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\markboth{\hfil  Stable  multiple-layer stationary solutions \hfil EJDE--1997/22}%
{EJDE--1997/22\hfil Arnaldo Simal do Nascimento \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1997}(1997), No.\ 22, pp. 1--17. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
Stable  multiple-layer stationary solutions of a semilinear parabolic
equation in two-dimensional domains
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35K20,  35K57, 35B25.
\hfil\break\indent
{\em Key words and phrases:} Diffusion equation, Gamma-convergence,
transition layers, \hfil\break\indent stable equilibria.
\hfil\break\indent
\copyright 1997 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted May 13, 1997. Published December 1, 1997.} }
\date{}
\author{Arnaldo Simal do Nascimento\\ \\
Dedicated to the memory of Ennio De Giorgi (1928-1996)}
\maketitle

\newcommand{\diver}{\mathop{\rm div}}

\begin{abstract} 
We use $\Gamma$--convergence to prove existence of stable multiple--layer
stationary solutions (stable patterns) to a reaction--diffusion equation.
Given nested simple closed curves in ${\Bbb R}^2$, we give sufficient conditions 
on their curvature so that the reaction--diffusion problem possesses
a family of stable patterns.
In particular, we extend to two-dimensional domains and to a spatially 
inhomogeneous source term, a previous result by Yanagida and Miyata.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{coro}{Corollary}[section]
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\newcommand{\quina}{\mathbin{\hbox{\vrule height8pt}\vbox{\hrule width 5pt}}}

\section{Introduction}

This paper is a contribution to the investigation of some diffusion
processes, wherein the interplay between diffusivity and a source term
gives rise to stable multiple-layer stationary solutions. This kind of
solution will be referred to as a multiple-layer pattern.
At the same time, along with \cite{N2}  and \cite{N3}, it consolidates the use of
the $\Gamma$--convergence technique to show the existence of multiple-layer
patterns of some semilinear parabolic equations which involve a small
parameter.

Specifically we will be focusing on processes governed  by the evolution problem
\begin{eqnarray}
\frac{\partial v_\varepsilon}{\partial t} &=& \varepsilon^2\, \diver\left(
       k_1(x)\nabla  v_\varepsilon\right) + k_2(x)(v_\varepsilon -\alpha)
(\beta-v_\varepsilon)
       (v_\varepsilon -\gamma_\varepsilon(x))\,,\nonumber \\
&& \mbox{ for } (x,t)\in\Omega\times{\Bbb R}^+ \\
&&v_\varepsilon(x,0) = v_0 \quad
\frac{\partial v_\varepsilon}{\partial \widehat{n}} = 0\,, \quad\mbox{for }
x\in \partial\Omega\,, \ t >0\nonumber
\end{eqnarray}
where $\widehat{n}$ is the inward normal to $\partial\Omega$; $\alpha,\beta\in{\Bbb R}$,
$\alpha<\beta$, with $\varepsilon$ a small positive parameter; $k_i(x)$, $i=1,2$,
are positive functions in $C^1(\Omega)$; $\Omega\subset{\Bbb R}^2$ has smooth
boundary $\partial\Omega$, say of class $C^3$; and
\begin{eqnarray}
&\gamma_\varepsilon(x) =\left(\frac{\alpha+\beta}{2}\right)+g_\varepsilon(x),
 \ \alpha< \beta, \ \gamma_\varepsilon\in C(\overline{\Omega})& \nonumber \\
&g_\varepsilon(x) = o(\varepsilon), \ 
             \mbox{ uniformly in $\Omega$ as $\varepsilon\to 0$}\,. &
\end{eqnarray}


In \cite{MY}, the existence of multiple-layer patterns has been proved for the
one-dimensional case, i.e, $\Omega=[0,1]$,  with the hypotheses that
$k_2(x)\equiv 1$, $\gamma_\varepsilon=1/2 + a_1\varepsilon + o(\varepsilon^2)$,$a_1$ constant, $\alpha=0$,  and $\beta=1$.
Here, besides considering any two-dimensional domain $\Omega$, we allow for
spatially inhomogeneous perturbations of the state
$v = \left[ (\alpha+\beta)/2\right]$. 

The procedure used in \cite{MY}  was to construct super and  sub-solutions by
modifying a traveling wave solution of (1.1), with the aforementioned
restrictions, to obtain a multiple-layer pattern. However, this procedure is
not suitable for a generalization to two-dimensional domains.
Herein a technique known as $\Gamma$--convergence devised by De Giorgi
\cite{GiF}  and developed by many others is used.

When seeking stable stationary solutions to some classes of  semilinear
parabolic equations, the $\Gamma$--convergence approach turns out to be
very useful. This approach replaces the original problem of minimizing a family
of functionals by a more tractable problem in the space of functions of
bounded variation, $BV(\Omega$), which  usually
yields more precise information on the geometric structure of the
minimizers. 

The prototype for  the source term in (1.1) is the case
$\alpha=0$, $\beta=1$ and $0<\gamma_\varepsilon<1$, which stems from the theory of
population genetics where $\alpha,\beta$ and $\gamma_\varepsilon$ denote some
probability measures.
The case in which $\Omega=[0,1]$, $\alpha=0$, $\beta=1$ and
$\gamma_\varepsilon(x)=a(x)$, $0<a(x)<1$, is studied in \cite{AMP}. 

There are many works dealing with the existence of multiple or
double-layer patterns for equations similar to (1.1). Most of these deal
with the unidimensional case. Among those which bear more resemblance to (1.1) are
\cite{FH,HR,HS}.

In \cite{FiH}, the generation and propagation of internal layers for a  
related problem are studied for the case $\Omega = {\Bbb R}$. See also \cite{ES}.
For a physical background on Problem (1.1) the interested reader is
referred to \cite{AMP,H}.

\section{Main Result}
\setcounter{equation}{0}

Let $\gamma_i$ be smooth simple closed curves whose
traces lie inside $\Omega$ and which are nested, in the sense that if $O_i$
denotes the open region enclosed by $\gamma_i$, i.e.\ $\gamma_i = \partial  O_i$,
$i=1,\ldots,p$, then 
$O_1\subset O_2\subset \cdots \subset O_{p+1}\stackrel{\rm def}{=}\Omega$\, and 
$\partial  O_i \cap \partial O_{i+1}=\emptyset\;,\;i=1,\ldots,p$.  
We abuse the notation and denote by $\gamma_i$ the map as well as the trace
of the curve itself and set throughout 
\[\Omega_1 \stackrel{\rm def}{=} O_1\,, \  \Omega_2 = O_2\backslash \overline{O}_1,
 \dots, \Omega_p = O_p\backslash \overline{O}_{p-1}\,, \ 
\Omega_{p+1} = \Omega\backslash \overline{O}_p\ . \]
For future reference, we consider the following function
\begin{equation}
v_0 = \alpha\chi_{\Omega_{\alpha}^0} +\beta\chi_{\Omega_{\beta}^0}
\end{equation}
where $\chi_A$ stands for the characteristic function of the set $A$ and
 \begin{equation}
 \Omega_{\alpha}^0 \stackrel{\rm def}{=}
\bigcup_{\stackrel{1\le j\le p+1}{j\mbox{\scriptsize\  is odd}} } \Omega_j\,,\qquad
 \Omega_{\beta}^0 \stackrel{\rm def}{=}
\bigcup_{\stackrel{1\le j\le p+1}{j\mbox{\scriptsize\  is even}} } \Omega_j\,.
\end{equation}
Let $\gamma_i$ be arc--length parametrized, i.e., $\gamma_i(s)$, $0\leq
s\leq L_i$, where \ $L_i$ is the total arc length of $\gamma_i$.
Let $\widehat{n}_i$ be the unit inner normal to $\gamma_i$,  and 
$\kappa_i(y)$, $y\in\gamma_i$, its signed curvature. Around each narrow enough
tubular neighbourhood of $\gamma_i$, we set a principal coordinate system as
follows. See \cite{GT}  for more details.

