A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 383-400.

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 2 the functional is I ( u ) = 1 2 Ω ϵ - 1 | 1 - | D u | 2 | 2 + ϵ | D 2 u | 2 d z where u belongs to the subset of functions in W 0 2,2 (Ω) whose gradient (in the sense of trace) satisfies D u ( 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 u n W 0 2,2 (Ω) with I ϵ n ( 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 = B 1 ( 0 ) without the requiring the trace condition on D u .

DOI : 10.1051/cocv/2010102
Classification : 49N99, 35J30
Mots clés : aviles giga functional
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Lorent, Andrew. A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 383-400. doi : 10.1051/cocv/2010102. http://www.numdam.org/articles/10.1051/cocv/2010102/

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