# Regularity of the Eikonal equation with two vanishing entropies

created by lorent on 02 Oct 2016
modified on 09 Jan 2018

[BibTeX]

Accepted Paper

Inserted: 2 oct 2016
Last Updated: 9 jan 2018

Journal: Ann. Inst. Poincare Anal. Non Lineaire.
Year: 2016

Abstract:

The Aviles-Giga functional $I_{\epsilon}$ 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 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 and $u$ satisfies the Eikonal equation and if $\nabla\cdot\left(\Sigma_{e_1 e_2}(\nabla u^{\perp})\right)=0\text{ and }\nabla\cdot\left(\Sigma_{\epsilon_1 \epsilon_2}(\nabla u^{\perp})\right)=0$ distributionally in $\Omega$ 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. This condition being fairly natural this result could also be considered a contribution to the study of the regularity of solutions of the Eikonal equation. The method of proof is to transform any solution of the Eikonal equation satisfying into a differential inclusion $DF\in K$ where $K\subset M^{2\times 2}$ is a connected compact set of matrices without Rank-$1$ connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set $K$ is also non-elliptic in the sense of Sverak. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat, DeLellis, Ignat as well as methods of Sverak, regularity is established.