{
  "abstract": "We study a quadrature-surface free boundary problem driven by the Riemannian $p$-Laplacian on a smooth compact finite-dimensional Riemannian manifold. We formulate the problem as a shape optimization problem and develop an intrinsic admissible class based on a uniform Riemannian $RC$-$GNP$ condition~\\cite{DS3}. We establish compactness of admissible domains and stability of the associated Dirichlet problems under strong $RC$-$GNP$ convergence. In particular, the states converge strongly in $W^{1,p}$ and the associated $p$-torsional energy is continuous. Under additional boundary regularity, we derive the first variation of the shape functional and obtain the free-boundary Euler--Lagrange condition \\[ |\\nabla_g u_Ω|_g^p=σH_{\\partialΩ}+k^p. \\] We also formulate the contact optimality inequality, prove the Riemannian comparison of second fundamental forms at tangential contact, and obtain a sufficient condition on a reference domain which rules out contact and yields a solution of the free-boundary problem. Explicit radial examples on geodesic balls of the round sphere are included.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17978",
  "author_total": 2,
  "authors": [
    {
      "name": "Ababacar Sadikhe Djite"
    },
    {
      "name": "Diaraf Seck"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22npdbfnrc2z1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17978",
  "fulltext_url": null,
  "id": "22npdbfnrc2z1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17978",
  "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.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213493353",
  "pdf_url": "https://arxiv.org/pdf/2609.17978",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22npdbfnrc2z1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17978",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17978"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On Quadrature Surface Free Boundary Problems for the Riemannian $p$-Laplacian",
  "title_text": "On Quadrature Surface Free Boundary Problems for the Riemannian p-Laplacian",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/22npdbfnrc2z1",
  "venue": "arXiv",
  "versions": []
}