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Sharp Lower Bound for the First Eigenvalue of the Weighted p-Laplacian I

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Abstract

We establish sharp lower bounds for the first nonzero eigenvalue of the weighted p-Laplacian with \(1< p< \infty \) on a compact Bakry–Émery manifold \((M^n,g,f)\) satisfying \({\text {Ric}}+\nabla ^2 f \ge \kappa \, g\), provided that either \(1<p \le 2\) or \(\kappa \le 0\). For \(1<p \le 2\), we provide a simple proof via the modulus of continuity estimates. The proof for the \(\kappa \le 0\) case is based on a sharp gradient comparison theorem for the eigenfunction together with a careful analysis of the underlying one-dimensional model equation.

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Acknowledgements

The authors would like to thank Professors Ben Andrews, Zhiqin Lu, Lei Ni, Guofang Wei and Richard Schoen for their interests in this work.

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  1. Xiaolong Li

    Present address: Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, Irvine, CA, 92697, USA

    Xiaolong Li

  2. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Kui Wang

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  1. Xiaolong Li
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Correspondence to Kui Wang.

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K. Wang research is supported by NSFC No. 11601359.

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Li, X., Wang, K. Sharp Lower Bound for the First Eigenvalue of the Weighted p-Laplacian I. J Geom Anal 31, 8686–8708 (2021). https://doi.org/10.1007/s12220-021-00613-4

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