{
  "abstract": "We study the $p$-Laplacian quadrature surface free boundary problem on a compact Riemannian manifold $(\\M,g)$ of dimension $N \\geq 2$, for $1<p<\\infty$. Given a nonnegative source $f \\in L^p_g(\\M)$ with compact support $K$, and $k>0$, one seeks a domain $Ω\\supset K$ on which the overdetermined problem $-Δ_{p,g} u = f$ in $Ω$, $u=0$ and $|\\nabla_g u|_g = k$ on $\\partialΩ$ admits a solution. Following the geometric and variational framework of Djité--Seck, we prove that the integral condition \\[ (NS)_{p,g} \\qquad \\int_C f\\,dv(g) > k^{p-1}\\,|\\partial C|_g, \\] where $C$ is the totally convex hull of $K$, is \\emph{sufficient} for the existence of a solution strictly containing $C$. We then show, by an explicit counterexample on the round sphere $S^n_R$, that $(NS)_{p,g}$ is \\emph{not necessary} in general under nonnegative curvature: the mechanism is that the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis---perimeter monotonicity on the admissible class inside a normal convex neighborhood of $C$---under which necessity is restored, and we discuss the critical equality case $\\int_C f\\,dv(g) = k^{p-1}|\\partial C|_g$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08260",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammed Barkatou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jp0dhfk64be8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08260",
  "fulltext_url": null,
  "id": "1jp0dhfk64be8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08260",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7212077026",
  "pdf_url": "https://arxiv.org/pdf/2609.08260",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jp0dhfk64be8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.08260",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08260"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the $p$-Laplacian Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient condition of existence",
  "title_text": "On the p-Laplacian Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient condition of existence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1jp0dhfk64be8",
  "venue": "arXiv",
  "versions": []
}