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By methods of stochastic analysis on Riemannian manifolds, we develop twoapproaches to determine an explicit constant $c(D)$ for an $n$-dimensionalcompact manifold $D$ with boundary such that$\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq \|{\rm Hess}\ \phi\|_{\infty}\leqc(D)\lambda \,\|\phi\|_{\infty}$ holds for any Dirichlet eigenfunction $\phi$of $-\Delta$ with eigenvalue $\lambda$. Our results provide the sharp Hessianestimate $\|{\rm Hess}\ \phi\|_{\infty}\lesssim \lambda^{\frac{n+3}{4}}$.Corresponding Hessian estimates for Neumann eigenfunctions are derived in thesecond part of the paper.

Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian