{
  "id": "2007.03043",
  "version": "v1",
  "published": "2020-07-06T20:10:00.000Z",
  "updated": "2020-07-06T20:10:00.000Z",
  "title": "Criterion for the functional dissipativity of second order differential operators with complex coefficients",
  "authors": [
    "Alberto Cialdea",
    "Vladimir Maz'ya"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In the present paper we consider the Dirichlet problem for the second order differential operator $E=\\nabla(A \\nabla)$,where $A$ is a matrix with complex valued $L^\\infty$ entries. We introduce the concept of dissipativity of $E$ with respect to a given function $\\varphi:R^+ \\to R^+$. Under the assumption that the $Im\\, A$ is symmetric, we prove that the condition $|s\\, \\varphi'(s)| \\, | \\langle Im\\, A (x)\\, \\xi,\\xi\\rangle |\\leq 2\\, \\sqrt{\\varphi(s)\\, [s\\, \\varphi(s)]'}\\, \\langle Re\\, A(x) \\, \\xi,\\xi\\rangle $ (for almost every $x\\in\\Omega\\subset R^N$ and for any $s>0$, $\\xi\\in R^N$) is necessary and sufficient for the functional dissipativity of $E$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-06T20:10:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B44",
      "35L30"
    ],
    "keywords": [
      "second order differential operator",
      "functional dissipativity",
      "complex coefficients",
      "dirichlet problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}