{
  "id": "1008.2089",
  "version": "v2",
  "published": "2010-08-12T10:51:32.000Z",
  "updated": "2011-01-15T15:57:40.000Z",
  "title": "Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures",
  "authors": [
    "Filip Rindler"
  ],
  "comment": "48 pages, 3 figures (revised version)",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We establish a general weak* lower semicontinuity result in the space $\\BD(\\Omega)$ of functions of bounded deformation for functionals of the form $$\\Fcal(u) := \\int_\\Omega f \\bigl(x, \\Ecal u \\bigr) \\dd x + \\int_\\Omega f^\\infty \\Bigl(x, \\frac{\\di E^s u}{\\di \\abs{E^s u}} \\Bigr) \\dd \\abs{E^s u} + \\int_{\\partial \\Omega} f^\\infty \\bigl(x, u|_{\\partial \\Omega} \\odot n_\\Omega \\bigr) \\dd \\Hcal^{d-1}$$, $u \\in \\BD(\\Omega)$. The main novelty is that we allow for non-vanishing Cantor-parts in the symmetrized derivative $Eu$. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, which is based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result without an Alberti-type theorem in $\\BD(\\Omega)$, which is not available at present. We also include existence and relaxation results for variational problems in $\\BD(\\Omega)$, as well as a complete discussion of some differential inclusions for the symmetrized gradient in two dimensions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-01-15T15:57:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J45",
      "35J50",
      "28B05",
      "49Q20",
      "74B05",
      "74C10"
    ],
    "keywords": [
      "young measures",
      "bounded deformation",
      "integral functionals",
      "lower semicontinuity result",
      "differential inclusions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00205-011-0408-0",
      "journal": "Archive for Rational Mechanics and Analysis",
      "year": 2011,
      "month": "Oct",
      "volume": 202,
      "number": 1,
      "pages": 63
    },
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011ArRMA.202...63R"
    }
  }
}