Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

Abstract

We prove that in Riemannian manifolds the k-th Steklov eigenvalue on a domain and the square root of the k-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min–max variational characterization of both eigenvalues are important ingredients.