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

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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⊂ R² the functional is Iε(u)=1/2∫Ω ε⁻¹|1-|Du|²|²+ε|D² u|² where u belongs to the subset of functions in W²,²₀(Ω) whose gradient (in the sense of trace) satisfies Du(x)· ηₓ=1 where ηₓ is the inward pointing unit normal to ∂ Ω at x. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences uₙ∈ W²,²₀(Ω) with Iεₙ(uₙ)→ 0 as εₙ→ 0. A corollary to their work is that if there exists such a sequence (uₙ) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u:=ₙ→ ∞ uₙ =dist(·,∂Ω). 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 Iε 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=B₁(0) without the requiring the trace condition on Du.