{
  "id": "2401.05550",
  "version": "v1",
  "published": "2024-01-10T21:30:51.000Z",
  "updated": "2024-01-10T21:30:51.000Z",
  "title": "Counterexamples to maximal regularity for operators in divergence form",
  "authors": [
    "Sebastian Bechtel",
    "Connor Mooney",
    "Mark Veraar"
  ],
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such operators admit maximal $L^2$-regularity on $H^{-1}$ under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal $L^p$-regularity on $H^{-1}(\\mathbb{R}^d)$ or $L^2$-regularity on $L^2(\\mathbb{R}^d)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-10T21:30:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K90",
      "35B65"
    ],
    "keywords": [
      "divergence form",
      "maximal regularity",
      "counterexamples",
      "operators admit maximal",
      "parabolic pde"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}