{
  "id": "2007.04064",
  "version": "v1",
  "published": "2020-07-08T12:29:23.000Z",
  "updated": "2020-07-08T12:29:23.000Z",
  "title": "Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions",
  "authors": [
    "Andrea Gentile"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \\begin{equation*} \\min \\left\\{\\int_{\\Omega}f(x, Dv(x))\\,:\\, v\\in \\mathcal{K}_\\psi(\\Omega)\\right\\}, \\end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\\mathcal{K}_\\psi(\\Omega)$ is the class of admissible functions $v\\in u_0+W^{1, p}_0(\\Omega)$ such that $v\\ge\\psi$ a. e. in $\\Omega$, where $u_0\\in W^{1,p}(\\Omega)$ is a fixed boundary datum. Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the gradient of the obstacle $\\psi$ transfers to the gradient of the solution, provided the partial map $x\\mapsto D_\\xi f(x,\\xi)$ belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i. e. $f(x, \\xi)\\approx a(x)|\\xi|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev or Besov-Lipschitz space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-08T12:29:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J87",
      "49J40",
      "47J20"
    ],
    "keywords": [
      "higher differentiability results",
      "non-autonomous obstacle problems",
      "sub-quadratic growth conditions",
      "subquadratic growth conditions",
      "besov-lipschitz regularity assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}