Acta mathematica scientia, Series B >
ESTIMATES ON THE FIRST TWO POLY-LAPLACIAN EIGENVALUES ON SPHERICAL DOMAINS
Received date: 2010-01-15
Revised date: 2010-10-27
Online published: 2012-03-20
Supported by
This research is supported by NSFC (11171368).
In this article, we study the first two eigenvalues of the higher order buckling problem on a domain in the unit sphere. We obtain an estimate on the second eigenvalue in terms of the first eigenvalue. In particular, the estimate on first two eigenvalues of the higher order buckling problem of Huang, Li and Qi [5] is included in our results.
Key words： Eigenvalue; spherical domain; buckling problem
MA Bing-Qing . ESTIMATES ON THE FIRST TWO POLY-LAPLACIAN EIGENVALUES ON SPHERICAL DOMAINS[J]. Acta mathematica scientia, Series B, 2012 , 32(2) : 745 -751 . DOI: 10.1016/S0252-9602(12)60054-X
[1] Ashbaugh M S. Isoperimetric and universal inequalities for eigenvalues//Davies E B, Safarov Yu, eds. Spectral Theory and Geometry. Vol 273. Edinburgh: London Math Soc Lecture Notes, 1999: 95–139
[2] Chen Z C, Qian C L. On the upper bound of eigenvalues for elliptic equations with higher orders. J Math Anal Appl, 1994, 186: 821–834
[3] Cheng Q M, Huang G Y, Wei G X. Estimates for lower order eigenvalues of a clamped plate problem. Cal Var PDE, 2010, 38: 409–416
[4] Cheng Q M, Ichikawa T, Mametsuka S. Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere. Cal Var PDE, 2009, 36: 507–523
[5] Huang G Y, Li X X, Qi X R. Estimates on the first two buckling eigenvalues on spherical domains. J Geom Physics, 2010, 60: 714–719
[6] Huang G Y, Li X X, Cao L F. Universal bounds on eigenvalues of the buckling problem on spherical domains. J of Math (PRC), 2011, 31: 840–846
[7] Huang G Y, Chen W Y. Universal bounds for eigenvlaues of Laplacian operator with any order. Acta Mathematica Scientia, 2010, 30B(3): 939–948
[8] Huang G Y, Chen W Y. Ineqalities of eigenvalues for bi-kohn Laplacian on Heisenberg group. Acta Mathematica Scientia, 2010, 30B(1): 125–131
[9] Jost J, Li-Jost X Q, Wang Q L, Xia C Y. Universal bounds for eigenvalues of the polyharmonic operator. Trans Amer Math Soc, 2011, 263: 1821–1854
[10] Jost J, Li-Jost X Q, Wang Q L, Xia C Y. Universal inequalities for eigenvalues of the buckling problem of arbitrary order. Comm PDE, 2010, 35: 1563–1589
[11] Payne L E, P´olya G, Weinberger H F. On the ratio of consecutive eigenvalues. J Math Phys, 1956, 35: 289–298
[12] Hile G N, Yeh R Z. Inequalities for eigenvalues of the biharmonic operator. Pacific J Math, 1984, 112: 115–133
[13] Cheng Q M, Yang H C. Universal bounds for eigenvalues of a buckling problem. Commun Math Phys, 2006, 262: 663–675
[14] Wang Q L, Xia C Y. Universal inequalities for eigenvalues of the buckling problem on spherical domains. Commun Math Phys, 2007, 270: 759–775
[15] Wang Q L, Xia C Y. Inequalities for eigenvalues of the clamped problem. Cal Var PDE, 2011, 40: 273–289
[16] Wang Q L, Xia C Y. Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifold. J Funct Anal, 2007, 245: 334–352
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