Abstract
The Szegö–Weinberger inequality asserts that the second Neumann eigenvalue \(\mu _2\) of Laplace operator on a bounded domain in \(\mathbb {R}^n\) is bounded from above by that of a ball of the same volume. In this note, we prove an upper bound for \(\mu _2\) on domains in Riemannian manifolds, which can be viewed as a Sezgö–Weinberger type inequality.
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Acknowledgements
The author would like to thank anonymous referees for helpful suggestion on the improvement of this paper, and for suggesting Corollary 1.4. This work is partly supported by NSFC(11601359), NSF of Jiangsu Province (BK20160301), and China Postdoctoral Foundation Grants (2016M591900 and 2017T100394).
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Wang, K. An upper bound for the second Neumann eigenvalue on Riemannian manifolds. Geom Dedicata 201, 317–323 (2019). https://doi.org/10.1007/s10711-018-0394-6
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DOI: https://doi.org/10.1007/s10711-018-0394-6