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

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 $s=1/2$.