Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

Abstract

We consider the first eigenvalue λ1(Ω,σ) of the Laplacian with Robin boundary conditions on a compact Riemannian manifold Ω with smooth boundary, σ∈R being the Robin boundary parameter. When σ>0 we give a positive, sharp lower bound of λ1(Ω,σ) in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of Ω, a lower bound of the mean curvature of ∂Ω and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean domains are obtained, which improve known estimates.Then, we extend a monotonicity result for λ1(Ω,σ) obtained in Euclidean space by Giorgi and Smits [10], to a class of manifolds of revolution which include all space forms of constant sectional curvature. As an application, we prove that λ1(Ω,σ) is uniformly bounded below by (n−1)24 for all bounded domains in the hyperbolic space of dimension n, provided that the boundary parameter σ≥n−12 (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed.