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

Sanghoon Lee; Taehun Lee

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arXiv · 2601.05015

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

Sanghoon Lee·Taehun Lee

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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.

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Sanghoon Lee, Taehun Lee. 2026-01-08. Nodal set comparison for Allen--Cahn solutions with conical asymptotics. https://arxiv.org/abs/2601.05015

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