Preprint

Inserted: 20 sep 2025
Last Updated: 20 sep 2025

Year: 2025

Abstract:

Weak solutions $m\colon\Omega\subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \[
m
=1\text{ a.e. and }\mathrm{div}\: m =0\,, \] 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 \[ m\in B^{1/p}_{p,\infty} \quad \Leftrightarrow \quad \sup_{h\in \mathbb R^2\setminus\lbrace 0\rbrace} \frac{\
m(\cdot +h)-m\
_{L^p}}{
h
^{1/p}} <\infty\,, \] for any $1\leq p<3$, thus going well beyond the BV setting ($p=1$) and leaving only the borderline case $p=3$ open.

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• LM eikonal supercritical.pdf