{
  "id": "2209.01462",
  "version": "v1",
  "published": "2022-09-03T16:46:37.000Z",
  "updated": "2022-09-03T16:46:37.000Z",
  "title": "On functions with given boundary data and convex constraints on the gradient",
  "authors": [
    "Camilla Brizzi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega\\subset\\mathbb{R}^{d}$ be an open set. Given a boundary datum $g$ on $\\partial\\Omega$ and a function $K:\\bar {\\Omega} \\to\\mathcal{K}$, the family of all compact convex subsets of $\\mathbb{R}^{d}$, we prove the existence of functions $u:\\Omega\\to\\mathbb{R}$ such that $u=g$ on $\\partial\\Omega$ and $\\nabla u(x)\\in K(x)$ a.e. and we investigate the regularity of such solutions on the set $\\mathcal{U} \\subset \\bar{\\Omega}$ of points at which they all coincide.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-03T16:46:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49K24",
      "49N15",
      "49N60",
      "35R70",
      "58B20"
    ],
    "keywords": [
      "boundary datum",
      "convex constraints",
      "compact convex subsets",
      "open set",
      "regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}