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

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 $Ω\subset R^2$ the functional is $I_ε(u)=1/2\int_Ω ε^{-1}|1-|Du|^2|^2+ε|D^2 u|^2$ where $u$ belongs to the subset of functions in $W^{2,2}_{0}(Ω)$ whose gradient (in the sense of trace) satisfies $Du(x)\cdot η_x=1$ where $η_x$ is the inward pointing unit normal to $\partial Ω$ at $x$. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences $u_n\in W^{2,2}_0(Ω)$ with $I_{ε_n}(u_n)\to 0$ as $ε_n\to 0$. 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) $u:=\lim_{n\to \infty} u_n =\mathrm{dist}(\cdot,\partialΩ)$. 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 $I_ε$ 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 $Ω=B_1(0)$ without the requiring the trace condition on $Du$.