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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Nondegeneracy of the saddle solution of the Allen-Cahn equation
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by Michał Kowalczyk and Yong Liu
Proc. Amer. Math. Soc. 139 (2011), 4319-4329
DOI: https://doi.org/10.1090/S0002-9939-2011-11217-6
Published electronically: July 26, 2011

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.
References
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Bibliographic 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
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2823077