A probabilistic approach to the first Dirichlet eigenvalue on non-compact Riemannian manifold

Abstract

By using diffusion process with absorbing boundary, some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+∇h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean's estimate for ∇h=0. Moreover, the first Dirichlet eigenvalue for elliptic operators onR d and the first mixed eigenvalue are also studied. Some examples show that our estimates can be sharp even for ∇h≠0.