Some energy estimates for stable solutions to fractional Allen–Cahn equations

Abstract

In this paper we study stable solutions to the fractional equation 0.1$$\begin{aligned} (-\Delta )^s u =f(u), \quad |u| \max \{0, 1-2s\}$$. 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 to (0.1) are 1-D solutions. The scheme used in this paper is inspired by Cinti–Serra–Valdinoci [16] which deals with stable nonlocal sets, and Figalli–Serra [26] which studies stable solutions to (0.1) for the case $$s=1/2$$.