Half space theorem for the Allen-Cahn equation and related problems

TL;DR

This paper proves that bounded solutions of the Allen-Cahn equation with level sets in a half-space are either one-dimensional or closely related to one-dimensional solutions, with results depending on the dimension.

Contribution

It establishes new rigidity results for solutions of the Allen-Cahn equation with half-space level sets, extending understanding of their structure in various dimensions.

Findings

For n ≤ 3, solutions are one-dimensional.

In n ≥ 4, solutions are either one-dimensional or asymptotically one-dimensional.

Generalizations to free boundary problems are also provided.

Abstract

In this paper we obtain rigidity results for a bounded non-constant entire solution $u$ of the Allen-Cahn equation in $\mathbb{R}^n$, whose level set $\{u=0\}$ is contained in a half-space. If $n\leq 3$ we prove that the solution must be one-dimensional. In dimension $n\geq 4$, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.