Abstract
In this paper, we study two eigenvalue problems of the weighted Laplacian and get the Reilly-type bounds and isoperimetric type bounds for the first nonzero $n$ eigenvalues on hypersurfaces of the Euclidean space. Besides, we give lower bounds for the first eigenvalue of weighted $p$-Laplacian on submanifolds with locally bounded weighted mean curvature. Meanwhile, several applications of these estimates have also been given.
Funding Statement
F. Du was partially supported by NSF of Hubei Provincial
Department of Education (Grant No. D20184301) and Hubei Key Laboratory of
Applied Mathematics (Hubei University).
J. Mao was supported in part by NSF of China (Grant No.
11801496), China Scholarship Council, the Fok Ying-Tung Education Foundation
(China), and Hubei Key Laboratory of Applied Mathematics (Hubei
University).
Q. Wang was supported by CNPq, Brazil (Grant No.
307089/2014-2).
C. Xia was supported by CNPq, Brazil (Grant No.
306146/2014-2).
Acknowledgement
The authors would like to thank the anonymous referee for his or her careful reading and valuable comments. The corresponding author, Prof. Jing Mao, wants to thank the Department of Mathematics, IST, University of Lisbon for its hospitality during his visit from September 2018 to September 2019.
Citation
Feng Du. Jing Mao. Qiaoling Wang. Changyu Xia. "Estimates for eigenvalues of weighted Laplacian and weighted $p$-Laplacian." Hiroshima Math. J. 51 (3) 335 - 353, November 2021. https://doi.org/10.32917/h2020086
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