frac{g(u_\tau) - g(u_0)}{u_\tau - u_0} \to g' (u_0) \quad \text{ a.e. in } \Omega. \end{equation} Via a C\'esaro mean argument we have that the weak $L^2$ limit and pointwise limit coincide. Hence, passing to the limit in $L^2 (\Omega)$ \begin{equation} 0 = \int_{\Omega} \frac{du_\tau}{d\tau }\Big|_{\tau = 0} \left( - \Delta \varphi + g^{\prime }(u_0) \varphi \right), \quad \varphi \in \mathcal C_c ^\infty (\Omega). \end{equation} Therefore $\frac{du_\tau}{d\tau}$ is the unique solution of \eqref{eq:u' in terms of theta}. \section{Proof of Lemma \ref{thm:approximation}} By considering the difference of the weak formulations we have \begin{equation*} \int_ \Omega \nabla (w_m - w) \nabla \varphi + \int_{\Omega } (\beta_m (w_m) - \beta_m (w)) \varphi = \int_{ \Omega } (\beta(w) - \beta_m (w) ) \varphi . \end{equation*} Taking $\varphi = w_m - w$, and using the monotonicity of $\beta_m$ we have \begin{equation*} \| \nabla (w_m - w) \|_{L^2} ^2 \le \| \beta_m - \beta \|_{L^\infty} \| w_m - w \|_{L^1 (\Omega)}. \end{equation*} Using Poincar\'e inequality and the embedding $L^1 \hookrightarrow L^2$ we have \begin{equation*} \| w_m - w \|_{L^2} \le C \| \beta_m - \beta \|_{L^\infty}. \end{equation*} By considering the equation \begin{align*} \| \Delta (w_m - w) \|_{L^2} &= \| \beta(w) - \beta_m (w_m) \|_{L^2} \\ &\le \| \beta(w) - \beta(w_m) \|_{L^2} + \| \beta(w_m) - \beta_m (w_m) \|_{L^2} \\ &\le \| \beta' \|_{L^\infty } \| w_m - w \|_{L^2} + \| \beta_m - \beta \|_{L^\infty}. \end{align*} Hence, to deduce \eqref{eq:H2 norm wm minus w} we apply that \begin{align*} \| w_m - w \|_{H^2 } &\le C ( \| \Delta( w_m - w ) \|_{L^2 } + \|w_m - w \|_{L^2}). \end{align*} Considering the difference of the weak formulations of the problems for $v_m$ and $v$ we have \begin{equation} \begin{aligned} \int _ \Omega \nabla (v_m - v) \nabla \varphi &= \int _ \Omega (\beta'(w) v - \beta_m'(w_m) v_m) \varphi \\ & = \int _\Omega (\beta' (w) - \beta_m'(w_m)) v_m \varphi + \int _ \Omega \beta' (w)(v - v_m) \varphi \\ & = \int_ \Omega (\beta' (w) - \beta' (w_m)) v_m \varphi + \int _ \Omega (\beta' (w_m) - \beta_m' (w_m)) v_m \varphi \\ &\quad + \int _ \Omega \beta' (w) (v - v_m) \varphi \end{aligned} \label{eq:weak formulation difference vs} \end{equation} for all $\varphi \in H_0^1 (\Omega)$. Considering the test function $\varphi = v_m - v + \nabla (w_m - w) \cdot \theta \in H_0^1 (\Omega)$ we have, applying \eqref{eq:H2 norm wm minus w}, \begin{align*} \int _ \Omega |\nabla (v_m - v)|^2 &\le C (1 + \| w_m- w\|_{H^2}) \Big( (1 + \| \beta' (w)\|_{L^\infty}) \| w_m- w\|_{H^2} \\ &\quad + \|v_m\|_{L^2} (\| \beta_m' + \beta' \|_{L^\infty} + \| \beta' (w_m) - \beta'(w)\|_{L^\infty} )\Big ) . \end{align*} We cannot guaranty that $\|\beta' (w_m) - \beta' (w)\|_\infty $ goes to zero. However it is, indeed, bounded by $2 \| \beta' \|_{L^\infty}$. On