Characterization of minimizers of Aviles-Giga functionals in special domains

We consider the singularly perturbed problem $F_\varepsilon (u,\Omega):=\int_\Omega \varepsilon |\nabla^2u| + \varepsilon^{-1}|1-|\nabla u|^2|^2$onboundeddomains$\Omega \subset$\mathbb{R}$^2$. Under appropriate boundary conditions, we prove that if $\mathrm{\Omega }$ is an ellipse then the minimizers of $F_{\epsilon }(\cdot ,\mathrm{\Omega })$ converge to the viscosity solution of the eikonal equation $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=1$ as $\epsilon \to 0$.