Upper bound for the first nonzero eigenvalue related to 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 nonzero eigenvalue of the following problems:

• (1) Closed eigenvalue problem:

  $$\begin{aligned} \Delta _p u = \lambda _{p} \ |u|^{p-2} \ u \quad \text{ on } {M}. \end{aligned}$$

• (2) Steklov eigenvalue problem:

  $$\begin{aligned} {\begin{array}{ll} \Delta _{p}u = 0 &{} \text{ in } \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu } = \mu _{p} \ |u|^{p-2} \ u &{} \text{ on } M . \end{array}} \end{aligned}$$

These bounds are given in terms of the first nonzero eigenvalue of the usual Laplacian on the geodesic ball of the same volume as of $\Omega$.