Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Universal bounds for eigenvalues of the polyharmonic operators


Authors: Jürgen Jost, Xianqing Li-Jost, Qiaoling Wang and Changyu Xia
Journal: Trans. Amer. Math. Soc. 363 (2011), 1821-1854
MSC (2010): Primary 35P15, 53C20
DOI: https://doi.org/10.1090/S0002-9947-2010-05147-5
Published electronically: November 8, 2010
MathSciNet review: 2746667
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $ k$th eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is sharper than the known Payne-Pólya-Weinberg type inequality and also covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. We also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a Euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by Ashbaugh. Finally, we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere.


References [Enhancements On Off] (What's this?)

  • [A1] M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Edinburgh, 1998), E. B. Davies and Yu Safalov, eds., London Math. Soc. Lecture Notes, vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 95-139. MR 1736867 (2001a:35131)
  • [A2] M. S. Ashbaugh, The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang. Spectral and inverse spectral theory (Goa, 2000). Proc. Indian Acad. Sci. Math. Sci. 112 (2002), 3-30. MR 1894540 (2004c:35302)
  • [AB1] M. S. Ashbaugh and R. D. Benguria, Proof of the Payne-Pólya-Weinberger conjecture, Bull. Amer. Math. Soc. 25 (1991), 19-29. MR 1085824 (91m:35173)
  • [AB2] M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. of Math. (2) 135 (1992), 601-628. MR 1166646 (93d:35105)
  • [AB3] M. S. Ashbaugh and R. D. Benguria, A second proof of the Payne-Pólya-Weinberger conjecture, Commun. Math. Phys. 147 (1992), 181-190. MR 1171765 (93k:33002)
  • [AB4] M. S. Ashbaugh and R. D. Benguria, More bounds on eigenvalue ratios for Dirichlet Laplacians in $ n$ dimensions, SIAM J. Math. Anal. 24 (1993), 1622-1651. MR 1241161 (94i:35139)
  • [AH] M. S. Ashbaugh and L. Hermi, A unified approach to universal inequalities for eigenvalues of elliptic operators, Pacific J. Math. 217 (2004), 201-219. MR 2109931 (2005k:35305)
  • [Ch] I.  Chavel, Eigenvalues in Riemannian geometry, Academic Press, 1984. MR 768584 (86g:58140)
  • [CIM] Q. M. Cheng, T. Ichikawa, and S. Mametsuka, Estimates for eigenvalues of Laplacian with any order in a unit sphere, Calc. Var. PDE 36 (2009), 507-523. MR 2558327 (2010k:35332)
  • [CY1] Q. M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian. Math. Ann. 331 (2005),445-460. MR 2115463 (2005i:58038)
  • [CY2] Q. M. Cheng and H. C. Yang, Universal bounds for eigenvalues of a buckling problem, Comm. Math. Phys. 262 (2006), 663-675. MR 2202307 (2007f:35056)
  • [CY3] Q. M. Cheng and H. C. Yang, Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc., 358 (2006), 2625-2635. MR 2204047 (2006m:35263)
  • [CY4] Q. M. Cheng and H. C. Yang, Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces, J. Math. Soc. Japan, 58 (2006), 545-561. MR 2228572 (2007k:58051)
  • [CQ] Z. C. Chen and C. L. Qian, Estimates for discrete spectrum of Laplacian operator with any order, J. China Univ. Sci. Tech. 20 (1990), 259-266. MR 1077287 (92c:35087)
  • [Co] R. Courant, Über die Schwingungen eingespannter Platten, Math. Zeitschr. 15 (1922), 195-200. MR 1544567
  • [CH] R. Courant and D. Hilbert, Methoden der mathematischen Physik I, 3rd ed., Springer, 1968. MR 0344038 (49:8778)
  • [HLP] G. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Reprint of the 1952 edition, Cambridge University Press, 1988. MR 0944909 (89d:26016)
  • [Ha] E. M. Harrell, Some geometric bounds on eigenvalue gaps, Commun. Part. Differ. Equ. 18 (1993), 179-198. MR 1211730 (94c:35135)
  • [HM1] E. M. Harrell and P. L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Commun. Part. Differ. Equ. 19 (1994), 2037-2055. MR 1301181 (95i:58182)
  • [HM2] E. M. Harrell and P. L. Michel, Commutator bounds for eigenvalues of some differential operators, Lecture Notes in Pure and Applied Mathematics, vol. 168 (G. Ferreyra, G. R. Goldstein and F. Neubrander, eds.) (New York: Marcel Dekker) (1995) pp. 235-244. MR 1300432 (95j:47028)
  • [HS] E. M. Harrell and J. Stubbe, On trace inequalities and the universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809. MR 1401772 (97i:35129)
  • [HP] G. N. Hile and M. H. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J. 29 (1980), 523-538. MR 578204 (82c:35052)
  • [HY] G. N. Hile and R. Z. Yeh, Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math. 112 (1984), 115-133. MR 739143 (85k:35170)
  • [H] S. M. Hook, Domain-independent upper bounds for eigenvalues of elliptic operators, Trans. Amer. Math. Soc. 318 (1990), 615-642. MR 994167 (90h:35075)
  • [LeP] M. Levitin and L. Parnovski, Commutators, spectral trace identities and universal estimates for eigenvalues, J. Funct. Anal. 192 (2002), 425-445. MR 1923409 (2003g:47040)
  • [PPW1] L. E. Payne, G. Pólya and H. F. Weinberger, Inequalities for eigenvalues of plates and membranes, J. Rational Mech. Anal. 4 (1955), 517-529. MR 0070834 (17:42a)
  • [PPW2] L. E. Payne, G. Pólya and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289-298. MR 0084696 (18:905c)
  • [WX1] Q. Wang and C. Xia, Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Anal. 245 (2007), 334-352. MR 2311628 (2008e:58033)
  • [WX2] Q. Wang and C. Xia, Universal inequalities for eigenvalues of the buckling problem on spherical domains, Comm. Math. Phys. 270 (2007), 759-775. MR 2276464 (2007m:35180)
  • [We] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1912), 441-469. MR 1511670
  • [WC] F. E. Wu and L. F. Cao, Estimates for eigenvalues of Laplacian operator with any order, Science in China A: Mathematics 50 (2007), 1078-1086. MR 2370015 (2009c:35076)
  • [Y] H. C. Yang, Estimates of the difference between consecutive eigenvalues, preprint, 1995 (revision of International Centre for Theoretical Physics preprint IC/91/60, Trieste, Italy, April 1991).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35P15, 53C20

Retrieve articles in all journals with MSC (2010): 35P15, 53C20


Additional Information

Jürgen Jost
Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
Email: jost@mis.mpg.de

Xianqing Li-Jost
Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
Email: xli-jost@mis.mpg.de

Qiaoling Wang
Affiliation: Departamento de Matemática, University of Brasilia, 70910-900, Brasília-DF, Brazil
Email: wang@mat.unb.br

Changyu Xia
Affiliation: Departamento de Matemática, University of Brasilia, 70910-900, Brasília-DF, Brazil
Email: xia@mat.unb.br

DOI: https://doi.org/10.1090/S0002-9947-2010-05147-5
Keywords: Universal bounds, eigenvalues, polyharmonic operator, Riemannian manifolds, Euclidean space, spheres
Received by editor(s): August 19, 2008
Published electronically: November 8, 2010
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society