{
  "abstract": "We study boundary vanishing orders and asymptotic expansions of planar harmonic functions satisfying the sublinear Robin condition $\\partial_νu=λu-|u|^{p-1}u,$ with $λ\\geq0$ and $0<p<1$. For nontrivial global solutions on bounded connected Lipschitz domains, we bound the number of nodal domains by the number of Steklov eigenvalues strictly below $λ$ and establish finite vanishing order at every boundary point. The boundary scaling gives the critical exponent $\\frac{1}{1-p}$. On $C^1$ boundaries, every positive subcritical vanishing order is an integer, and the leading term is a unique homogeneous Neumann profile in conformal coordinates. At a noninteger critical order, the rescaled solution converges in normalized graph coordinates to an explicit homogeneous profile determined by $p$ up to sign. Every supercritical bulk vanishing order $m$ is an integer, with a unique homogeneous Dirichlet profile in conformal coordinates. The boundary trace has vanishing order $\\frac{m-1}{p}>m$. Integer critical vanishing orders are excluded if the boundary is $C^1$ with Dini mean oscillation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15465",
  "author_total": 3,
  "authors": [
    {
      "name": "Hairong Liu"
    },
    {
      "name": "Long Tian"
    },
    {
      "name": "Xiaoping Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yhc9d1t74pyr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15465",
  "fulltext_url": null,
  "id": "7yhc9d1t74pyr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15465",
  "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": "W7221757366",
  "pdf_url": "https://arxiv.org/pdf/2610.15465",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yhc9d1t74pyr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15465",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15465"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition",
  "title_text": "Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7yhc9d1t74pyr",
  "venue": "arXiv",
  "versions": []
}