Nodal set comparison for Allen--Cahn solutions with conical asymptotics

TL;DR

This paper proves a comparison principle for solutions of the Allen--Cahn equation with nodal sets asymptotic to hypercones, showing how phase inclusion determines solutions and their nodal set arrangements.

Contribution

It introduces a maximum principle for the linearized operator on unbounded, possibly singular domains, extending the understanding of phase ordering in Allen--Cahn solutions.

Findings

Positive phase inclusion enforces a global ordering of solutions

The positive phase uniquely determines the solution

Strict phase inclusion implies disjoint nodal sets

Abstract

We establish a comparison principle for entire solutions of the Allen--Cahn equation whose nodal sets, possibly singular, are asymptotic to a regular minimizing hypercone. We show that inclusion of the positive phases enforces a global ordering of the solutions. As a consequence, the positive phase uniquely determines the solution, and strict phase inclusion implies that the corresponding nodal sets are disjoint. Our analysis relies on a maximum principle for the linearized operator on unbounded domains that are not necessarily smooth, and yields an Allen--Cahn analogue of the strong maximum principle for minimal hypersurfaces.