{
  "abstract": "We construct a function $W:\\mathbb{R}^{3\\times3}\\to\\mathbb{R}$ which is frame-indifferent, $W(QF)=W(F)$ for all $Q\\in\\mathrm{SO}(3)$, rank-one convex, but not quasiconvex. In a neighbourhood of $\\mathrm{SO}(3)$ the function is an explicit polynomial of degree four in the strain $F^TF-\\mathrm{Id}$; the failure of quasiconvexity occurs at $F=\\mathrm{Id}$ through a third-order mechanism of Šverák type, with test functions whose gradients are arbitrarily close to $\\mathrm{Id}$. The polynomial is obtained from a geometrically linear example, a rank-one convex, non-quasiconvex quartic polynomial of the symmetric part of the gradient, by evaluating it at the strain $\\frac12(F^TF-\\mathrm{Id})$, adding a uniaxial prestress and a cubic correction, and enlarging the quartic term. The local function is extended to all of $\\mathbb{R}^{3\\times3}$ by adding a frame-indifferent polyconvex function which vanishes near $\\mathrm{SO}(3)$. This work was carried out with the assistance of the AI model Claude Opus 5.5; the proofs of the main results have been formally verified in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31886",
  "author_total": 3,
  "authors": [
    {
      "name": "Sergio Conti"
    },
    {
      "name": "Georg Dolzmann"
    },
    {
      "name": "Patrick Dondl"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dq6wcda3k071g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31886",
  "fulltext_url": null,
  "id": "dq6wcda3k071g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31886",
  "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",
      "2204|Elasticity and Material Modeling"
    ]
  ],
  "openalex": "W7223681532",
  "pdf_url": "https://arxiv.org/pdf/2610.31886",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dq6wcda3k071g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.31886",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31886"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A frame-indifferent rank-one convex function in three dimensions which is not quasiconvex",
  "title_text": "A frame-indifferent rank-one convex function in three dimensions which is not quasiconvex",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/dq6wcda3k071g",
  "venue": "arXiv",
  "versions": []
}