Universal Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains

Wang, Qiaoling; Xia, Changyu

doi:10.1007/s00220-006-0170-6

Cited by 24 publications

(40 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…With the notations as above, we consider now the special case that Ω is a bounded domain in R n . Denote by x 1 , · · · , x n the coordinate functions of R n and let us decompose the vectorvalued functions x α ∇u i as and from the discussions in [11] and [34] we know that div W αi = 0, (2.9) where for a vector field Z on Ω, div Z denotes the divergence of Z.…”

Section: Proofs Of the Resultsmentioning

confidence: 99%

“…For a function g on Ω, we have (cf. (2.31) in [34]) ∆ ∇x α , ∇g = −2x α ∆g + ∇x α , ∇((∆ + n − 2)g) . For each q = 0, 1, · · ·, thanks to (2.43) and (2.50), there are polynomials F q and G q of degree q such that…”

Section: Proof Of Corollary 12 By Induction One Can Show Thatmentioning

confidence: 99%

“…then c αij = −c αji (cf. Lemma in [34]). By using the same arguments as in the proof of (2.37) in [34], we have…”

Section: Proof Of Corollary 12 By Induction One Can Show Thatmentioning

confidence: 99%

See 2 more Smart Citations

Inequalities for eigenvalues of the buckling problem of arbitrary order

Cheng

Wang

et al. 2017

Annali di Matematica

Self Cite

View full text Add to dashboard Cite

This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimates [16] on the buckling eigenvalues of order two to arbitrary order.2000 Mathematics Subject Classification : 35P15, 53C20, 53C42, 58G25 Key words and phrases: Universal inequality for eigenvalues, the buckling problem of arbitrary order, Euclidean space, sphere.

show abstract

Section: Proofs Of the Resultsmentioning

confidence: 99%

Section: Proof Of Corollary 12 By Induction One Can Show Thatmentioning

confidence: 99%

See 1 more Smart Citation

Inequalities for eigenvalues of the buckling problem of arbitrary order

Cheng

Wang

et al. 2017

Annali di Matematica

Self Cite

View full text Add to dashboard Cite

show abstract

“…(2 [14] with V = 0 and ρ = 1. For the recent developments about universal inequalities for eigenvalues of the Laplace operator on Riemannian manifolds, we refer to [5,[9][10][11][12][22][23][24][25][26][27][28] and the references therein.…”

Section: A General Inequality For Eigenvalues Of the Operator L (Ft )mentioning

confidence: 99%

Estimates for eigenvalues of ℒr operator on self-shrinkers

Huang

2017

Int. J. Math.

View full text Add to dashboard Cite

Let x : M → R N be an n-dimensional compact self-shrinker in R N with smooth boundary ∂Ω. In this paper, we study eigenvalues of the operator Lr on M , where Lr is defined by Lr = e |x| 2 2 div(e − |x| 2 2 T r ∇·) with T r denoting a positive definite (0,2)-tensor field on M . We obtain "universal" inequalities for eigenvalues of the operator Lr. These inequalities generalize the result of Cheng and Peng in [8]. Furthermore, we also consider the case that equalities occur.2000 Mathematics Subject Classification. Primary 53C40, Secondary 58C40.

show abstract

“…The inequalities on the higher eigenvalues of the Laplacian on a connected bounded domain in R n obtained by Payne-Pólya-Weinberger, Hile-Protter, Yang have also been extended to some other eigenvalue problems (cf. [1][2][3][4][5][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23]26,27,[32][33][34][35]37], etc). Here let us consider the case of the eigenvalue problem for the Dirichlet biharmonic operator or the clamped plate problem which describes the characteristic vibrations of a clamped plate.…”

mentioning

confidence: 99%

Inequalities for eigenvalues of a clamped plate problem

Wang

Xia

2010

Calc. Var.

Self Cite

View full text Add to dashboard Cite

In this paper we study eigenvalues of a clamped plate problem on compact domains in complete manifolds. For complete manifolds admitting special functions, we prove universal inequalities for eigenvalues of clamped plate problem independent of the domains of Payne-Pólya-Weinberger-Yang type. These manifolds include Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifolds and manifolds admitting eigenmaps to a sphere. In the case of warped product manifolds, our result implies a universal inequality on hyperbolic space proved by Cheng-Yang. We also strengthen an inequality for eigenvalues of clamped plate problem on submanifolds in a Euclidean space obtained recently by Cheng, Ichikawa and Mametsuka.

show abstract

Universal Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains

Cited by 24 publications

References 15 publications

Inequalities for eigenvalues of the buckling problem of arbitrary order

Inequalities for eigenvalues of the buckling problem of arbitrary order

Estimates for eigenvalues of ℒr operator on self-shrinkers

Inequalities for eigenvalues of a clamped plate problem

Contact Info

Product

Resources

About