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arXiv · 1901.07671

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

Francois Hamel·Yong Liu·Pieralberto Sicbaldi·Kelei Wang·Juncheng Wei
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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.

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Francois Hamel, Yong Liu, Pieralberto Sicbaldi, Kelei Wang, Juncheng Wei. 2019-07-27. Half space theorem for the Allen-Cahn equation and related problems. https://arxiv.org/abs/1901.07671

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