{
  "id": "math/0507079",
  "version": "v1",
  "published": "2005-07-04T18:57:54.000Z",
  "updated": "2005-07-04T18:57:54.000Z",
  "title": "Gradient Bounds for Solutions of Elliptic and Parabolic Equations",
  "authors": [
    "Vladimir I. Bogachev",
    "Giuseppe Da Prato",
    "Michael Röckner",
    "Zeev Sobol"
  ],
  "comment": "9 pages; BiBoS-Preprint 04-12-169; (BiBoS: http://www.physik.uni-bielefeld.de/bibos/)",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $L$ be a second order elliptic operator on $R^d$ with a constant diffusion matrix and a dissipative (in a weak sense) drift $b \\in L^p_{loc}$ with some $p>d$. We assume that $L$ possesses a Lyapunov function, but no local boundedness of $b$ is assumed. It is known that then there exists a unique probability measure $\\mu$ satisfying the equation $L^*\\mu=0$ and that the closure of $L$ in $L^1(\\mu)$ generates a Markov semigroup $\\{T_t\\}_{t\\ge 0}$ with the resolvent $\\{G_\\lambda\\}_{\\lambda > 0}$. We prove that, for any Lipschitzian function $f\\in L^1(\\mu)$ and all $t,\\lambda>0$, the functions $T_tf$ and $G_\\lambda f$ are Lipschitzian and |\\nabla T_tf(x)| \\leq T_t|\\nabla f|(x) and |\\nabla G_\\lambda f(x)| \\leq \\frac{1}{\\lambda} G_\\lambda |\\nabla f|(x). An analogous result is proved in the parabolic case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-07-04T18:57:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "35K10"
    ],
    "keywords": [
      "parabolic equations",
      "gradient bounds",
      "second order elliptic operator",
      "constant diffusion matrix",
      "unique probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7079B"
    }
  }
}