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

TL;DR

This paper proves a Bernstein-type theorem for solutions of the Allen-Cahn equation in a half-space, showing that under certain boundary and monotonicity conditions, solutions are necessarily one-dimensional with flat level sets intersecting the boundary at a fixed angle.

Contribution

It establishes a half-space Bernstein theorem for Allen-Cahn solutions, characterizing their flatness and geometric properties under boundary and monotonicity assumptions.

Findings

Solutions are one-dimensional under given conditions.

Level sets are flat and intersect the boundary at a fixed angle.

Solutions exhibit monotonicity and boundary behavior consistent with one-dimensional profiles.

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 $\overline{\mathbb{R}^n_+}:=\{(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n:\,x_1\geq 0\}$ with $|u|\leq 1$, 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 $\lim_{x_n\to\pm\infty}u(x',x_n)=\pm 1$ must itself be one-dimensional, and the parallel flat level sets and $\{x_1=0\}$ intersect at the same fixed angle in $(0, \frac{\pi}{2}]$.