Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds
Feng DU1,Jing MAO2,*()
1. School of Mathematics and Physics Science, Jingchu University of Technology, Jingmen 448000, China 2. Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China
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.
Chavel I. Eigenvalues in Riemannian Geometry. New York: Academic Press, 1984
3
Chen B Y. Total Mean Curvature and Submanifolds of Finite Type. Singapore: World Scientific, 1984
https://doi.org/10.1142/0065
4
Chen D G, Cheng Q M. Extrinsic estimates for eigenvalues of the Laplace operator. J Math Soc Japan, 2008, 60: 325-339
https://doi.org/10.2969/jmsj/06020325
5
Chen D G, Li H Z. The sharp estimates for the first eigenvalue of Paneitz operator in 4-manifold. arXiv: 1010.3102v1
6
Grosjean J F. Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds. Pacific J Math, 2002, 206: 93-112
https://doi.org/10.2140/pjm.2002.206.93
7
Mao J. Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel. J Math Pures Appl, 2014, 101(3): 372-393
https://doi.org/10.1016/j.matpur.2013.06.006
8
Reilly R. On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comm Math Helv, 1977, 52: 525-533
https://doi.org/10.1007/BF02567385
9
El Soufi A, Harrell ll E M, Ilias S. Universal inequalities for the eigenvalues of Laplace and Schr?dinger operators on submanifolds. Trans Amer Math Soc, 2009, 361: 2337-2350
https://doi.org/10.1090/S0002-9947-08-04780-6
10
Veron L. Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds. Colloquia Mathematica Societatis Janos Bolyai, Vol 62, P D E. Budapest, 1991, 317-352
