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

TL;DR

This paper demonstrates the existence of multi-layer solutions to the Allen-Cahn equation in hyperbolic space, with solutions closely aligned to given hyperplanes, expanding understanding beyond Euclidean cases.

Contribution

It establishes the existence of multi-layer solutions in hyperbolic space for arbitrary separated hyperplanes, without additional constraints, unlike in Euclidean space.

Findings

Solutions exist near any collection of separated hyperplanes in hyperbolic space.

No constraints are needed beyond the separation of hyperplanes.

The solutions' nodal sets closely follow the hyperplanes.

Abstract

In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn equation in hyperbolic space: \[ -\Delta_{\mathbb H} u+F'(u)=0; \] 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 $\RR^n$, there are no constraints beyond the separation parameter.