Upper bounds for the Steklov eigenvalues of warped products

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.