Harnack Inequalities on Manifolds with Boundary and Applications

Abstract

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $μ$ and for $K$ a constant, the curvature condition $\Ric-\nn Z\ge K$ together with the convexity of the boundary is equivalent to the heat kernel entropy inequality $$\int_M p_t(x,z)\log \ff{p_t(x,z)}{p_t(y,z)} μ(\d z)\le \ff{K\rr(x,y)^2}{2(\e^{2Kt}-1)}, t>0, x,y\in M,$$ where $\rr$ is the Riemannian distance. The main result is partly extended to manifolds with non-convex boundary and applied to derive the HWI inequality.