×

Inequalities for eigenvalues of the biharmonic operator with weight on Riemannian manifolds. (English) Zbl 1200.53042

Given a compact Riemannian manifold with boundary, the authors consider the eigenvalues of the biharmonic operator with weight on \(M\). They prove a general inequality involving these eigenvalues. Using this inequality, they consider these eigenvalues when \(M\) is a compact domain of the following three space: 1) a complex projective space, 2) a minimal submanifold of a Euclidean space and 3) a minimal submanifold of a unit sphere.

MSC:

53C20 Global Riemannian geometry, including pinching
58C40 Spectral theory; eigenvalue problems on manifolds
35P15 Estimates of eigenvalues in context of PDEs
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry, Edingurgh, 1998, (eds. E. B. Davies and Yu Safalov), London Math. Soc. Lecture Notes, 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. India Acad. Sci. Math. Sci., 112 (2002), 3-30. · Zbl 1199.35261
[3] M. S. Ashbaugh and R. D. Benguria, Proof of the Payne-Pólya-Weinberger conjecture, Bull. Amer. Math. Soc., 25 (1991), 19-29. · Zbl 0736.35075
[4] M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacian and extensions, Ann. of Math., 135 (1992), 601-628. · Zbl 0757.35052
[5] M. S. Ashbaugh and R. D. Benguria, A second proof of the Payne-Pólya-Weinberger conjecture, Commun. Math. Phys., 147 (1992), 181-190. · Zbl 0758.34075
[6] 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
[7] Q. M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann., 331 (2005), 445-460. · Zbl 1122.35086
[8] 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. · Zbl 1127.35026
[9] 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
[10] A. El Soufi, E. M. Harrell II and S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operator on submanifolds, Tran. Amer. Math. Soc., 361 (2009), 2337-2350. · Zbl 1162.58009
[11] Evans M. Harrell, II, Some geometric bounds on eigenvalue gaps, Comm. Partial Differential Equations, 18 (1993), 179-198. · Zbl 0810.35067
[12] Evans M. Harrell, II, Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, Comm. Partial Differential Equations, 32 (2007), 401-413. · Zbl 1387.35136
[13] Evans M. Harrell, II and P. L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Comm. Partial Differential Equations, 19 (1994), 2037-2055. · Zbl 0815.35078
[14] Evans M. Harrell, II and P. L. Michel, Commutator bounds for eigenvalues of some differential operators, Lecture Notes in Pure and Appl. Math., 168 , Marcel Dekker, New York, 1995, pp.,235-244. · Zbl 0816.35094
[15] Evans M. Harrell, II and J. Stubbe, On trace inequalities and the universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc., 349 (1997), 1797-1809. · Zbl 0887.35111
[16] G. N. Hile and M. H. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J., 29 (1980), 523-538. · Zbl 0454.35064
[17] G. N. Hile and R. Z Yeh, Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math., 112 (1984), 115-133. · Zbl 0541.35059
[18] S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc., 318 (1990), 615-642. · Zbl 0727.35096
[19] P. F. Leung, On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere, J. Austral. Math. Soc. (Series A), 50 (1991), 409-416. · Zbl 0728.53035
[20] P. Li, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helv., 55 (1980), 347-363. · Zbl 0451.53036
[21] L. E. Payne, G. Pólya and H. F. Weinberger, Sur le quotient de deux fréquences propres cosécutives, Comptes Rendus Acad. Sci. Paris, 241 (1955), 917-919. · Zbl 0065.08801
[22] L. E. Payne, G. Pólya and H. F. Weiberger, On the ratio of consecutive eigenvalues, J. Math. and Phys., 35 (1956), 289-298. · Zbl 0073.08203
[23] R. Schoen and S. T. Yau, Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994, v+235 pp. · Zbl 0830.53001
[24] Q. Wang and C. Xia, Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Ana., 245 (2007), 334-352. · Zbl 1113.58013
[25] Q. Wang and C. Xia, Universal bounds for eigenvalues of Schrödinger operator on Riemannian manifolds, Ann. Acad. Sci. Fen. Math., 33 (2008), 319-336. · Zbl 1171.35091
[26] H. C. Yang, An estimate of the difference between cosecutive eigenvalues, preprint IC/91/60 of ICTP, Trieste, 1991.
[27] P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 55-63. · Zbl 0446.58017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.