Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda }{n}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\le c(D)\lambda \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$ holds for any Dirichlet eigenfunction $\phi$ of $-\mathrm{\Delta }$ with eigenvalue $\lambda$. Our results provide the sharp Hessian estimate $\mathrm{\parallel }\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\lesssim {\lambda }^{\frac{n+3}{4}}$. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.