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Annals of Global Analysis and Geometry

, Volume 48, Issue 3, pp 255–268 | Cite as

Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons

  • Feng Du
  • Jing MaoEmail author
  • Qiaoling Wang
  • Chuanxi Wu
Article

Abstract

In this paper, we study the eigenvalue problem of poly-drifting Laplacian and get a general inequality for lower order eigenvalues on compact smooth metric measure spaces with boundary (possibly empty). Applying this general inequality, we obtain some universal inequalities for lower order eigenvalues for the eigenvalue problem of poly-drifting Laplacian on bounded connected domains in Euclidean spaces or unit spheres. Moreover, we separately get some universal inequalities for the eigenvalue problem of poly-drifting Laplacian on bounded connected domains in the Gaussian and cylinder shrinking solitons.

Keywords

Eigenvalues Universal inequalities Poly-drifting Laplacian  Gaussian shrinking soliton Cylinder shrinking soliton 

Mathematics Subject Classification

35P15 53C20 53C42 

Notes

Acknowledgments

F. Du was partially supported by CNPq, Brazil and the NSF of China (Grant No. 11401131). J. Mao was partially supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the NSF of China (Grant No. 11401131). Q.-L. Wang was partially supported by CNPq, Brazil. The authors would like to thank the anonymous referee for his or her careful reading and valuable comments.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematics and Physics ScienceJingchu University of TechnologyJingmenChina
  2. 2.Departamento de MatemáticaUniversidade de BrasiliaBrasíliaBrazil
  3. 3.Department of MathematicsHarbin Institute of Technology (Weihai)WeihaiChina
  4. 4.School of Mathematics and StatisticsHubei UniversityWuhanChina

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