Transition layer for the heterogeneous Allen–Cahn equation

Abstract

We consider the equation

where $\Omega$ is a smooth and bounded domain in ${\mathbb{R}}^n$, $\nu$ the outer unit normal to $\partial \Omega$, and $a$ a smooth function satisfying $-1<a(x)<1$ in $\overline{\Omega }$. We set $K$, ${\Omega }_{+}$ and ${\Omega }_{-}$ to be respectively the zero-level set of $a$, $\{a>0\}$ and $\{a<0\}$. Assuming $\nabla a\ne 0$ on $K$ and $a\ne 0$ on $\partial \Omega$, we show that there exists a sequence ${\varepsilon }_j\to 0$ such that Eq. (1) has a solution $u_{{\varepsilon }_j}$ which converges uniformly to $\pm 1$ on the compact sets of ${\Omega }_{\pm }$ as $j\to +\infty$. This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103–131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal. 38 (5) (2007) 1542–1564] only for $n=2$.