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

Minimizers and gradient flows are studied for the functional ∫ Ω W(u) + ϵ 2 ∣∇ u ∣ 2 d x , Ω ⊆ R n , ϵ > 0, where u satisfies a Dirichlet condition u = h ϵ 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 ∂ t u ϵ = 2 ϵ ∆ u ϵ — ϵ -1 W' ( u ϵ ), u ϵ ( x , 0) = g ( x ), u ϵ ( x, t ) = h ϵ 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.