{
  "abstract": "We study low-order eigenvalue ratio estimates for the Laplacian with Muckenhoupt $A_2$ weights. For weights bounded above and below, we prove a universal upper bound for $λ_2/λ_1$ with constant $1 + \\frac{4}{n}\\,\\frac{C}{c}$, linear in the ratio of the weight bounds, and recover the classical Payne-Pólya-Weinberger bound as a special case. For radial decreasing weights, we prove that the ball maximizes $λ_2/λ_1$ among domains of equal volume, assuming that an admissible radial test function exists (Definition 5.8) and that on the ball the second eigenvalue is attained by the $l=1$ mode. The argument adapts the method of Ashbaugh--Benguria to the weighted setting via a weighted Chiti comparison theorem and a monotonicity property of the admissible radial test function (Definition 5.8). Higher-order eigenvalue ratios are estimated in terms of the $(n-1)$-Hausdorff measure of the boundary, under an explicit balance condition on the weight at the covering scale. The generalized Payne-Pólya-Weinberger conjecture for eigenvalue sums remains open in general; conditional results are presented for restricted families of domains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02701",
  "author_total": 3,
  "authors": [
    {
      "name": "Jing Mao"
    },
    {
      "name": "He-Jun Sun"
    },
    {
      "name": "Lingzhong Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/at356kgfsvs6q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02701",
  "fulltext_url": null,
  "id": "at356kgfsvs6q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02701",
  "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.DG"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02701",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/at356kgfsvs6q.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02701",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02701"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Estimates for Ratios of Low-Order Eigenvalues of the Laplacian with Muckenhoupt Weights on Euclidean Domains",
  "title_text": "Estimates for Ratios of Low-Order Eigenvalues of the Laplacian with Muckenhoupt Weights on Euclidean Domains",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/at356kgfsvs6q",
  "venue": "arXiv",
  "versions": []
}