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

Mazzeo, Rafe; Saez, Mariel

doi:10.1090/S0002-9939-2014-11986-1

Multiple-layer solutions to the Allen-Cahn equation on hyperbolic space
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by Rafe Mazzeo and Mariel Saez PDF

Proc. Amer. Math. Soc. 142 (2014), 2859-2869 Request permission

Abstract:

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

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