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.

12 pages. Accepted version. Accepted by J. Funct. Anal. The condition "simply connected" in the main theorem was removed, as suggested by the referee. This version was submitted to the journal in August, 2018

References in corpus (1)

• Spectral geometry of the Steklov problem

Cited by in corpus (2)

• Compact manifolds with fixed boundary and large Steklov eigenvalues
• Some sharp bounds for Steklov eigenvalues