Gradient estimates of Poisson equations on Riemannian manifolds and applications
Let \(\mathcal L\) be the reflected diffusion generated by \(\Delta -\nabla V \cdot \nabla\) over a connected and complete Riemannian manifold \(M\) with empty or convex boundary. The author establishes some sharp estimates for \(\sup_{x\in M}|\nabla G|(x)\) in terms of the dimension, the diameter and the lower bound of curvature, where \(\mathcal G\) is the solution to the Poisson equation \(-\mathcal LG=g.\) Applications are given to the transportation-information inequality, to Cheeger's isoperimetric inequality and to the Gaussian concentration inequality.
scientific article; zbMATH DE number 5649986
scientific article