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

TL;DR

This paper establishes bounds relating Steklov eigenvalues of a domain to Laplacian eigenvalues on its boundary in Riemannian manifolds, using curvature conditions and variational methods, and derives a Weyl-type upper bound.

Contribution

It provides new inequalities linking Steklov and Laplacian eigenvalues in Riemannian geometry, incorporating curvature constraints and variational principles.

Findings

Mutual control of Steklov and Laplacian eigenvalues under curvature conditions

Derivation of a Weyl-type upper bound for Steklov eigenvalues

Use of Pohozaev identity and variational characterization in proofs

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.