{
  "abstract": "We give a simpler proof of Nadirashvili's theorem (Rayleigh's conjecture) that among vibrating clamped plates of prescribed area in the plane, the disk has the lowest fundamental tone. The proof uses a one-shot Talenti type comparison for sign-changing solutions of the Poisson equation with mean-zero data, established in a companion paper. The comparison considers just two rearranged sources, rather than optimizing over all admissible sources as in the traditional two-ball technique. This ``paired comparison'' simplification yields immediate nonlinear extensions: the disk minimizes the clamped quotient $\\|Δu\\|_q/\\|u\\|_p$ among bounded open sets of prescribed area, for every $1\\le p,q\\le\\infty$. Equality is characterized for $q>1$, while exceptional behavior at $q=1$ is identified. In higher dimensions, we prove ball minimality for $p=\\infty$ and conjecture minimality for all $p,q$ within the compact embedding range. We also establish the corresponding interval inequality in one dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29840",
  "author_total": 2,
  "authors": [
    {
      "name": "Dorin Bucur"
    },
    {
      "name": "Richard S. Laugesen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6x3s7ppmp6ehm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29840",
  "fulltext_url": null,
  "id": "6x3s7ppmp6ehm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29840",
  "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.SP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223223833",
  "pdf_url": "https://arxiv.org/pdf/2610.29840",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6x3s7ppmp6ehm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29840",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29840"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Rayleigh's conjecture for vibrating plates: simpler planar proof and nonlinear extensions",
  "title_text": "Rayleigh's conjecture for vibrating plates: simpler planar proof and nonlinear extensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/6x3s7ppmp6ehm",
  "venue": "arXiv",
  "versions": []
}