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

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 $I_{\epsilon }(u)=1/2{\int }_{\Omega }{\epsilon }^{-1}|1-|Du{|}^2{|}^2+\epsilon |D^{2}u{|}^2$ where $u$ belongs to the subset of functions in $W_0^{2,2}(\Omega )$ 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 $u_n\in W_0^{2,2}(\Omega )$ with $I_{{\epsilon }_n}(u_n)\to 0$ as ${\epsilon }_n\to 0$. 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) $u:={\lim }_{n\to \infty }{u}_n=\mathrm{dist}(\cdot ,\partial \Omega )$. 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 $Du$.

16 pages, 1 figure