{
  "abstract": "In this paper, we establish symmetry results for solutions of overdetermined problems arising in optimal transportation. These problems involve the Monge-Ampère equation with a Dirichlet boundary condition $u=0$ on $\\partial Ω$ and the natural boundary condition $Du(Ω)=Ω^{*}$. We show that symmetry holds when the target is the unit ball $B$. For more general target domains, including the cases $Ω^*=Ω$ and arbitrary $Ω^{*}$, symmetry is retained under an additional volume constraint on $Ω$ and a boundary condition on $|Du|$. Finally, we introduce a curvature-type overdetermined problem for the Monge-Ampère equation and obtain ellipsoidal or spherical symmetry under an additional integral normalization condition. Our proofs rely on distinct techniques in different contexts: optimal transport and convex analysis for some cases, integral identities and isoperimetric inequalities for others, and P-function methods for the remaining cases. As a byproduct, in the $τ=2$ case, a new maximum principle for the $P$-function $ϕ(x)=\\sum_{k,l=1}^{n}{\\frac{\\partial{S_τ(D^2{u})}}{\\partial{u_{kl}}}u_{k}u_{l}}-2\\binom{n-1}{τ-1}\\int_{0}^{u}{f^{\\fracτ{n}}(t)\\,dt}$ is established for arbitrary positive and nondecreasing $f$, which is of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39782",
  "author_total": 2,
  "authors": [
    {
      "name": "Qing Zhao"
    },
    {
      "name": "Feida Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6j3grd2kk9hnf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39782",
  "fulltext_url": null,
  "id": "6j3grd2kk9hnf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39782",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7215035973",
  "pdf_url": "https://arxiv.org/pdf/2609.39782",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6j3grd2kk9hnf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39782",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39782"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Overdetermined problem for optimal transportation",
  "title_text": "Overdetermined problem for optimal transportation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/6j3grd2kk9hnf",
  "venue": "arXiv",
  "versions": []
}