\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{amssymb} \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