# Some Energy Estimates for Stable Solutions to Fractional Allen-Cahn Equations

## TL;DR

This paper derives energy estimates for stable solutions of fractional Allen-Cahn equations, proving that in two dimensions, such solutions are effectively one-dimensional, with methods inspired by recent nonlocal set and stable solution studies.

## Contribution

It provides sharp energy estimates for solutions with fractional order s<1/2 and rough estimates for s≥1/2, offering a new proof of one-dimensionality in 2D.

## Key findings

- Sharp energy estimates for 0<s<1/2
- Rough energy estimates for 1/2≤s<1
- Stable solutions in 2D are 1-D solutions

## Abstract

In this paper we study stable solutions to the fractional equation \begin{align} (-\Delta)^s u =f(u), \quad |u| < 1 \quad \mbox{in $\mathbb{R}^d$}, \end{align}where $0<s<1$ and $f:[-1,1] \rightarrow \mathbb{R}$ is a $C^{1,\alpha}$ function for $\alpha>\max\{0, 1-2s\}$. We obtain sharp energy estimates for $0<s<1/2$ and rough energy estimates for $1/2 \le s <1$. These lead to a different proof from literature of the fact that when $d=2, \, 0<s<1$, entire stable solutions are $1$-D solutions. The scheme used in this paper is inspired by Cinti-Serra-Valdinoci[CSV17] which deals with stable nonlocal sets, and Figalli-Serra[FS17] which studies stable solutions for the case $s=1/2$.