The Aviles Giga functional is a well known second order functional that forms a model forblistering and in a certain regime liquid crystals, a related functional models thinmagnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functionalis \hbox{$I_{\ep}(u)=\frac{1}{2}\int_{\Omega}\ep^{-1}\lt|1-\lt|Du\rt|^2\rt|^2+\ep\lt|D^2 u\rt|^2 {\rm d}z$}${\mathit{I}}_{\mathit{ϵ}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int }}}_{\mathit{\Omega }}{\mathit{ϵ}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{ϵ}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ whereu belongs to the subset of functions in\hbox{$W^{2,2}_{0}(\Omega)$}${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ whose gradient (in thesense of trace) satisfiesDu(x)·ηx = 1where ηx is the inward pointing unit normalto ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl.Sci. 1 (2002) 187–202] Jabin et al. characterizeda class of functions which includes all limits of sequences\hbox{$u_n\in W^{2,2}_0(\Omega)$}${\mathit{u}}_{\mathit{n}}\mathrm{\in }{\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ withIϵn(un) → 0as ϵn → 0. A corollary to their work is thatif there exists such a sequence (un) for abounded domain Ω, then Ω must be a ball and (up tochange of sign)u: = limn → ∞un = dist(·,∂Ω).Recently [Lorent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted),http://arxiv.org/abs/0902.0154v1] we provided a quantitative generalizationof this corollary over the space of convex domains using ‘compensated compactness’inspired calculations of DeSimone et al. [Proc. Soc. Edinb. Sect.A 131 (2001) 833–844]. In this note we use methods of regularitytheory and ODE to provide a sharper estimate and a much simpler proof for the case whereΩ = B1(0) without the requiring the tracecondition on Du.