The Steklov spectrum of warped products

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, under assumptions on the $L^p$-norm of the warping function $h$. The upper bounds that we obtain 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. This is joint work with Jade Brisson, Bruno Colbois and Katie Gittins.