{
  "abstract": "Let \\(u>0\\) be the first Dirichlet \\(p\\)-eigenfunction on a bounded convex domain \\(Ω\\subset\\mathbb R^N\\), and set \\[ X_Ω:=|\\nabla\\log u|^{p-2}\\nabla\\log u. \\] We study whether the sharp one-dimensional two-point modulus for \\(X_Ω\\), which for \\(p=2\\) reduces to the logarithmic-gradient estimate of Andrews and Clutterbuck, persists for \\(p\\neq2\\). We prove sharp estimates on intervals and balls for every \\(p>1\\), with the radial modulus on balls strictly larger than the one-dimensional one. In dimensions \\(N\\ge2\\), however, the one-dimensional modulus fails on general convex domains for every \\(p\\neq2\\). For \\(1<p<2\\), at every sufficiently small fixed scale there are smooth uniformly convex domains for which the corresponding two-point flux tends to zero. For \\(p>2\\), on thin domains \\(Ω_\\varepsilon=D\\times(-\\varepsilon,\\varepsilon)\\), with \\(D\\subset\\mathbb R^{N-1}\\) bounded and convex, the normalized first eigenfunctions satisfy \\[ u_\\varepsilon(x,\\varepsilon z)\\longrightarrow \\frac{ϕ(z)}{ϕ(0)} \\left(\\frac{G_D(x)}{G_D(x_0)}\\right)^{2/p} \\] locally uniformly in \\(D\\times(-1,1)\\), where \\(ϕ\\) and \\(G_D\\) are the first Dirichlet eigenfunctions of the \\(p\\)-Laplacian on \\((-1,1)\\) and of the Laplacian on \\(D\\), respectively. This yields the failure for \\(p>2\\). Finally, for arbitrary \\(C^2\\) functions, positivity of the symmetric differential of the \\(p\\)-gradient implies convexity, and the converse holds universally if and only if \\(p=2\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29195",
  "author_total": 1,
  "authors": [
    {
      "name": "Rui Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/edgcqrb30a1vk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29195",
  "fulltext_url": null,
  "id": "edgcqrb30a1vk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29195",
  "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": "W7204887769",
  "pdf_url": "https://arxiv.org/pdf/2608.29195",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/edgcqrb30a1vk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.29195",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29195"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction",
  "title_text": "Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/edgcqrb30a1vk",
  "venue": "arXiv",
  "versions": []
}