A strong min-max property for level sets of Phase Transitions

TL;DR

This paper establishes a strong min-max principle for the Allen--Cahn energy related to functions near minimal hypersurfaces, drawing an analogy to classical minimal hypersurface theory.

Contribution

It introduces a novel min-max principle for functions with nodal sets near minimal hypersurfaces, extending White's results to the Allen--Cahn energy context.

Findings

Proves a strong min-max property for certain functions near minimal hypersurfaces.

Establishes an analogy between Allen--Cahn energy and minimal hypersurface theory.

Extends classical minimal surface results to phase transition models.

Abstract

We show that certain functions whose nodal sets lie near a fixed nondegenerate minimal hypersurface satisfy a strong min-max principle for the Allen--Cahn energy which is analogous to the strong min-max principle for non-degenerate minimal hypersurfaces first proved by Brian White.