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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 101, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2003 Texas State University-San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2003/101\hfil Variational characterization]
{Variational characterization of interior interfaces in phase
transition models on convex plane domains}
\author[C. E. Garza-Hume \& P. Padilla\hfil EJDE--2003/101\hfilneg]
{Clara E. Garza-Hume \& Pablo Padilla} % in alphabetical order
\address{Clara E. Garza-Hume \hfill\break
Department of Applied Mathematics, UNAM, Mexico City, Mexico }
\email{clara@mym.iimas.unam.mx}
\address{Pablo Padilla \hfill\break
Department of Applied Mathematics, UNAM, Mexico City, Mexico }
\email{pablo@mym.iimas.unam.mx}
\date{}
\thanks{Submitted July 15, 2003. Published October 2, 2003.}
\subjclass[2000]{49Q20, 35J60, 82B26}
\keywords{Phase transition, singularly perturbed Allen-Cahn equation,
\hfill\break\indent
convex plane domain, variational methods, transition layer, Gauss map,
geodesic, varifold}
\begin{abstract}
We consider the singularly perturbed Allen-Cahn equation on a
strictly convex plane domain. We show that when the perturbation
parameter tends to zero there are solutions having a transition
layer that tends to a straight line segment. This segment
can be characterized as the shortest path intersecting the
boundary orthogonally at two points.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[theorem]
\section{Introduction}
We consider the equation
\begin{equation} \label{1}
\begin{gathered}
-\epsilon^2 \Delta u + W' (u)=0 \quad\mbox{in }\Omega \\
\frac{\partial u}{\partial \nu} =0 \quad \mbox{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a strictly convex subset of $\mathbb{R}^2$ with $C^1$
boundary and $W$ is a double-well potential.
In the case $W=(1-u^2)^2$ this corresponds to
the scalar steady state Ginzburg-Landau equation. It arises in
phase transition models, super conductivity, material science, etc. (see
\cite{HT} for more references).
Finding solutions of (\ref{1}) is equivalent to finding critical points
of the functional
\begin{equation}
E_\epsilon(u)=\int_{\Omega} \big(\frac{\epsilon}{2}
|\nabla u|^2+\frac{1}{\epsilon} W(u)\big)\, \label{2}dS,
\end{equation}
in a suitable function space.
This problem, with and without volume constraint, has been studied by
Alikakos, Bates, Chen, Fusco, Kowalczyk, Modica, Sternberg and
Wei among many other authors (see \cite{HT}). Of particular interest is the
characterization of solutions when $\epsilon$ tends to zero. In
this situation, nontrivial solutions typically exhibit transition
layers, which in the case where there is no volume constraint are
expected to be staight lines.
Indeed, it is well known that the value of the Lagrange multiplier corresponds
to the curvature of the interface (see for instance \cite{HT}.)
In a recent paper, Kowalczyk (\cite{K})
has made these assertions precise by
applying the Implicit Function Theorem to construct special solutions.
It is the purpose of this paper to show that similar solutions can be
found using variational techniques. Moreover, the variational
characterization provides a natural way of describing the transition
along the lines of $\Gamma$-convergence methods.
We will find minima, $u_\epsilon$, of this functional subject to a
constraint which is
different from the standard volume constraint and was
motivated by a theorem by Poincar\'e and the results of \cite{GP}.
Problem (\ref{1}) was considered in \cite{GP} for the case when $\Omega$
is an oval surface embedded in $\mathbb{R}^3$ and $\Delta u$ represents
the Lapace-Beltrami operator with respect to the metric inherited from
$\mathbb{R}^3$.
In that paper it is shown that the constraint
\begin{equation}
G(u)=\int_{S^2} u(g^{-1}(y))\, dy=0 \label{G}
\end{equation}
for $\epsilon$ small and $g$ the Gauss map $g:\Omega \to S^2$
can be used to obtain nontrivial solutions whose interface tends to
a minimal closed geodesic. Roughly speaking the idea is the following.
Restriction (\ref{G}) for functions having uniformly bounded energy as
$\epsilon\to 0$ has a natural geometric interpretation. Namely, since
such functions are necessarily close to $\pm 1$ except for a small
set (the transition), then the restriction guarantees that this
transition divides, under the Gauss map, the unit sphere into two
components of equal area. Assuming that the transition takes place on a
regular curve $\gamma=u^{-1}(0)$ and using a result stated by Poincar\'e
in \cite{P} and proved by Berger and Bombieri in \cite{BB}
it is natural to expect that minimal solutions concentrate on minimal
closed geodesics, which is the content of the main result in \cite{GP}.
\begin{theorem}[Berger and Bombieri \cite{BB}] \label{thm1}
Let $\Gamma$ be the class of smooth curves $\gamma$ on an oval surface
such that under the Gauss map, $g(\gamma)$ divides $S^2$ into two
components of equal area. The curve $\gamma^*$ which minimizes the arc
length among curves in $\Gamma$ is a minimal closed geodesic.
\end{theorem}
Recalling that the energy $E_\epsilon$ as $\epsilon\to 0$ is proportional
to the length of the transition (see \cite{To} for details) we see that
minimizing $E_\epsilon$ subject to the constraint (\ref{G}) for
$\epsilon$ small is equivalent to the geometric problem of minimizing
arc length in $\Gamma$ and that indeed the interface should be a minimal
closed geodesic. We refer to \cite{GP} for precise statements of these
facts.
