Hessian Estimates for Dirichlet and Neumann Eigenfunctions of Laplacian

By methods of stochastic analysis on Riemannian manifolds, we develop an approach to determine an explicit constant $c(D)$ for an n-dimensional compact manifold D with smooth boundary such that λ /n\,\|φ \|∞ ≤ \|Hessφ \|∞ ≤ c(D)λ \,\|φ \|∞ holds for any Dirichlet eigenfunction φ of -Δ on D with eigenvalue λ. Our results provide the sharp Hessian estimate \|Hess φ \|∞ λ n+3/4\|φ \|_L². Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.