{
  "abstract": "We study the asymptotic behaviour, as $\\varepsilon\\to 0^+$, of the $p$-Dirichlet energy of maps $u\\colonΩ\\subset\\mathbb{R}^2 \\to \\mathbb{R}^2$ constrained to take values in the rescaled periodic network $\\varepsilon\\mathcal Z$, where $\\mathcal Z$ is a standard square network in $\\mathbb{R}^2$. The oscillations are carried by the constraint in the target, not by the energy density, and their effect depends on $p$. For $1<p<2$ the constraint disappears in the limit: the $Γ$-limit is an integral functional on the whole of $\\rm{W}^{1,p}(Ω, \\mathbb{R}^2 )$, whose density is given by an asymptotic cell formula on tilted cells. This density is equal to $(|a_1|+|a_2|)^p$ on rank-one matrices $a\\otimesν$, $|ν|=1$, and exceeds its convex envelope by a quantity comparable to $|\\det A|\\,|A|^{p-2}$, with constants of order $(2-p)^{-1}$. Consequently, although the original energy density is convex, the limit density is quasiconvex but neither convex nor polyconvex: quasiconvexity is generated by the geometry of the constraint. For $p\\ge2$ the constraint survives in the limit as $\\det\\nabla u=0$, and the $Γ$-limit is the degenerate energy $\\int_Ω(|\\nabla u_1|+|\\nabla u_2|)^p\\, \\rm{d} x$, that is, the $p$-energy of $u$ considered as a map into the normed plane $(\\mathbb{R}^2,|\\cdot|_1)$, the metric limit of the networks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32444",
  "author_total": 3,
  "authors": [
    {
      "name": "Andrea Braides"
    },
    {
      "name": "Jan Kristensen"
    },
    {
      "name": "Sergio Scalabrino"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f9hpnq0tke8aq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32444",
  "fulltext_url": null,
  "id": "f9hpnq0tke8aq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32444",
  "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|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223689248",
  "pdf_url": "https://arxiv.org/pdf/2610.32444",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f9hpnq0tke8aq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32444",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32444"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Energies with values in networks and geometric quasiconvexity",
  "title_text": "Energies with values in networks and geometric quasiconvexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/f9hpnq0tke8aq",
  "venue": "arXiv",
  "versions": []
}