{
  "id": "2406.16735",
  "version": "v1",
  "published": "2024-06-24T15:43:27.000Z",
  "updated": "2024-06-24T15:43:27.000Z",
  "title": "The $L^p$ Poisson-Neumann problem and its relation to the Neumann problem",
  "authors": [
    "Joseph Feneuil",
    "Linhan Li"
  ],
  "comment": "49 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\\rm{div }A\\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $\\Omega$, which considers solutions to $Lu=h-\\rm{div}\\vec{F}$ in $\\Omega$ with zero Neumann data on the boundary for $h$ and $\\vec F$ in some tent spaces. We give different characterizations of solvability of the $L^p$ Poisson-Neumann problem and its weaker variants, and in particular, we show that solvability of the weak $L^p$ Poisson-Neumann probelm is equivalent to a weak reverse H\\\"older inequality. We show that the Poisson-Neumman problem is closely related to the $L^p$ Neumann problem, whose solvability is a long-standing open problem. We are able to improve the extrapolation of the $L^p$ Neumann problem from Kenig and Pipher by obtaining an extrapolation result on the Poisson-Neumann problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-24T15:43:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25"
    ],
    "keywords": [
      "poisson-neumann problem",
      "chord arc domain",
      "zero neumann data",
      "solvability",
      "extrapolation result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}