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 1113.58013References:
| [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. |
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