\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 217, 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 2015 Texas State University - San Marcos.}
\vspace{7mm}}
\begin{document}
\title[\hfilneg EJDE-2015/217\hfil Limit of nonlinear elliptic equations]
{Limit of nonlinear elliptic equations with concentrated terms
and varying domains: the non uniformly Lipschitz case}
\author[G. S. Arag\~ao, S. M. Bruschi \hfil EJDE-2015/217\hfilneg]
{Gleiciane da Silva Arag\~ao, Simone Mazzini Bruschi}
\address{Gleiciane da Silva Arag\~ao \newline
Departamento de Ci\^encias
Exatas e da Terra, Universidade Federal de S\~ao Paulo,
Rua Professor Artur Riedel, 275, Jardim Eldorado,
Cep 09972-270, Diadema-SP, Brazil}
\email{gleiciane.aragao@unifesp.br, Phone (+55 11) 33193300}
\address{Simone Mazzini Bruschi \newline
Departamento de Matem{\'a}tica, Universidade de Bras\'ilia,
Campus Universit\'ario Darcy Ribeiro, ICC Centro, Bloco A,
Asa Norte, Cep 70910-900, Bras\'ilia-DF, Brazil}
\email{sbruschi@unb.br, Phone (+55 61) 31076389}
\thanks{Submitted February 28, 2015. Published August 19, 2015.}
\subjclass[2010]{35J60, 30E25, 35B20, 35B27}
\keywords{Nonlinear elliptic equation; boundary value problem;
\hfill\break\indent varying boundary; oscillatory behavior;
concentrating term; upper semicontinuity}
\begin{abstract}
In this article, we analyze the limit of the solutions of nonlinear
elliptic equations with Neumann boundary conditions, when nonlinear terms
are concentrated in a region which neighbors the boundary of domain and
this boundary presents a highly oscillatory behavior which is non
uniformly Lipschitz. More precisely, if the Neumann boundary conditions
are nonlinear and the nonlinearity in the boundary is dissipative, then we
obtain a limit equation with homogeneous Dirichlet boundary conditions.
Moreover, if the Neumann boundary conditions are homogeneous, then we obtain
a limit equation with nonlinear Neumann boundary conditions, which captures
the behavior of the concentration's region. We also prove the upper
semicontinuity of the families of solutions for both cases.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction} \label{introd}
In this article we analyze the limit of the solutions of nonlinear elliptic
equations with terms concentrating on the boundary and Neumann boundary conditions
for a family of domains $\Omega_\epsilon$ when the boundary
$\partial\Omega_\epsilon$ presents a behavior which is not uniformly
Lipschitz, as the parameter $\epsilon \to 0$, although $\Omega_\epsilon \to \Omega$
and $\partial\Omega_\epsilon \to \partial\Omega$. This fact can be understood
considering that in each point $x \in \partial\Omega$ the measure
$|\partial\Omega_\epsilon \cap B(x,r)| \to \infty$ when $\epsilon\to 0$,
where $B(x,r)$ is an open ball centre in $x$ with radius $r$ and $|\cdot |$
is the $(N-1)$-dimensional measure. For instance, our case can treat the family
$\Omega_\epsilon$ such that part of $\partial\Omega_\epsilon$ is parameterized
by $\psi_\epsilon(\theta)
= (r_\epsilon(\theta)\cos(\theta),r_\epsilon(\theta)\sin(\theta))$
and $r_\epsilon(\theta)=r_0(\theta)\epsilon\rho(\frac{\theta}{\epsilon^\alpha})$,
for $\theta \in [0,2\pi]$, where $\alpha >1$, $r_0(\cdot)$ is a continuous
function and $\rho(\cdot)$ is a periodic function. In this example,
the period of oscillations is much smaller than its amplitude.
We consider two types of boundary conditions. In the first case,
we study the behavior of the solutions of a concentrated elliptic equation
with nonlinear Neumann boundary conditions of the type
\begin{equation}\label{nbc}
\begin{gathered}
-\Delta u_{\epsilon}+u_{\epsilon}= \frac{1}{\epsilon}
\mathcal{X}_{\omega_{\epsilon}}f(x,u_{\epsilon})+h(x,u_{\epsilon}), \quad\text{in }
\Omega_\epsilon \\
\frac{\partial u_{\epsilon}}{\partial n}+g(x,u_{\epsilon})=0,
\quad\text{on } \partial \Omega_\epsilon
\end{gathered}
\end{equation}
and after the case that $g\equiv 0$, that is, the case with homogeneous
Neumann boundary conditions of the type
\begin{equation}\label{hbc}
\begin{gathered}
-\Delta u_{\epsilon}+u_{\epsilon}
= \frac{1}{\epsilon} \mathcal{X}_{\omega_{\epsilon}}f(x,u_{\epsilon})
+h(x,u_{\epsilon}), \quad\text{in } \Omega_\epsilon \\
\frac{\partial u_{\epsilon}}{\partial n}=0, \quad\text{on }
\partial \Omega_\epsilon.
\end{gathered}
\end{equation}
To describe the problem, we consider a family of bounded smooth domains
$\Omega_\epsilon \subset \mathbb{R}^N$, with $N\geq 2$ and
$0\leq\epsilon \leq \epsilon_0$, for some $\epsilon_0>0$ fixed.
We assume that $\Omega \equiv \Omega_0 \subset \Omega_{\epsilon}$
and we refer to $\Omega$ as the unperturbed domain and
$\Omega_\epsilon$ as the perturbed domains. We also assume that the
nonlinearities $f,g,h:U\times\mathbb{R}\to\mathbb{R}$ are continuous
in both variables and $C^2$ in the second one, where $U$ is a fixed and
smooth bounded domain containing all $\overline{\Omega}_\epsilon$,
for all $0\leq\epsilon\leq\epsilon_0$.
For sufficiently small $\epsilon$, $\omega_\epsilon$ is the region between
the boundaries of $\partial \Omega$ and $\partial \Omega_{\epsilon}$.
Note that $\omega_{\epsilon}$ shrinks to $\partial \Omega$ as $\epsilon\to 0$
and we use the characteristic function $\mathcal{X}_{\omega_{\epsilon}}$
of the region $\omega_{\epsilon}$ to express the concentration
in $\omega_{\epsilon}$. Figure \ref{fig1} illustrates the oscillating
set $\omega_{\epsilon} \subset \overline{\Omega}_{\epsilon}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1} % omega
\end{center}
\caption{The set $\omega_{\epsilon}$.} \label{fig1}
\end{figure}
The existing literature analyses separately concentrated terms and non
uniformly Lipschitz deformation. We consider a problem where these two
issues interact.
In \cite{AB1} the authors consider non uniformly Lipschitz deformation
without concentrated terms. It is proved that the interaction's effect
of non uniformly Lipschitz deformation with a strongly dissipative
nonlinear Neumann boundary condition results in a limit problem with
homogeneous Dirichlet boundary condition. On the other hand, the behavior
of the solutions of elliptic and parabolic problems with reaction and potential
terms concentrated in a neighborhood of the boundary of the domain was
initially studied in \cite{arrieta,anibal}, when the neighborhood is
a strip of width $\epsilon$ and has a base in the boundary, without oscillatory
behavior and inside of $\Omega$. In \cite{arrieta,anibal} the domain $\Omega$
is $C^2$ in $\mathbb{R}^N$.
Also, considering only perturbation of domain, in \cite{AC}
it was proved that the homogeneous Neumann boundary condition is preserved
in the limit problem for a large class of perturbations of domains
in which the non uniformly Lipschitz deformation is included.
In \cite{gam1} some results of \cite{arrieta} were adapted to a nonlinear
elliptic problem posed on an open square $\Omega$ in $\mathbb{R}^2$,
considering $\omega_\epsilon \subset \Omega $ and with highly oscillatory
behavior in the boundary inside of $\Omega$.
