Regularity of the Eikonal equation with two vanishing entropies

The Aviles-Giga functional $I_ε(u)=\int_Ω \frac{\left|1-\left|\nabla u\right|^2\right|^2}ε+ε\left|\nabla^2 u\right|^2 \, dx$ 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 $\lim_{n\rightarrow \infty} I_{ε_n}(u_n)=0$ for some sequence $u_n\in W^{2,2}_0(Ω)$ 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" $Φ$ function $u$ satisfies $\nabla\cdot\left[Φ(\nabla u^{\perp})\right]=0$ distributionally in $Ω$ then $\nabla u$ is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result by showing that if $Ω$ 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 $Σ_{e_1 e_2}$ and $Σ_{ε_1 ε_2}$ are the entropies introduced by Ambrosio, DeLellis, Mantegazza, Jin, Kohn, then $\nabla u$ is locally Lipschitz continuous outside a locally finite set.

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