Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

TL;DR

This paper proves the existence of stable solutions to the Allen-Cahn equation in dimensions 8 and higher, with zero sets asymptotic to minimal cones, extending understanding of phase transition models in high dimensions.

Contribution

It demonstrates the existence of stable solutions with non-hyperplane zero sets in dimensions ≥8, linked to minimal cones, which was previously unknown.

Findings

Stable solutions exist in dimension 8 and higher.

Zero sets are asymptotic to minimal cones, not hyperplanes.

Extends the class of known stable solutions in high-dimensional Allen-Cahn equations.

Abstract

In this paper, we are interested in bounded, entire, solutions of the Allen-Cahn equation which are defined in Euclidean space and whose zero set is asymptotic to a given minimal cone. In particular, in dimension larger than or equal to 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.