Gradient Bounds for Solutions of Elliptic and Parabolic Equations

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.