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 $\frac{\lambda }{n}\,\|\phi \|_{\infty } \leq \|\operatorname{Hess}\phi \|_{\infty } \leq c(D)\lambda \,\|\phi \|_{\infty } $ holds for any Dirichlet eigenfunction $\phi $ of $-\Delta $ on $D$ with eigenvalue $\lambda $. Our results provide the sharp Hessian estimate $\|\operatorname{Hess} \phi \|_{\infty }\lesssim \lambda ^{\frac{n+3}{4}}\|\phi \|_{L^{2}}$. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.