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arXiv · 1904.07443

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

Changfeng Gui·Qinfeng Li
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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 $0 \max\{0, 1-2s\}$. 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 $s=1/2$.

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Changfeng Gui, Qinfeng Li. 2019-04-16. Some Energy Estimates for Stable Solutions to Fractional Allen-Cahn Equations. https://arxiv.org/abs/1904.07443

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