On Aviles-Giga limit states with $L^p$ entropy productions

The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colon\Omega \subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(\Omega),\quad |m|=1\text{ a.e. in }\Omega\,, \end{align*} whose entropy productions $\mathrm{div}\,\Phi(m)$ are Radon measures in $\Omega$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $\Phi\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,\Phi(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions.