×

Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds. (English) Zbl 1191.35192

Summary: We study eigenvalues of a clamped plate problem on a bounded domain in an \(n\)-dimensional complete Riemannian manifold. By making use of Nash’s theorem and introducing \(k\) free constants, we derive a universal bound for eigenvalues, which solves a problem proposed by Q. Wang and Ch. Xia [J. Funct. Anal. 245, No. 1, 334–352 (2007; Zbl 1113.58013)].

MSC:

35P15 Estimates of eigenvalues in context of PDEs
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 1113.58013
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in spectral theory and geometry, Edinburgh, 1998, London Math. Soc. Lecture Note Ser., 273 , Cambridge Univ. Press, Cambridge, 1999, pp.,95-139. · Zbl 0937.35114
[2] M. S. Ashbaugh, Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H. C. Yang, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 3-30. · Zbl 1199.35261 · doi:10.1007/BF02829638
[3] D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan, 60 (2008), 325-339. · Zbl 1147.35060 · doi:10.2969/jmsj/06020325
[4] 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. · Zbl 0748.35022
[5] Q.-M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann., 331 (2005), 445-460. · Zbl 1122.35086 · doi:10.1007/s00208-004-0589-z
[6] Q.-M. Cheng and H. C. Yang, Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc., 358 (2006), 2625-2635. · Zbl 1096.35095 · doi:10.1090/S0002-9947-05-04023-7
[7] Q.-M. Cheng and H. C. Yang, Bounds on eigenvalues of Dirichlet Laplacian, Math. Ann., 337 (2007), 159-175. · Zbl 1110.35052 · doi:10.1007/s00208-006-0030-x
[8] Q.-M. Cheng and H. C. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, to appear in Proc. Amer. Math. Soc., 2010. · Zbl 1209.35089 · doi:10.1090/S0002-9939-2010-10484-7
[9] A. El Soufi, E. M. Harrell II and S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Amer. Math. Soc., 361 (2009), 2337-2350. · Zbl 1162.58009 · doi:10.1090/S0002-9947-08-04780-6
[10] E. M. Harrell II, Commutators, eigenvalue gaps and mean curvature in the theory of Schrödinger operators, Comm. Part. Diff. Eqns., 32 (2007), 401-413. · Zbl 1387.35136 · doi:10.1080/03605300500532889
[11] G. N. Hile and M. H. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J., 29 (1980), 523-538. · Zbl 0454.35064 · doi:10.1512/iumj.1980.29.29040
[12] S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc., 318 (1990), 615-642. · Zbl 0727.35096 · doi:10.2307/2001323
[13] M. Levitin and L. Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues, J. Funct. Anal., 192 (2002), 425-445. · Zbl 1058.47022 · doi:10.1006/jfan.2001.3913
[14] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math., 63 (1956), 20-63. · Zbl 0070.38603 · doi:10.2307/1969989
[15] L. E. Payne, G. Polya and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phis., 35 (1956), 289-298. · Zbl 0073.08203
[16] Q. Wang and C. Xia, Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Anal., 245 (2007), 334-352. · Zbl 1113.58013 · doi:10.1016/j.jfa.2006.11.007
[17] H. C. Yang, An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of ICTP, Trieste, 1991.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.