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

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.