{
  "id": "1212.3779",
  "version": "v1",
  "published": "2012-12-16T11:55:59.000Z",
  "updated": "2012-12-16T11:55:59.000Z",
  "title": "Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope",
  "authors": [
    "Luigi Ambrosio",
    "Maria Colombo",
    "Simone Di Marino"
  ],
  "comment": "The occasion for writing the paper has been the course given by the first author in Sapporo (July-August 2012). 46 pages. arXiv admin note: substantial text overlap with arXiv:1111.3730",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\\sfd,\\mm)$, $1<q<\\infty$, in metric measure spaces $(X,\\sfd,\\mm)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\\Gamma$-convergence; this result extends Cheeger's work because no Poincar\\'e inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $\\mm$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-16T11:55:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J52",
      "49M25",
      "49Q20",
      "58J35",
      "35K90",
      "31C25"
    ],
    "keywords": [
      "metric measure spaces",
      "sobolev space",
      "lower semicontinuity",
      "reflexivity",
      "result extends cheegers work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.3779A"
    }
  }
}