Upper bounds for the Steklov eigenvalues of the $p$-Laplacian

In this note we present upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian on domains of $\mathbb R^n$, $n\geq 2$. We show that for $1<p\leq n$ the variational eigenvalues $\sigma_{p,k}$ are bounded above in terms of $k,p,n$ and $|\partial\Omega|$ only. In the case $p>n$ upper bounds depend on a geometric constant $D(\Omega)$, the $(n-1)$-distortion of $\Omega$ which quantifies the concentration of the boundary measure. We prove that the presence of this constant is necessary in the upper estimates for $p>n$ and that the corresponding inequality is sharp, providing examples of domains with boundary measure uniformly bounded away from zero and infinity and arbitrarily large variational eigenvalues.