{
  "abstract": "The notion of variational mean curvature (VMC) arises from Massari's variational approach to hypersurfaces of prescribed mean curvature and expresses a minimizing property for Massari's functional. Here, we complement the theory of VMCs with a new sharp relation between hypersurface regularity and VMC integrability. In fact, our main result asserts that every bounded open set in $\\mathbb{R}^n$ with $\\mathrm{C}^{1,α}$ boundary, $0<α<1$, has a global VMC in $\\mathrm{L}^p(\\mathbb{R}^n)$ for every $p<\\frac{1+α}{1-α}$. In addition, we underpin sharpness of this conclusion by constructing examples of bounded open sets in $\\mathbb{R}^n$ with $\\mathrm{C}^{1,α}$ boundary which do not admit a VMC in $\\mathrm{L}^p(\\mathbb{R}^n)$ for any $p>\\frac{1+α}{1-α}$. Our proofs are based on Barozzi's level-set construction of global VMCs and determine integrability of these VMCs by geometric investigation of their superlevel sets and of related unions of balls.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24224",
  "author_total": 2,
  "authors": [
    {
      "name": "Thomas Schmidt"
    },
    {
      "name": "Jule Helena Schütt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85jwvcqcn68a4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24224",
  "fulltext_url": null,
  "id": "85jwvcqcn68a4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24224",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223062198",
  "pdf_url": "https://arxiv.org/pdf/2610.24224",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85jwvcqcn68a4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24224",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24224"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Variational mean curvatures for general $\\mathrm{C}^{1,α}$ domains",
  "title_text": "Variational mean curvatures for general C^1,α domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/85jwvcqcn68a4",
  "venue": "arXiv",
  "versions": []
}