Later, \cite{gam2} it proved the continuity of attractors for a nonlinear
parabolic problem posed on a $C^2$ domain $\Omega$ in $\mathbb{R}^2$,
when some terms are concentrated in a neighborhood of the boundary and the
``inner boundary'' of this neighborhood presents a highly oscillatory behavior.
It is important to note that these previous works with terms concentrating
in a neighborhood of the boundary treat with non varying domain and since
$\omega_\epsilon$ is inside of $\Omega$ then all the equations are defined
in the same domain.
We consider the concentration and varying domains simultaneously, therefore
it is necessary to investigate how these two effects interact and what is
the combined result.
In this line, in \cite{aragaobruschi} we consider varying domains and
the region of concentration $\omega_\epsilon$ is outside of $\Omega$
in which the main assumption was that $\partial\Omega_\epsilon$
is expressed in local charts as a Lipschitz deformation of $\partial\Omega$
with the Lipschitz constant uniformly bounded in $\epsilon$.
In \cite{aragaobruschi} it was proved that the limiting equation of \eqref{nbc},
with $h \equiv 0$, is given by
\begin{gather*}
-\Delta u+u=0, \quad\text{in } \Omega \\
\frac{\partial u}{\partial n}+\gamma(x)g(x,u)=\beta(x)f(x,u), \quad
\text{on } \partial \Omega
\end{gather*}
where the function $\gamma\in L^\infty(\partial\Omega)$ is related to
the behavior of the measure $(N-1)$-dimensional of the
$\partial\Omega_\epsilon$ and $\beta \in L^\infty(\partial\Omega)$
is related to the behavior of the measure $N$-dimensional of the
region of concentration $\omega_\epsilon$.
Since $\omega_{\epsilon}$ shrinks to $\partial \Omega$ as $\epsilon\to 0$,
it is reasonable to expect that the family of solutions of \eqref{nbc}
will converge to a solution of an equation with a nonlinear boundary condition
on $\partial \Omega$ that inherits the information about the region
$\omega_{\epsilon}$. Moreover, the oscillations at the boundary amplify
the effect of the nonlinearity $g(x,u)$ at the point $x\in \partial\Omega$
by a factor $\gamma(x)$. Hence, if $g(x,u)$ is strongly dissipative so
that energy is lost through the boundary, then the oscillations increase
the energy loss. While if the effect of the nonlinearity is to drive energy
into the system through the boundary, the oscillations increase the intake
of energy.
In this work, we continue the analysis initiated in \cite{aragaobruschi}
when $\partial\Omega_\epsilon$ is expressed in local charts as a Lipschitz
deformation of $\partial\Omega$ with the Lipschitz constant non uniformly
bounded in $\epsilon$. In this case,
if the nonlinearity $g(x,u)$ is strongly dissipative, we prove that the
family of solutions of \eqref{nbc} will converge to a solution of an
equation with most dissipative boundary condition, which is the
homogeneous Dirichlet boundary condition $u=0$, and that it does not
inherit information about the region of concentration $\omega_\epsilon$.
More precisely, we will show that the limiting equation of \eqref{nbc} is given by
\begin{equation} \label{nbc_limite_gamma_I}
\begin{gathered}
-\Delta u+u=h(x,u), \quad \text{in } \Omega \\
u=0, \quad \text{on } \partial \Omega.
\end{gathered}
\end{equation}
Also, we show that the limiting equation of \eqref{hbc} is an equation
with nonlinear Neumann boundary condition that inherits the information
about the region $\omega_\epsilon$ which is given by
\begin{equation} \label{hbc_limite_gamma_I}
\begin{gathered}
-\Delta u+u=h(x,u), \quad \text{in } \Omega \\
\frac{\partial u}{\partial n}=\beta(x) f(x,u), \quad \text{on } \partial \Omega
\end{gathered}
\end{equation}
where $\beta \in L^\infty(\partial\Omega)$ is related to the behavior of
the measure $N$-dimensional of the region $\omega_\epsilon$.
In both cases, we will prove the upper semicontinuity of the families of
solutions of \eqref{nbc} and \eqref{hbc} in $H^1(\Omega_{\epsilon})$.
This paper is organized as follows: in Section \ref{perdominio},
we define the domain perturbation and state our main results
(Theorems \ref{th-main} and \ref{th-main-homog}).
In Section \ref{concentrated}, we analyze the limit of concentrated integrals
and interior integrals. In Section \ref{uppernonlinear}, we prove
the upper semicontinuity of the family of solutions of
\eqref{nbc} in $H^{1}(\Omega_\epsilon)$.
In Section \ref{upperhomog}, we prove the upper semicontinuity of
the family of solutions of \eqref{hbc} in $H^{1}(\Omega_\epsilon)$.
In the last section, we state additional results.
We leave out some the proofs, but they can be obtained upon request to the authors.
\section{Setting of the problem and main results}
\label{perdominio}
We consider a family of smooth bounded domains
$\Omega_\epsilon \subset \mathbb{R}^N$, with $N\geq 2$ and
$0 \leq \epsilon \leq \epsilon_0$, for some $\epsilon_0 >0$ fixed, and we
regard $\Omega_\epsilon$ as a perturbation of the fixed domain
$\Omega\equiv\Omega_0$. We consider the following hypothesis on the domains
\begin{itemize}
\item[(H1)] There exists a finite open cover $\{U_i\}_{i=0}^m$ of $\Omega$
such that $\overline{U}_0 \subset \Omega$, $\partial\Omega \subset \cup_{i=1}^m U_i$
and for each $i=1, \ldots, m$, there exists a Lipschitz diffeomorphism
$\Phi_{i}: Q_N \to U_i$, where $Q_N=(-1,1)^N\subset \mathbb{R}^N$, such that
$$
\Phi_{i}(Q_{N-1} \times (-1,0)) = U_i \cap \Omega \quad \text{and} \quad
\Phi_{i}(Q_{N-1} \times \{0\}) = U_i \cap \partial \Omega.
$$
We assume that $\overline{\Omega}_\epsilon \subset \cup_{i=0}^m U_i\equiv U$.
For each $i=1, \ldots, m$, there exists a Lipschitz function
$\rho_{i,\epsilon}: Q_{N-1} \to (-1,1)$ such that
$\rho_{i,\epsilon} (x') \to 0$ as $\epsilon \to 0$, uniformly in $Q_{N-1}$.
Moreover, we assume that $\Phi_{i}^{-1}(U_i \cap \partial \Omega_\epsilon)$
is the graph of $\rho_{i,\epsilon}$ this means
$$
U_i \cap \partial \Omega_\epsilon=\Phi_{i}(\{(x',\rho_{i,\epsilon}(x')) : x'=(x_1,...,x_{N-1}) \in Q_{N-1}\}).
$$
\end{itemize}
We consider the following mappings: $T_{i,\epsilon}:Q_N \to Q_N$ defined by
$$
T_{i,\epsilon}(x',s) = \begin{cases}
(x',s+s\rho_{i,\epsilon}(x')+\rho_{i,\epsilon}(x')),&\text{for }s \in (-1,0) \\
(x',s-s\rho_{i,\epsilon}(x')+\rho_{i,\epsilon}(x')),&\text{for }s \in [0,1).
\end{cases}
$$
Also,
\begin{gather*}
\Phi_{i,\epsilon}:=\Phi_{i}\circ T_{i,\epsilon}:Q_N \to U_i; \\
\Psi_{i,\epsilon}:=\Phi_{i}\circ T_{i,\epsilon} \circ \Phi_{i}^{-1}:
U_i \cap \partial \Omega \to U_i \cap \partial \Omega_\epsilon.
