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

TL;DR

This paper establishes an upper bound for the first non-zero eigenvalue of the $p$-Laplacian on closed hypersurfaces and bounded domains, extending spectral estimates to nonlinear eigenvalue problems in geometric analysis.

Contribution

It provides new upper bounds for the first non-zero eigenvalues of the $p$-Laplacian for both closed hypersurfaces and Steklov problems, generalizing classical results to nonlinear settings.

Findings

Derived upper bounds for the first non-zero eigenvalue of the $p$-Laplacian on hypersurfaces.

Extended spectral estimates to the Steklov eigenvalue problem for the $p$-Laplacian.

Results applicable to nonlinear eigenvalue problems in geometric analysis.

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} \Delta_p u = \lambda_{p} \ |u|^{p-2} \ u \qquad \mbox{ on } \quad {M}. \end{align*} \item Steklov eigenvalue problem: \begin{align*} \begin{array}{rcll} \Delta_{p}u &=& 0 & \mbox{ in } \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} &=& \mu_{p} \ |u|^{p-2} \ u &\mbox{ on } M . \end{array} \end{align*} \end{itemize}