{
  "abstract": "The purpose of this paper is to study Faber--Krahn inequality of Dirichlet eigenvalues $\\{ λ^{\\ln,μ}_{k}(Ω)\\}_{k\\in \\N}$ for the logarithmic Hardy operator \\[ \\mathcal{L}_μ=(-Δ)^{\\ln}+2μ\\log\\frac{1}{|x|} \\] on a bounded Lipschitz domain $Ω\\subset\\mathbb{R}^{N}$ containing the origin, with $μ>-1$. We establish a Faber--Krahn inequality for $μ\\in(-1,0)$, the ball centered at the origin minimizes the first eigenvalue at fixed volume, with equality only for that ball. Finally, we show this inequality fails for $μ>0$ via an explicit two-ball-and-tube construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09578",
  "author_total": 3,
  "authors": [
    {
      "name": "Ying Wang"
    },
    {
      "name": "Tao Zhang"
    },
    {
      "name": "Wanwan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4drmtn0ydecwn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09578",
  "fulltext_url": null,
  "id": "4drmtn0ydecwn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09578",
  "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"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7221017814",
  "pdf_url": "https://arxiv.org/pdf/2610.09578",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4drmtn0ydecwn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09578",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09578"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator",
  "title_text": "Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/4drmtn0ydecwn",
  "venue": "arXiv",
  "versions": []
}