{
  "id": "1811.12619",
  "version": "v1",
  "published": "2018-11-30T05:24:30.000Z",
  "updated": "2018-11-30T05:24:30.000Z",
  "title": "Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains",
  "authors": [
    "Hyunseok Kim",
    "Hyunwoo Kwon"
  ],
  "comment": "33 pages, 4 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: $$-\\triangle u +\\operatorname{div}(u\\mathbf{b}) =f \\quad\\text{ and }\\quad -\\triangle v -\\mathbf{b} \\cdot \\nabla v =g$$ in a bounded Lipschitz domain $\\Omega$ in $\\mathbb{R}^n$ $(n\\geq 3)$, where $\\mathbf{b}:\\Omega \\rightarrow \\mathbb{R}^n$ is a given vector field. Under the assumption that $\\mathbf{b} \\in L^{n}(\\Omega)^n$, we first establish existence and uniqueness of solutions in $L_{\\alpha}^{p}(\\Omega)$ for the Dirichlet and Neumann problems. Here $L_{\\alpha}^{p}(\\Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(\\alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995, JFA) and Fabes-Mendez-Mitrea (1998, JFA) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(\\partial\\Omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-30T05:24:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "35J25"
    ],
    "keywords": [
      "neumann problems",
      "singular drifts",
      "second-order linear elliptic equations",
      "bessel potential space",
      "bounded lipschitz domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}