Upper bounds for the Steklov eigenvalues of warped products

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