{
  "abstract": "In this paper, we study two classes of partially overdetermined elliptic problems on planar annular domains in $\\mathbb{R}^2$, with a fixed inner circle and a free outer boundary. The Dirichlet values are prescribed to be $1$ and $0$ on the inner and outer boundaries, respectively, while the squared normal derivative is constant on the outer boundary. For the linear law $Δu=-μu$, we establish local bifurcation under a Dirichlet nonresonance condition. For the power law $Δu=u-u^p$ with $p>1$, we take the mean outer radius as the bifurcation parameter and construct large positive radial solutions on thin annuli. The limiting profile solves $-V''=V^p$ on an interval, and the principal mixed Dirichlet--Neumann eigenvalue of its linearization determines a constant $κ_p>0$. For every fixed inner radius $λ>0$ and every sufficiently large integer $K$, we obtain a critical radius $R_K=λ+λκ_p/K+o(K^{-1})$ and a smooth local curve of positive solutions with noncircular, even, $2π/K$-periodic outer boundaries. The proof combines shape derivatives, spectral estimates, and the Crandall--Rabinowitz bifurcation theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24397",
  "author_total": 3,
  "authors": [
    {
      "name": "Jun Wang"
    },
    {
      "name": "Fei Xu"
    },
    {
      "name": "Yong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6n9wmxxfgmr29.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24397",
  "fulltext_url": null,
  "id": "6n9wmxxfgmr29",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24397",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223070788",
  "pdf_url": "https://arxiv.org/pdf/2610.24397",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6n9wmxxfgmr29.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24397",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24397"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the annular overdetermined problems with linear and power nonlinearities in $\\mathbb{R}^2$",
  "title_text": "On the annular overdetermined problems with linear and power nonlinearities in ℝ^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/6n9wmxxfgmr29",
  "venue": "arXiv",
  "versions": []
}