the other hand, taking into account the boundary condition \begin{equation} \begin{aligned} \| v_m - v \|_{L^2 (\partial \Omega)} &\le C \| \nabla (w_m - w)\|_{L^2 (\partial \Omega)} \\ &\le C \| w_m - w \|_{H^2 (\Omega)} \le C \| \beta_m - \beta \|_{L^2} \to 0. \end{aligned}\label{eq:convergence of boundary condition difference vs} \end{equation} Hence, there is a weak limit $\widehat v \in H^1(\Omega)$, \begin{equation} v_m - v \rightharpoonup \widehat v \quad \text{in } H^1 (\Omega). \end{equation} By \eqref{eq:convergence of boundary condition difference vs} we have that $\widehat v \in H_0^1 (\Omega)$. Taking into account \eqref{eq:weak formulation difference vs} and the fact that $\beta'(w_m) \to \beta' (w)$ a.e. in $\Omega$, have \begin{equation} \int_\Omega \nabla \widehat v \nabla \varphi + \int_ \Omega \beta' (w) \widehat v \varphi = 0 \quad \forall \varphi \in H_0^1 (\Omega). \end{equation} Taking $\varphi = \widehat v \in H_0^1 (\Omega)$ as a test function we deduce that $\widehat v = 0$. \section{Proof of Theorem \ref{thm:existence of derivative}} We start by pointing out that, from condition on $f$ we have $0 \le w_m \le 1$. Since $\beta_m \nearrow \beta$ in $[0,1]$ we have $w_m$ is pointwise decreasing (see \cite{Evans1998}). Hence, there exists a pointwise limit $w$ such that $w_m \searrow w$ a.e. in $\Omega$. In particular $0 \le w \le 1$. By the Dominated Convergence Theorem we have \begin{equation} w_m \to w \text{ in } L^p (\Omega) \quad \forall 1 \le p < +\infty. \end{equation} Let $U \subset \Omega$ be an open neighbourhood of $\partial \Omega$ such that $\overline U \cap N_\Omega = \emptyset$ and $\partial U \in \mathcal C^2$. Then \begin{equation} \underline w_U = \inf _{U} w > 0. \end{equation} We have that $w_m \ge w \ge \underline w_U$. We have that $\beta \in \mathcal C^1([\underline w_U, 1])$ and, hence, $\beta_m \to \beta$ in $\mathcal C^1([\underline w_U, 1])$. Therefore \begin{equation} \beta_m(w_m) \to \beta(w) \text{ in } L^p (\Omega \setminus \overline U) \quad \forall 1 \le p < +\infty, \end{equation} Since $\| w_m\|_{H^1} \le C(1 + \| \beta_m (w_m)\|_{L^2} + \| f\|_{L^2} )$, we have $w_m \rightharpoonup w$ in $H^1 (\Omega)$, and thus $w$ is the unique solution of \eqref{eq:model problem w}. Applying this, \begin{equation} \Delta w_m = \beta_m (w_m) - f \to \beta (w) - f = \Delta w \text{ in } L^p (\Omega \setminus \overline U). \end{equation} Thus \begin{equation} \| w_m - w\| _{H^2 (\Omega \setminus \overline U)} \le C (\| \Delta (w_m - w) \|_{L^2 (\Omega \setminus \overline U)} + \|w_m - w\|_{L^2(\Omega \setminus \overline U)}) \to 0. \end{equation} Hence $ w_m \to w \text{ in } H^2(\Omega \setminus \overline U)$. In particular $$ \nabla w_m \to \nabla w \text{ in } H^{1/2}(\partial \Omega)^n. $$ Since $\beta_m' \in L^\infty (\mathbb R)$ we take the ``shape