Eigenvalue bounds of mixed Steklov problems

TL;DR

This paper establishes sharp bounds on the Riesz means of eigenvalues for mixed Steklov problems, including the sloshing problem, using variational principles and fractional Laplacian inequalities.

Contribution

It provides the first two-term asymptotically sharp bounds for Riesz means of mixed Steklov eigenvalues, advancing spectral theory in this area.

Findings

Two-term asymptotic lower bounds for sloshing eigenvalues

Asymptotically sharp upper bounds for Steklov-Dirichlet eigenvalues

Discussion of asymptotic sharpness of bounds in the appendix

Abstract

We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain $\Omega$ in $\mathbb{R}^n$. The Steklov-Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov-Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov-Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the appendix which in particular show the asymptotic sharpness of the bounds we obtain.