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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Yang-type inequalities for weighted eigenvalues of a second order uniformly elliptic operator with a nonnegative potential


Author: He-Jun Sun
Journal: Proc. Amer. Math. Soc. 138 (2010), 2827-2837
MSC (2010): Primary 35P15, 58C40; Secondary 58J50
Published electronically: March 16, 2010
MathSciNet review: 2644896
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Abstract: In this paper, we investigate the Dirichlet weighted eigenvalue problem of a second order uniformly elliptic operator with a nonnegative potential on a bounded domain $ \Omega \subset \mathbb{R}^n$. First, we prove a general inequality of eigenvalues for this problem. Then, by using this general inequality, we obtain Yang-type inequalities which give universal upper bounds for eigenvalues. An explicit estimate for the gaps of any two consecutive eigenvalues is also derived. Our results contain and extend the previous results for eigenvalues of the Laplacian, the Schrödinger operator and the second order elliptic operator on a bounded domain $ \Omega \subset \mathbb{R}^n$.


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Additional Information

He-Jun Sun
Affiliation: Department of Applied Mathematics, College of Science, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China
Email: hejunsun@163.com

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10321-9
PII: S 0002-9939(10)10321-9
Keywords: Eigenvalue, universal inequality, elliptic operator, Laplacian, Schr\"{o}dinger operator
Received by editor(s): August 22, 2009
Received by editor(s) in revised form: November 18, 2009
Published electronically: March 16, 2010
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.