A simple proof of the characterization of functions of low Aviles Giga energy on a ballviaregularity
Andrew Lorent · ESAIM Control Optimisation and Calculus of Variations · 2011
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)·ηx = 1 where ηx 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 Iϵn(un) → 0 as ϵn → 0. A corollary to their work is that if there exists such a sequence (un) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u: = limn → ∞un = 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.