摘要
For a compact Riemannian manifold M immersed into a higher dimensional manifold which can be chosen to be a Euclidean space, a unit sphere, or even a projective space, we successfully give several upper bounds in terms of the norm of the mean curvature vector of M for the first non-zero eigenvalue of the p-Laplacian （1 〈 p 〈 ＋∞） on M. This result can be seen as an extension of Reilly＇s bound for the first non-zero closed eigenvalue of the Laplace operator.
For a compact Riemannian manifold M immersed into a higher dimensional manifold which can be chosen to be a Euclidean space, a unit sphere, or even a projective space, we successfully give several upper bounds in terms of the norm of the mean curvature vector of M for the first non-zero eigenvalue of the p-Laplacian （1 〈 p 〈 ＋∞） on M. This result can be seen as an extension of Reilly＇s bound for the first non-zero closed eigenvalue of the Laplace operator.
基金
Acknowledgements The authors would like to thank the anonymous referees for their careful reading and valuable comments such that the article appears as its present version. The first author was partially supported by the Key Laboratory of Applied Mathematics of Hubei Province and the Research Project of Jingchu University of Technology. The second author was partially supported by the Starting-up Research Fund （Grant No. HIT（WH）201320） supplied by Harbin Institute of Technology （Weihai）, the Project （Grant No. HIT. NSRIF. 2015101） supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology, and the National Natural Science Foundation of China （Grant No. 11401131）.
