Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds
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.