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research
Eigenvalue bounds of mixed Steklov problems
Authors
Asma Hassannezhad
Ari Laptev
Publication date
6 September 2018
Publisher
Doi
Cite
View
on
arXiv
Abstract
We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain
Ω
\Omega
Ω
in
R
n
\mathbb{R}^n
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.Comment: An appendix by by F. Ferrulli and J. Lagac\'e is added; some changes in the introduction are mad
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Last time updated on 17/09/2019