GRADIENT AND HARNACK INEQUALITIES ON NONCOMPACT MANIFOLDS WITH BOUNDARY

By using the reflecting diffusion process and a conformal change of metric, a generalized maximum principle is established for (unbounded) time-space functions on a class of noncompact Riemannian manifolds with (nonconvex) boundary. As applications, Li-Yau-type gradient and Harnack inequalities are derived for the Neumann semigroup on a class of noncompact manifolds with (nonconvex) boundary. These generalize some previous ones obtained for the Neumann semigroup on compact manifolds with boundary. As a byproduct, the gradient inequality for the Neumann semigroup derived by Hsu on a compact manifold with boundary is confirmed on these noncompact manifolds.