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EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN

Published online by Cambridge University Press:  08 December 2010

SUN HEJUN  and
QI XUERONG
SUN HEJUN
Affiliation:
Department of Applied Mathematics, College of Science, Nanjing University of Science and Technology, Nanjing 210094, P.R. China e-mail: hejunsun@163.com
QI XUERONG
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan e-mail: qixuerong609@gmail.com
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Abstract

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For a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

Keywords

35P1558C4053C42
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 2 , May 2011 , pp. 321 - 332
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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