\end{gather*}
We also denote
$$
\begin{array}{rl}
\psi_{i,\epsilon}:Q_{N-1}&\to U_i\cap \partial\Omega_\epsilon \\
x'& \mapsto \Phi_{i,\epsilon}(x',0)
\end{array}
\qquad \text{and} \qquad
\begin{array}{rl}
\psi_{i}:Q_{N-1}&\to U_i\cap \partial \Omega \\
x'& \mapsto \Phi_{i}(x',0).
\end{array}
$$
Notice that $\psi_{i,\epsilon}$ and $\psi_{i}$ are local parameterizations
of $\partial\Omega_\epsilon$ and $\partial\Omega$, respectively.
Furthermore, observe that all the maps above are Lipschitz, although
the Lipschitz constant may not be bounded as $\epsilon \to 0$.
Figure \ref{fig2} illustrates the parameterizations.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} % parametrization
\end{center}
\caption{The parameterizations.} \label{fig2}
\end{figure}
With the notation above, we define
$$
\omega_{\epsilon}=\cup_{i=1}^{m}\Phi_{i}\left( \left\{(x',x_N)\in \mathbb{R}^N :
0\leq x_N < \rho_{i,\epsilon}(x') \text{ and } x'\in Q_{N-1} \right\} \right),
$$
for $0<\epsilon \leq \epsilon_0$.
To state the hypothesis to deal with the concentration in $\omega_\epsilon$,
and to analyze the behavior of the solutions of \eqref{nbc} and \eqref{hbc},
as $\epsilon \to 0$, we need the following definition
\begin{definition} \label{definition-jacobian} \rm
Let $\eta: A \subset \mathbb{R}^{N-1} \to \mathbb{R}^N$ almost everywhere
differentiable, we define the $(N-1)$-dimensional Jacobian of $\eta$ as
$$
J_{N-1}\eta \equiv \Big|{\partial \eta \over \partial x_1}\wedge
\ldots\wedge {\partial \eta\over \partial x_{N-1}}\Big|
= \sqrt{\sum_{j=1}^N (\det(\operatorname{Jac}\eta)_j)^2},
$$
where $v_1 \wedge\ldots\wedge v_{N-1}$ is the exterior product of the $(N-1)$
vectors $v_1,\ldots,v_{N-1} \in \mathbb{R}^{N}$
and $(\operatorname{Jac}\eta)_j$ is the $(N-1)$-dimensional matrix obtained
by deleting the $j$-th row of the Jacobian matrix of $\eta$.
\end{definition}
We use $J_N$ for the absolute value of the $N$-dimensional Jacobian determinant.
Now, we are ready to give the hypothesis.
\begin{itemize}
\item[(H2)] For each $i=1, \ldots, m$, $\rho_{i,\epsilon} (x')$ is
$O(\epsilon)$ as $\epsilon \to 0$, uniformly in $Q_{N-1}$, that means
$\| \frac{\rho_{i,\epsilon}}{\epsilon}\|_{L^\infty(Q_{N-1})}\leq C$,
with $C>0$ independent of $\epsilon$, $i=1,\ldots,m$.
And there exists a function $\tilde{\beta}_i \in L^\infty(Q_{N-1})$ such that
$$
\frac{\rho_{i,\epsilon}}{\epsilon}
\rightharpoonup \tilde{\beta}_i \quad
\text{in $L^1(Q_{N-1})$, as $\epsilon \to 0$}.
$$
\end{itemize}
\begin{definition} \label{definition-of-beta} \rm
For $x\in U_i\cap\partial\Omega$, let $(x',0) =\Phi_i^{-1} (x) \in Q_{N}$,
we define $\beta:\partial\Omega \to \mathbb{R}$ as
$$
\beta(x)=\frac{\tilde{\beta}_i(x')(J_N\Phi_i)(x',0)}{J_{N-1}\psi_i(x')}.
$$
\end{definition}
The function $\beta$ is independent of the charts $U_i$ and the maps
$\Phi_i$ and $\rho_{i,\epsilon}$. This was proved in
\cite[Corollary 3.7]{aragaobruschi}.
Now we give an example of the function $\rho_{i,\epsilon}$ satisfying
the hypothesis (H2) and its correspondent function $\tilde{\beta}_i$.
\begin{example} \rm
For each $i=1, \ldots, m$, let $\rho_{i}: \mathbb{R}^{N-1} \to \mathbb{R}^+$
be a $Y$-periodic Lipschitz function, where
$Y=(0,l_1)\times \dots \times (0, l_{N-1}) \in \mathbb{R}^{N-1}$ with
$l_1,\ldots,l_{N-1} \in \mathbb{R}^{+}$ (a function $\rho_{i}$ is called
$Y$-periodic if and only if $\rho_{i}(x'+kl_je_j)=\rho_{i}(x')$ on $Q_{N-1}$,
for all $k\in \mathbb{Z}$ and all $j\in \{1,\ldots,N-1\}$, where
$x'=(x_1, \ldots, x_{N-1})\in \mathbb{R}^{N-1}$ and $\{e_1,\ldots,e_{N-1}\}$
is the canonical basis of $\mathbb{R}^{N-1}$), and we define
$\rho_{i,\epsilon}: Q_{N-1} \to [0,1)$ by
$$
\rho_{i,\epsilon}(x')= \epsilon \varphi(x') \rho_{i}
\Big(\frac{x'}{\epsilon^\alpha}\Big),
$$
for $x'\in Q_{N-1}$ and sufficiently small $\epsilon$, say
$0<\epsilon \leq \epsilon_0$, where $\alpha >0$ and
$\varphi: Q_{N-1} \to \mathbb{R}$ is a continuous function.
From \cite[Theorem 2.6]{cioranescu}, we obtain
$$
\frac{\rho_{i,\epsilon}}{\epsilon} \rightharpoonup \varphi M_Y(\rho_i)
=\tilde{\beta}_i \quad \text{in $L^1(Q_{N-1})$, as $\epsilon \to 0$},
$$
where $M_{Y}(\rho_i)$ is the mean value of $\rho_i$ over $Y$ given by
$$
M_{Y}(\rho_i)=\frac{1}{|Y|}\int_{Y} \rho_i(x')dx'.
$$
\end{example}
The behavior of $J_{N-1}\psi_{i,\epsilon}$, as $\epsilon\to 0$, will be very
important to decide the behavior of the solutions of \eqref{nbc},
as $\epsilon\to 0$. Then, we will consider the hypothesis
\begin{itemize}
\item[(H3)] For each $t>1$, the set
$\{x' \in Q_{N-1}:J_{N-1}\psi_{i,\epsilon}(x')\leq t\}$ satisfies that its
$(N-1)$-dimensional measure goes to zero as $\epsilon\to 0$, for all $i=1,\dots,m$.
\end{itemize}
Now, with respect to the equations, we will be interested in studying
the behavior of the solutions of the elliptic equations \eqref{nbc}
and \eqref{hbc} where, as we mentioned in the introduction,
the nonlinearities $f, g,h:U\times\mathbb{R}\to\mathbb{R}$ are
continuous in both variables and $C^2$ in the second one, where $U$
is a bounded domain containing $\overline{\Omega}_\epsilon$,
for all $0\leq\epsilon\leq\epsilon_0$.
