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

Abstract

In this note we present upper bounds for the variational eigenvalues of the Steklov p-Laplacian on domains of R n , n ≥ 2. We show that for 1 < p ≤ n the variational eigenvalues σ p,k are bounded above in terms of k, p, n and |∂Ω| only. In the case p > n upper bounds depend on a geometric constant D(Ω), the (n − 1)-distortion of Ω 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.