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 $W^{ 2,2}_{ 0}(\Omega)$ 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 $I_{ \epsilon_n}(u_n)\to 0$ 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 $Du$.
Categories: math.AP