Consider the family of spaces $H^1(\Omega_\epsilon)$ and $H^1(\Omega)$
with their usual norms. Since we will need to compare functions defined
in $\Omega_\epsilon$ with functions defined in the unperturbed domain
$\Omega\equiv\Omega_0$, we will need a tool to compare functions which
are defined in different spaces. The appropriate notion for this is
the concept of $E$-convergence and a key ingredient for this will be
the use of the extension operator $E_\epsilon:H^1(\Omega)\to H^1(\Omega_\epsilon)$,
which is defined as $E_\epsilon = R_\epsilon \circ P$, where
$P:H^1(\Omega) \to H^1(\mathbb{R}^N)$ is an extension operator and
$R_\epsilon$ is the restriction operator from functions defined in
$\mathbb{R}^N$ to functions defined in $\Omega_\epsilon$,
$R_\epsilon w=w_{|\Omega_\epsilon}$. Observe that we also have
$E_\epsilon:L^p(\Omega)\to L^p(\Omega_\epsilon)$ and
$E_\epsilon:W^{1,p}(\Omega)\to W^{1,p}(\Omega_\epsilon)$, for all
$1\leq p\leq \infty$. Considering $X_\epsilon=H^1(\Omega_\epsilon)$
or $L^p(\Omega_\epsilon)$ or $W^{1,p}(\Omega_\epsilon)$,
for $\epsilon \geq 0$, from \cite{AB0} we have
$$
\|E_\epsilon u\|_{X_\epsilon} \to \|u\|_{X_0}.
$$
The concept of $E$-convergence is defined as follows:
$u_\epsilon {\stackrel{E}{\longrightarrow}} u$ if
$\|u_\epsilon-E_\epsilon u\|_{H^1(\Omega_\epsilon)}\to 0$, as
$\epsilon \to 0$. We also have a notion of weak $E$-convergence,
which is defined as follows:
$u_\epsilon {\stackrel{E}{\rightharpoonup}} u$, if
$(u_\epsilon,w_\epsilon)_{H^1(\Omega_\epsilon)}\to (u,w)_{H^1(\Omega)}$,
as $\epsilon \to 0$, for any sequence
$w_\epsilon {\stackrel{E}{\longrightarrow}} w$, where
$(\cdot,\cdot)_{H^1(\Omega_{\epsilon})}$ and
$(\cdot,\cdot)_{H^1(\Omega)}$ denote the inner product in
$H^1(\Omega_{\epsilon})$ and $H^1(\Omega)$, respectively.
More details about $E$-convergence can be found in \cite[Subsection 3.2]{AB0}.
Since $\Omega \subset \Omega_\epsilon$, $\Omega_\epsilon$ is an
exterior perturbation of $\Omega$, we consider the restriction operator
$R_\Omega: H^1(\Omega_\epsilon) \to H^1(\Omega)$ given by
$R_\Omega(u)=u_{|\Omega}$.
Our main results are stated in the following theorems
\begin{theorem}\label{th-main}
Assume that {\rm (H1)--(H3)} are satisfied and that the
nonlinearity $g$ satisfies a dissipative condition:
\begin{equation} \label{condition-on-g}
\exists\, b>0,\, d\geq 1, \text{ s.t. }g(x,s)s\geq b|s|^{d+1}, \quad
\forall |s|\leq R+1 \quad \text{and}\quad \forall x\in U.
\end{equation}
Let $\{u^*_\epsilon\}$, $0<\epsilon\leq \epsilon_0$, be a family of
the solutions of problem \eqref{nbc} satisfying
$\|u^*_\epsilon\|_{L^\infty(\Omega_\epsilon)}\leq R$, for some constant
$R>0$ independent of $\epsilon$. Then, there exist a subsequence
$\{u^*_{\epsilon_k}\}$ and a function $u^*_0\in H^1_0(\Omega)$,
with $\|u^*_0\|_{L^\infty(\Omega)}\leq R$, solution of \eqref{nbc_limite_gamma_I}
satisfying $u^*_{\epsilon_k} {\stackrel{E}{\longrightarrow}} u^*_0$.
\end{theorem}
\begin{theorem} \label{th-main-homog}
Assume that {\rm (H1)--(H2)} are satisfied.
Let $\{u^*_\epsilon\}$, $0<\epsilon\leq \epsilon_0$, be a family of the
solutions of problem \eqref{hbc} satisfying
$\|u^*_\epsilon\|_{L^\infty(\Omega_\epsilon)}\leq R$, for some constant
$R>0$ independent of $\epsilon$. Then, there exist a subsequence
$\{u^*_{\epsilon_k}\}$ and a function $u^*_0\in H^1(\Omega)$, with
$\|u^*_0\|_{L^\infty(\Omega)}\leq R$, solution of \eqref{hbc_limite_gamma_I}
satisfying $u^*_{\epsilon_k} {\stackrel{E}{\longrightarrow}} u^*_0$.
\end{theorem}
\begin{remark} \label{rm-cut-off} \rm
Since in Theorems \ref{th-main} and \ref{th-main-homog} we are concerned
with solutions that are uniformly bounded in $L^\infty(\Omega_\epsilon)$,
we may perform a cut-off in the nonlinearities $f, g$ and $h$ outside
the region $|u|\leq R$, without modifying any of these solutions,
in such a way that
\begin{equation}\label{hip-fgh-bounded}
\begin{gathered}
|f(x,u)|+|\partial_u f(x,u)|\leq M,\quad \forall x\in U, \; \forall u\in \mathbb{R} \\
|g(x,u)|+|\partial_u g(x,u)|\leq M,\quad \forall x\in U, \; \forall u\in \mathbb{R}\\
|h(x,u)|+|\partial_u h(x,u)|\leq M,\quad \forall x\in U, \; \forall u\in \mathbb{R}
\end{gathered}
\end{equation}
and we may also assume that the cut-off is performed so that the following
also holds
\begin{equation} \label{hip-disip-g}
g(x,s)s\geq b|s|, \quad \forall |s|\geq R+1,\; \forall x\in U.
\end{equation}
\end{remark}
\section{Concentrated integrals and interior integrals}
\label{concentrated}
In this section, we will analyze how the concentrated integrals
converge to boundary integrals and the convergence of the interior integrals,
as $\epsilon \to 0$. These convergence results will be needed to analyze
the limit of the solutions of \eqref{nbc} and \eqref{hbc}, as $\epsilon \to 0$.
Initially, we have
\begin{lemma} \label{lemconcentrat1}
Assume that {\rm (H1)--(H2)} are satisfied. Suppose that
$v_{\epsilon}\in W^{1,q}(\Omega_{\epsilon})$ with $\frac{1}{q}0$ independents of $\epsilon$ and $v_{\epsilon}$ such that
for any $0<\epsilon\leq\epsilon_0$, we have
\begin{equation}
\label{ineqconcentrat1}
\frac{1}{\epsilon}\int_{\omega_{\epsilon}}|v_{\epsilon}|^{q} d\xi
\leq L \epsilon^{q/q'}\|v_{\epsilon}\|^{q}_{W^{1,q}(\Omega_{\epsilon})}
+\tilde{L} \|v_{\epsilon}\|^{q}_{H^{s,q}(\Omega)}, \quad
\text{where }\frac{1}{q}+\frac{1}{q'}=1.
\end{equation}
\end{lemma}
\begin{proof}
Consider the finite cover $\{U_i\}_{i=0}^m$ such that
$\overline{\Omega}_{\epsilon} \subset \cup_{i=0}^m U_i\equiv U$ given in (H1).
