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


Authors: Rafe Mazzeo and Mariel Saez
Journal: Proc. Amer. Math. Soc. 142 (2014), 2859-2869
MSC (2010): Primary 53A10, 35J61
DOI: https://doi.org/10.1090/S0002-9939-2014-11986-1
Published electronically: April 15, 2014
MathSciNet review: 3209339
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Abstract: In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn equation in hyperbolic space:

$\displaystyle -\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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Additional Information

Rafe Mazzeo
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: mazzeo@math.stanford.edu

Mariel Saez
Affiliation: Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago, Chile
Email: mariel@mat.puc.cl

DOI: https://doi.org/10.1090/S0002-9939-2014-11986-1
Received by editor(s): January 30, 2012
Received by editor(s) in revised form: August 9, 2012
Published electronically: April 15, 2014
Additional Notes: The first author was supported by the NSF Grant DMS-1105050
The second author was supported by CONYCIT under grants Fondecyt de Iniciación 11070025, Fondecyt regular 1110048 and proyecto Anillo ACT-125, CAPDE
Communicated by: Michael Wolf
Article copyright: © Copyright 2014 American Mathematical Society