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

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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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 (2021). https://doi.org/10.1007/s12220-021-00613-4

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Keywords

  • Eigenvalue estimates
  • Weighted p-Laplacian
  • Bakry–Émery manifolds
  • and modulus of continuity
  • Gradient comparison theorem

Mathematics Subject Classification

  • 35P15
  • 35P30
  • 58J50