Uniqueness and stability of the saddle-shaped solution to the fractional Allen–Cahn equation

Abstract

In this paper we prove the uniqueness of the saddle-shaped solution $u:{\mathbb{R}}^{2m}\to \mathbb{R}$ to the semilinear nonlocal elliptic equation $(-\Delta {)}^{\gamma }u=f(u)$ in ${\mathbb{R}}^{2m}$, where $\gamma \in (0,1)$ and $f$ is of Allen–Cahn type. Moreover, we prove that this solution is stable if $2m\ge 14$. As a consequence of this result and the connection of the problem with nonlocal minimal surfaces, we show that the Simons cone $\{(x^{\prime },x^{\prime \prime })\in {\mathbb{R}}^m\times {\mathbb{R}}^m:|x^{\prime }|=|x^{\prime \prime }|\}$ is a stable nonlocal $(2\gamma )$-minimal surface in dimensions $2m\ge 14$.

Saddle-shaped solutions of the fractional Allen–Cahn equation are doubly radial, odd with respect to the Simons cone, and vanish only in this set. It was known that these solutions exist in all even dimensions and are unstable in dimensions 2, 4, and 6. Thus, after our result, the stability remains an open problem only in dimensions 8, 10, and 12.

The importance of studying this type of solution is due to its relation with the fractional version of a conjecture by De Giorgi. Saddle-shaped solutions are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.