Transition Layer for the Heterogeneous Allen-Cahn Equation

Abstract

We consider the equation $\e^{2}\Delta u=(u-a(x))(u^2-1)$ in $\Omega$, $\frac{\partial u}{\partial \nu} =0$ on $\partial \Omega$, where $\Omega$ is a smooth and bounded domain in $\R^n$, $\nu$ the outer unit normal to $\pa\Omega$, and $a$ a smooth function satisfying $-1<a(x)<1$ in $\ov{\Omega}$. We set $K$, $\Omega_+$ and $\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\nabla a \neq 0$ on $K$ and $a\ne 0$ on $\partial \Omega$, we show that there exists a sequence $\e_j \to 0$ such that the above equation has a solution $u_{\e_j}$ which converges uniformly to $\pm 1$ on the compact sets of $\O_{\pm}$ as $j \to + \infty$.