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

Weak solutions m\colon\Omega\subset\mathbb{R}^{2}\to\mathbb{R}^{2} of the eikonal equation, \begin{align*} |m|=1\text{ a.e.} \quad \text{and} \quad \mathrm{div} 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_{\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 \begin{align*} m\in B^{1/p}_{p,\infty} \quad \Longleftrightarrow \quad \sup_{h\in \mathbb R^2\setminus\lbrace 0\rbrace} \frac{\|m(\cdot +h)-m\|_{L^p}}{|h|^{1/p}}