Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds |
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Authors: | Qiaoling Wang |
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Institution: | Departamento de Matemática-IE, Universidade de Brasília, 70910-900 Brasília-DF, Brazil |
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Abstract: | In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k+1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang. |
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Keywords: | Universal bounds Eigenvalues Biharmonic operator Sphere Euclidean space Minimal submanifolds |
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