Transition Layer for the Heterogeneous Allen-Cahn Equation

Abstract

We consider the equation $\e^{2}Δu=(u-a(x))(u^2-1)$ in $\Omega$, $\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega$, where $\Omega$ is a smooth and bounded domain in ${\mathbb{R}}^n$, $\nu$ the outer unit normal to $\paΩ$, and $a$ a smooth function satisfying $-1<a(x)<1$ in $\ovΩ$. 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 $\e_j \to 0$ such that the above equation has a solution $u_{\e_j}$ which converges uniformly to $\pm 1$ on the compact sets of ${Ø}_{\pm }$ as $j\to +\infty$.

25 pages

Cited by in corpus (1)

• Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain