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

TL;DR

This paper establishes upper bounds for the Steklov $p$-Laplacian eigenvalues on domains in $ ^n$, showing how these bounds depend on geometric properties and the parameters $p$, $k$, and $n$, with sharpness demonstrated.

Contribution

It introduces new upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian, highlighting the role of geometric constants for $p>n$ and proving their necessity.

Findings

Upper bounds depend on boundary measure for $1<p extless n$

Bounds depend on geometric constant $D( ext{domain})$ for $p>n$

Bounds are sharp with examples of large eigenvalues

Abstract

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.