Gradient estimates of Poisson equations on Riemannian manifolds and applications

Abstract

For the reflected diffusion generated by L = Δ − ∇ V ⋅ ∇ on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of sup x ∈ M | ∇ G | ( x ) of the Poisson equation − L G = g in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger's isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided.