A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

Mathematics Department, University of Cincinnati, 2600 Clifton Ave., Cincinnati, Ohio 45221, USA
lorentaw@uc.edu

Received: 13 April 2010
Revised: 6 September 2010

Abstract

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ² the functional is where u belongs to the subset of functions in whose gradient (in the sense of trace) satisfies Du(x)·η_x = 1 where η_x is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202] Jabin et al. characterized a class of functions which includes all limits of sequences with I_ϵn(u_n) → 0 as ϵ_n → 0. A corollary to their work is that if there exists such a sequence (u_n) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u: = lim_n → ∞u_n = dist(·,∂Ω). Recently [Lorent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted), http://arxiv.org/abs/0902.0154v1] we provided a quantitative generalization of this corollary over the space of convex domains using ‘compensated compactness’ inspired calculations of DeSimone et al. [Proc. Soc. Edinb. Sect. A 131 (2001) 833–844]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B₁(0) without the requiring the trace condition on Du.