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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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  • 1. M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Edinburgh, 1998), E. B. Davies and Yu Safarov, eds., London Math. Soc. Lecture Notes, Vol. 273, Cambridge Univ. Press, Cambridge, 95-139, 1999. MR 1736867 (2001a:35131)
  • 2. M. S. Ashbaugh, Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H. C. Yang, Proc. Indian Acad. Sci. (Math. Sci.) 112 (2002), 3-30. MR 1894540 (2004c:35302)
  • 3. D. G. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008), 325-339. MR 2421979
  • 4. Q.-M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005), 445-460. MR 2115463 (2005i:58038)
  • 5. Q.-M. Cheng and H. C. Yang, Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces, J. Math. Soc. Japan 58 (2006), 545-561. MR 2228572 (2007k:58051)
  • 6. Q. -M. Cheng and H.C. Yang, Bounds on eigenvalues of Dirichlet Laplacian, Math. Ann. 337 (2007), 159-175. MR 2262780 (2007k:35064)
  • 7. E. M. Harrell II and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809. MR 1401772 (97i:35129)
  • 8. G. N. Hile and M. H. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J. 29 (1980), 523-538. MR 578204 (82c:35052)
  • 9. S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc. 318 (1990), 615-642. MR 994167 (90h:35075)
  • 10. P.-F. Leung, On the consecutive eigenvalues of the Laplacain of a compact minimal submanifold in a sphere, J. Austral. Math. Soc. 50 (1991), 409-426. MR 1096895 (92d:58212)
  • 11. P. Li, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helve. 55 (1980), 347-363. MR 593051 (81k:58067)
  • 12. L. Ljung, Recursive identification, in Stochastic systems: The mathematics of filtering and identification and applications, M. Hazewinkel and J. C. Willems, eds., Reidel, 1981, 247-283. MR 674319 (83m:93001)
  • 13. P. S. Maybeck, Stochastic models, estimation, and control. III, Academic Press, London, 1982. MR 0690418 (85a:93134b)
  • 14. L. E. Payne, G. Polya and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289-298. MR 0084696 (18:905c)
  • 15. C. L. Qian and Z. C. Chen, Estimates of eigenvalues for uniformly elliptic operator of second order, Acta Math. Appli. Sinica 10 (1994), 349-355. MR 1324579 (96b:35158)
  • 16. A. El Soufi, E. M. Harrell II and S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Amer. Math. Soc. 361 (2009), 2337-2350. MR 2471921
  • 17. H. J. Sun, Q.-M. Cheng and H. C. Yang, Lower order eigenvalues of Dirichlet Laplacian, Manuscripta Math. 125 (2008), 139-156. MR 2373079 (2009i:58042)
  • 18. Q. L. Wang and C. Y. Xia, Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Anal. 245 (2007), 334-352. MR 2311628 (2008e:58033)
  • 19. H. C. Yang, An estimate of the difference between consecutive eigenvalues, preprint IC/91/60, 1991, Trieste, Italy.
  • 20. P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemannian surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa CI. Sci. 7 (1980), 55-63. MR 577325 (81m:58084)

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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

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.

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