Upper bounds for the Steklov eigenvalues of warped products

TL;DR

This paper establishes upper bounds for Steklov eigenvalues on warped product manifolds, relating geometric and spectral properties, with some bounds being optimal and providing stability insights.

Contribution

It introduces new upper bounds for Steklov eigenvalues on warped products involving volume, eigenvalues of the fiber, and warping function norms, with dimension-dependent results.

Findings

Derived bounds depend on fiber dimension and warping function norms.

Some bounds are proven to be optimal.

Provided stability estimates for certain cases.

Abstract

We obtain upper bounds for the Steklov eigenvalues of warped products $\Omega\times_h\Sigma$, where $\Omega$ is a compact Riemannian manifold with boundary and $\Sigma$ is a closed Riemannian manifold. These bounds involve the volume of $\Omega$ and of $\partial\Omega$ as well as the eigenvalues of the Laplace operator on the fiber $\Sigma$ and the $L^p$-norm of the warping function $h$. The bounds are very different depending on the dimension $n$ of the fiber $\Sigma$ and the value of $p$. In some cases, we obtain optimal upper bounds and stability estimates.