If $d(x,\gamma_i)$ denotes the usual signed distance function which is positive 
inside $\Omega_i$ and negative outside $\Omega_i$, we set
\[ 
N_{\delta,i} \stackrel{\rm def}{=} \left\{ x\in\Omega : \left| d(x,\gamma_i)\right| 
< \delta \right\}\,. 
\]
For $\delta$  small enough, the change of coordinate map
\[ \Sigma_i : \gamma_i\times(-\delta,\delta) \longrightarrow  N_{\delta,i} 
\]
defined by $\Sigma_i(s,d) = \gamma_i(s)+d\,\widehat{n}_i(s)\,, \ 0\leq s < L_i\,, \ 
-\delta < d < \delta$, is a diffeomorphism. 
Moreover the Jacobian of $\Sigma_i(s,d)$ is given by 
\[ J_{\Sigma_i}(s,d) = (1-d\,\kappa_i(s)), \ 0\leq s\leq L_i\,, \ 
i=1,\ldots,p\ . \]
Note that for $\delta$ small enough $J_{\Sigma_i} > 0$ and $\Sigma_i(s,0)=\gamma_i(s)$.

Let us also set $\widetilde{k}_1(s,d)=k_1(\Sigma(s,d))$ ,
$\widetilde{k}_2(s,d)=k_2(\Sigma(s,d))$ and regarding (1.1) put
$$k(x)\stackrel{\rm def}{=} \left[ k_1(x)k_2(x)\right]^{1/2}\,.$$
We now state our main result.

\begin{theorem}
Suppose that $\gamma_i$, $i=1,\ldots,p$, is a $\nu_i$--level curve
of $k$, \  i.e.,
\[ k(x) = \nu_i\,, \ \ \mbox{for} \ \ x\in \gamma_i\,, \ i=1,\ldots,p\,. 
\]
Let $\widetilde{k}(s,d)=k(\Sigma(s,d))$ and $\Lambda_i(s,d)=\widetilde{k}(s,d)
J_{\Sigma_i}(s,d)$,
$(s,d)\in [0,L_i]\times (-\delta,\delta)$. Suppose that
$$ 
   \Lambda_i(s,d) > \Lambda_i(s,0) = \nu_i\,, \ \ \mbox{for} \ \ d\in(-\delta,\delta), \
   d\neq 0, \ i=1,\ldots,p  \eqno(H1) $$ 
Then there is a family ${\{ v_\varepsilon\}}_{0<\varepsilon\leq\varepsilon_0}$, 
with $\varepsilon_0$ small, of stationary solutions of (1.1) such that for each
 $\varepsilon\in (0,\varepsilon_0)$: 
\begin{description}
\item[{\rm (2.1.i)}] $v_\varepsilon\in C^{2,\sigma}(\overline{\Omega})$, $0<\sigma <1$ 
 and $\alpha < v_\varepsilon(x) < \beta$, \ $\forall \, x\in\overline{\Omega}$. 
 
\item[{\rm (2.1.ii)}] $\| v_\varepsilon - v_0\|_{L^1(\Omega)}\longrightarrow 0$, as
$\varepsilon\to 0$.

\item[{\rm (2.1.iii)}] For each $\lambda\in (0,(\beta-\alpha)/2)$  and 
$\Omega^\lambda_\varepsilon = \left\{ x\in \Omega : \alpha +\lambda < v_\varepsilon(x) 
< \beta-\lambda\right\}$ it holds that  
$\left| \Omega^\lambda_\varepsilon\right| \longrightarrow 0$, as
$\varepsilon\to 0$, 
 where  $|\cdot |$ is the 2-dimensional Lebesgue measure.

\item[{\rm (2.1.iv)}] $v_\varepsilon$ is a stable stationary solution of (1.1).
\end{description} 
\end{theorem}
Next we give necessary conditions for (H1) to be satisfied. 

\begin{lemma}
Suppose that  each $\gamma_i$, $i=1,\ldots,p$, satisfies
 $$ 
\begin{array}{rl} \displaystyle
\frac{\partial \widetilde{k}(s,0)}{\partial d} = \nu_i\,\kappa_i(s),
 & 0\leq s \leq L_i \\ 
\displaystyle
\frac{\partial^2\,\widetilde{k}(s,0)}{\partial d^2} > 2\,\nu_i\,\kappa^2_i(s),
 & 0\leq s \leq L_i 
\end{array}  \eqno(H2)
$$
Then hypothesis (H1) is satisfied.
\end{lemma}

\paragraph{Remark.}
An example of a function which satisfies (H2) can be easily constructed.
For instance, let 
$\gamma_i\subset\Omega\subset{\Bbb R}^2$ with $\gamma_i=\{ x\in\Omega :\;
 \| x\| =1\}$. Then the function 
$\tilde{k}(s,t)=1 + t + 2\,t^2$ with $t\in(-\delta,\delta)$, $\delta$ small, 
and $s\in[0,2\pi)$, satisfies (H2) with $\nu_i =1$.

\paragraph{Remark.}
The hypothesis on $\gamma_i$ could have been slightly more general. It would
suffice to have $\gamma_i$, $i=1,\ldots,p$, a subset of a $\nu^1_i$--level
set of $k_1$ and also a subset of a $\nu^2_i$--level set of $k_2$. Then
$\gamma_i$ would be a $\nu_i$--level curve of $k(x)$ where 
$\nu_i=\left( \nu^1_i\nu^2_i\right)^{1/2}$.

\paragraph{Remark.}
The two conditions in (H2) dictate the behavior of $k$ in a
neighbourhood of the limiting transition-phase curve $\gamma_i$. The first
condition requires that $k(x)$ increase or decrease as $x$ crosses
$\gamma_i$ along the direction of $\widehat{n}_i$ according to the sign of the
curvature $\kappa_i$, its slope being proportional to $|\kappa_i|$ at the
crossing-point, provided $\nu_i \neq 0$. 
The second one, for a fixed $s$ relates the concavity of
$k(\Sigma(s,d))$ with the curvature $\kappa_i(s)$ of $\gamma_i$ at $s$. Note that
 $\frac{\partial\widetilde{k}}{\partial  d}$ changes sign whenever the 
curvature of $\gamma_i$ does so.

Each limiting phase-transition curve $\gamma_i$ acts through (H2) as
a barrier which prevents a, say, diffusing substance whose initial
concentration evolves in time according to (1.1), to spread homogeneously
in space and eventually settling down in a uniform concentration.

Actually this would be the case if for instance we had $\Omega$ convex,
$k_1(x)\equiv\mbox{const.}$, $k_2(x)\equiv\mbox{const.}$ and
$g_\varepsilon(x)\equiv 0$. See \cite{M,CH}  for this matter. An example
where this occurs, in despite of the fact that the diffusion function
$k_2$ is not constant, can be found in \cite{N1}.

