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We consider the singular limit of a perturbed Allen-Cahn model on a bounded two dimensional domain: ut = "2 u 2(u "a)(u2 1); x2 R2 @nu = 0; x2 @ where " is a small parameter and a is an O(1) quantity. We study equilibrium solutions that have the form of a curved interface. Using singular perturbation techniques, we fully characterize the stability of such an equilibrium in terms of a certain geometric eigenvalue problem, and give a simple geometric interpretation of our stability results. Full numerical computations of the time-dependent PDE as well as of the associated two-dimensional eigenvalue problem are shown to be in excellent agreement with the analytical predictions.

Stability of Curved Interfaces in the Perturbed Two-Dimensional Allen–Cahn System