{
  "id": "1701.08332",
  "version": "v1",
  "published": "2017-01-28T23:28:13.000Z",
  "updated": "2017-01-28T23:28:13.000Z",
  "title": "Boundary value problems in Lipschitz domains for equations with drifts",
  "authors": [
    "Georgios Sakellaris"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-{\\rm div}(A\\nabla\\cdot)+b\\nabla\\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{\\rm div}(A^t\\nabla\\cdot)-{\\rm div}(b\\,\\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on ${\\rm div}b$ in order to obtain solvability. On the other hand, we show that $R_2$ for $L$ and $D_2$ for $L^t$ are uniquely solvable, assuming only that $A$ is Lipschitz continuous (and not necessarily symmetric) and $b$ is bounded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-28T23:28:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary value problems",
      "second order elliptic operators",
      "adjoint operators",
      "solvability",
      "regularity problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}