\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 62, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/62\hfil Existence and nonexistence of stable equilibria] {Sufficient conditions on diffusivity for the existence and nonexistence of stable equilibria with nonlinear flux on the boundary} \author[J. Crema, A. S. do Nascimento, M. Sonego \hfil EJDE-2012/62\hfilneg] {Janete Crema, Arnaldo Simal do Nascimento, Maicon Sonego} % in alphabetical order \address{Janete Crema \newline Instituto de Ci\^encias Matem\'aticas e de Computa\c c\~ao - USP, S. Carlos, S. P., Brasil} \email{janete@icmc.sc.usp.br} \address{Arnaldo S. do Nascimento \newline Departamento de Matem\'atica, Universidade Federal de S\~ao Carlos, S. Carlos, S. P., Brasil} \email{arnaldon@dm.ufscar.br} \address{Maicon Sonego \newline Departamento de Matem\'atica, Universidade Federal de S\~ao Carlos, S. Carlos, S. P., Brasil} \email{sonego@dm.ufscar.br} \thanks{Submitted June 27, 2011. Published April 12, 2012.} \subjclass[2000]{35J25, 35B35, 35B36} \keywords{Reaction-diffusion equation; stable stationary solutions; \hfill\break\indent variable diffusivity; implicit function theorem} \begin{abstract} A reaction-diffusion equation with variable diffusivity and nonlinear flux boundary condition is considered. The goal is to give sufficient conditions on the diffusivity function for nonexistence and also for existence of nonconstant stable stationary solutions. Applications are given for the main result of nonexistence. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this work, we study the nonlinear boundary-value evolution problem \begin{equation}\label{parabolica1} \begin{gathered} u_t = \operatorname{div} (a(x)\nabla u) +f(u) , \quad (t,x) \in \mathbb{R}^{+} \times \Omega \\ a(x)\partial_{\nu} u = g(u), \quad (t,x) \in \mathbb{R}^{+}\times \partial\Omega \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N$ $(N \geq 1) $. We assume $\Omega$ to be a smooth bounded domain, and $\nu$ denotes the exterior unit normal to $\partial \Omega $. We assume that $g$ and $f$ are \emph{bistable} type nonlinearities, and $a \in C^{1,\theta}(\overline{\Omega}, \mathbb{R}^+)$. Typically \eqref{parabolica1} models the time evolution of the concentration of a diffusing substance or heat in a medium whose diffusivity function is $a$ with the flux on the boundary being proportional to a prescribed function of the concentration. Roughly speaking, once $f$ and $g$ are fixed, non-constant stable stationary solutions to \eqref{parabolica1} (herein occasionally referred to as patterns, for short) arise from specific properties of the geometry of the domain and/or of the diffusivity function $a$. This work should be seen as an attempt to understand the role played by $a$ on existence and nonexistence of patterns to \eqref{parabolica1}. By stationary solutions to \eqref{parabolica1} we mean $C^2(\Omega) \cap C^0(\overline{\Omega})$ solutions to the nonlinear boundary value problem \begin{equation}\label{equacaog(u)} \begin{gathered} \operatorname{div} (a(x)\nabla u) +f(u) =0 , \quad x \in \Omega \\ a(x)\partial_{\nu} u= g(u), \quad x\in \partial\Omega . \end{gathered} \end{equation} Let us define the set of bi-stable functions $\mathcal{B}$ as the class of $C^1$ functions $h:\mathbb{R} \to \mathbb{R}$ such that \begin{itemize} \item There exist $\alpha,\beta \in \mathbb{R}$, $\alpha<0<\beta$ such that $h(\alpha)=h(\beta)=h(0)=0$, \item $h(s)\neq 0$ in $(\alpha,0)\cup (0,\beta)$, \item $h'(\alpha)<0$, $h'(\beta)<0$, $h'(0)>0$. \end{itemize} Throughout this work we assume that $f,g \in \mathcal{B}$ but eventually we need either $f\equiv0$ or $g\equiv0$; this will be explicitly mentioned wherever is needed. Let us briefly state our main results and applications. Suppose that there exists $\bar{a} \in C^{1,\theta}(\overline{\Omega})$ such that the only stable stationary solutions to \begin{equation}\label{parabolica2} \begin{gathered} u_t = \operatorname{div} (\bar{a}(x)\nabla u) +f(u) , \quad (t,x) \in \mathbb{R}^{+} \times \Omega \\ \bar{a}(x)\partial_{\nu} u = g(u), \quad (t,x) \in \mathbb{R}^{+}\times \partial\Omega . \end{gathered} \end{equation} are $\bar{u}=\alpha$ and $\bar{u}=\beta$. We prove that if $\|a-\bar{a}\|_{C^{1,\theta}(\overline{\Omega})}$ is small enough then the only stable stationary solutions to \eqref{parabolica1} are $\bar{u}=\alpha$ and $\bar{u}=\beta$. As applications we have the following 4 items: (1) If $\overline{a}(x)=c$ (with $c$ a positive constant) is large enough and $a \in C^{1,\theta}(\overline{\Omega})$ is a function satisfying $\|a-c\|_{C^{1,\theta}(\overline{\Omega})}$ sufficiently small then the only stable stationary solutions to \eqref{parabolica1} are $\bar{u}=\alpha$ and $\bar{u}=\beta$. (2) Suppose that $\Omega$ is a $N-$dimensional ball, $f\equiv0$ and $\overline{a}(x)=c$ (with $c$ a positive constant). If $\|a-c\|_{C^{1,\theta}(\overline{\Omega})}$ is sufficiently small then the only stable stationary solutions to \eqref{parabolica1} are $\bar{u}=\alpha$ and $\bar{u}=\beta$. Note that here, as opposed to (1), it was not required that $c$ be large. This condition is not required in the next application as well. (3) Suppose that $\Omega$ is a smooth convex domain, $g\equiv0$ and $\overline{a}(x)=c$ (c a positive constant). If $\|a-c\|_{C^{1,\theta}(\overline{\Omega})}$ is sufficiently small then the same conclusion of (1) holds. (4) Another interesting application is when the domain is a ball, $\Omega=B_R(0)$ say, $g\equiv0$, $\overline{a}(x)=\overline{a}(r)$ where $r=|x|$, i.e., $\overline{a}$ is radially symmetric and satisfies $r^2 (\sqrt{\overline{a}})''+(N-1)r(\sqrt{\overline{a}})' \leq (N-1)a$ for $0q$. Then $u\in W^{2,p}(\Omega)$. \end{lemma} For a proof the reader is referred to \cite[p. 114]{Grisvard}, for instance. Next results, regarding regularity of solutions to \eqref{equacaog(u)}, will also play a important role in the sequel. \begin{lemma}\label{regularidadesol.g(u)} Assume $g \in C^2(\mathbb{R})$, $f \in C^1(\mathbb{R})$ and let $a \in C^{1,\theta}(\overline{\Omega})$ be a positive function. If $u\in H^1(\Omega)\cap L^{\infty}(\Omega)$ is a solution to \eqref{equacaog(u)} then $u\in W^{2,N+1}(\Omega)$. Moreover $u\in C^{2,\theta}(\overline{\Omega})$. \end{lemma} \begin{proof} We start by proving that if $N>2k$, for some $k\in \mathbb{N}$, then $u\in W^{2,p_k}{\Omega})$ where $p_k=\frac{2N}{N-2k}$. The proof is by induction on $k$. Let $k=0$; thus $p_0=2$ and by hypothesis on $u$ and $a$ we have $\psi (x)= (\frac{g(u)}{a(x)}+u) \in H^1(\Omega)$, $\varphi (x)=-{f(u)}\in L^2(\Omega)$ and we see that $u$ is also a solution to \begin{equation}\label{equacaog(u)+u} \begin{gathered} \operatorname{div} (a(x)\nabla u)=\varphi (x) , \quad x \in \Omega \\ \partial_{\nu} u +u= \psi(x), \quad x \in \partial\Omega . \end{gathered} \end{equation} However, if $\Omega$ is a $C^{2,\theta}$ domain, $\psi \in C^{1,\theta}(\partial\Omega)$ and $\varphi\in C^{\theta}(\overline{\Omega})$, then \eqref{equacaog(u)+u} has only one solution in $C^{2,\theta}(\overline{\Omega})$ (cf. \cite[Chapter 6]{GT}, for instance). Hence from regularity of $a,\,f$ and $g$ as well as density of the inclusions $C^{1,\theta}(\overline{\Omega})\subset H^1(\Omega)\subset L^2(\Omega)$, one easily proves that $u\in H^2(\Omega)$. Assuming the result is true for $N>2k$ let us take $N>2(k+1)$. By induction hypothesis $u\in W^{2,p_k}(\Omega)$ with $p_k=\frac{2N}{N-2k}$. But $W^{2,p_k}(\Omega)\subset W^{1, p_{k+1}}(\Omega)$ since $2-\frac{N}{p_k}= 1-\frac{N}{p_{k+1}}$ for $N>2(k+1)$. By hypothesis on $f,g,a$ we have $f(u)/a\in L^{p_{k+1}}(\Omega)$, $a^{-1} \nabla a \cdot \nabla u\in L^{p_{k+1}}(\Omega)$ and $ \frac{g(u)}{a}\in W^{1-\frac{1}{p_{k+1}},p_{k+1}}(\partial \Omega)$. Moreover $p_{k+1}=\frac{Np_k}{N-p_k}$ and $p_k2k$ and as such, from the argument above, $u\in W^{2,p_k}(\Omega)$ with $p_k=2N>N$. In case $N$ is even $\exists\,k\geq 0$, $N=2k+2>2k$. Again the same argument implies $u\in W^{2,p_k}(\Omega)$ with $p_k=N$. On the account that $W^{2,N}(\Omega)\subset W^{2, (N-1/10)}(\Omega)\cap W^{1, N+1}(\Omega)$, Lemma \ref{lemaGrisvard} yields once more $u\in W^{2, N+1}(\Omega)$. Then in any case it follows that $u\in W^{2,N+1}(\Omega)$. But Sobolev continuous imbedding assures us that $W^{2,N+1}(\Omega)\subset C^{1,\theta}(\overline{\Omega})$ for $\theta =\frac{1}{N+1}$. Then $u$ is the solution to \eqref{equacaog(u)+u} with $\varphi \in C^{\theta}(\overline{\Omega})$ and $\psi \in C^{1,\theta}(\partial{\Omega})$. Given that $\Omega$ is $C^{2,\theta}$ domain we conclude $u\in C^{2,\theta}(\overline{\Omega})$. \end{proof} Before establishing our main results in this section we present an application of the Implicit Function Theorem in a specific setting that suits our purposes; it is a generalization of \cite{BTW} where the case $a$ is constant and $g\equiv 0$ was treated. \begin{lemma}\label{TFI} Suppose $g \in C^2(\mathbb{R})$ and $u_0=\alpha$ or $u_0=\beta$. Then for any positive function $\bar{a}\in C^{1,\theta}(\overline{\Omega})$ there are neighborhoods $\mathcal{V}_{\bar{a}}$ of $\bar{a}$ in $C^{1,\theta}(\overline{\Omega})$ and $ \mathcal{U}_{u_0}$ of $u_0$ in $ W^{2,p}(\Omega)\,(p>N)$ such that if \ $a \in \mathcal{V}_{\bar{a}}$ then \ $u_0$ is the only solution to \eqref{equacaog(u)} in $\mathcal{U}_{u_0}$. Moreover if either $f\equiv0$ and $g\neq0$ or $g\equiv0$ and $f\neq 0$ the result is still valid. \end{lemma} \begin{proof} First