Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian

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