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

Lorent, Andrew

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 383-400

Résumé

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 $\Omega \subset {ℝ}^2$ the functional is $I_{}\left(u\right)=\frac{1}{2}{\int }_{\Omega }{ϵ}^{-1}{|1-|Du|}^2{|}^2+ϵ{\left|D^{2}u\right|}^2\mathrm{d}z$ where $u$ belongs to the subset of functions in $W_0^{2,2}\left(\Omega \right)$ whose gradient (in the sense of trace) satisfies $Du\left(x\right)·{\eta }_x=1$ where ${\eta }_x$ is the inward pointing unit normal to $\partial \Omega$ 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 $u_n\in W_0^{2,2}\left(\Omega \right)$ with $I_{{ϵ}_n}\left(u_n\right)\to 0$ as ${ϵ}_n\to 0$. A corollary to their work is that if there exists such a sequence $\left(u_n\right)$for a bounded domain $\Omega$, then $\Omega$ must be a ball and (up to change of sign) $u:={lim}_{n\to \infty }{u}_n=\mathrm{dist}(·,\partial \Omega )$. 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 $\Omega =B_1\left(0\right)$ without the requiring the trace condition on $Du$.

Détail

DOI : 10.1051/cocv/2010102

Classification : 49N99, 35J30
Keywords: aviles giga functional