{
  "id": "1111.1468",
  "version": "v1",
  "published": "2011-11-07T01:42:56.000Z",
  "updated": "2011-11-07T01:42:56.000Z",
  "title": "The mixed problem in Lipschitz domains with general decompositions of the boundary",
  "authors": [
    "Justin L. Taylor",
    "Katharine A. Ott",
    "Russell M. Brown"
  ],
  "comment": "36 pages",
  "journal": "Trans. Amer. Math. Soc., 365 (2013), 2895-2930",
  "doi": "10.1090/S0002-9947-2012-05711-4",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\\Omega\\subset \\reals^n$, $n\\geq2$, with boundary that is decomposed as $\\partial\\Omega=D\\cup N$, $D$ and $N$ disjoint. We let $\\Lambda$ denote the boundary of $D$ (relative to $\\partial\\Omega$) and impose conditions on the dimension and shape of $\\Lambda$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $\\Omega$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev space $W^ {1,p}(D) $ has a unique solution with the non-tangential maximal function of the gradient of the solution in $L^p(\\partial\\Omega)$. We also obtain results for $p=1$ when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-07T01:42:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "35J05"
    ],
    "keywords": [
      "mixed problem",
      "lipschitz domain",
      "general decompositions",
      "sobolev space",
      "non-tangential maximal function"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.1468T"
    }
  }
}