of all for simplicity in notation $\mathcal{H}^{N}$ stands for the $N$-dimensional Hausdorff measure which in our case, according to the dimension, corresponds to the usual area or volume measure. Let us define $$ E_p := \{(v,w)\in L^p(\Omega)\times W^{1-\frac{1}{p},p}(\partial\Omega);\, \int_{\Omega}v=\int_{\partial\Omega}w\} $$ and the operator $F:C^{1,\theta}(\overline{\Omega})\times W^{2,p}(\Omega) \to E^p\times \mathbb{R}$ by \begin{equation} F({a},u)=\begin{pmatrix} \operatorname{div}(a(x)\nabla u)+f(u)-\frac{1}{\mathcal{H}^N(\Omega)} \big[ \int_{\partial\Omega} g(u) +\int_{\Omega}f(u)\big],\\ a(x) \partial_{\nu} u -g(u), \\ \frac{1}{\mathcal{H}^{N}(\Omega)}\big(\int_{\partial\Omega} g(u) +\int_{\Omega}f(u)\big) \end{pmatrix}. \end{equation} Note that $F$ is a $C^1$ operator by regularity of $a,f,g$ and on the account that $p>N$. Moreover $F(a,u)=(0,0,0)$ if and only if $u$ is a solution to \eqref{equacaog(u)}. In particular for any $\bar{a} \in C^{1,\theta}(\overline{\Omega})$ and any constant solution $u_0\in\{\alpha, \beta\}$ to \eqref{equacaog(u)} we have $F(\bar{a},u_0)=(0,0,0)$. Claim: $D_u F(\bar{a},u_0):W^{2,p}(\Omega)\longmapsto E_p\times \mathbb{R}$ is an isomorphism for any positive $\bar{a}\in C^{1,\theta}(\overline{\Omega})$. Note that this will be the case if for each $(v,w,t)\in E^p\times \mathbb{R}$ there is only one solution $\phi \in W^{2,p}(\Omega)$ to \begin{equation}\label{equacaog'(u_0)} \begin{gathered} \operatorname{div}({\bar{a}} \nabla\phi)+f'(u_0)\phi - \frac{1}{\mathcal{H}^{N}(\Omega)} \Big[\int_{\partial\Omega} g'(u_0)\phi +\int_{\Omega}f'(u_0)\phi\Big]= v , \quad x \in \Omega \\ \bar{a} \partial_{\nu} \phi -g'(u_0)\phi=w , \quad x \in \partial\Omega\\ \frac{1}{{\mathcal{H}^{N}(\Omega)}} \Big(\int_{\partial\Omega} g'(u_0)\phi +\int_{\Omega}f'(u_0)\phi\Big)=t \end{gathered} \end{equation} To prove that the application \eqref{equacaog'(u_0)} above is an isomorphism it suffices to show that \begin{equation}\label{equacaog'(u0)media0} \begin{gathered} \operatorname{div}({\bar{a}}\nabla \varphi)+f'(u_0)\varphi = v , \quad x \in \Omega \\ \bar{a} \partial_{\nu} \varphi -g'(u_0)\varphi= w+t\frac{g'(u_0)}{f'(u_0)} , \quad x \in \partial\Omega \end{gathered} \end{equation} has a unique solution $\varphi$. Indeed if this is the case then, keeping in mind that $(v,w) \in E_p$, the function $\phi=\varphi +\frac{t}{f'(u_0)}$ will be the only solution to \eqref{equacaog'(u_0)}. To prove existence and uniqueness of solutions to \eqref{equacaog'(u0)media0} we start by defining the operator $T:W^{2,p}(\Omega)\to L^p(\Omega)\times W^{1-1/p,p}(\partial \Omega)$ by $$ T(\varphi)=(\operatorname{div}(\bar{a} \nabla \varphi)+f'(u_0)\varphi, \bar{a}\,\partial_{\nu} \varphi-g'(u_0)\varphi ). $$ It is well known that $T$ is a Fredholm operator with index zero. And for $u_0=\alpha$ or $u_0=\beta$ we have that $\ker T=\{0\}$ since $f'(u_0)<0$ and $ g'(u_0)<0$. So $T$ is an isomorphism and hence $D_u F(\bar{a},u_0)$ is an isomorphism from $W^{2,p}(\Omega)$ to $E_p\times \mathbb{R}$. Finally we conclude from the Implicit Function Theorem (see \cite{TFI} for instance) the existence of a neighborhood $\mathcal{U}_{u_0} \in W^{2,p}(\Omega)$ of $u_0$ and a neighborhood $\mathcal{V}_{\bar{a}}\in C^1(\overline{\Omega})$ of $\bar{a}$ such that if $a \in \mathcal{V}_{\bar{a}}$, $u\in \mathcal{U}_{u_0}$ and $F(a, u)=(0,0,0)$ then $u=u_0$; i.e., $u_0$ is the only solution to \eqref{equacaog(u)} in $\mathcal{U}_{u_0}$. The cases $g=0$ and $f\neq 0$ or $f=0$ and $g\neq0$ are similar and will be omitted. \end{proof} Now we are ready to show the next result. \begin{theorem}\label{agrande} Let $\Omega \subset \mathbb{R}^N$ be a $C^{2,\theta}$ bounded domain, $a\in C^{1,\theta}(\overline{\Omega})$ with $\theta=1/(N+1)$ and $g\in C^2(\mathbb{R})$. Let $\bar{a}\in C^{1,\theta}(\overline{\Omega})$ be a positive function and suppose that $u_0=\alpha$ and $u_0=\beta$ are the unique stable stationary solutions to {parabolica2}. Then there is $\rho>0$ such that whenever $\|a-\bar{a}\|_{C^{1,\theta}(\overline{\Omega})}<\rho$, any stable stationary solution $u$ to \eqref{parabolica1} satisfying $\alpha\leq u \leq \beta$ in $ {\Omega}$ must be constant, i.e., $u=\alpha$ or $u=\beta$. Moreover if $f\equiv0$ and $g\neq0$ or $g\equiv0$ and $f\neq 0$ the result still holds true. \end{theorem} \begin{proof} Arguing by contradiction we obtain a sequence $\{a_j\}_{j=1}^{\infty}$ satisfying $a_j\to \bar{a}$ in $C^{1,\theta}(\overline{\Omega})$, as $j \to \infty,$ and a sequence of corresponding nonconstant stable solutions $\{u_j\}$ to \eqref{parabolica1} satisfying $\alpha\leq u_j(x)\leq \beta$ and \begin{equation}\label{equacaog(u)a_j} \begin{gathered} \operatorname{div}(a_j(x)\nabla