Multiple-layer solutions to the Allen-Cahn equation on hyperbolic space

2012/01/30 by Rafe Mazzeo, Mazzeo, Rafe, MarielSaez · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1201.6170

openalex publication_date 2012/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn equation in hyperbolic space: -Δ_{\mathbb H} u+F'(u)=0; here F is a nonnegative double-well potential with nondegenerate minima. We prove that for any collection of widely separated, non-intersecting hyperplanes in \mathbb H, there is a solution to this equation which has nodal set very close to this collection of hyperplanes. Unlike the corresponding problem in \RRⁿ, there are no constraints beyond the separation parameter.

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