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 $R^n$, $ngeq 2$ . We show that for $1n$ 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.