• Upper bound for the first nonzero eigenvalue related to the $p$-Laplacian

• # Fulltext

Permanent link:
https://www.ias.ac.in/article/fulltext/pmsc/130/0021

• # Keywords

$p$-Laplacian; closed eigenvalue problem; Steklov eigenvalue problem; center-of-mass.

• # 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:

$$\Delta_{p}u = \lambda_{p} |u|^{p−2} u \quad {\rm on}\; M.$$

(2) Steklov eigenvalue problem:

$$\begin{array}{ll} \Delta_{p}u =0 \quad \;\;\;\;\;\;\rm{in} \;\Omega,\\ |\nabla u|^{p−2} \frac{\partial u}{\partial v} = \mu_{p}|u|^{p−2} u \quad {\rm on} \ M. \end{array}$$

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

• # Author Affiliations

1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208 016, India

• # Proceedings – Mathematical Sciences

Volume 131, 2021
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Posted on July 25, 2019

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