Simmetry for monotone solutions of the fractional Allen-Cahn equation
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Posted online: 2018-07-03 15:55:01Z by Enrico Valdinoci95
Cite as: P-180703.1
General Description View the group
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].
Problem's Description
Let $s\in(0,2)$ and $u\in C^\infty({\mathbb{R}}^n,[-1,1])$ be a solution of $(-\Delta)^{s/2} u=u-u^3$ in ${\mathbb{R}}^n$, with $\partial_{x_n}u>0$. For which $n$ and $s$ can one conclude that $u$ is necessarily one-dimensional, i.e. that there exist a function $u_0:{\mathbb{R}}\to[-1,1]$ and a direction $\omega\in {\mathbb{S}}^{n-1}$ such that $u(x)=u_0(\omega\cdot x)$ for every $x\in {\mathbb{R}}^n$?
This is a fractional variation of a classical conjecture by Ennio De Giorgi when $s=2$, see [De Giorgi, 1979].
Up to now, it is known that this question has a positive answer in dimension $n\le 3$, and also in dimension $4$ when $s=1$.
See [Cabré-Solà-Morales, 2005] for the case $n=2$, $s=1$, [Sire-Valdinoci, 2009] and [Cabré-Sire, 2015] for $n=2$, every $s\in(0,1)$, [Cabré-Cinti, 2010] for $n=3$, $s=1$, [Cabré-Cinti, 2014] for $n=3$, $s\in(1,2)$, [Dipierro-Farina-Valdinoci, 2018] for $n=3$, $s\in(0,1)$ (also relying on [Dipierro-Serra-Valdinoci]) and [Figalli-Serra] for $n=4$, $s=1$.
An example providing a negative answer to this question in dimensions $n \geq 9$ when $s\in(1,2)$ has been announced after Theorem 1.3 in [Chan-Liu-Wei], to be addressed in a forthcoming article with J. Dávila and M. del Pino.
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Created at: 2018-07-03 15:55:01Z
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