u)+f(u)=0 , \quad x \in \Omega \\ a_j(x) \partial_{\nu} u = g(u), \quad x \in \partial\Omega \end{gathered} \end{equation} Lemma \ref{regularidadesol.g(u)} yields $u_j\in C^{2,\theta}(\overline{\Omega})$. But for all $v\in H^1(\Omega)$ we have \begin{equation}\label{formulacao fraca} \int_{\Omega}a_j(x)\nabla v \nabla u_j - vf(u_j)dx - \int_{\partial\Omega} v g(u_j) d\sigma=0 \end{equation} For $j$ large enough there is $k>0$ such that $a_j(x)\geq k$ for all $x \in \overline{\Omega}$. Hence $$ k\int_{\Omega}|\nabla u_j|^2 dx \leq \int_{\partial\Omega} g(u_j)u_j\, d\sigma +\int_{\Omega}u_jf(u_j)dx $$ and given that the sequence $\{u_j\}$ is bounded in $L^{\infty}(\Omega)$ it is also bounded in $H^1(\Omega)$. Extracting a subsequence, still denoted by $\{u_j\}$, there is a function $\overline{u}\in H^1(\Omega)$ such that $u_j \rightharpoonup \bar{u}$ weakly in $H^1(\Omega)$ and strongly in $L^2(\Omega)$ as well as in $L^2(\partial\Omega)$. Given the uniform convergence of $\{a_j\}$ in $\overline{\Omega}$ we conclude from \eqref{formulacao fraca} that $\bar{u}$ is a weak solution to \begin{equation}\label{eqlimit} \begin{gathered} \operatorname{div}( \bar{a}\nabla \bar{u})+f(\bar{u})=0 , \quad x \in \Omega \\ \bar{a} \partial_{\nu} \bar{u} =\,g(\bar{u}), \quad x \in \partial\Omega. \end{gathered} \end{equation} where $\bar{u} \in C^{2,\theta}(\overline{\Omega})$ by Lemma \ref{regularidadesol.g(u)}. Moreover $\alpha \leq \bar{u} \leq \beta$ and $u_j\to \bar{u}$ in $W^{2,p}(\Omega)$. In fact, since $u_j$ and $\bar{u}$ are in $C^{2}(\overline{\Omega})$ we can utilize the classical Amann estimative, \begin{equation} \begin{aligned} \|u_j-\bar{u}\|_{H^1(\Omega)} & \leq C\big(\|\Delta(u_j-\bar{u})\|_{L^2(\Omega)} +\|\partial_{\nu}(u_j-\bar{u})\|_{L^2(\partial\Omega)}\big) \\ &\leq C\Big(\|\frac{1}{a_j} {\nabla a_j} \nabla u_j-\frac{1}{\bar{a}} {\nabla \bar{a}} \nabla \bar{u}\|_{L^2(\Omega)} +\|\frac{f(u_j)}{a_j}-\frac{f(\bar{u})}{\bar{a}}\|_{L^2(\Omega)} \\ &\quad + \|\frac{g(u_j)}{a_j}-\frac{g(\bar{u})}{\bar{a}}\|_{L^2(\partial\Omega)}\Big) \end{aligned} \end{equation} where $C$ is a constant, independent of $j$, to obtain strong convergence in $H^1(\Omega)$. Using the Agmon-Douglis-Nirenberg inequality (see \cite{ADN}, for instance) and previous convergence we conclude that $u_j\to \bar{u}$ in $ H^2(\Omega)$. Now similarly to the proof of Lemma \ref{regularidadesol.g(u)} we can prove that $u_j\to \bar u$ in $W^{2,p}(\Omega)$ for $p=N+1$. Claim: $\bar u$ is a stable stationary solution to {parabolica2}. Indeed since $\lambda_1(a_j, u_j)$ is the first eigenvalue of the linearized problem \begin{equation} \begin{gathered} \operatorname{div}(a_j(x)\nabla \phi)+f'(u_j)\phi=\lambda \phi , \ x \in \Omega \\ a_j(x) \partial_{\nu} \phi = g'(u_j)\phi, \ x \in \partial\Omega \end{gathered} \end{equation} then $$ \lambda_1(a_j,u_j) =\sup_{\phi\in H^1(\Omega), \, \phi\neq0} \Big\{ \frac{\int_{\Omega}-a_j|\nabla \phi|^2+\int_{\Omega}f'(u_j)\phi^2+\int_{\partial\Omega}g'(u_j)\phi^2} {\int_{\Omega}\phi^2} \Big\} $$ and $\lambda_1(a_j,u_j)\leq 0$ on the account that $u_j$ is stable. Since $u_j\to \bar{u} \in W^{2,p}(\Omega)$ and $a_j\to\bar{a} \in C^{1,\theta}(\overline{\Omega})$ we can pass to the limit to obtain $$ 0\geq\lambda_1(\bar{a},\bar{u})= \sup_{\phi\in H^1(\Omega), \, \phi\neq0}\Big\{\frac{\int_{\Omega}-\bar{a}|\nabla \phi|^2+\int_{\Omega}f'(\bar{u})\phi^2+\int_{\partial\Omega}g'(\bar{u}) \phi^2}{\int_{\Omega}\phi^2}\Big\}. $$ This implies that $\bar{u}$ is a stable stationary solution to {parabolica2}, which is the evolutionary equation corresponding to (2.7). Indeed if $\lambda_1(\bar{a},\bar{u})<0$ this result is very well-known. If $\lambda_1(\bar{a},\bar{u})=0$ the result still holds (see \cite[theorem 6.2.1]{H}). Roughly speaking in this case $0$ is a simple eigenvalue (having $\Omega$ smooth is crucial here) and therefore there is a local one-dimensional critical invariant manifold $W(\bar{u})$ tangent to the principal eigenfunction such that if $\bar{u}$ is stable in $W(\bar{u})$ then it also stable in $H^1(\Omega)$. As for the stability of $\bar{u}$ in $W(\bar{u})$ it follows from the existence of a Lyapunov functional and the fact the $W(\bar{u})$ is one-dimensional. Summing up, $\bar{u}$ is a stable stationary solution to {parabolica2} but by hypothesis the only stable stationary solution to {parabolica2} are $\bar{u}=\alpha$ or $\bar{u}=\beta$. Thus $u_j\to \bar{u}$ in $W^{2,p}(\Omega)$ for $p>N$ where $\bar{u}\equiv\text{constant}$. But according to Lemma \ref{TFI}, if $j$ large enough, this cannot happen given that from the contradiction hypothesis each $u_j$ is a nonconstant function. \end{proof} Aiming at future applications we now prove the