{
  "abstract": "Let $Ω\\subset\\mathbb R^2$ be a bounded connected Lipschitz domain, and let $\\{μ_j(Ω)\\}_{j\\geq1}$ and $\\{λ_j(Ω)\\}_{j\\geq1}$ denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that $$ μ_3(Ω)<λ_1(Ω), $$ thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05893",
  "author_total": 2,
  "authors": [
    {
      "name": "Quanyu Tang"
    },
    {
      "name": "Haiqi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1xaynr3z679hg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05893",
  "fulltext_url": null,
  "id": "1xaynr3z679hg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05893",
  "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",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7212055453",
  "pdf_url": "https://arxiv.org/pdf/2609.05893",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1xaynr3z679hg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05893",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05893"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane",
  "title_text": "Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/1xaynr3z679hg",
  "venue": "arXiv",
  "versions": []
}