Regularity of the Eikonal equation with two vanishing entropies

TL;DR

This paper extends regularity results for solutions to the Eikonal equation, showing that under certain entropy conditions, the gradient of the solution is Lipschitz continuous outside a finite set.

Contribution

It generalizes previous results by incorporating two specific entropy conditions, leading to improved regularity conclusions for solutions of the Eikonal equation.

Findings

Gradient of solutions is Lipschitz outside a finite set under entropy conditions.

Generalization to two entropy conditions broadens applicability.

Results connect entropy conditions with regularity of solutions.

Abstract

The Aviles-Giga functional $I_{\epsilon}(u)=\int_{\Omega} \frac{\left|1-\left|\nabla u\right|^2\right|^2}{\epsilon}+\epsilon \left|\nabla^2 u\right|^2 \, dx$ 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 $\lim_{n\rightarrow \infty} I_{\epsilon_n}(u_n)=0$ for some sequence $u_n\in W^{2,2}_0(\Omega)$ and $u=\lim_{n\rightarrow \infty} u_n$ 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(\Sigma_{e_1 e_2}(\nabla u^{\perp})\right)=0\text{and}\nabla\cdot\left(\Sigma_{\epsilon_1 \epsilon_2}(\nabla u^{\perp})\right)=0\text{distributionally in}\Omega, \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.