Simmetry for minimizers of the fractional Allen-Cahn equation

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

The fractional Allen-Cahn equation in a domain $\Omega\subset{\mathbb{R}}^n$ may be seen as the Euler-Lagrange equation of the energy functional $$ {\mathcal{E}}_\Omega(u):= c\,\iint_{Q_\Omega} \frac{|u(x)-u(y)|^2}{|x-y|^{n+s}}\,dx\,dy+\int_\Omega (1-u^2(x))^2\,dx,$$ where $c>0$ is an appropriate normalizing constant and $$ Q_\Omega:=\big(\Omega\times\Omega\big)\cup\big(\Omega\times\Omega^c\big)\cup\big(\Omega^c\times\Omega\big).$$ One says that $u:{\mathbb{R}}^n\to{\mathbb{R}}$ is a minimizer in $\Omega$ if ${\mathcal{E}}_\Omega(u)\leq {\mathcal{E}}_\Omega(u+\varphi)$ for any $\varphi\in C^\infty_0(\Omega)$. One also says that $u$ is a local minimizer when it is a minimizer in any domain $\Omega\subset{\mathbb{R}}^n$.

For which $n$ and $s$ can one conclude that a local minimizer $u$ is necessarily one-dimensional?

Till now, this is known up to dimension $n\leq7$, when $s\in (s_0(n),2)$, for a suitable $s_0(n)\in(0,1)$. The explicit value of $s_0(n)$ is not known in general. The case $s\in (s_0(n),1)$ is dealt with in Theorem 1.5 of [Dipierro-Serra-Valdinoci]. The case $s\in(1,2)$ is treated in Theorem 1.2 of [Savin, 2018] and $s=1$ in Theorem 1.2 of [Savin].

An example providing a negative answer to this question in dimensions $n\ge9$ when $s\in(1,2)$ should follow from the announced result 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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