{
  "abstract": "Let $Ω\\subset\\mathbb{R}^N$ ($N\\geq2$) be measurable with $0<|Ω|<\\infty$, and let $u_Ω$ be its logarithmic potential ($N=2$) or Newtonian potential ($N\\geq3$), normalized so that $Δu_Ω=χ_Ω$. We prove the sharp inequality \\[ \\int_Ω|\\nabla u_Ω|^2\\,\\mathrm{d} x \\leq\\frac{1}{2Nω_{N-1}} \\iint_{Ω\\timesΩ}\\frac{\\mathrm{d} x\\,\\mathrm{d} y}{|x-y|^{N-2}}, \\] where $ω_{N-1}=|\\mathbb{S}^{N-1}|$, with equality if and only if $Ω$ is a ball up to a null set. In dimension two this settles the $L^2$ Ahlfors--Beurling conjecture of Fleeman and Khavinson. For $N\\ge3$ it is equivalent to the scale-free bound that at most the fraction $(N-2)/(2N)$ of the total field energy lies inside $Ω$. We also obtain an explicit deficit identity, a sharp extension to bounded nonnegative densities, and consequences for torsional rigidity, Bergman analytic content, Brownian motion, and aggregation equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37886",
  "author_total": 2,
  "authors": [
    {
      "name": "Guofang Wang"
    },
    {
      "name": "Mingwei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11kd7b1a4zdsy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37886",
  "fulltext_url": null,
  "id": "11kd7b1a4zdsy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37886",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37886",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11kd7b1a4zdsy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37886",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37886"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An $L^2$ Ahlfors--Beurling inequality in every dimension",
  "title_text": "An L^2 Ahlfors--Beurling inequality in every dimension",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/11kd7b1a4zdsy",
  "venue": "arXiv",
  "versions": []
}