We have
\begin{align*}
\frac{1}{\epsilon}\int_{\omega_{\epsilon} \cap U_i}|v_{\epsilon}|^q d\xi
&= \frac{1}{\epsilon}\int_{Q_{N-1}} \int^{\rho_{i,\epsilon}(x')}_0
|v_{\epsilon}(\Phi_i(x',x_N))|^q J_N \Phi_i(x',x_N) dx_N\,dx'
\\
&\leq \big\|J_N\Phi_i \big\|_{L^{\infty}(Q_N)}\int_{Q_{N-1}}
\int^{1}_0|v_{\epsilon}(\Phi_i(x',s\rho_{i,\epsilon}(x')))|^q
\frac{\rho_{i,\epsilon}(x')}{\epsilon}ds \,dx'
\\
&\leq 2^q\big\|J_N\Phi_i \big\|_{L^{\infty}(Q_N)}
\big\| \frac{\rho_{i,\epsilon}}{\epsilon} \big\|_{L^{\infty}(Q_{N-1})}
\\
&\quad\times \Big[ \int_{Q_{N-1}} \int^{1}_0 | v_{\epsilon}
(\Phi_i(x',s\rho_{i,\epsilon}(x')))
- v_{\epsilon} (\Phi_i(x',0)) |^q ds \,dx' \\
&\quad + \int_{Q_{N-1}} \int^{1}_0 | v_{\epsilon}
(\Phi_i(x',0)) |^q ds \,dx'\Big],
\end{align*}
where we changed the variable using $x_N=s\rho_{i,\epsilon}(x')$ and,
by hypothesis (H2),
$\| \frac{\rho_{i,\epsilon}}{\epsilon}\|_{L^\infty(Q_{N-1})}\leq C$,
with $C>0$ independent of $\epsilon$, $i=1,\ldots,m$.
Using the trace Theorem for a fixed domain, we obtain
\begin{equation} \label{concent1}
\begin{aligned}
&\int_{Q_{N-1}} \int^{1}_0 | v_{\epsilon} (\Phi_i(x',0)) |^q ds \,dx'\\
&= \int_{Q_{N-1}} | v_{\epsilon} (\psi_i(x')) |^q dx' \\
&\leq \frac{1}{ \inf_{z\in Q_{N-1}} J_{N-1}\psi_i(z)}
\int_{Q_{N-1}} | v_{\epsilon} (\psi_i(x')) |^q J_{N-1}\psi_i(x') \,dx'\\
&\leq \tilde{C} \| v_\epsilon\|^{q}_{L^q(\partial \Omega)}
\leq \hat{C} \| v_\epsilon\|^{q}_{H^{s,q}(\Omega)} .
\end{aligned}
\end{equation}
Now, we estimate
\begin{align*}
&\int_{Q_{N-1}} \int^{1}_0 | v_{\epsilon}(\Phi_i(x',s\rho_{i,\epsilon}(x')))
- v_{\epsilon} (\Phi_i(x',0)) |^q ds \,dx'
\\
&= \int_{Q_{N-1}} \int^{1}_0 \Big| \int_0^{s\rho_{i,\epsilon}(x')}
\frac{\partial (v_\epsilon\circ\Phi_i)}{\partial
\tilde{x}_N}(x',\tilde{x}_{N}) \,d\tilde{x}_{N} \Big|^q ds \,dx'
\\
&\leq \int_{Q_{N-1}} \int^{1}_0 |s\rho_{i,\epsilon}(x')|^{q/q'}
\int_0^{s\rho_{i,\epsilon}(x')}
\Big| \frac{\partial (v_\epsilon\circ\Phi_i)}{\partial \tilde{x}_N}(x',
\tilde{x}_{N}) \Big|^q d\tilde{x}_{N} \,ds \,dx'
\\
&\leq \epsilon^{q/q'} C^{q/q'} \int_{Q_{N-1}}
\int^{1}_0 \int_0^{\rho_{i,\epsilon}(x')} |
\nabla (v_\epsilon\circ\Phi_i)(x',\tilde{x}_N)|^q d\tilde{x}_N \,ds \,dx'
\\
&\leq \epsilon^{q/q'} N^{q+1} C^{q/q'} \|D\Phi_i\|_{L^\infty(Q_N)}^q
\int_{Q_{N-1}} \int_0^{\rho_{i,\epsilon}(x')}
|\nabla v_\epsilon(\Phi_i(x',\tilde{x}_N))|^q d\tilde{x}_N \,dx'
\\
&\leq \frac{\epsilon^{q/q'} N^{q+1} C^{q/q'}
\|D\Phi_i\|_{L^\infty(Q_N)}^q}{ \inf_{z\in Q_N} J_N\Phi_i(z)}
\\
&\quad\times \int_{Q_{N-1}} \int_0^{\rho_{i,\epsilon}(x')}
|\nabla v_\epsilon(\Phi_i(x',\tilde{x}_N))|^q J_N\Phi_i(x',\tilde{x}_N)
\,d\tilde{x}_N \,dx'
\\
&= \frac{\epsilon^{q/q'} N^{q+1} C^{q/q'} \|D\Phi_i\|_{L^\infty(Q_N)}^q}
{ \inf_{z\in Q_N} J_N\Phi_i(z)} \int_{\omega_{\epsilon} \cap U_i}
|\nabla v_\epsilon|^q \,d\xi
\\
&\leq \frac{ N^{q+1} C^{q/q'}\|D\Phi_i\|_{L^\infty(Q_N)}^q}{ \inf_{z\in Q_N}
J_N\Phi_i(z)} \epsilon^{q/q'} \| v_\epsilon\|^{q}_{W^{1,q}(\Omega_\epsilon)}.
\end{align*} %\label{concent2}
From the above inequality and \eqref{concent1} we obtain \eqref{ineqconcentrat1}.
\end{proof}
In \cite[Lemma 3.6]{aragaobruschi} we proved that concentrated integrals
converge to boundary integrals, as $\epsilon \to 0$. We note that
this result of convergence is still true in the case of non uniformly Lipschitz
deformation since we did not use the hypothesis of
$\| \nabla \rho_{i,\epsilon}\|_{L^\infty(Q_{N-1}) }$ is uniformly bounded
for $\epsilon$. This result is stated in the following lemma
\begin{lemma} \label{convconcentanterior}
Assume that {\rm (H1)--(H2)} are satisfied. Then, for any functions
$h, \varphi \in H^{s}(U)$ with $\frac{1}{2} < s \leq 1$, we obtain
$$
\lim_{\epsilon \to 0}\frac{1}{\epsilon}\int_{\omega_{\epsilon}} h \varphi \, d\xi
= \int_{\partial \Omega} \beta h \varphi \, dS.
$$
\end{lemma}
Proceeding as in \cite[Proposition 4.3]{aragaobruschi}, we have the following lemma.
\begin{lemma} \label{lconvergencenonlinearity}
Assume that {\rm (H1)--(H2)} are satisfied. Let $\{u_\epsilon\}$ and
$\{z_\epsilon\}$ be bounded sequences in $H^1(\Omega_\epsilon)$ such that
$u_\epsilon {\stackrel{E}{-{\hspace{-2mm}}\rightharpoonup}} u$ and
$z_\epsilon {\stackrel{E}{-{\hspace{-2mm}}\rightharpoonup}} z$. Then
\[
\frac{1}{\epsilon}\int_{\omega_\epsilon} f(x,u_\epsilon)z_\epsilon\to
\int_{\partial \Omega}\beta f(x,u)z , \quad
\int_{\Omega_\epsilon} h(x,u_\epsilon)z_\epsilon\to
\int_{\Omega}h(x,u)z,
\]
as $\epsilon \to 0$.
