Some energy estimates for stable solutions to fractional Allen–Cahn equations
Some energy estimates for stable solutions to fractional Allen–Cahn equations
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DOI：
10.1007/s00526-020-1701-2
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发表时间：
2019-04
影响因子：
2.1
通讯作者：
C. Gui;Qinfeng Li
中科院分区：
文献类型：
--
作者：
C. Gui;Qinfeng Li
本文研究了分数阶方程$$\开始{aligned}（-\Delta）^s u =f（u），\quad的稳定解|u| < 1 \quad \hbox {in }\mathbb {R}^d，\end{aligned}$$whereand是的函数。我们获得了的精确能量估计和的粗略能量估计。这导致了与文献不同的证明，即当，（0.1）的整个稳定解是1-D的解决方案。本文中使用的方案受到Cinti-Serra-Valdinoci [16]和Figalli-Serra [26]的启发，前者处理稳定的非局部集，后者研究了（0.1）的稳定解。
In this paper we study stable solutions to the fractional equation $$\begin{aligned} (-\Delta )^s u =f(u), \quad |u| < 1 \quad \hbox {in }\mathbb {R}^d, \end{aligned}$$whereandis afunction for. We obtain sharp energy estimates forand rough energy estimates for. These lead to a different proof from literature of the fact that when, 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.