{
  "id": "0705.0836",
  "version": "v1",
  "published": "2007-05-07T01:39:44.000Z",
  "updated": "2007-05-07T01:39:44.000Z",
  "title": "Analyticity of layer potentials and $L^{2}$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\\infty}$ coefficients",
  "authors": [
    "M. Alfonseca",
    "P. Auscher",
    "A. Axelsson",
    "S. Hofmann",
    "S. Kim"
  ],
  "journal": "Adv. Math. 226 (2011), no. 5, 4533--4606",
  "doi": "10.1016/j.aim.2010.12.014",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We consider divergence form elliptic operators of the form $L=-\\dv A(x)\\nabla$, defined in $R^{n+1} = \\{(x,t)\\in R^n \\times R \\}$, $n \\geq 2$, where the $L^{\\infty}$ coefficient matrix $A$ is $(n+1)\\times(n+1)$, uniformly elliptic, complex and $t$-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on $L^2(\\mathbb{R}^{n})=L^2(\\partial\\mathbb{R}_{+}^{n+1})$, is stable under complex, $L^{\\infty}$ perturbations of the coefficient matrix. Using a variant of the $Tb$ Theorem, we also prove that the layer potentials are bounded and invertible on $L^2(\\mathbb{R}^n)$ whenever $A(x)$ is real and symmetric (and thus, by our stability result, also when $A$ is complex, $\\Vert A-A^0\\Vert_{\\infty}$ is small enough and $A^0$ is real, symmetric, $L^{\\infty}$ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with $L^2$ (resp. $\\dot{L}^2_1)$ data, for small complex perturbations of a real symmetric matrix. Previously, $L^2$ solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients $A_{j,n+1}=0=A_{n+1,j}$, $1\\leq j\\leq n$, which corresponds to the Kato square root problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-07T01:39:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25",
      "35J25"
    ],
    "keywords": [
      "divergence form elliptic equations",
      "boundary value problems",
      "layer potential",
      "solvability",
      "coefficient matrix"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.0836A"
    }
  }
}