Concentration of eikonal entropy production on jump curves
Let $\Omega \subset \mathbb{R}^2$ be open and let $m \in L^\infty(\Omega; \mathbb{R}^2)$ satisfy $|m| = 1$ almost everywhere and $\operatorname{div} m = 0$ in the sense of distributions. An entropy is a map $\Phi \in C^{1,1}(S^1; \mathbb{R}^2)$ with $$\frac{d}{d\theta} \Phi(e^{i\theta}) \cdot e^{i\theta} = 0 \qquad \text{for all } \theta;$$ for smooth $m$ the chain rule gives $\operatorname{div} \Phi(m) = 0$. Assume that every entropy production $\mu_\Phi = \operatorname{div} \Phi(m)$ is a locally finite signed Radon measure on $\Omega$. Let $J_m$ be the approximate jump set of $m$: the points at which, for some line through the point, $m$ has two distinct constant limiting values in mean on the two half-balls the line determines. If $m$ has bounded variation, the chain rule for $BV$ functions puts every $\mu_\Phi$ on $J_m$. For general $m$, De Lellis and Otto found an $\mathcal{H}^1$-rectifiable set $J$, at $\mathcal{H}^1$-almost every point of which $m$ has one-sided limits $m^\pm$ in this sense, such that for smooth entropies $\mu_\Phi$ agrees on $J$ with $\nu \cdot (\Phi(m^+) - \Phi(m^-))\,\mathcal{H}^1$, $\nu$ a unit normal to $J$, and vanishes on every subset of $\Omega \setminus J$ of finite length. Ghiraldin and Lamy characterised such $m$, locally, by Besov regularity $B^{1/3}_{3,\infty}$. Lamy and Marconi proved that if $\Omega$ is bounded and $m \in B^{1/p}_{p,\infty}(\Omega)$ for some $1 \le p < 3$, then $\mu_\Phi$ is carried by the jump set for every entropy with $\frac{d}{d\theta}\Phi(-e^{i\theta}) = -\frac{d}{d\theta}\Phi(e^{i\theta})$ (odd entropies plus constants).
Problem. Must $|\mu_\Phi|(\Omega \setminus J_m) = 0$ for every entropy $\Phi$, that is, does all entropy production lie on the one-dimensional jump set?
Source: De Lellis and Otto (2003), Conjecture 1(d′), for smooth entropies; in this form, with C^{1,1} entropies, Lamy and Marconi (2026), §1, who relate it to the Aviles–Giga energy.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Aviles–Giga functionals, eikonal equations, entropy productions
References
@article{delellis2003,
author = {De Lellis, Camillo and Otto, Felix},
title = {Structure of entropy solutions to the eikonal equation},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {5},
number = {2},
year = {2003},
pages = {107--145},
doi = {10.1007/s10097-002-0048-7},
}
@article{lamy2026,
author = {Lamy, Xavier and Marconi, Elio},
title = {Rectifiability of entropy productions for weak solutions of the {2D} eikonal equation with supercritical regularity},
journal = {Rev. Mat. Iberoam.},
year = {2026},
doi = {10.4171/RMI/1614},
eprint = {2509.16692},
archivePrefix = {arXiv},
}
@article{ghiraldin2020,
author = {Ghiraldin, Francesco and Lamy, Xavier},
title = {Optimal {B}esov differentiability for entropy solutions of the eikonal equation},
journal = {Comm. Pure Appl. Math.},
volume = {73},
number = {2},
year = {2020},
pages = {317--349},
doi = {10.1002/cpa.21868},
eprint = {1802.01373},
archivePrefix = {arXiv},
}
@misc{aim-eikonal-entropy-concentration,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Concentration of eikonal entropy production on jump curves}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/156-eikonal-entropy-concentration.md},
note = {Entry 156 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00856, "Concentration of eikonal entropy production on jump curves", clopen.solutions.