Gradient Estimates on Dirichlet and Neumann Eigenfunctions

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants c(1) (D) and c(2) (D) for a d-dimensional compact Riemannian manifold D with boundary such that c(1) (D)root parallel to phi parallel to(infinity) <= parallel to del phi parallel to(infinity) <= c(2) (D) root lambda parallel to phi parallel to(infinity) holds for any Dirichlet eigenfunction phi of -Delta with eigenvalue lambda. In particular, when D is convex with nonnegative Ricci curvature, the estimate holds for c(1) (D) = 1/de and c(2) (D) = root e (root 2/root pi + root pi/4 root 2). Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.