{
  "abstract": "Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of \\[ -Δu=f(u)\\quad\\hbox{in }κΩ, \\qquad u+α\\partial_νu=0\\quad\\hbox{on }\\partial(κΩ),\\] when $α$ is fixed, and remarked that the dilation threshold should not depend on $α$. We show that it does not, including at the Dirichlet and Neumann endpoints, for possibly unbounded uniformly $C^{2,γ}$ domains. The half-space linearizations have a common positive spectral gap over the compactified boundary parameter. The Dirichlet end requires a separate compactness argument because the Robin coefficient diverges there. After rescaling by its inverse, the equation has a harmonic half-space limit. A Liouville lemma rules out a nonzero limiting trace, and the resulting endpoint compactness, together with the half-space gap and localization, yields the uniform uniqueness statement. When $\\partialΩ\\neq\\varnothing$, we further obtain convergence of the spectral bottom to its half-space value with error $O(κ^{-1/2})$. For bounded $C^{4,γ}$ domains the boundary layer also has a first mean-curvature correction, with remainder $O(κ^{-1-γ}+κ^{-2})$ on each fixed boundary strip.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30211",
  "author_total": 2,
  "authors": [
    {
      "name": "Sophie Sun"
    },
    {
      "name": "Xuanrui Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cthx6sqymyfwb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30211",
  "fulltext_url": null,
  "id": "cthx6sqymyfwb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30211",
  "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": "W7204913122",
  "pdf_url": "https://arxiv.org/pdf/2608.30211",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cthx6sqymyfwb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2608.30211",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30211"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Parameter-uniform Robin uniqueness on large dilations",
  "title_text": "Parameter-uniform Robin uniqueness on large dilations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/cthx6sqymyfwb",
  "venue": "arXiv",
  "versions": []
}