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

Abstract

The Aviles-Giga energy provides sequences of maps converging to weak solutions $m:\Omega \subset {\mathbb{R}}^2\to {\mathbb{R}}^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(Ω),\quad |m|=1\text{ a.e. in }Ω\,, \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 $\Phi :{\mathbb{S}}^1\to {\mathbb{R}}^2$ such that $\mathrm{div}\Phi (m_{\ast })=0$ 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.

14 pages