The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H C Yang

  • Mark S. Ashbaugh

Abstract

In this paper we present a unified and simplified approach to the universal eigenvalue inequalities of Payne—Pólya—Weinberger, Hile—Protter, and Yang. We then generalize these results to inhomogeneous membranes and Schrödinger’s equation with a nonnegative potential. We also show that Yang’s inequality is always better than HileProtter’s (and hence also better than Payne—Pólya—Weinberger’s). In fact, Yang’s weaker inequality (which deserves to be better known),
$$\lambda _{k + 1}< \left( {1 + \frac{4}{n}} \right)\frac{1}{k}\sum\limits_{i = 1}^k {\lambda _i } $$
, is also strictly better than Hile—Protter’s. Finally, we treat Yang’s (and related) inequalities for minimal submanifolds of a sphere and domains contained in a sphere by our methods.

Keywords

Eigenvalues of the Laplacian universal inequalities for eigenvalues eigenvalue ratios the Payne—Pólya—Weinberger inequality 

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References

  1. [1]
    Allegretto W, Lower bounds on the number of points in the lower spectrum of elliptic operators,Can. J. Math. 31 (1979) 419–426MATHMathSciNetGoogle Scholar
  2. [2]
    Anghel N, Extrinsic upper bounds for eigenvalues of Dirac-type operators,Proc. Am. Math. Soc. 117 (1993) 501–509MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Ashbaugh M S and Benguria R D, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions,Ann. Math. 135 (1992) 601–628CrossRefMathSciNetGoogle Scholar
  4. [4]
    Ashbaugh M S and Benguria R D, More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimensions,SIAM J. Math. Anal. 24 (1993) 1622–1651MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Ashbaugh M S and Benguria R D, Isoperimetric inequalities for eigenvalue ratios, Partial Differential Equations of Elliptic Type, Cortona, 1992,Symposia Mathematica, vol. 35 (eds) A Alvino, E Fabes and G Talenti (Cambridge: Cambridge University Press) (1994) pp. 1–36.Google Scholar
  6. [6]
    Ashbaugh M S and Benguria R D, Bounds for ratios of the first, second, and third membrane eigenvalues, Nonlinear Problems in Applied Mathematics: in Honor of Ivar Stakgold on his Seventieth Birthday (eds) T S Angell, L Pamela Cook, R E Kleinman and W E Olmstead (Philadelphia: Society for Industrial and Applied Mathematics) (1996) pp. 30–42Google Scholar
  7. [7]
    Ashbaugh M S and Benguria R D, A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of Sn,Trans. Am. Math. Soc. 353 (2001) 1055–1087MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Ashbaugh M S and Hermi L, On extending the inequalities of Payne, Pólya, and Weinberger using spherical harmonics (2000) preprintGoogle Scholar
  9. [9]
    Brands J J A M, Bounds for the ratios of the first three membrane eigenvalues,Arch. Rational Mech. Anal. 16 (1964) 265–268MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Chavel I, Eigenvalues in Riemannian Geometry (New York: Academic Press) (1984)MATHGoogle Scholar
  11. [11]
    Chen Z-C, Inequalities for eigenvalues of polyharmonic operator Δ6,Kexue Tongbao (English Ed.) 30 (1985) 869–876MathSciNetGoogle Scholar
  12. [12]
    Cheng S-Y, Eigenfunctions and eigenvalues of Laplacian,Proc. Symp. Pure Math., vol. 27, part 2, Differential Geometry (eds) S S Chern and R Osserman (Providence, Rhode Island: Am. Math. Soc.) (1975) pp. 185–193Google Scholar
  13. [13]
    Chiti G, A bound for the ratio of the first two eigenvalues of a membrane,SIAM J. Math. Anal. 14 (1983) 1163–1167MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Harrell E M, Some geometric bounds on eigenvalue gaps,Commun. Part. Differ. Equ. 18 (1993) 179–198MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Harrell E M and Michel P L, Commutator bounds for eigenvalues, with applications to spectral geometry,Commun. Part. Differ. Equ. 19 (1994) 2037–2055MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Harrell E M and Michel P L, Commutator bounds for eigenvalues of some differential operators, Evolution Equations, Lecture Notes in Pure and Applied Mathematics, vol. 168 (eds) G Ferreyra, G R Goldstein and F Neubrander (New York: Marcel Dekker) (1995) pp. 235–244Google Scholar
