One-dimensional symmetry for the fractional Allen-Chan equation

Problem's Description

Given $s\in(0,2)$, one takes into account the fractional Allen-Cahn equation $(-\Delta)^{s/2} u=u-u^3$. This equation, at a large scale, is related to minimizers of the classical perimeter functional when $s\in[1,2)$ and to minimizers of the fractional $s$-perimeter functional when $s\in(0,1)$, see [Savin-Valdinoci, 2012].

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1. Open Simmetry for monotone solutions of the fractional Allen-Cahn equation

2. Open Simmetry for minimizers of the fractional Allen-Cahn equation

3. Open Simmetry for monotone solutions of the fractional Allen-Cahn equation under a limit equation