Eigenvalue bounds of mixed Steklov problems

Research paper by Asma Hassannezhad, Ari Laptev

Indexed on: 03 Dec '17Published on: 03 Dec '17Published in: arXiv - Mathematics - Spectral Theory

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 such 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.