Inequalities for eigenvalues of a clamped plate problem
Qing-Ming Cheng,Hongcang Yang +1 more
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In this article, an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, is obtained.Abstract:
Let D be a connected bounded domain in an n-dimensional Euclidean space R n . Assume that 0 < λ 1 < λ 2 ≤ ··· ≤λ k ≤··· are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: Then, we give an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, that is, we prove the following: Further, a more explicit inequality of eigenvalues is also obtained.read more
Citations
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Bounds on eigenvalues of Dirichlet Laplacian
TL;DR: In this article, the eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space Rn was investigated, and it was shown that λk+1 is the (k + 1)th eigen value of DLA on Ω.
Journal ArticleDOI
Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds
Qiaoling Wang,Changyu Xia +1 more
TL;DR: In this paper, the (k+1)th eigenvalue of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) was studied.
Journal ArticleDOI
Extrinsic estimates for eigenvalues of the Laplace operator
Daguang Chen,Qing-Ming Cheng +1 more
TL;DR: For a compact spin manifold M isometrically embedded into Euclidean space, the authors derived the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding.
Journal ArticleDOI
Inequalities for eigenvalues of a clamped plate problem
Qiaoling Wang,Changyu Xia +1 more
TL;DR: In this article, the eigenvalues of a clamped plate problem on complete manifolds are studied and a universal inequality for the case of warped product manifolds is shown. But the universal inequalities are not applicable to the complete manifold with Ricci curvatures.
Proceedings ArticleDOI
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References
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On the ratio of consecutive eigenvalues
Journal ArticleDOI
A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions
TL;DR: The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia.
Book ChapterDOI
Isoperimetric and universal inequalities for eigenvalues
TL;DR: A review of the known inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians on bounded domains in Euclidean space can be found in this paper.
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Proof of the Payne-Pólya-Weinberger conjecture
TL;DR: In this paper, it was shown that the optimal upper bound for A2/Aj was its value for the disk, approximately 2.539, which is the same upper bound as the conjecture of Payne, Pólya, and Weinberger.
Journal ArticleDOI
Domain-independent upper bounds for eigenvalues of elliptic operators
TL;DR: In this paper, the authors apply the method used by Hile and Protter [2] to a variety of second-order elliptic problems, in particular, to all constant coefficient problems, where the Laplacian is replaced by a more general operator in a Hilbert space.