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arXiv · 1712.00753

Eigenvalue bounds of mixed Steklov problems

Asma Hassannezhad·Ari Laptev
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Abstract

We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain $Ω$ 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.

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Asma Hassannezhad, Ari Laptev. 2018-09-06. Eigenvalue bounds of mixed Steklov problems. https://arxiv.org/abs/1712.00753

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