Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition

Abstract

Minimizers and gradient flows are studied for the functional ∫Ω W(u) + ϵ2∣∇u∣2dx, Ω ⊆ Rn, ϵ > 0, where u satisfies a Dirichlet condition u = hϵ on ∂Ω. Here W is taken to be a double-well potential with minimum value zero attained at u = a and u = b. Questions of existence and structure of minimizers for small ϵ are resolved through the identification of a limiting variational problem, the so-called Γ-limit. A formal asymptotic solution is then constructed for the gradient flow ∂tuϵ = 2ϵ∆uϵ—ϵ-1W'(uϵ), uϵ(x, 0) = g(x), uϵ(x, t) = hϵ on ∂Ω, valid when ϵ is small. Using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity ϵk, where k is mean curvature. At the intersection of a front with ∂Ω, the Dirichlet condition is shown to imply a contact angle condition for the front. This asymptotically correct evolution process represents gradient flow for the Γ-limit.