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

We consider the singular limit of a perturbed Allen–Cahn model on a bounded twodimensional domain: { ut = ε2Δu− 2(u− εa)(u2 − 1), x ∈ Ω ⊂ R2 ∂nu = 0, x ∈ ∂Ω where ε 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.