Characterization of minimizers of Aviles-Giga functionals in special domains

TL;DR

This paper studies the behavior of minimizers of the Aviles-Giga functional on elliptical domains, showing they converge to solutions of the eikonal equation as the perturbation parameter approaches zero.

Contribution

It proves convergence of minimizers to the viscosity solution of the eikonal equation on elliptical domains, extending understanding of Aviles-Giga functionals in special geometries.

Findings

Minimizers converge to the viscosity solution of | abla u|=1 as ε→0.

Convergence holds under appropriate boundary conditions on elliptical domains.

Provides rigorous analysis of the asymptotic behavior of the functional.

Abstract

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 $\varepsilon \to 0$.