\end{lemma}
\begin{proof}
From \cite[Lemma 3.1 (iii)]{AB1} we obtain the convergence of the
interior integrals. Now, using \eqref{hip-fgh-bounded},
Cauchy-Schwarz and Lemma \ref{lemconcentrat1}, we have
\begin{align*}
&\Big|\frac{1}{\epsilon}\int_{\omega_\epsilon} f(x,u_\epsilon)z_\epsilon-
\int_{\partial\Omega} \beta f(x,u)z \Big|
\\
&\leq \tilde{C} \Big( \frac{1}{\epsilon}\int_{\omega_\epsilon}|u_\epsilon
- E_\epsilon u|^2 \Big)^{1/2}
\Big(\frac{1}{\epsilon}\int_{\omega_\epsilon}|z_\epsilon|^2\Big)^{1/2}
+C\Big(\frac{1}{\epsilon}\int_{\omega_\epsilon} |z_\epsilon
- E_\epsilon z|^2\Big)^{1/2} \\
&\quad + \Big|\frac{1}{\epsilon}\int_{\omega_\epsilon}
f(x,E_\epsilon u)E_\epsilon z-\int_{\partial\Omega}\beta f(x,u)z\Big| \\
&\leq \tilde{K} \Big( \epsilon \|u_{\epsilon}-E_\epsilon u\|^2_{H^{1}
(\Omega_{\epsilon})} + \|u_{\epsilon}-E_\epsilon u\|^2_{H^{s}(\Omega)} \Big)^{1/2}
\|z_{\epsilon}\|_{H^{1}(\Omega_{\epsilon})} \\
&\quad + K \Big(\epsilon \|z_{\epsilon}-E_\epsilon z\|^2_{H^{1}(\Omega_{\epsilon})}
+ \|z_{\epsilon}-E_\epsilon z\|^2_{H^{s}(\Omega)} \Big)^{1/2} \\
&\quad + \Big|\frac{1}{\epsilon}\int_{\omega_\epsilon} f(x,E_\epsilon u)E_\epsilon z
-\int_{\partial\Omega}\beta f(x,u)z\Big| \to 0, \quad \text{as }\epsilon \to 0,
\end{align*}
where $s\in \mathbb{R}$ such that $\frac{1}{2}0$ independent of $\epsilon$, then there exists $C>0$
independent of $\epsilon$ such that
\begin{equation} \label{boundednesscondition}
\| \nabla z_\epsilon\|^2_{L^2(\Omega_\epsilon)}
+ \|z_\epsilon\|^2_{L^2(\Omega_\epsilon)}
+\int_{\partial \Omega_\epsilon}|z_\epsilon|^{d(z_\epsilon(x))}
\leq C, \quad 0<\epsilon \leq \epsilon_0,
\end{equation}
for some sufficiently small $\epsilon_0$, where
\begin{equation}\label{def-d}
d(s)=\begin{cases}
d+1,&\text{if } |s|\leq R+1 \\
1,&\text{if } |s|\geq R+1
\end{cases}
\end{equation}
where $d$ and $R$ are defined in \eqref{condition-on-g}.
\end{lemma}
\begin{proof}
Multiplying the equation \eqref{nbc} by $z_\epsilon$ and integrating by parts,
we obtain
$$
\int_{\Omega_\epsilon} |\nabla z_\epsilon|^2
+ \int_{\Omega_\epsilon} |z_\epsilon|^2
+ \int_{\partial\Omega_\epsilon} g(x,z_\epsilon)z_\epsilon
= \frac{1}{\epsilon} \int_{\omega_\epsilon} f(x,z_\epsilon) z_\epsilon
+\int_{\Omega_\epsilon} h(x,z_\epsilon) z_\epsilon.
$$
By Cauchy-Schwarz and Young inequalities, \eqref{hip-fgh-bounded} and
Lemma \ref{lemconcentrat1} with $s=1$, we have
\begin{equation}\label{limitconcent1}
\begin{aligned}
\frac{1}{\epsilon}\int_{\omega_\epsilon} f(x,z_\epsilon) z_\epsilon
& \leq \frac{\delta}{\epsilon} \|z_\epsilon\|^2_{L^2(\omega_\epsilon)}
+ \frac{C_\delta}{\epsilon}
\|f(\cdot,z_\epsilon(\cdot))\|^2_{L^2(\omega_\epsilon)} \\
& \leq \delta K_2 \|z_\epsilon\|^2_{H^1(\Omega_\epsilon)} +C_\delta K_1,
\end{aligned}
\end{equation}
where $K_1$ and $K_2$ are independent of $\epsilon$, with
$0<\epsilon \leq \epsilon_0$ for some sufficiently small $\epsilon_0$.
Again, using Cauchy-Schwarz and Young inequalities and \eqref{hip-fgh-bounded},
we obtain
\begin{equation} \label{limitconcent2}
\int_{\Omega_\epsilon} h(x,z_\epsilon) z_\epsilon
\leq \delta \|z_\epsilon\|^2_{L^2(\Omega_\epsilon)}
+ C_\delta \|h(\cdot,z_\epsilon(\cdot))\|^2_{L^2(\Omega_\epsilon)}
\leq \delta \|z_\epsilon\|^2_{H^1(\Omega_\epsilon)} + C_\delta K_3.
\end{equation}
Now, using \eqref{condition-on-g}, \eqref{hip-disip-g} and \eqref{def-d}, we obtain
\begin{equation}\label{limitconcent3}
\int_{\partial\Omega_\epsilon}g(x,z_\epsilon)z_\epsilon
\geq b\int_{\partial\Omega_\epsilon}|z_\epsilon|^{d(z_\epsilon(x))}.
\end{equation}
Therefore, using \eqref{limitconcent1}, \eqref{limitconcent2} and
\eqref{limitconcent3} and taking $\delta$ such that $\delta (K_2+1)<1$, we obtain
$$
\min\{1-\delta (K_2+1),b\} \Big(\|\nabla z_\epsilon\|^2_{L^2(\Omega_\epsilon)}
+ \|z_\epsilon\|^2_{L^2(\Omega_\epsilon)}
+ \int_{\partial\Omega_\epsilon}|z_\epsilon|^{d(z_\epsilon(x))}\Big)
\leq C_\delta (K_1+K_3).
$$
This shows \eqref{boundednesscondition}.
\end{proof}
Now, we can prove the upper semicontinuity of the solutions of \eqref{nbc}.
\begin{proof}[Proof of Theorem \ref{th-main}]
Let $\{u^*_\epsilon\}$, $0<\epsilon\leq \epsilon_0$, be a family of the
solutions of \eqref{nbc} satisfying
$\|u^*_\epsilon\|_{L^\infty(\Omega_\epsilon)}\leq R$, for some constant
$R>0$ independent of $\epsilon$. Applying Lemma \ref{lemboundedsolution},
the sequence $\{u^*_\epsilon\}$ is bounded in $H^1(\Omega_{\epsilon})$.
By \cite[Lemma 3.1 (i)]{AB1}, there exist a subsequence $\{u^*_{\epsilon_k}\}$
and a function $u^*_0 \in H^1(\Omega)$ such that
$u^*_{\epsilon_k} {\stackrel{E}{-{\hspace{-2mm}}\rightharpoonup}} u^*_0$
and $\|u^*_{\epsilon_k} - E_{\epsilon_k}u^*_0\|_{L^2(\Omega_{\epsilon_k})}\to 0$.
We give a brief proof of $u^*_0 \in H^1_0(\Omega)$ in the case
$\Omega \subset \Omega_\epsilon$ for $0<\epsilon\leq \epsilon _0$.
The complete proof is in \cite[Proposition 4.2]{AB1}.
The trace operator from $H^1(\Omega)$ to
$L^2(U_i \cap \partial\Omega)$, $i=1,2,\ldots, n$, is continuous and compact, then
${u^*_{\epsilon_k}}_{| U_i \cap \partial\Omega}$ converges to
${u^*_0}_{| U_i \cap \partial\Omega}$ in $L^2(U_i \cap \partial\Omega)$.
Hence, given $\beta>0$
small, there exists $\epsilon_0$ such that
\begin{equation}\label{ineq-1}
\int_{U_i \cap \partial\Omega}|u^*_{\epsilon_k}-u^*_0|
\leq \beta, \quad \text{for } 0< \epsilon\leq\epsilon_0.
\end{equation}
Using \cite[Lemma 3.2]{AB1} for $\eta=0$, we obtain
that for each $\beta >0$ fixed, we
can choose an even smaller $\epsilon_0$ such that
\begin{equation}\label{ineq-2}
\int_{U_i \cap \partial\Omega}|u^*_{\epsilon_k} \circ\Psi_{i,\epsilon_k}
- u^*_{\epsilon_k}| \leq \beta,\quad \text{ for } 0<\epsilon\leq \epsilon_0.