\paragraph{Remark.}
The stability referred to in (2.1.iv) should be understood in the
following sense: a stationary solution $v(x)$ of (1.1) is said to be
stable in the $H^1(\Omega)$-norm, say, if for any $\mu>0$ there exists $\delta>0$ 
such that $T(t)\psi$ exists for all $t>0$ (here $T(t)$ denotes the
nonlinear semigroup generated by (1.1)\,) and
\[ \| T(t)\psi - v\|_{H^1(\Omega)} < \mu\,, \ \ 0<t<\infty\,, \]
for any $\psi\in H^1(\Omega)$ which satisfies $\|\psi -v\|_{H^1(\Omega)}< \delta$.
Here $T(0)\psi = \psi$. If in addition 
\[ \lim_{t\to\infty} \|T(t)\psi - v\|_{H^1(\Omega)} = 0 \]
then $v$ is said to be strongly stable.
 
\paragraph{Remark.}
It might be worthwhile to mention that each limiting phase-transition curve $\gamma_i$,
$i=1,\ldots,p$, is a $\nu_i$--level curve of $\widetilde{k}$ and a curve of minima of 
$\Lambda_i(s,d)=\widetilde{k}(s,d)(1-d\,\kappa_i(s))$. Of course if $\gamma_i$ contains a
segment of straight line $\ell_i$, say, then $\Lambda_i(s,d)=\widetilde{k}(s,d)$ on
$\ell_i$ and the two notions coincide there.


\section{Preliminaries on $BV$-functions}

Before proving the main result we recall some notation on 
measures, and results on functions of bounded variation. 
The reader is referred to \cite{Z, EG,G} for further background.
The Lebesgue measure in ${\Bbb R}^n$ is denoted by $|\cdot|$ and the 
$m$--dimensional Hausdorff measure by ${\cal H}^m$, $m\in[0,2]$.

If $\mu$ is a Borel measure on $\Omega$ with values in $[0,+\infty[$ or in 
${\Bbb R}^k$, $k\geq 1$, its total variations is denoted by $|\mu|$. If $F$ 
is a Borel subset of $\Omega$, the measure $\mu \quina F$ is defined by 
$(\mu \quina F)(B) = \mu
(B\cap F)$ for any Borel set $B\subset\Omega$. For every $\mu$-integrable function 
$f$, the measure $f\mu$ is defined by $(f\mu)(B) = \int_B f d\mu$ for every Borel 
set $B\subset\Omega$. We shall use also the notation $\int_B f\mu$.

The space $BV(\Omega)$ of functions of bounded variation in $\Omega$ is defined as 
the set of all functions $u\in L^1(\Omega)$ whose distributional gradient $Du$ is a 
Radon measure with bounded total variation in $\Omega$.
We denote by $BV(\Omega;\{\alpha,\beta\})$ the class of all $u\in BV(\Omega)$ which 
take values $\alpha,\beta$ only.

The essential boundary of a set $E\subset{\Bbb R}^n$ is the set $\partial_*E$ of all 
points in $\Omega$ where $E$ has neither density 1 nor density 0. A set 
$E\subset\Omega$ has finite perimeter in $\Omega$ if its characteristic function 
$\chi_E$ belongs to $BV
(\Omega)$. In this case $\partial_*E$ is rectifiable, and we may endow it with a 
measure theoretic normal $\nu_E$ so that the measure derivative $D\chi_E$ is 
represented as 
$$D\chi_E(B) = \int\limits_{B\cap\partial_*E}{\hspace*{-3mm}\nu_E d {\cal H}^1}
\quad \mbox{for every Borel set}\quad B\subset \Omega\,.$$

The following result is a simple generalization of the total variation of a $BV$ 
function, and we omit the proof.

\begin{lemma} Let $A\subset\Omega$ be an open set and $v\in BV(\Omega)$. Let $\{v_j\}$ 
be a sequence of functions in $BV(\Omega)$ converging to $v$ in $L^1(\Omega)$. Then
\[ \left | \frac{\partial  v}{\partial  x_i}\right |(A)\leq\lim\inf_{\hspace*{-6mm}
   j\to +\infty}\,\left| \frac{\partial  v_j}{\partial  x_i}\right |(A)\; 
\mbox{ for }\;   i=1,2. \]
\end{lemma}

The next lemma plays an important role in the proof of the main results. For 
simplicity, since we consider $\gamma_i$ one of the curves introduced in Section 2, 
we drop the index $i$ in the statement and in the proof.

\begin{lemma} Let $v\in BV(\Omega)$ and set $\mu = Dv$. Let $\Sigma$,
$\delta$, $N_{\delta}$, $J_{\Sigma}$ be defined as in Section 2. 
Let $\tilde{v} = v(\Sigma) : \gamma\times(-\delta,\delta)\to{\Bbb R}$ 
for $\delta$ small enough, and $\tilde{N}_{\delta} = \Sigma^{-1}(N_{\delta})$. 
Then, for any Borel set $B\subset N_{\delta}$, we have
\[
|\mu|(B)\geq\left((|\widetilde{\mu}_1|(\widetilde{B}))^2+
(|\widetilde{\mu}_2|(\widetilde{B}))^2\right)^{1/2}\,,
\]
where $\tilde{B} = \Sigma^{-1}(B)$,
$\displaystyle \tilde{\mu}_1 = \frac{\partial\tilde{v}}{\partial s}$, 
and $\displaystyle \tilde{\mu}_2 = J_{\Sigma}\frac{\partial\tilde{v}}{\partial d}$.
\end{lemma}

\paragraph{ Proof:} Let ${v_j}$ be a sequence of functions of class 
$C^{\infty}(N_{\delta})$ converging in $L^1(N_{\delta})$ to $v$ and such that 
$|\bigtriangledown v_j|(N_{\delta})\to |Dv|(N_{\delta})$ as $j\to +\infty$. 
Following the notation of Section 2, 
a direct computation yields
\[ \left| M^{-1}_\Sigma\nabla \widetilde{v}_j\right| = 
   \left[ \left( \frac{\partial\widetilde{v}_j}{\partial  s} \right)^2\,J^{-2}_{\Sigma}
 + \left( \frac{\partial\widetilde{v}_j}{\partial  d}\right)^2\right]^{1/2}\,,
\]
where $M_{\Sigma}$ denotes the Jacobian matrix corresponding to the coordinate map 
$\Sigma$, and $\tilde{v}_j = v_j(\Sigma)$. If $B\subset N_{\delta}$ is a Borel set, 
we then have
\begin{eqnarray*}
       \int_B |Dv| & = &\lim_{j\to +\infty}\,\int_B |\nabla  v_j|
        =  \lim_{j\to +\infty}\,\int_{\widetilde{B}} J_\Sigma\left|
                 M^{-1}_\Sigma \nabla \,\widetilde{v}_j\right|  \\
      & = &  \lim_{j\to +\infty}\,\int_{\widetilde{B}} \left[ 
               \left(\frac{\partial\widetilde{v}_j}{\partial  s}\right)^2 + J^2_\Sigma
            \left(\frac{\partial\widetilde{v}_j}{\partial  d}\right)^2\right]^{1/2} \\
     &\geq & \lim_{j\to +\infty}\,\left[\left( \int_{\widetilde{B}}  
          \left|\frac{\partial\widetilde{v}_j}{\partial  s}\right|\right)^2 + 
          \left( \int_{\widetilde{B}} J_\Sigma
         \left|\frac{\partial\widetilde{v}_j}{\partial  d}\right|\right)^2\right]^{1/2}
\end{eqnarray*}
where the last inequality follows by the inequality $\int |z| \geq |\int z |$, 
$z\in {\Bbb R}^2$. Then, using Lemma 3.1, we conclude this proof.


