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

Sanghoon Lee, Taehun Lee

TL;DR

This work establishes a strong maximum-principle-type framework for Allen–Cahn solutions whose nodal sets are asymptotic to regular minimizing cones, leading to a global ordering principle: if one solution's positive phase contains another's, then the two solutions are ordered everywhere and their nodal sets are disjoint unless they coincide. The authors develop a maximum principle for the linearized operator on unbounded, possibly non-smooth domains and analyze asymptotics to a cone-driven 1D profile $U(x) = u_0(\mathrm{dist}(x, \mathcal{C}))$ with $u_0(z)=\tanh(z/\sqrt{2})$, establishing convergence to this profile at infinity. This yields uniqueness of the solution determined by its positive phase and broad compares with saddle-shaped and Hardt–Simon–leaf-type configurations, contributing to the broader understanding of the Allen–Cahn–minimal-hypersurface correspondence. The results generalize prior uniqueness theorems to settings where nodal sets are only asymptotic to cones and may possess singularities, with potential implications for foliations by 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.

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

TL;DR

This work establishes a strong maximum-principle-type framework for Allen–Cahn solutions whose nodal sets are asymptotic to regular minimizing cones, leading to a global ordering principle: if one solution's positive phase contains another's, then the two solutions are ordered everywhere and their nodal sets are disjoint unless they coincide. The authors develop a maximum principle for the linearized operator on unbounded, possibly non-smooth domains and analyze asymptotics to a cone-driven 1D profile with , establishing convergence to this profile at infinity. This yields uniqueness of the solution determined by its positive phase and broad compares with saddle-shaped and Hardt–Simon–leaf-type configurations, contributing to the broader understanding of the Allen–Cahn–minimal-hypersurface correspondence. The results generalize prior uniqueness theorems to settings where nodal sets are only asymptotic to cones and may possess singularities, with potential implications for foliations by 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.
Paper Structure (4 sections, 12 theorems, 57 equations)

This paper contains 4 sections, 12 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic behavior of solutions
  4. Proof of Theorem \ref{['thm:main-order']}

Key Result

Theorem 1.1

Let $\mathcal{C} \subset \mathbb{R}^n$ be a regular minimizing hypercone. Suppose that $u_1$ and $u_2$ are nonconstant solutions of eq:main whose nodal sets are asymptotic to $\mathcal{C}$ in the $C^1$ sense (see Definition def:asymptotic_C1). If $\{u_1 > 0\} \subseteq \{u_2 > 0\}$, then $u_1 \le u_

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2: Uniqueness of solutions
  • Definition 2.1: Saddle-shaped solution
  • Definition 2.2: Asymptotic to a cone in the $C^1$ sense
  • Lemma 2.3: Strong maximum principle
  • Lemma 2.4: Unique continuation principle
  • Proposition 2.5: Stability of positive solutions
  • proof
  • Proposition 2.6
  • proof
  • ...and 12 more