Catenoidal layers for the Allen-Cahn equation in bounded domains

TL;DR

This paper constructs solutions to the Allen-Cahn equation in three-dimensional bounded domains, where the solutions' level sets approximate minimal surfaces intersecting the boundary orthogonally, revealing geometric structures in phase transition models.

Contribution

It introduces a new family of solutions with level sets collapsing onto minimal surfaces that meet the boundary orthogonally, expanding understanding of phase transition interfaces in bounded domains.

Findings

Level sets collapse onto minimal surfaces with finite total curvature

Solutions exhibit orthogonal intersection with domain boundary

Explicit examples of applicable minimal surfaces provided

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, \quad \hbox{in }\Omega\subset \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.