\section{Local Minimizers Via $\Gamma$--convergence} \setcounter{equation}{0}

A stationary solution of (1.1) satisfies the boundary value problem
\begin{eqnarray}
   &\varepsilon^2\mbox{ div }\left[ k_1(x)\nabla  v\right] +
       k_2(x)f_\varepsilon(x,v)=0\,, \ \ x\in\Omega &\nonumber \\ 
   &\nabla  v(x)\cdot\widehat{n}(x)=0\,, \  \mbox{ for }\ x\in\partial\Omega 
\end{eqnarray}
where 
$ f_\varepsilon(x,v)=(v-\alpha)(\beta -v)(v-(\theta + g_\varepsilon(x))$
and, for convenience, we set $\theta = (\alpha+\beta)/2$.

If $F_\varepsilon(x,v) = \int^v_\theta f_\varepsilon(x,\xi)d\xi$ then 
$F_\varepsilon(x,v) = F^0(v) - g_\varepsilon(x)F^1(v)$  where
$$F^0(v) = \int^v_\theta (\xi-\alpha)(\beta-\xi)(\xi-\theta)d\xi \ \mbox{ and } \
F^1(v) = \int^v_\theta (\xi-\alpha)(\beta-\xi)d\xi\,.$$
Next we define a family of functionals $E_\varepsilon : L^1(\Omega) \to {\Bbb R}$,
\begin{equation}
E_\varepsilon(v)=
  \left\{\begin{array}{l}\displaystyle
\int_\Omega\big\{\frac{\varepsilon\,k_1(x)}{2}\,|\nabla  v|^2+ 
\varepsilon^{-1} k_2(x)\left[ F(\alpha)-F^0(v)\right]\\
\quad + \varepsilon^{-1}g_\varepsilon(x)k_2(x)F^1(v)
\big\}\,dx\,, \quad \mbox{ if $v\in H^1(\Omega)$} \\
                 \infty\,, \quad  \mbox{ otherwise.}
 \end{array}\right.
\end{equation}

The term $\varepsilon^{-1}k_2(x)F(\alpha)$ has been artificially added since it does
not affect $E_\varepsilon$, as long as existence of minimizers is concerned
and what is more important, the potential function
\[ \widetilde{F}(v) \stackrel{\rm def}{=} F(\alpha)-F^0(v) \]
satisfies 
\begin{eqnarray}
 &\widetilde{F}(\alpha) = \widetilde{F}(\beta) = 0\,, \ \ 
\widetilde{F}\in C^2({\Bbb R}) & \nonumber\\
& \widetilde{F}(v) > 0  \ \ \mbox{for any} \ \ v\in{\Bbb R}\,, \ v\neq\alpha\,, \ v
\neq\beta &\\
&\widetilde{F}'(\alpha) = \widetilde{F}'(\beta) = 0\,,
 \ \widetilde{F}''(\alpha) > 0\,, \
            \widetilde{F}''(\beta) > 0\,.\nonumber
\end{eqnarray}
It is easy to see that any local minimizer $v_\varepsilon$ of $E_\varepsilon$
will be a solution to (H1) and, by regularity, $v_\varepsilon\in
C^{2,\nu}(\Omega), \ 0<\nu <1$.

Therefore, it is our aim now to find a family of local minimizers of
$E_\varepsilon$. To that end we will need the concept of $\Gamma$--convergence and
of a local isolated minimizer of $E_\varepsilon$. See for example  \cite{Ma}.

\paragraph{Definition}
A family $\{ E_\varepsilon\}_{0<\varepsilon\leq\varepsilon_0}$ of real-extended functionals
defined in $L^1(\Omega)$ is said to $\Gamma$--converge, as $\varepsilon\to 0$, to a
functional $E_0$, at $v$ and we write
\[ \Gamma(L^1(\Omega)^-) - \lim_{\varepsilon\to 0} E_\varepsilon(v)=E_0(v) \]
if
\begin{itemize}
\item For each $v\in L^1(\Omega)$ and for any sequence $\{ v_\varepsilon\}$ in
$L^1(\Omega)$ such that $v_\varepsilon\to v$ in $L^1(\Omega)$, as $\varepsilon\to 0$, it
holds that $E_0(v)\leq\displaystyle\liminf_{\vspace*{-6mm}\varepsilon\to 0} E_\varepsilon(v_\varepsilon)$.
\item For each $v\in L^1(\Omega)$ there is a sequence $\{ w_\varepsilon\}$ in 
$L^1(\Omega)$ such that $w_\varepsilon\to v$ in $L^1(\Omega)$, as $\varepsilon\to 0$ and also
$E_0(v)\geq \displaystyle \limsup_{\vspace*{-6mm}\varepsilon\to 0} E_\varepsilon(w_\varepsilon)$.
\end{itemize}

\paragraph{Definition.} 
We say that $v_0\in L^1(\Omega)$ is an $L^1$-local minimizer of $E_0$ if
there is $\rho>0$ such that
\[ E_0(v_0)\leq E_0(v) \ \ \mbox{whenever} \ \ 
   0 < \| v-v_0\|_{L^1(\Omega)} < \rho\,.\]
Moreover, if $E_0(v_0)<E_0(v)$ for $0<\|v-v_0\|_{L^1(\Omega)}<\rho$, then
$v_0$ is called an isolated $L^1$-local minimizer of $E_0$. 
\medskip

De Giorgi's $\Gamma$--convergence provides, for equicoercive functionals, 
the convergence of global minimizers to a global minimizer of the $\Gamma$--limit.
Concerning convergence of local minimizers, the following theorem extends an 
observation made by Kohn and Sternberg in [K,S]; the proof is the same as the one 
in \cite{KS}.

\begin{theorem}
Suppose that a family of extended--real functionals $\{ E_\varepsilon\}$,
$\Gamma$--converges, as $\varepsilon\to 0$, to a extended--real functional $E_0$ and
the following hypotheses are satisfied:
\begin{description}
\item[{\rm (4.1.i)}] Any sequence $\{ u_\varepsilon\}_{\varepsilon>0}$ such that 
$E_\varepsilon(u_\varepsilon)\leq\mbox{const.}<\infty$, $\varepsilon>0$, is compact 
in $L^1(\Omega)$.
\item[{\rm (4.1.ii)}] There exists an isolated $L^1$-local minimizer $v_0$
of $E_0$. 
\end{description}
Then there exists $\varepsilon_0>0$ and a family 
$\{ v_\varepsilon\}_{0<\varepsilon\leq\varepsilon_0}$ such that:
\begin{itemize}
\item $v_\varepsilon$ is an $L_1$-local minimizer of $E_\varepsilon$
\item $\|v_\varepsilon-v_0\|_{L^1(\Omega)} \to 0$,  \, as \, $\varepsilon\to 0$.
\end{itemize}
\end{theorem}

Let us consider the family of functionals defined by (4.2). For the case
in which $g_\varepsilon(x)\equiv 0$ and $\widetilde{F}$ satisfies (4.3), the
$\Gamma$--convergence of this type of functionals has been treated in 
\cite{S,MM1,Mo}.

