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

TL;DR

This paper proves the uniqueness and stability of saddle-shaped solutions to the fractional Allen-Cahn equation in high dimensions, linking these solutions to stable nonlocal minimal surfaces and advancing understanding of the fractional De Giorgi conjecture.

Contribution

It establishes the uniqueness and stability of saddle-shaped solutions in dimensions $2m \, \geq \, 14$, and connects these solutions to stable nonlocal minimal surfaces, advancing the study of the fractional De Giorgi conjecture.

Findings

Uniqueness of saddle-shaped solutions in dimensions $2m \, \geq \, 14$

Stability of these solutions in dimensions $2m \, \geq \, 14$

Connection of solutions to stable nonlocal minimal surfaces in high dimensions

Abstract

In this paper we prove the uniqueness of the saddle-shaped solution 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 whenever $2m\geq 14$. As a consequence of this result and the connection of the problem with nonlocal minimal surfaces, we show that the Simons cone $\{(x', x'') \in \mathbb R^{m}\times \mathbb R^m \ : \ |x'| = |x''|\}$ is a stable nonlocal $(2\gamma)$-minimal surface in dimensions $2m\geq 14$. 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.