Rectifiability of entropy productions for weak solutions of the 2D eikonal equation with supercritical regularity

Abstract

Weak solutions $m:\Omega \subset {\mathbb{R}}^2\to {\mathbb{R}}^2$ of the eikonal equation,

arise naturally as sharp interface limits of bounded energy configurations in various physically motivated models, including the Aviles–Giga energy. The distributions ${\mu }_{\Phi }=\mathrm{div}\Phi (m)$, defined for a class of smooth vector fields $\Phi$ called entropies, carry information about singularities and energy cost. If these entropy productions are Radon measures, a long-standing conjecture predicts that they must be concentrated on the 1-rectifiable jump set of $m$ –as they do if $m$ has bounded variation (BV) thanks to the chain rule. We establish this concentration property, for a large class of entropies, under the Besov regularity assumption

for any $1\le p<3$, thus going well beyond the BV setting ($p=1$) and leaving only the borderline case $p=3$ open.