Catenoidal layers for the Allen-Cahn equation in bounded domains

Abstract

In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation ${\alpha }^2\Delta u+u(1-u^2)=0,in\; \Omega \subset {\mathbb{R}}^N$ where $N=3$, $\Omega$ is a smooth bounded domain and $\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as $\alpha \to 0$, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that intersects orthogonally $\partial \Omega$ of the domain and that is non-degenerate respect to $\Omega$. We provide explicit examples of surfaces to which our result applies.

30 pages