Estimates on eigenvalues of Laplacian

Mathematische Annalen

Volumn 331, Issue 2, 2005, Pages 445-460

  Cheng, Qing Ming ^a   Yang, Hongcang ^b  

^a SAGA UNIVERSITY   (Japan)

^b INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 15244354281     PISSN: 00255831     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00208-004-0589-z     Document Type: Article

References (26)

1. Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In: E.B. Davies, Yu Safalov (eds.), Spectral theory and geometry (Edinburgh, 1998), London Math. Soc. Lecture Notes, vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 95-139
2. Ashbaugh, M.S.: Universal eigenvalue bounds of Payne-Pólya- Weinberger, Hile-Prottter and H.C. Yang, Proc. Indian Acad. Sci. Math. Sci. 112, 3-30 (2002)
3. Ashbaugh, M.S., Benguria R.D.: Proof of the Payne-Pólya-Weinberger conjecture. Bull. Amer. Math. Soc. 25, 19-29 (1991)
4. Ashbaugh M.S., Benguria, R.D.: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. of Math. 135, 601-628 (1992)
5. Ashbaugh, M.S., Benguria R.D.: A second proof of the Payne-Pólya- Weinberger conjecture. Commun. Math. Phys. 147, 181-190 (1992)
6. Brands, J.J.A.M.: Bounds for the ratios of the first three membrane eigenvalues. Arch. Rational Mech. Anal. 16, 265-268 (1964)
7. Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, New York, 1984
8. Cheng, Q.-M., Yang, H.C.: Inequalities of eigenvalues for a clamped plate problem. Preprint
9. Cheng, Q.-M., Yang H.C.: Inequalities of eigenvalues on Laplacian on domains and compact hypersurfaces in complex projective spaces, Preprint
10. Cheng, S.Y.: Eigenfunctions and eigenvalues of Laplacian. In: S.S. Chern, R. Osserman (eds.), Differntial Geometry, Proc. Symp. Pure Math. vol. 27 part 2, Amer. Math. Soc., Providence, Rhode Island, 1975, pp. 185-193
11. Harrell II, E.M.: Some geometric bounds on eigenvalue gaps. Comm. Part. Diff. Eqs. 18, 179-198 (1993)
12. Harrell II, E.M., Michel, P.L.: Commutator bounds for eigenvalues with applications to spectral geometry. Comm. in Part. Diff. Eqs. 19, 2037-2055 (1994)
13. Harrell II, E.M., Stubbe, J.: On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Amer. Math. Soc. 349, 1797-1809 (1997)
14. Hill, G.N., Protter, M.H.: Inequalities for eigenvalues of Laplacian. Indiana Univ. Math. 29, 523-538 (1980)
15. Hile, G.N., Yeh, R.Z.: Inequalities for eigenvalues of the biharmonic operator. Pac. J. Math. 112, 115-133 (1984)
16. Hook, S.M.: Domain independent upper bounds for eigenvalues of elliptic operator. Trans. Amer. Math. Soc. 318, (1990)
17. Ivrii, V.Ya.: Second term of the spectrual asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary. Funct. Analy. Appl. 14(2), 98-105 (1980)
18. Lee, J.M.: The gaps in the spectrum of the Laplace-Beltrami operator. Houston J. Math. 17, 1-24 (1991)
19. Leung, P.-F.: On the consecutive eigenvalues of the Laplacain of a compact minimal submanifold in a sphere. J. Austral. Math. Soc. 50, 409-426 (1991)
20. Li, P.: Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helvetici 55, 347-363 (1980)
21. Payne, L.E., Polya, G., Weinberger, H.F.: Sur le quotient de deux fréquences propres consécutives. Comptes Rendus Acad. Sci. Paris 241, 917-919 (1955)
22. Payne, L.E., Polya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289-298 (1956)
23. Yang, H.C.: An estimate of the difference between consecutive eigenvalues. Preprint IC/91/60 of ICTP, Trieste, 1991
24. Yang, H.C.: Estimates of the difference between consecutive eigenvalues. Preprint of ICTP, Trieste, 1995
25. Yang, P.C., Yau, S.T.: Eigenvalues of the Laplacian of compact Riemannian surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa CI. Sci. 7, 55-63 (1980)
26. Yau, S.T., Schoen, R.: Differential Geometry. Science Press, Beijing, 1988, pp. 142-145