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Annali di Matematica Pura ed Applicata

, Volume 184, Issue 1, pp 17–52 | Cite as

On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions

  • Michał KowalczykEmail author
Article

Abstract

In this paper we study transition layers in the solutions to the Allen-Cahn equation in two dimensions. We show that for any straight line segment intersecting the boundary of the domain orthogonally there exists a solution to the Allen-Cahn equation, whose transition layer is located near this segment. In addition we analyze stability of such solutions and show that it is completely determined by a geometric eigenvalue problem associated to the transition layer. We prove the existence of both stable and unstable solutions. In the case of the stable solutions we recover a result of Kohn and Sternberg [13]. As for the unstable solutions we show that their Morse index is either 1 or 2.

Keywords

Line Segment Eigenvalue Problem Transition Layer Stable Solution Straight Line Segment 
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Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA

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