Hostname: page-component-7dd5485656-zklqj Total loading time: 0 Render date: 2025-10-31T21:53:28.805Z Has data issue: false hasContentIssue false
  • English
  • Français

EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN

Published online by Cambridge University Press:  08 December 2010

SUN HEJUN  and
QI XUERONG
SUN HEJUN
Affiliation:
Department of Applied Mathematics, College of Science, Nanjing University of Science and Technology, Nanjing 210094, P.R. China e-mail: hejunsun@163.com
QI XUERONG
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan e-mail: qixuerong609@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

Keywords

35P1558C4053C42

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 2 , May 2011 , pp. 321 - 332
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Ashbaugh, M. S., Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Davies, E. B. and Safarov, Y., Editors) (Edinburgh, 1998), London Math. Soc. Lecture Notes, Vol. 273 (Cambridge Univ. Press, Cambridge, 1999), 95–139.Google Scholar
2.Ashbaugh, M. S., Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H. C. Yang, Proc. Indian Acad. Sci. (Math. Sci.) 112 (2002), 3–30.Google Scholar
3.Ashbaugh, M. S. and Hermi, L., A unified approach to universal inequalities for eigenvalues of elliptic operators, Pacific J. Math. 217 (2004), 201–219.CrossRefGoogle Scholar
4.Chen, D. G. and Cheng, Q.-M., Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008), 325–339.CrossRefGoogle Scholar
5.Chen, Z. C. and Qian, C. L., Estimates for discrete spectrum of Laplacian operator with any order, J. China Univ. Sci. Tech. 20 (1990), 259–266.Google Scholar
6.Cheng, Q.-M., Ichikawa, T. and Mametsuka, S., Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds, J. Math. Soc. Japan, 62 (2010), 673–686.CrossRefGoogle Scholar
7.Cheng, Q.-M. and Yang, H. C., Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc. 358 (2006), 2625–2635.CrossRefGoogle Scholar
8.Cheng, Q.-M. and Yang, H. C., Estimates for eigenvalues on Riemannian manifolds, J. Differ. Equ. 247 (2009), 2270–2281.CrossRefGoogle Scholar
9.Hile, G. N. and Protter, M. H., Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J. 29 (1980), 523–538.CrossRefGoogle Scholar
10.Hile, G. N. and Yeh, R. Z., Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math. 112 (1984), 115–133.CrossRefGoogle Scholar
11.Hook, S. M., Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc. 318 (1990), 615–642.CrossRefGoogle Scholar
12.Ljung, L., Recursive identification, in Stochastic systems: The mathematics of filtering and identification and applications (Hazewinkel, M. and Willems, J. C., Editors) (Reidel, Dordrecht, 1981), 247–283.CrossRefGoogle Scholar
13.Maybeck, P. S., Stochastic models, estimation, and control III (Academic Press, New York, 1982).Google Scholar
14.Nash, J., The imbedding problem for Riemannian manifolds, Ann. Math. 63 (1956), 20–63.CrossRefGoogle Scholar
15.Payne, L. E., Polya, G. and Weinberger, H. F., On the ratio of consecutive eigenvalues, J. Math. Phys. 35 (1956), 289–298.CrossRefGoogle Scholar
16.El Soufi, A., Harrell, E. M. and Ilias, S., Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Am. Math. Soc. 361 (2009), 2337–2350.CrossRefGoogle Scholar
17.Wang, Q. L. and Xia, C. Y., Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Anal. 245 (2007), 334–352.CrossRefGoogle Scholar