{
  "abstract": "We study the fundamental gap of the Dirichlet $p$-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate $Γ_p(I_D,V)\\ge (p-1)(2^p-1)(π_p/D)^p$, $I_D=(-D/2,D/2)$, for all $1<p<\\infty$ and convex potentials $V$, with equality exactly for constant $V$. An analysis of collapsing convex domains shows that $p=2$ is exceptional: for $p\\ne2$, the higher-dimensional gap is not governed by the sharp one-dimensional constant. For $V\\equiv0$, define $G_{p,N}:=\\inf_{Ω\\subset\\mathbb R^N}\\operatorname{diam}(Ω)^pΓ_p(Ω,0)$ over bounded convex domains. We prove $G_{2,N}=3π^2$ and $G_{p,N}=0$ for $1<p<2$, $N\\ge2$, while for every $p>2$, $G_{p,N}>0$ and the infimum is attained. We obtain explicit dimension-dependent bounds and $G_{p,N}\\asymp_p N^{p-2}$ as $N\\to\\infty$, while for fixed $N$, $\\lim_{p\\downarrow2}G_{p,N}=3π^2$ and $\\lim_{p\\to\\infty}G_{p,N}^{1/p}=4$. We also characterize the collapse as $p\\downarrow2$: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals $2\\cos^2(πt)\\,dt$ on $(-1/2,1/2)$. The proof establishes log-concavity of the first $p$-eigenfunction with convex potential, a degenerate weighted Poincaré inequality for the ground-state measure, and a lower gap bound in terms of the first eigenvalue and the associated weighted Poincaré constant. These estimates also give the compactness and equality analysis for the collapsing minimizers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.13443",
  "author_total": 2,
  "authors": [
    {
      "name": "Rui Chen"
    },
    {
      "name": "Daniel Hauer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18am3yt10rnjs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.13443",
  "fulltext_url": null,
  "id": "18am3yt10rnjs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.13443",
  "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"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7203487615",
  "pdf_url": "https://arxiv.org/pdf/2608.13443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18am3yt10rnjs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2608.13443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.13443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains",
  "title_text": "Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/18am3yt10rnjs",
  "venue": "arXiv",
  "versions": []
}