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Moduli space theory for the Allen-Cahn equation in the plane


Authors: Manuel del Pino, Michał Kowalczyk and Frank Pacard
Journal: Trans. Amer. Math. Soc. 365 (2013), 721-766
MSC (2010): Primary 35B08, 35P99, 35Q80
DOI: https://doi.org/10.1090/S0002-9947-2012-05594-2
Published electronically: August 9, 2012
MathSciNet review: 2995371
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Abstract: In this paper we study entire solutions of the Allen-Cahn equation $ \Delta u-F^{\prime }(u)=0$, where $ F$ is an even, bistable function. We are particularly interested in the description of the moduli space of solutions which have some special structure at infinity. The solutions we are interested in have their zero set asymptotic to $ 2k$, $ k\geq 2$ oriented affine half-lines at infinity and, along each of these affine half-lines, the solutions are asymptotic to the one-dimensional heteroclinic solution: such solutions are called multiple-end solutions, and their set is denoted by $ \mathcal M_{2k}$. The main result of our paper states that if $ u \in \mathcal M_{2k}$ is nondegenerate, then locally near $ u$ the set of solutions is a smooth manifold of dimension $ 2k$. This paper is part of a program whose aim is to classify all $ 2k$-ended solutions of the Allen-Cahn equation in dimension $ 2$, for $ k \geq 2$.


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Additional Information

Manuel del Pino
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Email: delpino@dim.uchile.cl

Michał Kowalczyk
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Email: kowalczy@dim.uchile.cl

Frank Pacard
Affiliation: Centre de Mathématiques Laurent Schwartz UMR-CNRS 7640, École Polytechnique, 91128 Palaiseau, France
Email: frank.pacard@math.polytechnique.fr

DOI: https://doi.org/10.1090/S0002-9947-2012-05594-2
Received by editor(s): September 6, 2010
Received by editor(s) in revised form: March 2, 2011
Published electronically: August 9, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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