{
  "id": "2001.00815",
  "version": "v1",
  "published": "2020-01-03T13:11:32.000Z",
  "updated": "2020-01-03T13:11:32.000Z",
  "title": "Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth",
  "authors": [
    "Michał Łasica",
    "Piotr Rybka"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$ space. Under the assumption of convexity of the domain of integration, we prove that if the datum is in $W^{1,1}$, then the functional has a minimizer in $W^{1,1}$. In fact, the minimizer inherits $W^{1,p}$ regularity from the datum for any $p \\in [1, +\\infty]$. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth, possibly defined on vector-valued maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-03T13:11:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35B65",
      "35J60",
      "35J70",
      "35J75"
    ],
    "keywords": [
      "linear growth",
      "variational problems",
      "convex integral functionals",
      "fidelity term",
      "convex integrand"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}