A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime

摘要

We consider bounded solutions of the nonlocal Allen-Cahn equation $$(-\Delta)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ under the monotonicitycondition $\partial_{x_3}u>0$ and in the genuinely nonlocal regime inwhich~$s\in\left(0,\frac12\right)$. Under the limit assumptions $$\lim_{x_n\to-\infty} u(x',x_n)=-1\quad{\mbox{ and }}\quad \lim_{x_n\to+\infty}u(x',x_n)=1,$$ it has been recently shown that~$u$ is necessarily $1$D, i.e. itdepends only on one Euclidean variable. The goal of this paper is to obtain asimilar result without assuming such limit conditions. This type of results canbe seen as nonlocal counterparts of the celebrated conjecture formulated byEnnio De Giorgi.