{
  "abstract": "We construct the first convex planar counterexample to the Schiffer and Pompeiu conjectures: a bounded strictly convex non-disc domain with real-analytic boundary carrying a nonconstant solution of $Δu+u=0$ with $u=1$ and $\\partial_νu=0$ on the boundary. Together with the higher-dimensional constructions, this gives convex non-ball counterexamples in every dimension from two to eighteen, and in dimensions twenty and twenty-one, with real-analytic boundaries diffeomorphic to spheres and indicator Fourier transforms vanishing on the unit sphere. In dimensions $4$, $6$, $8$, $10$ and $14$, planar reductions through compact Lie algebras of rank two use Harish-Chandra's radial part formula and Kostant's convexity theorem. In dimensions $3$, $5$ and $7$, an axisymmetric formulation on the unit ball gives a quartic equation in weighted coefficient spaces. The exact two-sided inverse of the residual-corrected linearisation combines a Dirichlet Helmholtz inverse, a holomorphic boundary real-part problem and the material-derivative identity; it gives the planar Schiffer domain and, by one dimension-parametrised argument, the domains in dimensions $5$, $9$, $11$ to $13$, $15$ to $18$, $20$ and $21$. We also construct the first convex planar counterexample to Berenstein's conjecture: a strictly convex non-disc domain with real-analytic boundary carrying a sign-changing solution of $Δu+u=0$ with $u=0$ and $\\partial_νu=1$ on the boundary. Planar existence for this problem is due to Colbrook, Sadeghi and Stepaniants. All existence proofs in this paper are computer-assisted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35419",
  "author_total": 1,
  "authors": [
    {
      "name": "Jizhou Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b32vkmc0wqsnx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35419",
  "fulltext_url": null,
  "id": "b32vkmc0wqsnx",
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  "landing_url": "https://arxiv.org/abs/2609.35419",
  "language": null,
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      "2610|Numerical methods in inverse problems"
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  "pdf_url": "https://arxiv.org/pdf/2609.35419",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b32vkmc0wqsnx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.35419",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35419"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions two to eighteen",
  "title_text": "Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions two to eighteen",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/b32vkmc0wqsnx",
  "venue": "arXiv",
  "versions": []
}