A SIMPLE PROOF OF THE CHARACTERIZATION OF FUNCTIONS OF LOW AVILES GIGA ENERGY ON A BALL VIA REGULARITY

作者：Lorent Andrew^*

来源：ESAIM: Control, Optimisation and Calculus of Variations , 2012, 18(2): 383-400.

DOI：10.1051/cocv/2010102

• Aviles Giga functional

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 of R-2 the functional is I-epsilon(u) = 1/2 integral(Omega) epsilon(-1) vertical bar 1 - vertical bar Du vertical bar(2)vertical bar(2) + epsilon vertical bar D(2)u vertical bar(2) dz where u belongs to the subset of functions in W-0(2,2) (Omega) whose gradient (in the sense of trace) satisfies Du(x) . eta(x) = 1 where eta(x) is the inward pointing unit normal to partial derivative 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) epsilon W-0(2,2)(Omega) with I-epsilon n (u(n)) -%26gt; 0 as epsilon(n)-%26gt; 0. A corollary to their work is that if there exists such a sequence (u(n)) for a bounded domain Omega, then Omega must be a ball and (up to change of sign) u := lim(n -%26gt;infinity) u(n) = dist(., partial derivative 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%26apos; 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(0) without the requiring the trace condition on Du.

• 出版日期2012-4