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 $\frac{\lambda }{n}\Vert \varphi {\Vert }_{\infty }\le \Vert \mathrm{Hess}\varphi {\Vert }_{\infty }\le c(D)\lambda \Vert \varphi {\Vert }_{\infty }$ holds for any Dirichlet eigenfunction $\varphi$ of $-\Delta$ with eigenvalue $\lambda$. Our results provide the sharp Hessian estimate $\Vert \mathrm{Hess}\varphi {\Vert }_{\infty }\lesssim {\lambda }^{\frac{n+3}{4}}$. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.