{
  "id": "2301.11795",
  "version": "v1",
  "published": "2023-01-27T15:52:32.000Z",
  "updated": "2023-01-27T15:52:32.000Z",
  "title": "Higher regularity for weak solutions to degenerate parabolic problems",
  "authors": [
    "Andrea Gentile",
    "Antonia Passarelli di Napoli"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \\begin{equation*} u_t-\\div\\left(\\left(\\left|Du\\right|-1\\right)_+^{p-1}\\frac{Du}{\\left|Du\\right|}\\right)=f\\qquad\\mbox{ in }\\Omega_T, \\end{equation*} where $\\Omega$ is a bounded domain in $\\mathbb{R}^{n}$ for $n\\geq2$, $p\\geq2$ and $\\left(\\,\\cdot\\,\\right)_{+}$ stands for the positive part. We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that $f\\in L^{2}_{\\loc}\\left(\\Omega_T\\right)$. This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-27T15:52:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B45",
      "35B65",
      "35D30",
      "35K10",
      "35K65"
    ],
    "keywords": [
      "weak solutions",
      "degenerate parabolic problems",
      "higher regularity",
      "spatial gradient",
      "strongly degenerate parabolic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}