In (4.2) the presence of the term  $\varepsilon^{-1}g_\varepsilon(x)k_2(x)F^1(v)$ adds
no additional difficulty 
because its smoothness makes of it a continuous perturbation with respect to
$L^1$--convergence. 
Moreover by virtue of (1.2) the perturbation term
$\int_\Omega\varepsilon^{-1}g_\varepsilon(x)k_2(x)F^1(v_\varepsilon)dx$ vanishes 
when one takes the $\Gamma$--limit. Hence the above results can be evoked thus 
yielding the following theorem.

\begin{theorem}
Consider the family of functionals given by (4.2). Then
\[ \Gamma(L^1(\Omega)^-) - \lim_{\varepsilon\to 0} E_\varepsilon(v) = E_0(v) \]
where
\begin{equation}
E_0(v) =
  \left\{\begin{array}{ll}
       \displaystyle
C_0 \int_\Omega\,k\left | D{\chi}_{\{v=\beta\}}\right |, 
&\mbox{if } v\in BV(\Omega,\{\alpha,\beta\})\,, \\
& 0 < |\{v=\beta\}| < |\Omega| ;\\
\infty\, & \mbox{otherwise.}
 \end{array}\right.
\end{equation}
and \ $C_0 = (\beta -\alpha)\int^\beta_\alpha\,(\widetilde{F}(t))^{1/2}dt$.
\end{theorem}

Our next goal is to apply Theorem 4.1 to 
$\{ E_\varepsilon\}$, $0<\varepsilon\leq\varepsilon_0$ and to
$E_0$,  as defined in (4.2) and (4.4).
We only have to worry about (4.1.ii) since (4.1.i) has essentially been
proved in \cite{KS,Mo}. Their proof can be easily adapted to fit our
case. 

As for (4.1.ii) it has been proved in \cite{N2} and \cite{N3} for the case of
just one limiting transition--phase curve. We give here a neater and more
geometric proof for this case and then consider the case of $p$ limiting
transition--phase curves. 

For the single curve case the proof presented in
the references above uses the approximation result mentioned at the
beginning of the proof of Lemma 3.2. This approach is somehow cumbersome
since it has to deal with sequences and subsequences throughout the proof
and unnatural since it avoids to work with the geometry of $BV$--spaces which
is the setting our problem is naturally casted into. 
This approximation approach can be avoided by resorting to Lemma 3.1.

In the next theorem we will be dealing with the single curve case, i.e.,
$p=1$, and therefore we set for simplicity $\gamma=\gamma_1$. Thus
$N_\delta=N_\delta(\gamma)$ will denote its tubular neighbourhood,\, $\Sigma$\, the
corresponding coordinate map and we drop the subindex 1 in all other
definitions. The proof is a modification of a similar proof given in \cite{KS}.

\begin{theorem}
With the notation of Section 2, suppose that $\gamma\subset \Omega$ is a simple
closed $\nu$--level curve of $k$ and that {\rm (H1)} is
satisfied with $\Omega=\Omega_1\cup\gamma\cup\Omega_2$.
Then 
\[ v_0(x) = \alpha\,{\chi}_{\Omega_1}(x) + \beta\,{\chi}_{\Omega_2}(x)\,, \ \
x\in \Omega \]
is an $L^1$-local isolated minimizer of $E_0$.
\end{theorem}

\paragraph{ Proof:}
It suffices to prove that, if $v \in BV(\Omega ;\{\alpha,\beta\})$ and 
$0 < \|v - v_0\|_{L^1(N_\delta)} <\rho$ for a suitable $\rho > 0$, then
\[ \int_{N_\delta(\gamma)}k |Dv| > \int_{N_\delta(\gamma)}k|Dv_0|.\]

Let us start by computing $E_0(v_0)$. By the coarea formula 
(see \cite{F}) we obtain
\begin{eqnarray*}
E_0(v_0)&=&\int_{N_\delta(\gamma)} k|Dv_0| = 
   \int_{\alpha}^{\beta}\int_{N_\delta\cap\partial_*\{{v_0>\xi}\}}kd
   {\cal H}^1 d\xi  \\
&=& (\beta -\alpha)\int_{\gamma}kd{\cal H}^1  =  (\beta -\alpha)\nu\,L\,,
\end{eqnarray*}
where $L$ is the total arc--length of $\gamma$. For a fixed $d\in (-\delta,\delta)$ 
define
\[ \ell_d = \left\{ (s,d)\,,\,0 < s < L\right\}\,. \]
Then the trace of $\widetilde{v}(\cdot ,d)$ is well-defined on $\ell_d$, a.e. in
$(-\delta,\delta)$; because each $\ell_d$ is $C^1$
(actually Lipschitz would suffice). 

Suppose that \begin{description}
\item{i)} \ $\widetilde{v}=\widetilde{v}_0$, along
$\ell_{\widetilde{d}}\cup\ell_{-\widetilde{d}}$, in the sense of traces, for some
$\widetilde{d}\in (\delta/2,\delta)$.
\end{description}
Recall that $\Lambda(s,d) = \widetilde{k}(s,d)J_\Sigma(s,d)$. We say that $v$ is an admissible function if $\|v-v_0\|_{L^1(N_\delta)}>0$
and $v\in BV(N_\delta(\gamma);\{\alpha,\beta\})$. For any such
function $v$ and with the use of Lemma 3.2 we have:
\begin{eqnarray*}
E_0(v)&=&\int_{N_\delta}k|Dv| \\ 
&\geq &
\int_{\widetilde{N}_\delta}\widetilde{k}\left( |\widetilde{\mu}_1|^2+
|\widetilde{\mu}_2|^2\right)^{1/2} \geq 
\int_{\widetilde{N}_\delta}\widetilde{k}|\widetilde{\mu}_2| \\
   & \geq & 
\nu\,\int^L_0\int^\delta_{-\delta}\left|\frac{\partial\widetilde{v}}{\partial d}\right|
\; \mbox{(by $(H_1)$ and definition of $\widetilde{\mu}_2$)}  \\
   & \geq & 
\nu(\beta-\alpha)L\,, 
\end{eqnarray*}
since \ i) \ implies that $\int_{-\delta}^{\delta}\left |\frac{\partial\widetilde{v}}{\partial d}\right | \geq (\beta - \alpha)$ for any $s\in [a,b]$.