derivative'' $v_m$ solution of \eqref{eq:model vn}, which is well defined. Let us find their limit. Let us show that \begin{equation} \beta_m' (w_m ) \to \beta' (w) \text{ a.e. in } \Omega. \label{eq:pointwise convergence potential} \end{equation} First, let $x \notin N_\Omega$. Then $\beta$ is $C^1$ in $w(x)$. Therefore $\beta' (w_m(x)) \to \beta' (w(x))$. Hence, the sequence $\beta' (w_m (x))$ is bounded, so $ \beta' (w_m (x)) \le m_0$ for some $m_0$ large. Thus $\beta_m' (w_m (x)) = \beta' (w_m (x) )$ for $m \ge m_0$. Hence the convergence is proved for $x \notin N_\Omega$. Let $x \in N_\Omega$. Then $\beta' (w(x)) = +\infty$. Since $w_m (x) \to w (x)$, it follows then $\beta' (w_m (x)) \to +\infty$. In this case, we have \begin{equation*} \beta_m' (w_m (x)) = \beta (w_m (x)) \wedge m \to + \infty = \beta (w(x)). \end{equation*} This completes the proof of \eqref{eq:pointwise convergence potential}. Let us show that sequence $(v_m)$ is bounded in $H^1 (\Omega )$. There exist two open sets $U_0, U_1 \subset \Omega$ such that $\partial \Omega \subset U_1, N_\Omega \subset U_0$, $U_0 \cap U_1 = \emptyset$. There also exists a smooth transition function $\Psi$ such that $\Psi = 0$ in $U_0$ and $\Psi = 1$ in $U_1$. Let us define $g_m = \Psi \nabla w_m \cdot \theta \in H^1(\Omega)$. Then $\varphi = v_m + g_m \in H_0^1(\Omega)$ and it can be used as a test function in the weak formulation. Hence \begin{align*} \int _ \Omega \nabla v_m \nabla (v_m + g_m) + \int _ \Omega \beta_m' (w_m) v_m (v_m + g_m) = 0 . \end{align*} Therefore, through standard arguments, \begin{align*} &\int_ \Omega |\nabla v_m|^2 + \int_ \Omega \beta_m'(w_m) v_m^2 \\ &= - \int _ \Omega \nabla v_m \nabla g_m - \int _\Omega \beta'_m(w_m) v_m g_m \\ &\le \Big( \int _ \Omega |\nabla v_m|^2 \Big)^{1/2} \Big( \int _ \Omega |\nabla g_m|^2 \Big)^{1/2} + \Big( \int _\Omega \beta'_m(w_m) v_m^2 \Big)^{1/2} \Big( \int _\Omega \beta'_m(w_m) g_m^2 \Big)^{1/2} \\ & \le \frac{1}{2} \Big( \int_ \Omega |\nabla v_m|^2 + \int_ \Omega \beta_m'(w_m) v_m^2 \Big) + C \Big( \int_ \Omega |\nabla g_m|^2 + \int_ \Omega \beta_m'(w_m) g_m^2 \Big). \end{align*} Since $\beta'_m(w_m)$ is uniformly bounded in $L^\infty (\Omega \setminus \overline {U_0})$ we have that the sequence is bounded: \[ \Big( \int_ \Omega |\nabla v_m|^2 + \int_ \Omega \beta_m'(w_m) v_m^2 \Big) \le C \Big( \int_ \Omega |\nabla g_m|^2 + \int_ \Omega \beta_m'(w_m) g_m^2 \Big) \le C. \] In particular, there exists $v \in H^1 (\Omega)$ such that, up to a subsequence, $ v_m \rightharpoonup v$ in $H^1 (\Omega)$. Also, by Fatou's lemma, \begin{equation} \int_ \Omega \beta'(w) v^2 \le C. \end{equation} Since $\beta' (w) = + \infty$ in $N_\Omega$ we have that $v = 0$ a.e. in $N_\Omega$. For $\varphi \in W_c^{1,\infty} (\Omega \setminus N_\Omega)$ we have \begin{equation} \int_{\Omega \setminus N_\Omega} \nabla