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

Weak solutions $m\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\subset {\mathbb{R}}^2\to {\mathbb{R}}^2$ of the eikonal equation \begin{align*} |m|=1$\text{ a.e. and }$$\mathrm{d}\mathrm{i}\mathrm{v}$: m =0,, \end{align*} arise naturally as sharp interface limits of bounded energy configurations in various physically motivated models, including the Aviles-Giga energy. The distributions ${\mu }_{\mathrm{\Phi }}=\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Phi }(m)$, defined for a class of smooth vector fields $\mathrm{\Phi }$ 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 \begin{align*} m\in B^{1/p}{p,\infty} \quad \Leftrightarrow \quad \sup{h\in \mathbb R^2\setminus\lbrace 0\rbrace} \frac{|m(\cdot +h)-m|_{L^p }}{|h|^{1/p}} <\infty,, \end{align*} for any $1\le p<3$, thus going well beyond the BV setting ($p=1$) and leaving only the borderline case $p=3$ open.