{
  "id": "2305.19934",
  "version": "v1",
  "published": "2023-05-31T15:18:52.000Z",
  "updated": "2023-05-31T15:18:52.000Z",
  "title": "On the Lavrentiev gap for convex, vectorial integral functionals",
  "authors": [
    "Lukas Koch",
    "Matthias Ruf",
    "Mathias Schäffner"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form $$ F: g+W_0^{1,1}(\\Omega)^m\\to\\mathbb{R}\\cup\\{+\\infty\\},\\qquad F(u)=\\int_\\Omega W(x,\\mathrm{D} u)\\,\\mathrm{d}x, $$ where the boundary datum $g:\\Omega\\subset \\mathbb{R}^d\\to\\mathbb{R}^m$ is sufficiently regular, $\\xi\\mapsto W(x,\\xi)$ is convex and lower semicontinuous, satisfies $p$-growth from below and suitable growth conditions from above. More precisely, if $p\\leq d-1$, we assume $q$-growth from above with $q\\leq \\frac{(d-1)p}{d-1-p}$, while for $p>d-1$ we require essentially no growth conditions from above and allow for unbounded integrands. Concerning the $x$-dependence, we impose a well-known local stability estimate that is redundant in the autonomous setting, but in the general non-autonomous case can further restrict the growth assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-31T15:18:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49K40"
    ],
    "keywords": [
      "vectorial integral functionals",
      "lavrentiev gap",
      "well-known local stability estimate",
      "growth assumptions",
      "general non-autonomous case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}