{
  "abstract": "In this paper, we establish sharp gradient stability of the $L^{p}$-Poincaré inequality with explicit constants on bounded convex domains for any $1<p<\\infty$. These estimates control the distance to the optimizers in the strongest possible norm with the optimal stability exponent $\\max\\{2,p\\}$, exhibiting the same phenomenon initially observed by Figalli and Zhang [FZ22a] for the $L^p$-Sobolev inequality. In the case $1<p<2$, we also establish a sharp upper bound gradient stability estimate with an explicit constant and the optimal exponent $p$, matching the exponent obtained by Ignat and Zuazua [IZ26]. By extending the lower stability estimate to general bounded domains with $C^2$ boundary, we are able to confirm the conjecture posed by Fleckinger-Pellé and Takáč [FPT02, Section 8]. Finally, we apply the stability inequality to obtain positive fundamental gap estimates in terms of the inradius for the Dirichlet $p$-Laplacian.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27281",
  "author_total": 2,
  "authors": [
    {
      "name": "Nurgissa Yessirkegenov"
    },
    {
      "name": "Amir Zhangirbayev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0me3c1trch45k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27281",
  "fulltext_url": null,
  "id": "0me3c1trch45k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27281",
  "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",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223091403",
  "pdf_url": "https://arxiv.org/pdf/2610.27281",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0me3c1trch45k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27281",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27281"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Explicit sharp gradient stability of the $L^{p}$-Poincaré inequality and fundamental gaps for the Dirichlet $p$-Laplacian operator",
  "title_text": "Explicit sharp gradient stability of the L^p-Poincaré inequality and fundamental gaps for the Dirichlet p-Laplacian operator",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:14Z",
  "url": "https://research.draconitic.com/paper/0me3c1trch45k",
  "venue": "arXiv",
  "versions": []
}