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Gradient Estimates on Dirichlet Eigenfunctions

By using stochastic analysis on Riemannian manifold, we derive explicit constants c1(D) and c2(D) for a d-dimensional compact Riemannian manifold D with boundary such that holds for any Dirichlet eigenfunction φ on D with eigenvalue λ. In particular, when D is convex with nonnegative Ricci curvature, this estimate holds for

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