Flat level sets of Allen-Cahn equation in half-space

Abstract

We prove a half-space Bernstein theorem for Allen-Cahn equation. More precisely, we show that every solution $u$ of the Allen-Cahn equation in the half-space $\overline{{\mathbb{R}}_{+}^n}:=\{(x_1,x_2,\cdots ,x_n)\in {\mathbb{R}}^n:x_1\ge 0\}$ with $|u|\le 1$, boundary value given by the restriction of a one-dimensional solution on $\{x_1=0\}$ and monotone condition ${\partial }_{x_n}u>0$ as well as limiting condition ${\lim }_{x_n\to \pm \infty }u(x^{\prime },x_n)=\pm 1$ must itself be one-dimensional, and the parallel flat level sets and $\{x_1=0\}$ intersect at the same fixed angle in $(0,\frac{\ddot{I}}{€}2]$.

13 pages, 2 figures