Weak solutions $m\colon\Omega\subset\mathbb{ R}^2\to\mathbb{ R}^2$ of the eikonal equation \begin{ align*} |m|=1\text{ a.e. and }\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 \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 $1\leq p<3$, thus going well beyond the BV setting ($p=1$) and leaving only the borderline case $p=3$ open.
Categories: math.AP