We claim that $E_0(v_0)<E_0(v)$. If this were not the case then for any
admissible function $v$, the coarea formula would yield
\begin{eqnarray}
(\beta -\alpha)\nu L & = & \int_{N_\delta} k |Dv| =  
                            \int^\infty_{-\infty}\left(
                            \int_{N_\delta\cap\partial\{ v>\xi\}} 
                             k|Dv|\right) d\xi \nonumber \\
                & = & \int^\infty_{-\infty}\left(
                        \int_{N_\delta\cap\partial_*\{ v>\xi\}} 
                         kd{\cal H}^1\right) d\xi \nonumber \\ 
               & = & \int^\infty_{-\infty}\left(
                        \int_{\widetilde{N}_\delta\cap\partial_*\{\tilde{v}>\xi\}} 
                        \Lambda(s,t)d{\cal H}^1\right) d\xi \nonumber \\ 
               & = & (\beta -\alpha)\int_{\widetilde{N}_\delta\cap\partial_*
                        \{\widetilde{v}=\beta\}\cap\partial_*\{\widetilde{v}=\alpha\}} 
                         \Lambda(s,t)d{\cal H}^1 
\end{eqnarray}
To deduce (4.5) we use that
\[ 
|D\widetilde{v}| = {\cal H}^1 \quina \left( \partial_*\{\widetilde{v}=\alpha\}
\cap \partial_*\{\widetilde{v}=\beta\}\right)
 \]
taking into account that $\partial\{\widetilde{v}=\alpha\} 
\cap \partial\{\widetilde{v}=\beta\}$ is a
set of finite perimeter in $\widetilde{N}_\delta$.

Also our hypothesis implies
 
\[ \int_{\widetilde{N}_\delta}\widetilde{k}\left( |\widetilde{\mu}_1|^2 + |\widetilde{\mu}_2|^2
   \right)^{1/2} =
   \int_{\widetilde{N}_\delta}\widetilde{k}|\widetilde{\mu}_2|. \]
But the above equality holds if and only if
$|\widetilde{\mu}_1| \equiv 0$ on
$\widetilde{N}_\delta$, what is to say that $\widetilde{v}(\cdot , d)$ is ${\cal H}^1$-a.e.
constant along each $\ell_d$, for a.e. $d\in (-\delta,\delta)$.
Hence, 
$$\widetilde{N}_\delta\cap\partial_*\{\widetilde{v}=
\alpha\}\cap\partial_*\{\widetilde{v}=\beta\} =
\cup^m_{j=1}\ell_{d_{\scriptstyle j}}\,,$$
 for some $m\in{\Bbb N}$ and $-\delta <d_j <\delta$, $j=1,\ldots,m$. 
Note that 
$$\widetilde{N}_\delta\cap\partial\{\widetilde{v}_0=\alpha\}\cap
\partial\{\widetilde{v}_0=\beta\}=\ell_0\,.$$
Therefore by virtue of (H1), (4.5)
holds if and only if $m=1$ and $d_1=0$, i.e., $\widetilde{v}=\widetilde{v}_0$ a.e. in
$\widetilde{N}_\delta$. This is a contradiction since $v$ is an admissible
function and as such $\|\widetilde{v}-\widetilde{v}_0\|_{L^1(\widetilde{N}_\delta)}> 0$.

This takes care of our theorem in the case in which \, i-) \, holds. Now
if \, i) \, does not hold then one of the following cases would occur, in
the sense of traces of $BV$-functions:
\begin{description}
\item[ii)] $\widetilde{v}$ is not const. ${\cal H}^1$--a.e. along 
$\ell_{\widetilde{d}}$,
for a.e. $\widetilde{d}\in\left( \delta/2\,,\,\delta\right)$

\item[iii)] $\widetilde{v}$ is not const. ${\cal H}^1$--a.e. along 
$\ell_{-\widetilde{d}}$,
for a.e. $\widetilde{d}\in\left( \delta/2\,,\,\delta\right)$

\item[iv)] $\widetilde{v}\equiv\alpha\;{\cal H}^1$-a.e. along 
$\ell_{\widetilde{d}}$\, and $\widetilde{v}\equiv\beta\, ,{\cal H}^1$-a.e. along 
$\ell_{-\widetilde{d}}$\, , for a. e. $\widetilde{d}\in (\delta/2,\delta)$. 
\end{description}

Following ideas set forth in \cite{KS},  we define a set $\Delta \subset (0,\delta)$,
$$
\Delta\stackrel{\rm def}=\left\{ \begin{array}{l} \displaystyle
\widetilde{d}\in(0,\delta) : \int^L_0\big\{
|\widetilde{v}-\widetilde{v}_0|(s,\widetilde{d}) J_\Sigma(s,\widetilde{d}) 
 + |\widetilde{v}-\widetilde{v}_0|(s,-\widetilde{d})
J_\Sigma(s,-\widetilde{d})\big\}\, ds \\ \displaystyle
\quad > 4\rho/\delta, 
\mbox{ where }\int_{\widetilde{N}_\delta}|\widetilde{v}-\widetilde{v}_0|
 J_\Sigma < \rho \end{array}\right\}
$$
Hence $|\Delta| < \delta/4$. Now if \ iv)  does not hold, then by choosing
$ \rho < \delta L(\beta-\alpha)/ 2$ we obtain
\[ \int^L_0 \left\{
|\widetilde{v}-\widetilde{v}_0|(s,\widetilde{d}) J_\Sigma(s,\widetilde{d}) + 
|\widetilde{v}-\widetilde{v}_0|(s,-\widetilde{d}) 
J_\Sigma(s,-\widetilde{d})\right\} = 2\,L\,(\beta-\alpha) > (4\rho/\delta)\,,
\] 
which by definition implies that $\widetilde{d}\in \Delta$. But then for a.e. 
$\widetilde{d}\in \left( (\delta/2,\delta)\backslash\Delta\right)$, 
either \, ii)  or \, iii) holds. 

For any admissible $v$, Lemma 3.1 allows us to conclude that
\begin{eqnarray*}
E_0(v) & = & \int_{N_\delta}k\,|Dv| \geq 
\int_{N_\delta\backslash N_{\delta/2}}k\,|Dv| +
\int_{N_{\delta/2}}k\,|Dv| \\
 & = & \int_{\widetilde{N}_\delta\backslash \widetilde{N}_{\delta/2}}
\widetilde{k}\,\left(
|\widetilde{\mu}_1|^2 +|\widetilde{\mu}_2|^2\right)^{1/2} +
\int_{\widetilde{N}_{\delta/2}}\widetilde{k}\,\left(
|\widetilde{\mu}_1|^2 +|\widetilde{\mu}_2|^2\right)^{1/2} \\
&\geq & \int_{\widetilde{N}_\delta\backslash\widetilde{N}_{\delta/2}}\widetilde{k}
\,\left |\frac{\partial\widetilde{v}}{\partial s}\right | +
          \int_{\widetilde{N}_{\delta/2}}\Lambda\,
          \left |\frac{\partial\widetilde{v}}{\partial d}\right | \\
&=& I_1+I_2
\end{eqnarray*}
where $I_1$ and $I_2$ denote respectively the first and the second
integrals in the last term of the above inequalities.

