Gradient Bounds for Solutions of Elliptic and Parabolic Equations

Details Gradient Bounds for Solutions of Elliptic and Parabolic Equations Gradient Bounds for Solutions of Elliptic and Parabolic Equations

2005-07-04

arxiv.org/abs/math/0507079v1

Mathematics

Probability

9 pages; BiBoS-Preprint 04-12-169; (BiBoS: http://www.physik.uni-bielefeld.de/bibos/)

Scientific paper

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.

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