Acta mathematica scientia, Series B >
ESTIMATES ON EIGENVALUES FOR THE BIHARMONIC OPERATOR ON A BOUNDED DOMAIN IN Hn(−1)
Received date: 2009-08-13
Revised date: 2010-03-29
Online published: 2011-07-20
Supported by
This research is supported by NSFC (11001076); Project of Henan Provincial department of Sciences and Technology(092300410143); and NSF of Henan Provincial Education Department (2009A110010; 2010A110008).
In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the (k + 1)th eigenvalue in terms of the first kth eigenvalues independent of the domains.
Key words： universal inequality; eigenvalue; hyperbolic space; clamped plate problem
HUANG Guang-Yue , LI Xin-Xiao . ESTIMATES ON EIGENVALUES FOR THE BIHARMONIC OPERATOR ON A BOUNDED DOMAIN IN Hn(−1)[J]. Acta mathematica scientia, Series B, 2011 , 31(4) : 1383 -1388 . DOI: 10.1016/S0252-9602(11)60325-1
[1] Ashbaugh M S. Isoperimetric and universal inequalities for eigenvalues//Davies E B, Safarov Yu, eds. Spectral Theory and Geometry London Math Soc Lecture Notes, 273. Cambridge: Cambridge Univ Press, 1999: 95–139
[2] Ashbaugh M S. Universal eigenvalue bounds of Payne-Polya-Weinberger, Hile-Protter, and H. C. Yang. Proc Indian Acad Sci Math Sci, 2002, 112: 3–30
[3] Cheng Q M, Yang H C. Estimates on eigenvalues of Laplacian. Math Ann, 2005, 331: 445–460
[4] Cheng Q M, Yang H C. Inequalities for eigenvalues of a clamped plate problem. Trans Amer Math Soc, 2006, 358(6): 2625–2635
[5] Cheng Q M, Ichikawa T, Mametsuka S. Inequalities for eigenvalues of Laplacian with any order. Comm Contemp Math, 2009, 11: 639–655
[6] Cheng Q M, Yang H C. Estimates for eigenvalues on Riemannian manifolds. J Differ Equ, 2009, 247: 2270–2281
[7] Cheng Q M, Huang G Y, Wei G X. Estimates for lower order eigenvalues of a clamped plate problem. Calc Var Partial Diff Equ, 2010, 38: 409–416
[8] El Soufi A, Harrell II E M, Ilias S. Universal inequalities for the eigenvalues of Laplace and Schr¨odinger operators on submanifolds. Trans Amer Math Soc, 2009, 361: 2337–2350
[9] Hile G N, Protter M H. Inequalites for eigenvalues of the Laplacian. Indiana Univ Math J, 1980, 29: 523–538
[10] Huang G Y, Chen W Y. Universal bounds for eigenvlaues of Laplacian operator with any order. Acta Math Sci, 2010, 30B(3): 939–948
[11] Huang G Y, Chen W Y. Inequalities of eigenvalues for bi-Kohn Laplacian on Heisenberg group. Acta Math Sci, 2010, 30B(1): 125–131
[12] Huang G Y, Li X X. Estimates on eigenvalues for higher order Laplacians on spherical domains. J Math (PRC), 2009, 29(4): 449–453
[13] Huang G Y, Li X X, Xu R W. Extrinsic estimates for the eigenvalues of Schr¨odinger operator. Geom Dedicata, 2009, 143: 89–107
[14] Payne L E, P´olya G, Weinberger H F. On the ratio of consecutive eigenvalues. J Math Phys, 1956, 35: 289–298
[15] Taylor R L, Govindjee S. Solution of clamped rectangular plate problems. Communi Numer Meth Eng, 2004, 20: 757–765
[16] Wang Q L, Xia C Y. Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds. J Funct Anal, 2007, 245: 334–352
[17] Wang Q L, Xia C Y. Universal bounds for eigenvalues of Schr¨odinger operator on Riemannian manifolds. Ann Acad Sci Fenn Math, 2008, 33: 319–336
[18] Yang H C. An estimate of the difference between consecutive eigenvalues. Preprint IC/91/60 of ICTP Trieste, 1991
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