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Abstract 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 n\, \| \| \|Hess \| c (D) \, \| \| holds for any Dirichlet eigenfunction of - on D with eigenvalue. Our results provide the sharp Hessian estimate \|Hess \| ^n+3{4}\| \|₋^₂. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.
Cheng et al. (Mon,) studied this question.