Characterization of minimizers of Aviles-Giga functionals in special domains

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