RESEARCH ARTICLE

Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds

  • Feng DU 1 ,
  • Jing MAO , 2
Expand
  • 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

Received date: 14 Nov 2013

Accepted date: 22 Jul 2014

Published date: 01 Apr 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

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.

Cite this article

Feng DU , Jing MAO . Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds[J]. Frontiers of Mathematics in China, 2015 , 10(3) : 583 -594 . DOI: 10.1007/s11464-015-0422-x

1
Cao L F, Li H Z. r-Minimal submanifolds in space forms. Ann Global Anal Geom, 2007, 32: 311-341

DOI

2
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

DOI

4
Chen D G, Cheng Q M. Extrinsic estimates for eigenvalues of the Laplace operator. J Math Soc Japan, 2008, 60: 325-339

DOI

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

DOI

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

DOI

8
Reilly R. On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comm Math Helv, 1977, 52: 525-533

DOI

9
El Soufi A, Harrell ll E M, Ilias S. Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans Amer Math Soc, 2009, 361: 2337-2350

DOI

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

Outlines

/