The case of a planar convex domain can be naturally considered as the
limit of an oval surface that is gradually flattened in one direction.
We prove that up to a subsequence the solutions $u_\epsilon$ have
an interface that converges when $\epsilon\to 0$ to the shortest
straight line that intersects the boundary orthogonally. This line
would be the limit of the shortest closed geodesic on the surface.
In fact, this analogy had been used in the context of dynamical
systems to study the flow of billiards on convex plane
domains as the limit of the geodesic flow in oval surfaces (see \cite{Mo}).
We point out that this line has also a minimax characterization.
It is shown in \cite{KP} that the solution obtained via the
Mountain Pass Theorem has a transition along this segment (in the
$\epsilon\to 0$ limit).
The rest of the paper is organized as follows: in section 2 we
recall some well-known facts, introduce notation and
state the results on the convergence of solutions as $\epsilon$
tends to zero that we need.
The setting is very similar to that of \cite{GP} but we include
the essential parts for the convenience of the reader.
In section 3 we present the proof.
\section{Setting}
We make the following assumptions:
\begin{itemize}
\item[(A1)] The function $W:\mathbb{R}\to[0,\infty)$ is $\mathcal{C}^3$ and
$W(\pm 1)=0$. For some
$\gamma\in(-1,1)$, $W'<0$ on $(\gamma,1)$ and $W'>0$ on $(-1,\gamma)$. For some
$\alpha\in(0,1)$ and
$\kappa>0$, $W''(x)\geq\kappa$ for all $|x|\geq\alpha$.
\item[(A2)] There exist constants $20$ such that
\begin{gather*}
c|x|^k\le W(x)\le c^{-1} |x|^k \\
c|x|^{k-1}\le |W^{'}(x)|\le c^{-1}|x|^{k-1}
\end{gather*}
for sufficiently large $|x|$.
\item[(B1)]
The subset $U\subset \mathbb{R}^n$ is bounded, open and has Lipschitz boundary
$\partial U$.
A sequence of functions $\{u^i\}^\infty_{i=1}$ in $C^3(U)$ satisfies
\begin{equation}
\epsilon_i \Delta u^i = \epsilon_i^{-1} W^{'} (u^i)-\lambda_i
\end{equation}
on $U$. Here, $\lim_{i\to \infty} \epsilon_i =0$, and we assume there
exist $c_0$, $\lambda_0$ and $E_0$ such that $\sup_U |u^i|\le c_0$,
$|\lambda_i|\le \lambda_0$
and for all $i$,
$$
\int_U \frac{\epsilon_i |\nabla u^i|^2}{2} + \frac{ W(u^i)}{\epsilon_i} \le
E_0\,.
$$
\end{itemize}
Now, we recall some formalism from Geomtric Measure Theory that will be
used. As in \cite{HT}
let
$$\phi(s)=\int_0^s \sqrt{W(s)/2}\, ds$$
and define new functions
$$w^i=\phi\circ u^i$$
for each $i$
and
we associate to each function $w^i$ a varifold $V^i$ (\cite{Fe,Si})
defined as
$$
V^i(A)=\int_{-\infty}^\infty v(\{w^i=t\})(A)\, dt
$$
for each Borel set $A\subset G_{n-1}(U)$,
$G_{n-1}(U)=U\times G(n,n-1)$, where $G(n,n-1)$ is the Grassman manifold of
unoriented $(n-1)$-dimensional planes in $\mathbb{R}^n$.
By the compactness theorem for BV functions, there exists an a.e. pointwise
limit $w^\infty$. Let $\phi^{-1}$ be the inverse of $\phi$ and define
$$
u^\infty=\phi^{-1}(w^{\infty}).
$$
$u^{\infty}=\pm 1$ a.e. on $U$ and the sets $\{u^{\infty} =\pm 1\}$ have finite
perimeter in $U$.
The following theorem is proved in \cite{HT}.
\begin{theorem} \label{thm2}
Let $V^i$ be the varifold associated with $u^i$ (via $w^i$). On passing to a
subsequence we can assume
$$
\lambda_i \to \lambda_\infty, \quad u^i\to u^\infty\; a.e., \quad
V^i\to V \mbox{ in the varifold sense.}
$$
Moreover,
\begin{enumerate}
\item For each $\phi\in C_c(U)$,
\[
\|V\|(\phi)=\lim_{i\to\infty} \int \phi \frac{\epsilon_i |\nabla u^i|^2}{2}=
\lim_{i\to\infty} \int \phi \frac{W(u^i)}{\epsilon_i}
= \lim_{i\to\infty} \int \phi |\nabla w^i|.
\]
\item $\mathop{\rm supp}\|\partial\{u^\infty =1\}\|\subset
\mathop{\rm supp} \|V\|$,
and $\{u^i\}$ converges locally uniformly to $\pm 1$ in $U\setminus
\mathop{\rm supp}\|V\|$, where $\partial$ denotes the reduced boundary.
\item For each $\tilde{U}\Subset U$ and $0