  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.
