{
  "id": "1707.06804",
  "version": "v1",
  "published": "2017-07-21T08:58:16.000Z",
  "updated": "2017-07-21T08:58:16.000Z",
  "title": "Traces of functions of bounded A-variation and variational problems with linear growth",
  "authors": [
    "Dominic Breit",
    "Lars Diening",
    "Franz Gmeineder"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the space $BV^{A}(\\Omega)$ of functions of bounded $A$-variation. For a given first order linear homogeneous differential operator with constant coefficients $A$, this is the space of $L^1$--functions $u:\\Omega\\rightarrow R^N$ such that the distributional differential expression $A u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\\Omega\\subset R^{n}$, $BV^{A}(\\Omega)$--functions have an $L^1(\\partial\\Omega)$--trace if and only if $A$ is $C$-elliptic (or, equivalently, if the kernel of $A$ is finite dimensional). The existence of an $L^1(\\partial\\Omega)$--trace was previously only known for the special cases that $A u$ coincides either with the full or the symmetric gradient of the function $u$ (and hence covered the special cases $BV$ or $BD$). As an application we study quasiconvex variational functionals with linear growth depending on $A u$ and show the existence of a minimiser in $BV^{A}(\\Omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-21T08:58:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D10",
      "06D05"
    ],
    "keywords": [
      "linear growth",
      "variational problems",
      "bounded a-variation",
      "first order linear homogeneous differential",
      "study quasiconvex variational functionals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}