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.
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Mathematics Subject Classification (2000)
35J60, 35Q72, 35J20, 35P15, 35P20, 35B25, 35B35, 35B40, 35B41
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Kowalczyk, M. On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions. Annali di Matematica 184, 17–52 (2005). https://doi.org/10.1007/s10231-003-0088-y
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DOI: https://doi.org/10.1007/s10231-003-0088-y