{
  "abstract": "We study the relations between seven generalized notions of convexity for continuous integrands in the space of symmetric $3\\times 3$ matrices. Here, symmetric quasiconvexity is defined through symmetrized gradients, and symmetric rank-one convexity through symmetrized rank-one lines. We construct a symmetric quasiconvex integrand that is not polyconvex at symmetric matrices, and an integrand that is both symmetric rank-one convex and Hessian quasiconvex but not symmetric quasiconvex. We also establish a transfer principle that converts smooth rank-one convex functions that lack Hessian quasiconvexity into symmetric rank-one convex integrands while preserving this lack of of Hessian quasiconvexity. Applying this scheme to already existing counterexamples yields the existence of symmetric rank-one convex integrands that are not Hessian quasiconvex. Together with known examples, these results determine all pairwise implications among the seven notions considered here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35216",
  "author_total": 1,
  "authors": [
    {
      "name": "Florian Gammer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35dbr4nd4h9hg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35216",
  "fulltext_url": null,
  "id": "35dbr4nd4h9hg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35216",
  "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": "W7223937912",
  "pdf_url": "https://arxiv.org/pdf/2610.35216",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35dbr4nd4h9hg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35216",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35216"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the symmetric Morrey problem in three dimensions",
  "title_text": "On the symmetric Morrey problem in three dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/35dbr4nd4h9hg",
  "venue": "arXiv",
  "versions": []
}