Mathematische Annalen

, Volume 331, Issue 2, pp 445–460 | Cite as

Estimates on Eigenvalues of Laplacian

Article

Abstract.

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere S n (1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere S N (1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere S n (1).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. 95–139Google Scholar
  2. 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)Google Scholar
  3. 3.
    Ashbaugh, M.S., Benguria R.D.: Proof of the Payne-Pólya-Weinberger conjecture. Bull. Amer. Math. Soc. 25, 19–29 (1991)MATHGoogle Scholar
  4. 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)MATHGoogle Scholar
  5. 5.
    Ashbaugh, M.S., Benguria R.D.: A second proof of the Payne-Pólya-Weinberger conjecture. Commun. Math. Phys. 147, 181–190 (1992)Google Scholar
  6. 6.
    Brands, J.J.A.M.: Bounds for the ratios of the first three membrane eigenvalues. Arch. Rational Mech. Anal. 16, 265–268 (1964)CrossRefMATHGoogle Scholar
  7. 7.
    Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, New York, 1984Google Scholar
  8. 8.
    Cheng, Q.-M., Yang, H.C.: Inequalities of eigenvalues for a clamped plate problem. PreprintGoogle Scholar
  9. 9.
    Cheng, Q.-M., Yang H.C.: Inequalities of eigenvalues on Laplacian on domains and compact hypersurfaces in complex projective spaces, PreprintGoogle Scholar
  10. 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. 185–193Google Scholar
  11. 11.
    Harrell II, E.M.: Some geometric bounds on eigenvalue gaps. Comm. Part. Diff. Eqs. 18, 179–198 (1993)Google Scholar
  12. 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)Google Scholar
  13. 13.
    Harrell II, E.M., Stubbe, P.L.: On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Amer. Math. Soc. 349, 1797–1809 (1997)CrossRefGoogle Scholar
  14. 14.
    Hill, G.N., Protter, M.H.: Inequalities for eigenvalues of Laplacian. Indiana Univ. Math. 29, 523–538 (1980)Google Scholar
  15. 15.
    Hile, G.N., Yeh, R.Z.: Inequalities for eigenvalues of the biharmonic operator. Pac. J. Math. 112, 115–133 (1984)MATHGoogle Scholar
  16. 16.
    Hook, S.M.: Domain independent upper bounds for eigenvalues of elliptic operator. Trans. Amer. Math. Soc. 318, (1990)Google Scholar
  17. 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)Google Scholar
  18. 18.
    Lee, J.M.: The gaps in the spectrum of the Laplace-Beltrami operator. Houston J. Math. 17, 1–24 (1991)Google Scholar
  19. 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)MATHGoogle Scholar
  20. 20.
    Li, P.: Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helvetici 55, 347–363 (1980)MATHGoogle Scholar
  21. 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)MATHGoogle Scholar
  22. 22.
    Payne, L.E., Polya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289–298 (1956)MATHGoogle Scholar
  23. 23.
    Yang, H.C.: An estimate of the differance between consecutive eigenvalues. Preprint IC/91/60 of ICTP, Trieste, 1991Google Scholar
  24. 24.
    Yang, H.C.: Estimates of the differance between consecutive eigenvalues. Preprint of ICTP, Trieste, 1995Google Scholar
  25. 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)MATHGoogle Scholar
  26. 26.
    Yau, S.T., Schoen, R.: Differential Geometry. Science Press, Beijing, 1988, pp. 142–145Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and EngineeringSaga UniversitySagaJapan
  2. 2.Academy of Mathematics and Systematical SciencesBeijingChina

Personalised recommendations