If 
$ k_m = \min_{x\in\overline{\Omega}}\,k(x) $
then by virtue of \, ii)  and \, iii),  for a.e. $\widetilde{d}\in
(\delta/2,\delta)\backslash\Delta$ it holds that
\begin{eqnarray*}
\lefteqn{\int^L_0\left\{\widetilde{k}(s,d)\,|\widetilde{\mu}_1|
(s,\widetilde{d}) + \widetilde{k}(s,-\widetilde{d})\,|\widetilde{\mu}_1| 
(s,-\widetilde{d})\right\} }\\
& \geq & k_m\left\{ \mbox{ess }V^L_0\left[\widetilde{v}(\cdot,\widetilde{d})\right] +
\mbox{ess }V^L_0\left[\widetilde{v}(\cdot,-\widetilde{d})\right]\right\} \\
& \geq & k_m(\beta -\alpha),
\end{eqnarray*}
where ess $V^L_0\left[\widetilde{v}(\cdot,\widetilde{d})\right]$ stands for the
essential variation of $\widetilde{v}(\cdot,\widetilde{d})$ on $[0,L]$. 
Note that since $\tilde{v}\in BV(\tilde{N}_{\delta},\{\alpha,\beta\})$, the function
 $$
\tilde{d}\longrightarrow\;\mbox{ess}\;V^L_0 [\tilde{v}(\cdot,\tilde{d})] = 
\left |\frac{\partial \tilde{v}(\cdot,\tilde{d})}{\partial s} \right | (0,L)
$$
is integrable in $(\delta/2,\delta)$. See \cite{Z,EG}.
Then by integrating over $(\delta/2,\delta)\backslash\Delta$, 
\begin{eqnarray*}
I_1 & \geq & k_m\int^\delta_{\delta/2}\left\{ \mbox{ess }V^L_0\left[\widetilde{v}
(\cdot,\widetilde{d})\right]+\mbox{ess }V^L_0\left[\widetilde{v}(\cdot,-\widetilde{d})
\right]\right\} \\
& \geq & (\delta/4)k_m(\beta -\alpha).
\end{eqnarray*}

In order to obtain a lower estimate for $I_2$ we begin by remarking that since
\[ |(0,\delta/2)\backslash\Delta| \geq \delta/4 \]
and\, $\widetilde{v}$\, is an admissible function then there is 
$\overline{d}\in (0,\delta/2)\backslash\Delta$ such that $(s,\overline{d})$ and 
$(s,-\overline{d})$
are points of approximate continuity of $\widetilde{v}(s,\widetilde{d})$, for a.e. 
$s\in[0,L]$. Also by the definition of $\widetilde{v}_0$,
\[
 |\widetilde{v}(s,\widetilde{d})-\widetilde{v}(s,-\widetilde{d})| \geq (\beta-\alpha) 
- \left\{|\widetilde{v}_0-\widetilde{v}|(s,\widetilde{d}) + |\widetilde{v}_0-
\widetilde{v}|(s,-\widetilde{d})\right\} 
\]
for any $(s,\widetilde{d})\in\widetilde{N}_\delta$ such that $\widetilde{v}$ is 
approximately continuous at $(s,\widetilde{d})$. Hence
\begin{eqnarray*}
I_2 & = & \int_{\widetilde{N}_{\delta/2}}\Lambda\,
          \left |\frac{\partial\widetilde{v}}{\partial d}\right | \geq \nu
      \int^L_0\int^{\delta/2}_{-\delta/2}\left |\frac{\partial\widetilde{v}}{\partial d}
          \right | \\
    & = & \nu\int^L_0 \mbox{ess }V^{\delta/2}_{-\delta/2}\left[\widetilde{v}(s,\cdot)
           \right] ds \geq 
       \nu\int^L_0 |\widetilde{v}(s,\overline{d}) - \widetilde{v}(s,-\overline{d})|ds \\
 &\geq & \nu\int^L_0 \left\{ (\beta-\alpha)-\left[ |\widetilde{v}-
       \widetilde{v}_0|(s,\overline{d}) + 
          |\widetilde{v}-\widetilde{v}_0|(s,-\overline{d})\right]\right\} ds \\
 &\geq  & \left\{ \nu\,L(\beta-\alpha)-\frac{4\,\rho\,\nu}{\delta\,J_m(-\delta,\delta)}
          \right\} \,,
\end{eqnarray*}
where
\[ J_m(-\delta,\delta) = \min_{-\delta\leq d\leq\delta}\left\{ \min_{0\leq s\leq L}\, 
J_\Sigma(s,d)\,,\,\min_{0\leq s\leq L}\, J_\Sigma(s,-d)\right\}\,. \]
These estimates finally yield
\begin{eqnarray*}
E_0(v) \geq I_1+I_2 & \geq & \left[ \frac{\delta}{4}\,k_m(\beta-\alpha) +
\nu\,L(\beta-\alpha) - \frac{4\,\rho\,\nu}{\delta\,J_m(-\delta,\delta)}\right] \\
                    &  >   & \nu(\beta-\alpha)L = E_0(v_0)\,, 
\end{eqnarray*}
as long as we take
\begin{equation}
\rho < \min\left\{ \frac{k_m(\beta-\alpha)\delta^2J_m(-\delta,\delta)}{16\nu}\,,
\,{\delta\,L(\beta-\alpha)\over 2}\right\}
\end{equation}

Now our claim follows by extending $v_0$ to be constant on each connected
component of $\Omega\backslash\gamma$ and observing that 
$ |Dv_0|\left( \Omega\backslash\gamma\right)=0$.   \hfil$\Box$


The next task is to generalize Theorem 3.1 to the case of $p$ limiting 
phase-transition curves.
Recall the notation set forth in Section 2 where
$N_{\delta,i}$ is a $\delta$-tubular neighbourhood around each limiting
phase-transition curve $\gamma_i$, $i=1,2,\ldots,p$.

\begin{coro}
Let $\gamma_i$, $i=1,\ldots,p$, $\Omega_i$, $k$ be as in Section 2.
If (H1) is satisfied, then the function $v_0$ is a $L^1$--local minimizer of $E_0$.
\end{coro}

\paragraph{ Proof:}
For any $v\in BV(\Omega ,\{\alpha,\beta\})$, $||v - v_0||_{L^1(\Omega)}<
\rho$, where $\rho$ satisfies (4.6) with $\nu = \max\{\nu_i, i=1,...,p\}$, let
\[ \Omega_\alpha \stackrel{\rm def}{=}\{ x\in \Omega:v(x)=\alpha\}\,, \quad 
\Omega_\beta \stackrel{\rm def}{=}\{ x\in\Omega:v(x)=\beta\}
\] 
and consider the sets $ \Omega^0_\alpha \;\mbox{and}\; \Omega^0_\beta$ defined by (2.2).

Then the set $(\Omega_\alpha \cup \Omega_\beta )\cap \Omega$
has finite perimeter. By applying Theorem 4.3 to each $\gamma_i$, $i=1,\ldots,p$ we 
obtain, using the coarea formula
\begin{eqnarray*}
E_0(v) &=& \int_\Omega k |Dv|  =  
(\beta-\alpha)\,\int_{\partial_*\Omega_\alpha\cap\partial_*\Omega_\beta\cap\Omega} 
k\,d{\cal H}^1 \\
                          & \geq & 
(\beta-\alpha)\,\sum^p_{i=1}\int_{\partial_*\Omega_\alpha\cap\partial_*\Omega_\beta\
cap N_{\delta,i}}k\,d\,{\cal H}^1 \\
                        &  >  &
(\beta-\alpha)\sum^p_{i=1}\int_{\partial_*\Omega^0_\alpha\cap\partial_*\Omega^0_\alpha
\cap N_{\delta,i}}k\,d\,{\cal H}^1 \\
                        &   = & 
(\beta-\alpha)\int_{\bigcup^{p}_{i=1}\!\gamma_i}k\,d\,{\cal H}^1 
                           =  
\int_\Omega k\,|Dv_0| = E_0(v_0)\,.
\end{eqnarray*}

\section{Proof of Theorem 2.1}
\setcounter{equation}{0}

In this section we show how the previous sequence of lemmas can be used to
accomplish the proof of Theorem 2.1.

