{
  "abstract": "{Let $λ_i = ν_i(F)$ denote the three singular values of the deformation gradient $F \\in {\\rm GL}^+(3)$.} We consider the family of incompressible isotropic energies $ W_ψ(F)=\\sum_i ψ\\left(|\\!\\logλ_i|\\right)$ with $ψ:[0,\\infty)\\mapsto\\mathbb{R}$. Set $g(s)=ψ\\left({\\rm arcosh}\\frac{s}{2}\\right)$ for all $s\\geq2$. If $g$ has a convex and non-decreasing extension $\\bar g$ to $[0,\\infty)$, then $W_ψ$ is the restriction to ${\\rm SL}(3)$ of the explicit polyconvex function \\ {$ F\\mapsto\\sum_i \\bar g\\left(ν_i(F+{\\rm Cof} F)\\right). $} The proof uses the identity $ν_i(F+{\\rm Cof} F)=λ_i+λ_i^{-1}$, up to permutation, on ${\\rm SL}(3)$ and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in ${\\rm SL}(3)$ and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\\mapsto\\sum_i e^{\\log^2\\!λ_i} $ is rank-one convex on ${\\rm SL}(3)$ and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in $\\log λ_i$ alone is insufficient; the quadratic Hencky energy $\\sum_i \\log^2λ_i$ is not rank-one convex on ${\\rm SL}(3)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18660",
  "author_total": 3,
  "authors": [
    {
      "name": "Ionel-Dumitrel Ghiba"
    },
    {
      "name": "Maximilian P. Wollner"
    },
    {
      "name": "Patrizio Neff"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bz9sky1gpqcyy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18660",
  "fulltext_url": null,
  "id": "bz9sky1gpqcyy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18660",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18660",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bz9sky1gpqcyy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18660",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18660"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type",
  "title_text": "Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/bz9sky1gpqcyy",
  "venue": "arXiv",
  "versions": []
}