Characterization of minimizers of Aviles-Giga functionals in special domains

Abstract

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

References in corpus (1)

• Rectifiability of entropy defect measures in a micromagnetics model

Cited by in corpus (2)

• On Lebesgue points of entropy solutions to the eikonal equation
• Variational analysis of the $J_1$-$J_2$-$J_3$ model: a non-linear lattice version of the Aviles-Giga functional