Upper bound for the first non-zero eigenvalue of the $p$-Laplacian

Abstract

Let $M$ be a closed hypersurface in ${\mathbb{R}}^n$ and $\Omega$ be a bounded domain such that $M=\partial \Omega$. In this article, we obtain an upper bound for the first non-zero eigenvalue of the following problems. \begin{itemize} \item Closed eigenvalue problem: \begin{align*} %\label{eqn:closedep} Δ_p u = λ_{p} \ |u|^{p-2} \ u \qquad \mbox{ on } \quad {M}. \end{align*} \item Steklov eigenvalue problem: \begin{align*} \begin{array}{rcll} Δ_{p}u &=& 0 & \mbox{ in } Ω,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial ν} &=& μ_{p} \ |u|^{p-2} \ u &\mbox{ on } M . \end{array} \end{align*} \end{itemize}