following result. \begin{theorem}\label{Arnaldogeneralizado} In addition to the hypotheses mentioned in the Introduction assume $g\in C^2(\mathbb{R})$ and that $f(s)g(s)>0$ for $s\neq 0, \alpha, \beta$. Then for $\lambda>0$ small enough any stable solution to \begin{equation}\label{Arnaldo} \begin{gathered} u_t= \Delta u + \lambda f(u)=0 , \quad t>0, \; x \in \Omega \\ \partial_{\nu} u= \lambda g(u), \quad x\in \partial\Omega . \end{gathered} \end{equation} with $\alpha\leq u\leq \beta$ satisfies $u=\alpha$ or $u=\beta$. \end{theorem} \begin{proof} The proof is similar but simpler than those given in Lemma \ref{TFI} and Theorem \ref{agrande} and hence the details will be omitted. First we define a $C^1$ operator $T:\mathbb{R}\times W^{2,p}(\Omega) \to E^p\times \mathbb{R} $, with $p>N$, and a set of functions $E_p$ as in Lemma \ref{TFI}, $$ T(\lambda,u)=\begin{pmatrix} \Delta u-\lambda f(u)+\frac{\lambda}{\mathcal{H}^{N}(\Omega)} \big[\int_{\partial\Omega}g(u)d\sigma+\int_{\Omega}f(u)dx\big],\\ \partial_{\nu}u-\lambda g(u),\\ \frac{1}{\mathcal{H}^{N}(\Omega)} \big[\int_{\partial\Omega}g(u)d\sigma+\int_{\Omega}f(u)dx\big] \end{pmatrix} $$ We see that for $\lambda \neq 0$, $T(\lambda,u)=(0,0,0)$ if and only if $u$ is a solution to \eqref{Arnaldo} and $T(0,u)=(0,0,0)$ if and only if $u=\alpha$ or $u=\beta$ or $u=0$ since $f(s)g(s)>0$ for $s\neq 0$. It is easy to verify that for a constant function $u_0\in \{\alpha, \beta, 0\}$, the operator $D_uT(0,u_0): W^{2,p}(\Omega)\to E_p\times \mathbb{R}$, where $$ D_uT(0,u_0)\phi= \Big( \Delta\phi, \partial_{\nu}\phi, \frac{1}{\mathcal{H}^{N}(\Omega)} \Big[\int_{\partial\Omega}g'(u_0)\phi +\int_{\Omega}f'(u_0)\phi \Big] \Big), $$ is an isomorphism. Indeed the problem $$ \begin{gathered} \Delta\phi=v\in L^p(\Omega)\\ \partial_{\nu}\phi=w\in W^{1-1/p,p}(\partial\Omega) \end{gathered} $$ has a family of solutions $\{\phi_c=\varphi+c, c\in \mathbb{R}\}$ and then given $t\in\mathbb{R}$ there exists only one $c$ such that $$ t=\frac{1}{\mathcal{H}^{N}(\Omega)}\Big[\int_{\partial\Omega}g'(u_0)\phi _c +\int_{\Omega}f'(u_0)\phi_c \Big]. $$ Again as in Lemma \ref{TFI}, since $T(0,u_0)=(0,0,0)$, we conclude from the Implicit Function Theorem the existence of a neighborhood $\mathcal{U}_{u_0}\in W^{2,p}(\Omega) $ of $u_0$ and $\lambda_0>0$ such that if $|\lambda|<\lambda _0$, $u\in \mathcal{U}_{u_0}$ and $T(\lambda, u)=(0,0,0)$ then $u=u_0$, i.e., $u_0$ is the only solution to \eqref{Arnaldo} in $\mathcal{U}_{u_0}$. Now arguing by contradiction let us suppose that there is a sequence $\lambda _j\to 0$ and a corresponding sequence $\{u_j\}$ of nonconstant stable stationary solutions to \eqref{Arnaldo} satisfying $\alpha\leq u_j\leq \beta$ and \begin{equation}\label{equacaoestacionariaArnaldo} \begin{gathered} \Delta u_j= - \lambda_j f(u_j) , \quad t>0, \; x \in \Omega \\ \partial_{\nu} u_j= \lambda_j g(u_j), \quad x\in \partial\Omega . \end{gathered} \end{equation} First of all as in Theorem \ref{agrande} we can show the existence of $\bar{u}$ and a subsequence of non-constant functions, still denoted by $\{u_j\}$, such that $u_j\rightharpoonup \bar{u} \in H^1(\Omega)$. Moreover once $\lambda_j\to 0$ we have $|\nabla u_j|_{L^2(\Omega)}\to 0$ and then the convergence is strong and $\bar{u}$ is constant. Now using Agmon-Douglis-Niremberger inequality we conclude that $u_j\to \bar{u}$ in $W^{2,p}(\Omega)$ for some $p>N$. Since $\int_{\Omega} f(u_j)dx+\int_{\partial\Omega}g(u_j)d\sigma=0$ it holds that $$ \mathcal H^{N}({\Omega}) f(\bar{u})+\mathcal H^{N-1}({\partial\Omega})g(\bar{u})=0, $$ and on the account that we have $f(v)g(v)>0$ for $v\neq \alpha, 0, \beta$ then we must have $\bar{u}\in \{\alpha, 0, \beta\}$. Given that $u_j\to \bar{u}$ in $W^{2,p}(\Omega)$ ($p>N$) we conclude from the first part of the proof that for small $\lambda_j$ the corresponding solution to \eqref{equacaoestacionariaArnaldo} satisfies one of the following cases; ${u_j}=\alpha$, ${u_j}=\beta$ or $u_j=0$. This is a contradiction since each ${u_j}$ is a non-constant function. Then for $\lambda$ small enough any stationary stable solution to \eqref{Arnaldo} must be $u=\alpha$, $u=\beta$ or $u=0$. But if $u=0$ the first eigenvalue $\mu(\lambda,0)$ corresponding to the linearized problem \begin{equation} \begin{gathered} \Delta \phi+\lambda f'(0)\phi=\lambda \phi , \quad x \in \Omega \\ \partial_{\nu} \phi = \lambda g'(0)\phi, \quad x \in \partial\Omega \end{gathered} \end{equation} satisfies $$ \mu(\lambda,0)=\sup_{\phi\in H^1(\Omega), \ \phi\neq0}\frac{\int_{\Omega}-|\nabla \phi|^2+\int_{\Omega}\lambda f'(0)\phi^2+\int_{\partial\Omega}\lambda g'(0)\phi^2}{\int_{\Omega}\phi^2}> 0 $$ due to the fact that $f'(0), g'(0)>0$. Hence we must have $u=\alpha$ or $u=\beta$. \end{proof} The next result is direct consequence of the previous two theorems. \begin{theorem}\label{corolario} Let $\Omega \in \mathbb{R}^N$ be a $C^{2,\theta}$ bounded domain, $a\in C^{1,\theta}(\overline{\Omega})$ with $\theta=1/(N+1)$, $f,g\in \mathcal{B}$, $f(s)g(s)>0$ for $s\neq \alpha, 0, \beta$ and $g\in C^2(\mathbb{R})$. Then given any real number $\bar{a}$ large enough there is $\rho>0$ such that whenever $\|a(\cdot)-\bar{a}\|_{C^{1,\theta}(\overline{\Omega})}<\rho$, any stable stationary solution $u$ to \eqref{parabolica1} satisfying $\alpha\leq u \leq \beta$ in $ {\Omega}$ must be constant; i.e., $u=\alpha$ or $u=\beta$. \end{theorem} \section{Application to specific cases} In this section we illustrate how the above version of the Implicit Function theorem can be used to draw conclusions on non-existence of non-constant stable stationary solutions to some specific cases of \eqref{parabolica1}. \begin{corollary} In addition to the hypotheses of Theorem \ref{agrande} suppose that $\Omega$ is a $N-$dimensional ball and $f\equiv0$. Then given any real number $\bar{a}>0$ (not necessarily large) there is $\rho>0$ such that whenever $\|a(\cdot)-\bar{a}\|_{C^{1,\theta}(\overline{\Omega})}<\rho$, any stable stationary solution $u$ to \eqref{parabolica1} satisfying $\alpha\leq u \leq \beta$ in $ {\Omega}$ must be constant, i.e., $u=\alpha$ or $u=\beta$. \end{corollary} \begin{proof} Since $\Omega$ is a ball we know (see \cite{Neus2}, for instance) that if $ \bar{u}$ is a stable stationary solution to \begin{equation}\label{equacaoparabolica 2g(u)a_j} \begin{gathered} u_t=\bar{a} \triangle u , \quad x \in \Omega \\ \partial_{\nu} u= \bar{a}^{-1}g(u), \quad x \in \partial\Omega \end{gathered} \end{equation} then $ \bar{u}$ must be a constant function. Hence, since $u\equiv0$ is a unstable equilibrium, we conclude $\bar{u}=\alpha \text{ or } \bar{u}=\beta$ and Theorem \ref{agrande} can be applied to complete the proof. \end{proof} A similar nonexistence result can be obtained for $g\equiv0$ as long as $\Omega$ is smooth and convex. \begin{corollary} In addition to the hypotheses of Theorem \ref{agrande} suppose that $\Omega \subset \mathbb{R}^N$ is smooth and convex and $g\equiv0$. Then given any real number $\bar{a}>0$ there is $\rho>0$ such that whenever $\|a(\cdot)-\bar{a}\|_{C^{1,\theta}(\overline{\Omega})}<\rho$, any stable stationary solution $u$ to \eqref{parabolica1} satisfying $\alpha\leq u \leq \beta$ in $ {\Omega}$ must be constant; i.e., $u=\alpha$ or $u=\beta$. \end{corollary} \begin{proof} If $\bar{u}$ is a stable stationary solution to \begin{equation}\label{equacaoparabolica2f(u)} \begin{gathered} u_t=\bar{a} \triangle u +f(u) , \quad x \in \Omega \\ \frac{\partial u}{\partial \nu}=0, \quad x \in \partial\Omega \end{gathered} \end{equation} since $\Omega$ is smooth and convex, we can resort to \cite{CH} or \cite{M} to conclude that $\bar{u}$ must be constant. Hence, since $u\equiv0$ is a unstable equilibrium, we conclude $\bar{u}=\alpha \text{ or } \bar{u}=\beta$. Now the result is a immediate consequence of Theorem \ref{agrande}. \end{proof} \begin{corollary} Consider the problem \begin{equation}\label{parabolica11} \begin{gathered} u_t = \operatorname{div} (a(x)\nabla u) +f(u) , \quad (t,x) \in \mathbb{R}^{+} \times B_R(0) \\ \partial_{\nu} u = 0, \quad (t,x) \in \mathbb{R}^{+}\times \partial B_R(0) . \end{gathered} \end{equation} where $f \in C^1$ and $B_R(0)$ stands for the $N$-dimensional ball of radius $R$ and center at the origin. Let $r=\|x\|$ and suppose that $\overline{a} \in C^2(0,R) $ is a positive radially symmetric function satisfying $$ r^2 (\sqrt{\overline{a}})''+(N-1)r(\sqrt{\overline{a}})' \leq (N-1)a $$ for $0N$ define the twice continuously differentiable energy functional $E:W^{1,p}(\Omega)\longmapsto\mathbb{R}$ by $$ E(u)=\frac{1}{2}\int_{\Omega}a(x)\left|\nabla u\right|^2\ dx-\int_{\Omega}F(u)\ dx-\int_{\partial\Omega}G(u)\ d\sigma. $$ Before establishing the next result, we remember that the eigenvalues of the Steklov problem defined in a set $D\subset \mathbb{R}^N$, \begin{equation}\label{problemadeSteklov} \begin{gathered} \Delta\varphi=0, \quad x \in D\\ \frac{\partial \varphi}{\partial \eta}=\mu \varphi, \quad x \in D \end{gathered} \end{equation} satisfy $0=\mu_0<\mu_1\leq\mu_2\dots \to \infty$ and we recall the following well-know result which can be proved using variational characterization of the eigenvalues. The following result whose proof can be found in \cite[Lemma 3.1]{Neus} will play an important role in this section. \begin{lemma}\label{autovaloresdesteklov} Let $D \subset\mathbb{R}^N$ be a domain with Lipschitz continuous boundary. Then for any $v\in W^{1,2}(D)$ it holds that $$ \int_{\partial