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

Abstract

Minimizers and gradient flows are studied for the functional ∫ΩW(u)+ϵ2∣∇u∣2dx,Ω⊆Rn,ϵ> 0, whereusatisfies a Dirichlet conditionu=hϵon ∂Ω. HereWis taken to be a double-well potential with minimum value zero attained atu=aandu=b. Questions of existence and structure of minimizers for smallϵare 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) =hϵon ∂Ω, valid whenϵis small. Using multiple timescales we show that fronts rapidly develop and then propagate with normal velocityϵk, wherekis 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.