Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

Abstract

AbstractWe consider the singularly perturbed problem $$F_\varepsilon (u,\Omega ):=\int _\Omega \varepsilon |\nabla ^2u|^2 + \varepsilon ^{-1}|1-|\nabla u|^2|^2$$ F ε ( u , Ω ) : = ∫ Ω ε | ∇ 2 u | 2 + ε - 1 | 1 - | ∇ u | 2 | 2 on bounded domains $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . Under appropriate boundary conditions, we prove that if $$\Omega $$ Ω is an ellipse, then the minimizers of $$F_\varepsilon (\cdot ,\Omega )$$ F ε ( · , Ω ) converge to the viscosity solution of the eikonal equation $$|\nabla u|=1$$ | ∇ u | = 1 as $$\varepsilon \rightarrow 0$$ ε → 0 .