D}v^2d\sigma\leq \frac{1}{\mu_1}\int_{D}|\nabla v|^2\, dx+ \frac{1}{\mathcal{H}^{N-1}(\partial D)}\Big(\int_ {\partial D}vd\sigma \Big)^2. $$ Moreover if $S\subset \partial D$ is smooth with $\mathcal{H}^{N-1}(S)\neq 0$ then \begin{equation}\label{desigualdadeautovaloresdesteklov} \int_{\partial S}v^2d\sigma\leq \frac{1}{\mu_1}\int_{D}|\nabla v|^2\, dx+ \frac{1}{\mathcal{H}^{N-1}(S)}\Big(\int_ {S}vd\sigma \Big)^2. \end{equation} \end{lemma} A proof of the next lemma can be found in \cite{M}, where the case $a\equiv 1$ and $g\equiv0$ was treated. However given that this technique has since then been used in the related literature we decided to present here a much simpler and entirely variational proof. \begin{lemma}\label{lemaLambda} Let $\Omega\subset\mathbb{R}^N$ ($N\geq 2$) be a smooth bounded domain, $\Omega_l$ and $\Omega_r$ two disjoint sub-domains of $\Omega$ with smooth boundaries and $S_j=\partial\Omega\cap\partial\Omega_j$, $\mathcal{H}^{N-1}(S_j)>0$ $(j=l,r)$. For $p>N$, we define the set \begin{align*} \Lambda(\Omega_l, \Omega_r) &= \big\{ v\in W^{1,p}(\Omega): \alpha\leq v(x)\leq\beta,\, x\in\overline{\Omega},\, \int_{S_l}v\ d\sigma<0,\\ &\quad \int_{S_r}v\ d\sigma>0,\, E(v)<\varepsilon_0-G(\beta)\mathcal{H}^{N-1}(\partial\Omega) -F(\beta)\mathcal{H}^{N}(\Omega) \Big\}, \end{align*} where $$ \varepsilon_0=G(\beta)\min \{\mathcal{H}^{N-1}(S_l)\min\{1, \mu_1(\Omega_l)a_m^{l}\},\mathcal{H}^{N-1}(S_r)\min\{1, \mu_1(\Omega_r)a_m^{r}\}\}, $$ $a_m^{j}=\min_{x\in\Omega_j}a(x)$ $(j=l,r)$ and $\mu_1(\Omega_j)$ is the first positive eigenvalue of Steklov Problem \eqref{problemadeSteklov} defined in $\Omega_j$ $(j=l,r)$. If $\Lambda\neq\emptyset$ then \eqref{parabolica1} has at least one nonconstant stationary solution $u\in\Lambda$ which is stable in $W^{1,p}(\Omega)$. \end{lemma} \begin{proof} Let $T(t)u_0=u(t,x)$ be the solution to \eqref{parabolica1} with $u(0,x)=u_0$. The proof consists in showing that $\Lambda$ is invariant under $T(t)$ for $t\geq 0$ and then to use this fact to conclude that there is a stable stationary solution in the interior of $\Lambda$. Let us consider $u_0\in \Lambda$. Since $f,g\in \mathcal{B}$ an application of Maximum Principle yields $\alpha\leq T(t)u_0\leq \beta$. Moreover $\frac{d}{dt}E(u(t,x))=-\int_{\Omega}(u_t(t,x)^2)dx$ and hence $E(u(t,x))\leq E(u_0)<\varepsilon_0$. Let us show that $\int_{S_l}T(t)u_0\, d\sigma<0$ for $t\geq 0$. By contradiction let $t_1>0$ be such that $w_1=T(t_1)u_0$ and $\int_{S_l}w_1\, d\sigma=0$. But $$ \int_{S_l}w_1^2\, d\sigma \leq \frac{1}{\mu_1(\Omega_l)}\int_{\Omega_l}|\nabla w_1|^2dx+\Big(\int_{S_l}w_1\, d\sigma\Big)^2=\frac{1}{\mu_1(\Omega_l)}\int_{\Omega_l}|\nabla w_1|^2dx $$ and because $0\leq sg(s)\leq s^2$ for $s\in[\alpha, \beta]$ we have $0\leq G(s)\leq s^2/2$ and then $$ \int_{\Omega_l}\frac{a}{2}|\nabla w_1|^2dx\geq \int_{\Omega_l}\frac{a_m^l}{2}|\nabla w_1|^2dx\geq \int_{S_l}{a_m^l}{\mu_1(\Omega_l)}G( w_1)d\sigma. $$ Since $f,g\in\mathcal{B}$ we have $F(w_1)\leq F(\beta)$ as well as $G(w_1)\leq G(\beta)$ and then $$ E(w_1) \geq {a_m^l}{\mu_1(\Omega_l)}\int_{S_l}G( w_1)d\sigma- \int_{S_l}G(w_1)\, d\sigma-G(\beta)\mathcal{H}^{N-1}(\partial\Omega\backslash S_l)-F(\beta)\mathcal{H}^{N}({\Omega}). $$ We also have $E(w_1)\leq E(u_0)<\varepsilon_0-G(\beta)\mathcal{H}^{N-1}(\partial\Omega) -F(\beta)\mathcal{H}^{N}(\Omega) $. Hence $\varepsilon_0>(a_m^l\mu_1(\Omega_l)-1)\int_{S_l}G(w_1)\,d\sigma+ G(\beta)\mathcal{H}^{N-1}(S_l)$. If $a_m^l\mu_1(\Omega_l)<1$ we have $\varepsilon_0>a_m^l\mu_1(\Omega_l)G(\beta)\mathcal{H}^{N-1}(S_l)$. And if $a_m^l\mu_1(\Omega_l)\geq 1$ we have $\varepsilon_0>G(\beta)\mathcal{H}^{N-1}(S_l)$. In both cases we have a contradiction, so $\int_{S_l}T(t)u_0\, d\sigma<0$ for $t\geq 0$. Analogously we have $\int_{S_r}T(t)u_0\, d\sigma>0$ for $t\geq 0$. So we conclude that $\Lambda$ is invariant under $T(t)$. Now if $v\in \Lambda$ we have $\gamma(v)=\{T(t)v,\, t\geq 0\}\subset \Lambda$. Because the system is gradient, $\gamma (v)$ is compact and then the set $$ \omega(v)=\{u=\lim_{t_n\to \infty}T(t_n)v \text{ for some real sequence } (t_n)\} $$ is not empty. Moreover if $\mathcal{E}$ is the set of all equilibrium solutions to \eqref{parabolica1} then $\omega(v)\subset \mathcal{E}$. So if $u\in \omega(v)$ it is an equilibrium solution to \eqref{parabolica1}, $\alpha\leq u\leq \beta$, $E(u)\leq E(v)<\varepsilon_0-G(\beta)\mathcal{H}^{N-1}(\partial\Omega)F(\beta)\mathcal{H}^{N}(\Omega)$ and as before we conclude by contradiction that $\int_{S_l}u\, d\sigma<0$ and $\int_{S_r}u\, d\sigma>0$ and then $\omega(v)\subset{\Lambda}$. Hence if $v\in\Lambda$ then $\omega(v)\subset\overline{\Lambda}\cap\mathcal{E}$ which is a compact set. Since $E$ is continuous there is $e_0\in \overline{\Lambda}\cap\mathcal{E}$ such that $E(e_0)\leq E(v)$ for any $v\in \overline{\Lambda} \cap\mathcal{E}$. But in reality $e_0$ is a minimum of $E$ in $\Lambda$ since otherwise there would be $v_1\in \Lambda$ such that $E(v_1)0\},\\ \Lambda_4=\{ u \in W^{1,p}(\Omega): E(u)< \varepsilon_0 -\lambda G(\beta) \mathcal{H}^{N-1}(\partial\Omega)\} \end{gather*} are open in $W^{1,p}(\Omega)$ and that $e_0 \in \cap_{j=1,\dots,4}\Lambda_j$. We have \begin{itemize} \item $\Lambda_4$ is open in $W^{1,p}(\Omega)$ since $E$ is continuous in $W^{1,p}(\Omega)$. \item $\Lambda_3$ and $\Lambda_2$ are open by the continuity of the functionals $I_2(u) =\int_{S_l} u\, d\sigma$ and $I_3(u) =\int_{S_r} u\, d\sigma$, defined in $W^{1,p}(\Omega)$. \item Using that $W^{1,p}(\Omega) \hookrightarrow C(\overline{\Omega})$ ($p>N$), one can easily check that $\Lambda_1$ is also open in $W^{1,p}(\Omega)$. \end{itemize} As for the inclusion we have: \begin{itemize} \item Clearly $E(e_0) \leq \varepsilon_0 - G(\beta)\mathcal{H}^{N-1}(\partial \Omega)-F(\beta)\mathcal{H}^{N}(\Omega)$ and equality can be ruled out since in case it occurred we would have for any $w \in \Lambda$ (by hypothesis $\Lambda \neq \emptyset$), $$ E(w)<\varepsilon_0- \lambda G(\beta)\mathcal{H}^{N-1}(\partial\Omega)-F(\beta)\mathcal{H}^{N}(\Omega)= E(e_0), $$ which contradicts ${E}(e_0) \leq {E}(v)$ for all $v\in \Lambda$. \item We have $\int_{S_l} e_0\,d\sigma \leq 0$ and equality can be ruled out by contradicting the definition of $\varepsilon_0$, as it was given before. The other case is similar. \item Since $e_0\in \mathcal{E}$ an application of Maximum Principle yields $\alpha 0$ such that $\operatorname{dist}(\Omega_j, S)\geq{m}$ $(j=l,r)$. \end{itemize} We define the signed distance function in $\mathbb{R}^N$ by $$ d(x, S)=\begin{cases} \operatorname{dist}x, S)&\text{if } x\in \mathbb{R}^N_r,\\ -\operatorname{dist}x, S)&\text{if } x\in \mathbb{R}^N_l. \end{cases} $$ and, for $\delta>0$, the tubular neighborhood of $S$ by $$ Q_{\delta}=\{x\in\overline{\Omega}: |d(x,S)|<{\delta} \}. $$ For $S_l, S_r$ as in Lemma \ref{lemaLambda} we suppose $\mathcal{H}^{N-1}(S_l) \leq \mathcal{H}^{N-1}(S_r) $ and choose $ {\delta}< {m}$ small enough such that \begin{equation}\label{deltapequeno} G(\beta)\mathcal{H}^{N-1}(\partial Q_{\delta}\cap\partial\Omega)+F(\beta)\mathcal{H}^N(Q_{\delta})< G(\beta)\mathcal{H}^{N-1}(S_l) \end{equation} Consider a function $\xi:\mathbb{R}\longrightarrow\mathbb{R}$ defined by $$ \xi(t)=\begin{cases} \alpha,&\text{if } t\leq - \delta\\ {\alpha+\beta} + \frac{(\beta-\alpha)}{\delta}t,&\text{if } -\delta0$. Let $S^-$ and $S^+$ be portions of $\partial\Omega$ defined by $\partial\Omega\backslash (\partial Q_{\delta}\cap\partial\Omega)=S^-\cup S^+$. Then $S^-\cap S^+=\emptyset$, $S^-\cap S_l\neq\emptyset \neq S^+\cap S_r$. Since $w_0$ is constant on each connected component of $\Omega\backslash Q_{\delta}$, $G(\alpha)=G(\beta)$ and $F(\alpha)=F(\beta)$ we obtain \begin{align*} E(w_0)&\leq \frac{1}{2}\int_{Q_{\delta}}a(x)|\nabla w_0|^2\ dx -\int_{\partial Q_{\delta}\cap\partial\Omega}G(w_0)\ d\sigma\\ &\quad - G(\beta)\mathcal{H}^{N-1}(\partial\Omega\backslash(\partial Q_{\delta}\cap\partial\Omega))-F(\beta)\mathcal{H}^N(\Omega\backslash Q_{\delta}). \end{align*} Given that $\int_{\partial Q_{\delta}\cap\partial\Omega}G(w_0)d\sigma\geq 0$ in order to have \begin{equation}\label{desigualdadequecaracterizaLambda} E(w_0)<\varepsilon_0-G(\beta)\mathcal{H}^{N-1}(\partial\Omega)-F(\beta)\mathcal{H}^N(\Omega), \end{equation} where $$ \varepsilon_0=G(\beta)\min\{\mathcal{H}^{N-1}(S_l)\min\{1, \mu_1(\Omega_l)a_m^{\Omega_l}\},\mathcal{H}^{N-1}(S_r)\min\{1, \mu_1(\Omega_r)a_m^{\Omega_r}\}\} $$ it suffices to require \begin{equation}\label{DFF} \begin{split} &\varepsilon_0 + F(\beta)\mathcal{H}^N(\Omega\backslash Q_{\delta})\\ &>\frac{1}{2}\int_{Q_{\delta}}a(x)|\nabla w_0|^2\ dx + G(\beta)\mathcal{H}^{N-1}(\partial Q_{\delta}\cap\partial\Omega)+F(\beta)\mathcal{H}^N(\Omega). \end{split} \end{equation} Since the diffusivity function $a$ is to be chosen we set $a_m^{\Omega_j}=\min_{x\in\Omega_j}a(x)$ $(j=l,r)$ and take \begin{equation}\label{condicao1} a_m^{\Omega_l}>\frac{1}{\mu_1(\Omega_l)}, \quad a_m^{\Omega_r}>\frac{1}{\mu_1(\Omega_r)}. \end{equation} Hence \begin{equation}\label{epsilon0} \varepsilon_0=G(\beta)\mathcal{H}^{N-1}(S_l). \end{equation} Moreover setting $a_M^{\delta}=\max_{x\in Q_{\delta}}a(x)$, we have $$ \frac{1}{2}\int_{Q_{\delta}}a(x)|\nabla w_0|^2\ dx\leq \frac{a_M^{\delta}}{2}\frac{(\beta-\alpha)^2}{\delta^2}\mathcal{H}^N(Q_{\delta}). $$ Therefore, \eqref{DFF}, and consequently \eqref{desigualdadequecaracterizaLambda}, will be realized provided \begin{equation} \label{condicao2} 0