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First Robin eigenvalue of the p-Laplacian on Riemannian manifolds

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Abstract

We consider the first Robin eigenvalue \(\lambda _p(M,\alpha )\) for the p-Laplacian on a compact Riemannian manifold M with nonempty smooth boundary, with \(\alpha \in \mathbb {R}\) being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for \(\lambda _p(M,\alpha )\). Secondly, when \(\alpha >0\) we establish sharp lower bound of \(\lambda _p(M,\alpha )\) in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when \(\alpha <0\). Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the p-Laplacian when letting \(\alpha \rightarrow +\infty \).

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Acknowledgements

The first author would like to thank Professor Richard Schoen for his support and interest in this work. Both authors are grateful to Professor Lei Ni for his encouragement and helpful conversations.

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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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  2. Kui Wang
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Correspondence to Xiaolong Li.

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The research of the second author is supported by NSFC No. 11601359.

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Li, X., Wang, K. First Robin eigenvalue of the p-Laplacian on Riemannian manifolds. Math. Z. 298, 1033–1047 (2021). https://doi.org/10.1007/s00209-020-02645-y

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