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

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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. (1)

    Closed eigenvalue problem:

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

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Acknowledgements

The author would like to thank Prof. G. Santhanam for discussions and many helpful comments on the manuscript. She also wishes to thank Prof. Bruno Colbois for pointing out a mistake in Theorem 1 in the original manuscript.

Author information

Correspondence to Sheela Verma.

Additional information

Communicating Editor: S Kesavan

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Verma, S. Upper bound for the first nonzero eigenvalue related to the p-Laplacian. Proc Math Sci 130, 21 (2020). https://doi.org/10.1007/s12044-019-0529-1

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Keywords

  • p-Laplacian
  • closed eigenvalue problem
  • Steklov eigenvalue problem
  • center-of-mass

2010 Mathematics Subject Classification

  • 35P15
  • 58J50