A quantitative characterisation of functions with low Aviles Giga energy on convex domains

Abstract

Given a connected Lipschitz domain we let 3() be the set of functions in W 2,2 () with u = 0 on @ and whose gradient (in the sense of trace) satisfies ru(x) • ⌘ x = 1, where ⌘ x is the inward pointing unit normal to @ at x.dz, minimised over 3(), serves as a model in connection with problems in liquid crystals and thin film blisters.It is also the most natural higher order generalisation of the Modica and Mortola functional.In [16] Jabin, Otto and Perthame characterised a class of functions which includes all limits of sequences u n 2 3 () with I ✏ n (u n ) ! 0 as ✏ n !0. A corollary to their work is that if there exists such a sequence (u n ) for a bounded domain , then must be a ball and (up to change of sign) u := lim n!1 u n is equal dist(•, @).We prove a quantitative generalisation of this corollary for the class of bounded convex sets.Namely we show that there exists a positive constant 1 such that, if is a convex set of diameter 2 and u 2 3() with I ✏ (u) = , then |B 1 (x)4|  c 1 for some x and ZA corollary of this result is that there exists a positive constant 2 < 1 such that if is convex with diameter 2 and C 2 boundary with curvature bounded by ✏ 1 2 , then for any minimiser v of I ✏ over 3() we havewhere ⇣ (z) = dist(z, @).Neither of the constants 1 or 2 are optimal.

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