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

## TL;DR

This paper proves a symmetry result for solutions of a nonlocal phase transition equation in three dimensions, extending previous results by removing certain limit assumptions, and contributing to the nonlocal De Giorgi conjecture.

## Contribution

It establishes a three-dimensional symmetry result for the nonlocal Allen-Cahn equation without requiring limit conditions at infinity.

## Key findings

- Solutions are one-dimensional under the given conditions.
- Extends nonlocal De Giorgi conjecture results to broader settings.
- Provides new techniques for analyzing nonlocal phase transition equations.

## Abstract

We consider bounded solutions of the nonlocal Allen-Cahn equation $$ (-\Delta)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ under the monotonicity condition $\partial_{x_3}u>0$ and in the genuinely nonlocal regime in which~$s\in\left(0,\frac12\right)$. 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. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by Ennio De Giorgi.