\end{equation}
Putting together \eqref{ineq-1} and \eqref{ineq-2}, we obtain that for
$0<\epsilon\leq\epsilon_0$,
\begin{equation} \label{ineq-4}
\int_{U_i\cap\partial\Omega} |u^*_0|\leq
2\beta+\int_{U_i\cap\partial\Omega}|u^*_{\epsilon_k} \circ\Psi_{i,\epsilon_k}|.
\end{equation}
For each $t>1$, we consider the sets $A_t^{\epsilon_k}=\{x'\in Q_{N-1}:
J_{N-1}\psi_{i,\epsilon_k}(x')\leq t\}$ and
$B_t^{\epsilon_k}=\{x'\in Q_{N-1}: J_{N-1}\psi_{i,\epsilon_k}(x')>
t\}$ so that $Q_{N-1}=A_t^{\epsilon_k}\cup B_t^{\epsilon_k}$, $A_t^{\epsilon_k}\cap
B_t^{\epsilon_k}=\emptyset$ and, by (H3), $|A_t^{\epsilon_k}|\to
0$ as $\epsilon_k\to 0$. Moreover,
\begin{equation} \label{ineq-5}
\int_{U_i \cap \partial\Omega} |u^*_{\epsilon_k} \circ\Psi_{i,\epsilon_k}|=
\int_{\psi_{i}(A_t^{\epsilon_k})}|u^*_{\epsilon_k}\circ\Psi_{i,\epsilon_k}|+
\int_{\psi_{i}(B_t^{\epsilon_k})}|u^*_{\epsilon_k}\circ\Psi_{i,\epsilon_k}|.
\end{equation}
We analyze separately the two integrals in \eqref{ineq-5}.
Initially, for all $1
R+1\}$, with $R$ given as in the Lemma \ref{lemboundedsolution}, so that $$ \int_{B_t^{\epsilon_k}} |u^*_{\epsilon_k} \circ \psi_{i,\epsilon_k}(x')|dx' = \int_{C_t^{\epsilon_k}} |u^*_{\epsilon_k} \circ \psi_{i,\epsilon_k}(x')|dx'+\int_{D_t^{\epsilon_k}} |u^*_{\epsilon_k} \circ \psi_{i,\epsilon_k}(x')|dx'. $$ Using Holder inequality, $\frac{J_{N-1}\psi_{i,\epsilon_k}}{t} >1$ on ${B_t^{\epsilon_k}}$ and \eqref{boundednesscondition}, we obtain \begin{align*} &\int_{C_t^{\epsilon_k}} |u^*_{\epsilon_k} \circ \psi_{i,\epsilon_k}(x')|dx' \\ &\leq \frac{|C_t^{\epsilon_k}|^{\frac{d}{d+1}}}{t^\frac{1}{d+1}} \Big(\int_{C_t^{\epsilon_k}}|u^*_{\epsilon_k} \circ \psi_{i,\epsilon_k}(x')|^{d+1} J_{N-1}\psi_{i ,\epsilon_k}(x')dx'\Big)^{1\over d+1} \\ &\leq \frac{|C_t^{\epsilon_k}|^{\frac{d}{d+1}}}{t^\frac{1}{d+1}} \Big(\int_{U_i\cap\partial\Omega_{\epsilon_k}} |u^*_{\epsilon_k}|^{d(u^*_{\epsilon_k})}\Big)^\frac{1}{d+1}\leq Ct^{-\frac{1}{d+1}} \end{align*} and \begin{align*} %\label{parte>tand>R+1} \int_{D_t^{\epsilon_k}} |u^*_{\epsilon_k} \circ \psi_{i,\epsilon_k}(x')|dx' &\leq \frac{1}{t}\int_{D_t^{\epsilon_k}}|u^*_{\epsilon_k} \circ \psi_{i,\epsilon_k}(x')| J_{N-1}\psi_{i ,\epsilon_k}(x') dx'\\ &\leq \frac{1}{t}\int_{U_i\cap\partial\Omega_{\epsilon_k}} |u^*_{\epsilon_k}|^{d(u^*_{\epsilon_k})}\\ &\leq Ct^{-1}\leq Ct^{-\frac{1}{d+1}}. \end{align*} Since $t$ can be chosen arbitrarily large in the inequalities above and using \eqref{ineq-4}, \eqref{ineq-5} and \eqref{ineq-6}, we obtain $$ \int_{\partial\Omega\cap U_i}|u^*_0|=0,\quad i=1,2,\ldots, n, $$ which implies that $u^*_0\in H^1_0(\Omega)$. To show that $u^*_0$ is a weak solution of \eqref{nbc_limite_gamma_I}, we consider $\theta \in \mathcal{C}^{\infty}_{c}(\Omega)$. Multiplying \eqref{nbc} by $E_{\epsilon_k}\theta$ and integrating by parts, we obtain $$ \int_{\Omega_{\epsilon_k}}\nabla u^*_{\epsilon_k} \nabla E_{\epsilon_k}\theta + \int_{\Omega_{\epsilon_k}}u^*_{\epsilon_k} E_{\epsilon_k}\theta = \int_{\Omega_{\epsilon_k}}h(x,u^*_{\epsilon_k}) E_{\epsilon_k}\theta. $$ Taking the limit as $\epsilon_k \to 0$ and using that $u^*_{\epsilon_k} {\stackrel{E}{-{\hspace{-2mm}}\rightharpoonup}} u^*_0$ in $H^1(\Omega_{\epsilon_k})$ and Lemma \ref{lconvergencenonlinearity}, we obtain that $u^*_0$ satisfies $$ \int_{\Omega}\nabla u^*_0 \nabla \theta + \int_{\Omega}u^*_0\theta =\int_{\Omega}h(x,u^*_0)\theta. $$ Therefore, $u^*_0$ is a weak solution of \eqref{nbc_limite_gamma_I}. Now, we prove that $u^*_{\epsilon_k} {\stackrel{E}{\longrightarrow}} u^*_0$. In order to do this, we prove the convergence of the norms $\|u^*_{\epsilon_k}\|_{H^1(\Omega_{\epsilon_k})} \to \|u^*_0\|_{H^1(\Omega)}$. In fact, multiplying the equation \eqref{nbc} by $u^*_{\epsilon_k}$ and integrating by parts, we obtain \begin{align*} \|u^*_{\epsilon_k}\|^2_{H^1(\Omega_{\epsilon_k})} & = \frac{1}{\epsilon_k}\int_{\omega_{\epsilon_k}} f(x,u^*_{\epsilon_k}) u^*_{\epsilon_k}+ \int_{\Omega_{\epsilon_k}} h(x,u^*_{\epsilon_k})u^*_{\epsilon_k} - \int_{\partial\Omega_{\epsilon_k}} g(x,u^*_{\epsilon_k})u^*_{\epsilon_k} \\ & \leq \frac{1}{\epsilon_k}\int_{\omega_{\epsilon_k}} f(x,u^*_{\epsilon_k}) u^*_{\epsilon_k}+ \int_{\Omega_{\epsilon_k}} h(x,u^*_{\epsilon_k})u^*_{\epsilon_k}, \end{align*} where we have used that $g(x,u)u\geq 0$. Using Lemma \ref{lconvergencenonlinearity} and $u^*_0\in H^{1}_0(\Omega)$, we obtain that $ \lim_{\epsilon_k\to 0} \|u^*_{\epsilon_k}\|^2_{H^1(\Omega_{\epsilon_k})} \leq \|u^*_0\|^2_{H^1(\Omega)}$. By \cite[Proposition 3.2]{AB0}, we obtain $u^*_{\epsilon_k} {\stackrel{E}{\longrightarrow}} u^*_0$. This completes the proof. \end{proof} \begin{remark} \label{solutionzero} \rm If $h(x,u)=0$ in \eqref{nbc} then the limit problem of \eqref{nbc} is given by \begin{equation}\label{nbc_limite_gamma_I0} \begin{gathered} -\Delta u+u=0, \quad \text{in } \Omega \\ u=0, \quad \text{on } \partial \Omega. \end{gathered} \end{equation} By Lax-Milgram Theorem, the unique solution in $H^{1}(\Omega)$ of \eqref{nbc_limite_gamma_I0} is given by $u\equiv 0$. Hence, in Theorem \ref{th-main}, $u^*_{\epsilon_k} {\stackrel{E}{\longrightarrow}} 