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

TL;DR

This paper proves the uniqueness of saddle-shaped solutions to the Allen-Cahn equation in all even dimensions and establishes their stability in dimensions 14 and higher, advancing understanding of their mathematical properties.

Contribution

It demonstrates the uniqueness of saddle-shaped solutions in all even dimensions and proves their stability for dimensions 14 and above, extending prior results.

Findings

Uniqueness of saddle-shaped solutions in all even dimensions.

Stability of these solutions for dimensions ≥14.

Existence and properties of saddle-shaped solutions across dimensions.

Abstract

We establish the uniqueness of a saddle-shaped solution to the diffusion equation $-\Delta u = f(u)$ in all of $\mathbb{R}^{2m}$, where $f$ is of bistable type, in every even dimension $2m \geq 2$. In addition, we prove its stability whenever $2m \geq 14$. Saddle-shaped solutions are odd with respect to the Simons cone ${\mathcal C} = \{(x^1,x^2) \in \mathbb{R}^m \times \mathbb{R}^m : |x^1|=|x^2| \}$ and exist in all even dimensions. Their uniqueness was only known when $2m=2$. On the other hand, they are known to be unstable in dimensions 2, 4, and 6. Their stability in dimensions 8, 10, and 12 remains an open question. In addition, since the Simons cone minimizes area when $2m \geq 8$, saddle-shaped solutions are expected to be global minimizers when $2m \geq 8$, or at least in higher dimensions. This is a property stronger than stability which is not yet established in any dimension.