  17. [17]
    Harrell E M and Stubbe J, On trace identities and universal eigenvalue estimates for some partial differential operators,Trans. Am. Math. Soc. 349 (1997) 1797–1809MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Hile G N and Protter M H, Inequalities for eigenvalues of the Laplacian,Indiana Univ. Math. J. 29 (1980) 523–538MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Hile G N and Yeh R Z, Inequalities for eigenvalues of the biharmonic operator,Pac. J. Math. 112 (1984) 115–133MATHMathSciNetGoogle Scholar
  20. [20]
    Hook GN, Domain-independent upper bounds for eigenvalues of elliptic operators,Trans. Am. Math. Soc. 318 (1990) 615–642MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Lee J M, The gaps in the spectrum of the Laplace—Beltrami operator,Houston J. Math. 17 (1991) 1–24MATHMathSciNetGoogle Scholar
  22. [22]
    Leung P-F, On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere,J. Austral. Math. Soc. (Series A) 50 (1991) 409–416MATHGoogle Scholar
  23. [23]
    Li P, Eigenvalue estimates on homogeneous manifolds,Comment. Math. Helvetia 55 (1980) 347–363MATHCrossRefGoogle Scholar
  24. [24]
    Lorch L, Some inequalities for the first positive zeros of Bessel functions,SIAM J. Math. Anal. 24 (1993) 814–823MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Maeda M, On the eigenvalues of Laplacian,Sci. Rep. Yokohama Nat. Univ. Sect. I 24 (1977) 29–33Google Scholar
  26. [26]
    Payne L E, Pólya G and Weinberger H F, Sur le quotient de deux fréquences propres consécutives,Comptes Rendus Acad. Sci Paris 241 (1955) 917–919MATHGoogle Scholar
  27. [27]
    Payne L E, Pólya G and Weinberger H F, On the ratio of consecutive eigenvalues,J. Math. Phys. 35 (1956) 289–298Google Scholar
  28. [28]
    Protter M H, Can one hear the shape of a drum? Revisited,SIAM Rev. 29 (1987) 185–197MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    Protter M H, Universal inequalities for eigenvalues, Maximum Principles and Eigenvalue Problems in Partial Differential Equations,Pitman Research Notes in Mathematics Series vol. 175 (ed) P W Schaefer (Harlow, Essex, United Kingdom: Longman Scientific and Technical) (1988) pp. 111–120Google Scholar
  30. [30]
    Protter M H, Upper bounds for eigenvalues of elliptic operators, Partial Differential Equations and Applications, Collected Papers in Honor of Carlo Pucci,Lecture Notes in Pure and Applied Mathematics, vol. 177 (eds) P Marcellini, G Talenti and E Vesentini (New York: Marcel Dekker) (1996) pp. 271–277Google Scholar
  31. [31]
    Qian C-L and Chen Z-C, Estimates of eigenvalues for uniformly elliptic operator of second order,Acta Math. Appl. Sinica (English Ser.) 10 (1994) 349–355MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    Thompson C J, On the ratio of consecutive eigenvalues in n-dimensions,Stud, Appl. Math. 48 (1969) 281–283MATHGoogle Scholar
  33. [33]
    Yang H C, Estimates of the difference between consecutive eigenvalues (1995) preprint (revision of International Centre for Theoretical Physics preprint IC/91/60, Trieste, Italy, April 1991)Google Scholar
  34. [34]
    Yang P C and Yau S-T, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980) 55–63MATHMathSciNetGoogle Scholar
  35. [35]
    Yu Q-H, On the first and second eigenvalues of Schrödinger operator,Chinese Ann. Math. (Ser. B) 14 (1993) 85–92MATHMathSciNetGoogle Scholar

Copyright information

© Printed in India 2002

Authors and Affiliations

  • Mark S. Ashbaugh
    • 1
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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