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

The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\subset {\mathbb{R}}^2\to {\mathbb{R}}^2$ of the eikonal equation \begin{align*} $\mathrm{d}\mathrm{i}\mathrm{v}$, m=0$\text{ in }$\mathcal D’(Ω),\quad |m|=1$\text{ a.e. in }$Ω,, \end{align*} whose entropy productions $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Phi }(m)$ are Radon measures in $\mathrm{\Omega }$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:{\mathbb{S}}^1\to {\mathbb{R}}^2$ such that $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Phi }(m_{\ast })=0$ for any smooth solution $m_{\ast }$. 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.