Characterization of Minimizers of Aviles-Giga Functionals in Special Domains.

We consider the singularly perturbed problem F ε ( u , Ω ) : = ∫ Ω ε | ∇ 2 u | 2 + ε - 1 | 1 - | ∇ u | 2 | 2 on bounded domains Ω ⊂ R 2 . Under appropriate boundary conditions, we prove that if Ω is an ellipse, then the minimizers of F ε ( · , Ω ) converge to the viscosity solution of the eikonal equation | ∇ u | = 1 as ε → 0 .