In view of Corollary 4.1 and previous remarks, Theorem 4.1 yields a family
$\{v_\varepsilon\}_{0<\varepsilon \leq\varepsilon_0}$ of $L^1$--local minimizers 
of $E_\varepsilon$, where $E_\varepsilon$ is given by 4.2.
 Clearly any such minimizer $v_\varepsilon$ is a weak solution ($H^1$-sense) of
(4.1) and regularity theory implies $v_\varepsilon \in C^{2,\sigma}(\overline{\Omega})$,
$0<\sigma <1$.
An application of the maximum principle then gives 
$\alpha<v_\varepsilon(x)<\beta$, $\forall \, x\in\overline{\Omega}$ and (2.1.i) is 
proved.

As for (2.1.ii) it is obtained from Theorem 4.1. In order to prove (2.1.iii) we 
examine the potential function $ \widetilde{F}(v)=F(\alpha)-F^0(v)$.

Suppose by contradiction that there is a sequence 
$\varepsilon_j\,,\,\varepsilon_j\to 0$ as
$j\to\infty$, and $\tau >0$ such that
\[ \left|\Omega^\lambda_{\varepsilon_{\scriptstyle j}}\right| \geq \tau > 0 \,, \]
for a fixed $\lambda$ and $\forall \, j$. Since $\| v_{\varepsilon_{\scriptstyle j}} -
v_0\|_{L^1(\Omega)} \stackrel{j\to\infty}{\longrightarrow} 0$ it follows that 
$\{ \varepsilon_j\}$ has a subsequence, still denoted by $\{ \varepsilon_j\}$ such that
$ v_{\varepsilon_{\scriptstyle j}}\to v_0 , \ \ \ \mbox{a.e. in}\ \Omega$.
Thus, (4.3) allows us to invoke the Lebesgue convergence theorem to
conclude that
\[\lim_{j\to 0} \, \int_\Omega \widetilde{F}(v_{\varepsilon_{\scriptstyle j}})dx =0\,.
\]
But this is a contradiction to
\[ \int_\Omega \widetilde{F}(v_{\varepsilon_{\scriptstyle j}})dx  \geq \min \left\{
\widetilde{F}(\alpha+\lambda),\,\widetilde{F}(\beta+\lambda)\right\}\,
\left|\Omega^\lambda_
{\varepsilon_{\scriptstyle j}}\right| \geq \widetilde{F}(\alpha+\lambda)\,\tau > 0 
\,, \]
for a fixed $\lambda$ and $\forall \, j$. Note that $\widetilde{F}(\alpha+\lambda) = \widetilde{F}(\beta-\lambda)$.

It remains to prove \, 2.1.iv)\,, \, which concerns stability of the
family $\{ v_\varepsilon\}$, ${0<\varepsilon\leq\varepsilon_0}$.
From the fact that $v_\varepsilon$ is also a $H^1$-local minimizer of
$E_\varepsilon$, ${0<\varepsilon\leq\varepsilon_0}$, it follows that
\begin{equation}\langle E''_\varepsilon(v_\varepsilon)\psi\,,\,\psi\rangle_{H^1,H^*}\geq 0 \,, \ \ \
\forall \, \psi \in H^1(\Omega)
\end{equation}
where $H^*$ stands for the dual of $H^1(\Omega)$.

Consider now the linearized eigenvalue problem
\begin{eqnarray}
&\varepsilon^2\diver\left[ k_1(x)\nabla \psi\right] + 
k_2(x)f'_\varepsilon(x,v_\varepsilon)\psi =
\lambda\psi\,, \ \ x\in\Omega &\nonumber \\
&\frac{\partial\psi}{\partial n} = 0 \ \ \mbox{ on } \ \ \partial\Omega &
  \end{eqnarray}
where $\displaystyle f'_\varepsilon(x,v)=\frac{\partial f_\varepsilon(x,v)}{\partial v}$.

Denoting by $\{\lambda_n\}_n$, $n=1,2,\dots$, the sequence of eigenvalues of (5.2)
then, taking into account (5.1), and using the variational
characterization of the eigenvalues, we infer that $\lambda_n \leq 0$, 
$n=1,2,\ldots$. By well-known results from linearized stability and
semigroup theory we conclude that in the case that the first eigenvalue
$\lambda_1$ is negative then $v_\varepsilon$ is a strongly stable stationary
solution of (1.1).

Now if $\lambda_1=0$ a classical application of the
Krein-Rutman theorem gives that zero is a simple eigenvalue of (5.2). In
this case there is a local one dimensional critical manifold
$M_c(v_\varepsilon)$, tangent to $[\psi_1]$ (the eigenspace spanned by the
principal eigenfunction $\psi_1$\,), at $v_\varepsilon$, such that if $v_\varepsilon$ is
stable in $M_c(v_\varepsilon)$ then it is also stable in $H^1(\Omega)$.

For this matter we refer to Theorem 6.2.1 in \cite{H}, which proof can be adapted
to fit our case. But now the stability of $v_\varepsilon$ in $M_c(v_\varepsilon)$
follows from the fact that the semigroup $\{T(t)\}_{t\geq 0}$ generated by
(1.1) defines a gradient flow in $H^1(\Omega)$. To be more specific the
functionals $E_\varepsilon[v_\varepsilon(x,t)]$ defines a Lyapunov function and along
each solution $v_\varepsilon(x,t)$ it holds that 
\[ \frac{d}{dt}\,E_\varepsilon[v_\varepsilon(x,t)] \leq  0, \ \ \ t\geq 0 \;.\]
This concludes the proof of Theorem 2.1. \hfil$\Box$

\paragraph{Remark.}
Condition (2.1.iii) actually shows the multiple--layer profile of $v_\varepsilon$, 
for small $\varepsilon$, and it should hold that $v_\varepsilon\to v_0$, as 
$\varepsilon\to 0$,
uniformly on compact sets of $\Omega\backslash\cup^p_{i=1}\gamma_i$. This
should be done following ideas set forth in \cite{CC}.

\paragraph{Remark.}
The difficulties when trying to generalize our results to higher space dimensions are 
of technical nature and inherent to the proof of Theorem 4.3. For instance, in this case
one would no longer have a global parametrization of the limiting phase-transition 
hypersurface.  

\paragraph{Remark.}
Other types of patterns can be considered rather than just the case of nested
limiting phase--transition curves. For instance, let $\gamma_i$, $i=1,\dots,p$,
be smooth simple closed curves in $\Omega$ with $O_i$ 
denoting the open region enclosed by $\gamma_i$, and $\overline{O}_i$ its closure,
then $\cap_{i=1}^p \overline{O}_i$ is empty. Then Theorem 4.3 and Corollary
4.1 still hold with 
$$v_0 = \alpha\chi_{\cup_{i=1}^p O_i}+\beta\chi_{\Omega
\backslash\cup_{i=1}^p \overline{O}_i}\,.
$$ 
A suitable combination of the two patterns referred to above could also be considered.


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{\sc Arnaldo Simal do Nascimento}\\
Universidade Federal de S\~ao Carlos, D.M. \\
13565-905 - S\~ao Carlos, S.P.  Brazil \\
E-mail address: dasn@power.ufscar.br \\
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