{
  "abstract": "We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace--Neumann eigenvalue. This is the quadrilateral case of the Neumann analogue of the long-standing Pólya--Szegő conjecture, which asserts that the regular $n$-gon minimizes the first Laplace--Dirichlet eigenvalue among $n$-gons of equal area. The proof uses a Rayleigh--Ritz lower bound, obtained from the span of the first five nonconstant Neumann modes of the square, on a smooth auxiliary functional whose minimization implies the original maximization. Local maximality at the square follows from a symmetry-reduced Hessian, computed in closed form, together with a certified second-order difference-quotient test that establishes the local inequality on an explicit ball around the square. Global maximality is then established by a certified box covering of the parameter space, certifying on each box a direct upper bound on the first Rayleigh--Ritz eigenvalue. Both certifications operate entirely on integrals over a fixed reference square and avoid a posteriori finite-element bounds on the perturbed quadrilateral.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02784",
  "author_total": 2,
  "authors": [
    {
      "name": "Ryoki Endo"
    },
    {
      "name": "Braxton Osting"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ba3w2ggkhshne.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02784",
  "fulltext_url": null,
  "id": "ba3w2ggkhshne",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02784",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7207800231",
  "pdf_url": "https://arxiv.org/pdf/2609.02784",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ba3w2ggkhshne.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02784",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02784"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals",
  "title_text": "Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/ba3w2ggkhshne",
  "venue": "arXiv",
  "versions": []
}