Simmetry for monotone solutions of the fractional Allen-Cahn equation under a limit equation
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Posted online: 2018-07-03 15:55:01Z by Enrico Valdinoci80
Cite as: P-180703.3
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 $u \in C^\infty({\mathbb{R}}^n,[-1,1])$ be a solution of $(-\Delta)^{s/2}u=u-u^3$ in ${\mathbb{R}}^n$, with $\partial_{x_n}u>0$ and $$ \lim_{x_n\to\pm\infty} u(x',x_n)=\pm1,$$ for any $x'\in {\mathbb{R}}^{n-1}$. For which dimension $n$ and fractional exponent $s\in(0,2)$ can one conclude that $u$ is necessarily one-dimensional?
Till now, a positive answer is known up to dimension $n\leq8$, when $s\in(s_0(n),2)$, for a suitable $s_0(n)\in(0,1)$, whose explicit value is not known in general. The case $s\in(s_0(n),1)$ is dealt with in Theorem 1.6 of [Dipierro-Serra-Valdinoci]. The case $s\in(1,2)$ is treated in Theorem 1.3 of [Savin, 2018] and $s=1$ in Theorem 1.3 of [Savin].
An example providing a negative answer to this question in dimensions $n\geq9$ 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.
The cases $n\leq8$ with $s\in (0,s_0(n)]$ and $n\geq9$ with $s\in(0,1)$ (and also $s=1$ at the moment) remain open.
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Created at: 2018-07-03 15:55:01Z
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