Calculus of Variations and Geometric Measure Theory
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A. Lorent

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

created by lorent on 13 Apr 2010
modified on 18 Feb 2014

[BibTeX]

Published Paper

Inserted: 13 apr 2010
Last Updated: 18 feb 2014

Journal: ESIAM COCV
Volume: 18
Pages: 383-400
Year: 2012

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 $\Omega\subset R^2$ the functional is $I_{\epsilon}(u)=\int_{\Omega} \epsilon^{-1}|1-|Du|^2|^2+\epsilon |D^2 u|^2$ where $u$ belongs to the subset of functions in $W^{2,2}_{0}(\Omega)$ whose gradient (in the sense of trace) satisfies $Du(x)\cdot \eta_x=1$ where $\eta_x$ is the inward pointing unit normal to $\partial \Omega$ at $x$. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences $u_n\in W^{2,2}_0(\Omega)$ with $I_{\epsilon_n}(u_n)\rightarrow 0$ as $\epsilon_n\rightarrow 0$. 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) $u:=\lim_{n\rightarrow \infty} u_n =\mathrm{dist}(\cdot,\partial\Omega)$. Recently we provided a quantitative generalization of this corollary over the space of convex domains using `compensated compactness' inspired calculations originating from the proof of coercivity of $I_{\epsilon}$ by DeSimone, Muller, Kohn, Otto. 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 $Du$.

Keywords: Aviles Giga functional


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