{
  "id": "2211.15246",
  "version": "v1",
  "published": "2022-11-28T11:52:54.000Z",
  "updated": "2022-11-28T11:52:54.000Z",
  "title": "Borderline gradient continuity for the normalized $p$-parabolic operator",
  "authors": [
    "Murat Akman",
    "Agnid Banerjee",
    "Isidro H. Munive"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1904.13076",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we prove gradient continuity estimates for viscosity solutions to $\\Delta_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $\\Delta_{p}^N$ is the game theoretic normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for $u$ in terms of the modified parabolic Riesz potential $\\mathbf{P}^{f}_{n+1}$ as defined in (2.8) below. Moreover, for $f \\in L^{m}$ with $m>n+2$, we also obtain H\\\"older continuity of the spatial gradient of the solution $u$, see Theorem 2.6 below. This improves the gradient H\\\"older continuity result in [3] which considers bounded $f$. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in [9]. Moreover differently from that in [3], our approach is independent of the Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-28T11:52:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic operator",
      "constitutes borderline gradient continuity estimate",
      "main results theorem",
      "modified parabolic riesz potential",
      "game theoretic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}