Singular-set dimension of polyhedral partial-transport free boundaries
Let $n \ge 3$, and let $\Omega, \Omega^* \subset \mathbb{R}^n$ be bounded domains strictly separated by a hyperplane, where $\Omega^*$, possibly nonconvex, has boundary made of finitely many flat polygonal facets. Fix $0 < m < \min(|\Omega|, |\Omega^*|)$, and minimize $\int |x - y|^2 \, d\gamma$ over nonnegative measures $\gamma$ of mass $m$ whose marginals are bounded by $\mathbf{1}_\Omega \, dx$ and $\mathbf{1}_{\Omega^*} \, dy$. Let $U \subset \Omega$ be the open active source region of the optimal plan and $\mathcal{F} = \partial U \cap \Omega$ its free boundary. Caffarelli and McCann showed that when the supports are separated by a hyperplane, the transported part of the source is separated from the rest by a semiconcave Lipschitz graph over the hyperplane, which is a $C^{1,\alpha}_{\mathrm{loc}}$ hypersurface when the domains are strictly convex. For a nonconvex polygonal target in the plane, Chen, Li and Liu proved that the free boundary is smooth away from finitely many singular points.
Problem. Must $\mathcal{F}$ be locally a smooth embedded hypersurface outside a relatively closed set of Hausdorff dimension at most $n - 2$?
Source: Chen, Li and Liu (2026), Conjecture 5.2, in arXiv:2311.15655v4. Stated for n ≥ 3 and strictly separated domains by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, free boundaries, Monge–Ampère obstacle problems, optimal partial transport
References
@misc{chen2026polygonal,
author = {Chen, Shibing and Li, Yuanyuan and Liu, Jiakun},
title = {Optimal (partial) transport to non-convex polygonal domains},
year = {2026},
eprint = {2311.15655v4},
archivePrefix = {arXiv},
note = {Version 4, 17 May 2026; Conjecture 5.2},
}
@article{caffarelli2010,
author = {Caffarelli, Luis A. and McCann, Robert J.},
title = {Free boundaries in optimal transport and {M}onge--{A}mp{\`e}re obstacle problems},
journal = {Ann. of Math. (2)},
volume = {171},
number = {2},
year = {2010},
pages = {673--730},
doi = {10.4007/annals.2010.171.673},
}
@misc{aim-polyhedral-partial-transport-free-boundary,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Singular-set dimension of polyhedral partial-transport free boundaries}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/393-polyhedral-partial-transport-free-boundary.md},
note = {Entry 393 at commit 24edaad, 2026-10-15},
}
Resolution claims
We study optimal transport with a non-convex polyhedral target in $\mathbb{R}^n$. The closure of the singular set is locally a smooth hypersurface outside a relatively closed set of Hausdorff dimension at most $n-2$. For partial transport between domains separated by a hyperplane, the source free boundary has the same regularity.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.18756v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00820, "Singular-set dimension of polyhedral partial-transport free boundaries", clopen.solutions.