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The Quarterly Journal of Mathematics Advance Access published online on September 4, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap026
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

UNIVERSAL INEQUALITIES FOR EIGENVALUES OF THE VIBRATION PROBLEM FOR A CLAMPED PLATE ON RIEMANNIAN MANIFOLDS{dagger}

Changyu Xia{ddagger}

Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília-DF, Brazil

{ddagger} E-mail: xia{at}mat.unb.br

Received 21 October 2008;
  Abstract

Eigenvalues of the vibration problem for a clamped plate on compact Riemannian manifolds with boundary (possibly empty) are studied. Universal bounds on eigenvalues of the vibration problem for a clamped plate on compact domains in a complex projective space, a minimal submanifold of a Euclidean space or of a unit sphere are obtained and in particular, an explicit upper bound for the (k + 1)th eigenvalue of the vibration problem for a clamped plate on such objects in terms of its first k eigenvalues will be given.

{dagger} This work was done while the author was visiting MPI for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany.


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