{
  "abstract": "For $3\\le n\\le6$, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure $μ$ on $\\overlineΩ$, \\begin{equation*} \\overlineλ_1^N(Ω,μ)\\le\\frac{n(n-1)}{n-2}ω_n\\sin^2\\vartheta_n\\left(\\frac{|Ω|}{ω_n}\\right)^{(n-2)/n}, \\end{equation*} where $ω_n$ is the volume of the unit ball and $\\vartheta_n\\in(π/2,π)$ is the angle at the first stationary radius of the regular rotational harmonic-map profile. The bound is attained on balls by a smooth positive radial density. In fact, we affirmatively resolve Question~1.12 posed by Vinokurov \\cite{Vinokurov2026}. Furthermore, for admissible domains with finite boundary measure, we also prove the sharp strict Steklov bound \\begin{equation*} σ_1(Ω)|\\partialΩ|\\,|Ω|^{(2-n)/n}<\\frac{n(n-1)}{n-2}ω_n^{2/n}\\sin^2\\vartheta_n, \\end{equation*} whose constant is approached by $C^1$ perforated domains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33555",
  "author_total": 3,
  "authors": [
    {
      "name": "Daguang Chen"
    },
    {
      "name": "Hao Liu"
    },
    {
      "name": "Chengxi Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9j2h7ftgb52z8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33555",
  "fulltext_url": null,
  "id": "9j2h7ftgb52z8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33555",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214791244",
  "pdf_url": "https://arxiv.org/pdf/2609.33555",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9j2h7ftgb52z8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33555",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33555"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six",
  "title_text": "Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9j2h7ftgb52z8",
  "venue": "arXiv",
  "versions": []
}