{
  "id": "1201.1416",
  "version": "v2",
  "published": "2012-01-06T14:04:18.000Z",
  "updated": "2015-12-12T17:00:25.000Z",
  "title": "On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on ${\\mathbb R}^n$",
  "authors": [
    "Jean Bourgain",
    "Mikhail V. Korobkov",
    "Jan Kristensen"
  ],
  "journal": "J.Reine Angew.Math. 700 (2015), 93--112",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We establish Luzin N and Morse--Sard properties for functions from the Sobolev space $W^{n,1}({\\mathbb R}^{n})$. Using these results we prove that almost all level sets are finite disjoint unions of $C^1$--smooth compact manifolds of dimension $n-1$. These results remain valid also within the larger space of functions of bounded variation $BV_{n}({\\mathbb R}^{n})$. For the proofs we establish and use some new results on Luzin--type approximation of Sobolev and BV--functions by $C^k$--functions, where the exceptional sets have small Hausdorff content.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-06T14:04:18.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-12T17:00:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58C25",
      "46E35"
    ],
    "keywords": [
      "level sets",
      "morse-sard property",
      "sobolev functions",
      "finite disjoint unions",
      "smooth compact manifolds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}