{
  "id": "2207.10554",
  "version": "v1",
  "published": "2022-07-21T15:52:41.000Z",
  "updated": "2022-07-21T15:52:41.000Z",
  "title": "$L^p$-solvability of the Poisson-Dirichlet problem and its applications to the regularity problem",
  "authors": [
    "Mihalis Mourgoglou",
    "Bruno Poggi",
    "Xavier Tolsa"
  ],
  "comment": "65 pages",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\\operatorname{div} A\\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\\operatorname{div} F$, assuming that $H$ and $F$ lie in certain Carleson-type spaces, and that the domain $\\Omega\\subset\\mathbb R^{n+1}$, $n\\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular boundary. The $L^{p'}$-solvability of the Poisson problem (with an $L^{p'}$ estimate on the non-tangential maximal function) is new even when $L=-\\Delta$ and $\\Omega$ is the unit ball. In turn, we use this result to show that, in a bounded domain with uniformly $n$-rectifiable boundary that satisfies the corkscrew condition, $L^{p'}$-solvability of the homogeneous Dirichlet problem for an operator $L=-\\operatorname{div} A\\nabla$ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the $L^p$-regularity problem for the adjoint operator $L^*=-\\operatorname{div} A^T \\nabla$, where $1/p+1/p'=1$ and $A^T$ is the transpose matrix of $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-21T15:52:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J05",
      "35J15",
      "35J25",
      "35J08",
      "35B30",
      "31B20",
      "31B35",
      "42B25",
      "42B35",
      "42B37"
    ],
    "keywords": [
      "regularity problem",
      "solvability",
      "poisson-dirichlet problem",
      "homogeneous dirichlet problem",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}