Regularity of the Eikonal equation with two vanishing entropies

Abstract

The Aviles-Giga functional $I_{\epsilon }(u)={\int }_{\Omega }\frac{{|1-{|\nabla u|}^2|}^2}{\epsilon }+\epsilon {\left|{\nabla }^{2}u\right|}^{2}dx$ is a well known second order functional that models phenomena from blistering to liquid crystals. The zero energy states of the Aviles-Giga functional have been characterized by Jabin, Otto, Perthame. Among other results they showed that if ${\lim }_{n\to \infty }{I}_{{\epsilon }_n}(u_n)=0$ for some sequence $u_n\in W_0^{2,2}(\Omega )$ and $u={\lim }_{n\to \infty }{u}_n$ then $\nabla u$ is Lipschitz continuous outside a locally finite set. This is essentially a corollary to their theorem that if $u$ is a solution to the Eikonal equation $\left|\nabla u\right|=1$ a.e. and if for every "entropy" $\Phi$ function $u$ satisfies $\nabla \cdot \left[\Phi (\nabla u^{\perp })\right]=0$ distributionally in $\Omega$ then $\nabla u$ is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result by showing that if $\Omega$ is bounded and simply connected, $u$ satisfies the Eikonal equation and if \begin{equation} \label{eqi88} \nabla\cdot\left(Σ_{e_1 e_2}(\nabla u^{\perp})\right)=0\text{and}\nabla\cdot\left(Σ_{ε_1 ε_2}(\nabla u^{\perp})\right)=0\text{distributionally in}Ω, \end{equation} where ${\Sigma }_{e_1{e}_2}$ and ${\Sigma }_{{\epsilon }_1{\epsilon }_2}$ are the entropies introduced by Ambrosio, DeLellis, Mantegazza, Jin, Kohn, then $\nabla u$ is locally Lipschitz continuous outside a locally finite set.