0$. Moreover, by uniqueness of solutions of \eqref{nbc_limite_gamma_I0}, we obtain the E-convergence for the whole family $\{ u^*_{\epsilon}\}$ of solution of \eqref{nbc}, that is, $u^*_{\epsilon} {\stackrel{E}{\longrightarrow}} 0$. \end{remark} \section{Upper semicontinuity of solutions for the homogeneous boundary conditions \eqref{hbc}} \label{upperhomog} In this section, we provide a proof of Theorem \ref{th-main-homog}. Initially, we prove boundedness of the solutions of \eqref{hbc}. \begin{lemma} \label{lemboundedsolutionhomog} Assume that {\rm (H1)--(H2)} are satisfied. If $\{z_\epsilon\}$, $0<\epsilon\leq \epsilon_0$, is a family of the solutions of \eqref{hbc} satisfying $\|z_\epsilon\|_{L^\infty(\Omega_\epsilon)}\leq R$, for some constant $R>0$ independent of $\epsilon$, then there exists $C>0$ independent of $\epsilon$ such that \begin{equation} \label{boundednessconditionhomog} \|z_\epsilon\|_{H^1(\Omega_\epsilon)} \leq C, \quad 0<\epsilon \leq \epsilon_0, \end{equation} for some sufficiently small $\epsilon_0$. \end{lemma} \begin{proof} Multiplying \eqref{hbc} by $z_\epsilon$ and integrating by parts, we obtain $$ \|z_\epsilon\|^2_{H^1(\Omega_\epsilon)} =\int_{\Omega_\epsilon} |\nabla z_\epsilon|^2 + \int_{\Omega_\epsilon} |z_\epsilon|^2 = \frac{1}{\epsilon} \int_{\omega_\epsilon} f(x,z_\epsilon) z_\epsilon + \int_{\Omega_\epsilon} h(x,z_\epsilon) z_\epsilon. $$ Therefore, using \eqref{limitconcent1} and \eqref{limitconcent2}, we obtain \[ [1-\delta (K_2+1)]\|z_\epsilon\|^2_{H^1(\Omega_\epsilon)} \leq C_\delta (K_1+K_3). \] Now, taking $\delta$ such that $\delta(K_2+1)<1$, we obtain \eqref{boundednessconditionhomog}. \end{proof} Now, we can prove the upper semicontinuity of the solutions of \eqref{hbc}. \begin{proof}[Proof of Theorem \ref{th-main-homog}] Let $\{u^*_\epsilon\}$, $0<\epsilon\leq \epsilon_0$, be a family of the solutions of \eqref{hbc} satisfying $\|u^*_\epsilon\|_{L^\infty(\Omega_\epsilon)}\leq R$, for some constant $R>0$ independent of $\epsilon$. Applying Lemma \ref{lemboundedsolutionhomog}, the sequence $\{u^*_\epsilon\}$ is bounded in $H^1(\Omega_{\epsilon})$. By \cite[Lemma 3.1 (i)]{AB1}, there exist a subsequence $\{u^*_{\epsilon_k}\}$ and a function $u^*_0 \in H^1(\Omega)$ such that $u^*_{\epsilon_k} {\stackrel{E}{-{\hspace{-2mm}}\rightharpoonup}} u^*_0$ and $\|u^*_{\epsilon_k}-E_{\epsilon_k}u^*_0\|_{L^{2}(\Omega_{\epsilon_k})}\to 0$. To show that $u^*_0$ is a weak solution of \eqref{hbc_limite_gamma_I}, we consider $\theta \in H^1(\Omega)$. Multiplying \eqref{hbc} by $E_{\epsilon_k} \theta$ and integrating by parts, we obtain $$ \int_{\Omega_{\epsilon_k}}\nabla u^*_{\epsilon_k} \nabla E_{\epsilon_k}\theta + \int_{\Omega_{\epsilon_k}}u^*_\epsilon E_{\epsilon_k}\theta = \frac{1}{\epsilon_k}\int_{\omega_{\epsilon_k}} f(x,u^*_{\epsilon_k}) E_{\epsilon_k}\theta+\int_{\Omega_{\epsilon_k}} h(x,u^*_{\epsilon_k}) E_{\epsilon_k}\theta. $$ Taking the limit as $\epsilon_k \to 0$ and using that $u^*_{\epsilon_k} {\stackrel{E}{-{\hspace{-2mm}}\rightharpoonup}} u^*_0$ in $H^1(\Omega_{\epsilon_k})$ and Lemma \ref{lconvergencenonlinearity}, we obtain that $u^*_0$ is a weak solution of \eqref{hbc_limite_gamma_I}. Now, we prove that $u^*_{\epsilon_k} {\stackrel{E}{\longrightarrow}} u^*_0$. To do this, we prove the convergence of the norms $\|u^*_{\epsilon_k}\|_{H^1(\Omega_{\epsilon_k})} \to \|u^*_0\|_{H^1(\Omega)}$. In fact, multiplying the equation \eqref{hbc} by $u^*_{\epsilon_k}$, integrating by parts and again using Lemma \ref{lconvergencenonlinearity}, we obtain \begin{align*} \|u^*_{\epsilon_k}\|^2_{H^1(\Omega_{\epsilon_k})} &= \frac{1}{\epsilon_k}\int_{\omega_{\epsilon_k}} f(x,u^*_{\epsilon_k}) u^*_{\epsilon_k}+ \int_{\Omega_{\epsilon_k}} h(x,u^*_{\epsilon_k})u^*_{\epsilon_k} \\ &\to \int_{\partial \Omega} \beta f(x,u^*_0)u^*_0 +\int_{\Omega} h(x,u^*_0)u^*_0 = \|u^*_0\|^{2}_{H^1(\Omega)}, \quad \text{as }\epsilon_k \to 0. \end{align*} Hence, we obtain that $ \lim_{\epsilon_k\to 0} \|u^*_{\epsilon_k}\|^2_{H^1(\Omega_{\epsilon_k})} \leq \|u^*_0\|^2_{H^1(\Omega)}$. By \cite[Proposition 3.2]{AB0}, we obtain $u^*_{\epsilon_k} {\stackrel{E}{\longrightarrow}} u^*_0$. \end{proof} \subsection*{Conclusion} %\label{conclusion} With the results obtained in this work and proceeding analogously to \cite{AB1} and \cite{AC}, we can prove the lower semicontinuity of the families of solutions of \eqref{nbc} and \eqref{hbc} in $H^{1}(\Omega_\epsilon)$, in the case where the solutions of the limit problems \eqref{nbc_limite_gamma_I} and \eqref{hbc_limite_gamma_I} are hyperbolic, and then the convergence of the eigenvalues and eigenfunctions of the linearizations around the solutions. These results are important for understanding the behavior of the dynamics of the parabolic equations associated to the problems \eqref{nbc} and \eqref{hbc}. \subsection*{Acknowledgments} G. S. Arag\~ao is partially supported by CNPq 475146/2013-1, Brazil. S. M. Bruschi is partially supported by FEMAT, Brazil. \begin{thebibliography}{0} \bibitem{aragaobruschi} G. S. Arag\~ao, S. M. 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