v_m \nabla \varphi +\int_{\Omega \setminus N_\Omega} \beta'_m(w_m) v_m \varphi = 0. \end{equation} Let us consider the compact subset $K = \operatorname{supp} \varphi \subset \Omega \setminus N_\Omega$. Let us show that $\beta'(w_m) \to \beta'(w)$ in $L^2 (K)$. We have $0 < \underline w_K \le w \le w_m$ in $K$. By the Dominated Convergence Theorem we have that $\beta_m'(w_m) \to \beta' (w)$ strongly in $L^p (K)$ for $1 \le p < + \infty$. Hence, by passing to the limit we deduce that \begin{equation} \int_{\Omega \setminus N_\Omega} \nabla v \nabla \varphi +\int_{\Omega \setminus N_\Omega} \beta'(w) v \varphi = 0. \end{equation} This completes the proof. \section{Proof of Proposition \ref{prop:general nonlinearity}} Let us consider $x_0 \in \partial N_\Omega$ and \begin{equation} W(t) = w_\Omega(x_0 + t n (x_0)) \end{equation} where $n(x_0)$ represents the normal vector to $ \partial N_\Omega$ at $x_0$. By \cite[Theorem 1.24]{Diaz:1985}, we have \begin{equation} \frac 1 2 |\nabla w_\Omega (x)|^{2} \le \int_0^{w_\Omega(x)} \beta(s)ds + \alpha w_\Omega(x) \end{equation} for all $x \in \overline \Omega$. Hence \begin{align*} \frac{d W}{dt} &\le | \frac{d W}{dt} | = | \nabla w_\Omega (x_0 + t n(x_0)) \cdot n(x_0)| \\ &\le |\nabla w_\Omega (x_0 + t n(x_0)) | \le G(w_\Omega (x_0 + t n(x_0))) \\ &= G (W(t)). \end{align*} Thus, $W$ is a solution of the ordinary differential inequality \begin{equation} \begin{gathered} \frac{d W}{dt} (t) \le G(W(t)), \\ W(0) = 0. \end{gathered} \end{equation} Let us consider $W_\varepsilon$, the solution of \begin{equation} \begin{gathered} \frac{d W_\varepsilon}{dt} (t) = G(W_\varepsilon(t)), \\ v_\varepsilon(0) = \varepsilon. \end{gathered} \end{equation} This problem has a unique smooth solution, since $G \in \mathcal C^1(\mathbb R\setminus \{0\})\cap \mathcal C(\mathbb R)$ is strictly increasing and $G(0) = 0$. In fact, solving this simply separable O.D.E., we obtain \begin{equation} \label{eq:defn W eps} W_\varepsilon (t) = \Psi^{-1} ( t + \Psi (\varepsilon)). \end{equation} By the monotonicity of $G$ we have \begin{equation} W(t) \le W_\varepsilon (t) \quad \forall t \ge 0. \end{equation} Passing to the limit as $\varepsilon \to 0$ in \eqref{eq:defn W eps} we have \begin{equation} W(t) \le \Psi^{-1} ( t ). \end{equation} Hence, since we can parametrize a neighbourhood of $\partial N_\Omega$ by $(x, t) \in \partial N_\Omega \times (-\lambda_0, \lambda_0) \mapsto x + t n (x) $, we deduce that \begin{equation} w(x) \le \Psi ^{-1} ( d (x, N _\Omega) ) \end{equation} at least in a neighbournood of $\partial N_\Omega$. This proves the proposition. \subsection*{Acknowledgments} The author is thankful to Professor Jes\'us Ildefonso D\'iaz for the fruitful discussions in the preparation of this paper and his continued support. 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