Accepted Paper

Inserted: 14 apr 2021
Last Updated: 23 jul 2021

Journal: Arch. Ration. Mech. Anal.
Year: 2021

Abstract:

We consider the singularly perturbed problem $F_{\varepsilon} (u,\Omega):=\int_{\Omega} \varepsilon \vert\nabla^2u\vert^2 + \varepsilon^{-1}\vert 1-\vert \nabla u\vert^2\vert^2$ 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 $\varepsilon \to 0$.

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