Abstract

Given a connected Lipschitz domain Ω we let Λ be the set of functions in W2,2 with u=0 on ∂Ω and whose gradient satisfies Vu.nx=1, where nx is the inward pointing unit normal to ∂Ω at x. The functional Iϵ=1/2 ∫Ωϵ-1|1 -|Vu|2|2+ϵV2u|2dz, minimised over Λ, serves as a model in connection with problems in liquid crystals and thin film blisters. It is also the most natural higher order gerealisation of the Modica and Mortola functional. in [16] Jabin, Otto and Perthame characterised a class of functions which includes all limits of sequences un ϵ Λ with Iϵπ→ 0 as ϵn→ 0. A corollary to their work is that if there exists such a sequence for a bounded domain Ω, then Ω must be a ball and u := limn→∞un is equal dist. We prove a quantative generlisation of this corollary for the class of bounded convex sets. Namely we show that there exists a positive constant γ1 such that, if Ω is a convex set of diameter 2 and u ϵ Λ with Iϵ = B, then |B1ΔΩ|≤cBγ1 for some x and ∫Ω|Vu+z-x/|z-x|2 dz≤cBγ1. A corollary of this result is that there exists a positive constant γ2 <γ1 such that if Ω is convex with diameter 2 and C2 boundary with curvature bounded by ϵ-1/2, then for any minimiser v of Iϵ over Λ we have ||v-ζ||W1,2 ≤ c|)γ2, where ζ=dist. Niether of the constands γ1 or γ2 are optimal