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Nondegeneracy of the saddle solution of the Allen-Cahn equation


Authors: Michał Kowalczyk and Yong Liu
Journal: Proc. Amer. Math. Soc. 139 (2011), 4319-4329
MSC (2010): Primary 35B08, 35P99, 35Q80
DOI: https://doi.org/10.1090/S0002-9939-2011-11217-6
Published electronically: July 26, 2011
MathSciNet review: 2823077
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Abstract: A solution of the Allen-Cahn equation in the plane is called a saddle solution if its nodal set coincides with the coordinate axes. Such solutions are known to exist for a large class of nonlinearities. In this paper we consider the linear operator obtained by linearizing the Allen-Cahn equation around the saddle solution. Our result states that there are no nontrivial, decaying elements in the kernel of this operator. In other words, the saddle solution of the Allen-Cahn equation is nondegenerate.


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Additional Information

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

Yong Liu
Affiliation: Departamento de Ingeniería Matemática and CMM (UMI 2807, CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Address at time of publication: School of Mathematics and Physics, North China Electric Power University, Beijing, People’s Republic of China 102206
Email: yliu@dim.uchile.cl

DOI: https://doi.org/10.1090/S0002-9939-2011-11217-6
Keywords: Allen-Cahn equation, nondegeneracy
Received by editor(s): September 17, 2010
Published electronically: July 26, 2011
Additional Notes: The first author was partially supported by Chilean research grants Fondecyt 1090103, Fondo Basal CMM-Chile, and Project Añillo ACT-125 CAPDE
The second author was partially supported by Chilean research grants Fondecyt 3100011 and Fondo Basal CMM-Chile and doctoral grants of North China Electric Power University
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.