OP-00821 Solution claimed

Regularity of transport interfaces for nonconvex polyhedral targets

Let $n \ge 3$, let $\Omega \subset \mathbb{R}^n$ be a bounded domain, and let $\Omega^* \subset \mathbb{R}^n$ be a bounded nonconvex polyhedral domain with $|\Omega^*| = |\Omega|$, polyhedral meaning that its boundary is a finite union of flat polygonal facets. Let $u$ be a convex Brenier potential with $(\nabla u)_\# \mathbf{1}_\Omega = \mathbf{1}_{\Omega^*}$, and set $$A = \{x \in \Omega : u \text{ is not differentiable at } x\}, \qquad \Sigma = \overline{A} \cap \Omega.$$ For a nonconvex polygonal target in the plane, Chen, Li and Liu proved that $A$ is either finite or, away from finitely many points, locally a smooth curve. Chen and Liu developed a regularity theory for the singular set when the target is the union of two disjoint convex domains, without convexity of the source.

Problem. Is there a relatively closed set $S \subset \Sigma$ with $\dim_H S \le n - 2$ such that $\Sigma \setminus S$ is locally a smooth embedded hypersurface? The empty set $\Sigma$ is allowed.

Source: Chen, Li and Liu (2026), Conjecture 5.1, in arXiv:2311.15655v4. Stated for n ≥ 3 by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: AP — Analysis of PDEs
Secondary: OC
Keywords: AIM Open Applied Problems, Brenier maps, optimal transport, singular sets

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 5.1},
}
@article{chen2025singular,
  author        = {Chen, Shibing and Liu, Jiakun},
  title         = {Regularity of singular set in optimal transportation},
  journal       = {Invent. Math.},
  volume        = {242},
  number        = {1},
  year          = {2025},
  pages         = {1--44},
  doi           = {10.1007/s00222-025-01353-w},
  eprint        = {2210.13841},
  archivePrefix = {arXiv},
}
@misc{aim-polyhedral-transport-singular-interface,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Regularity of transport interfaces for nonconvex polyhedral targets}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/392-polyhedral-transport-singular-interface.md},
  note         = {Entry 392 at commit 24edaad, 2026-10-15},
}

Resolution claims

Optimal transport to non-convex polyhedral domains v1 arXiv ↗ AI-Verified

Shibing Chen; Qi-Rui Li; Yuanyuan Li; Jiakun Liu; Xiaotian Wu

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.

Full resolution · not verified · fidelity not established · posted on arXiv 13 Oct 2026

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-00821, "Regularity of transport interfaces for nonconvex polyhedral targets", clopen.solutions.