Advanced Search
Article Contents
Article Contents

Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • In this paper, we firstly study the eigenvalue problem of a systemof elliptic equations with drift and get some universal inequalities of PayneP′olya-Weinberger-Yang type on a bounded domain in Euclidean spaces and inGaussian shrinking solitons. Furthermore, we study two kinds of the clampedplate problems and the buckling problems for the bi-drifting Laplacian and getsome sharp lower bounds for the first eigenvalue for these eigenvalue problemon compact manifolds with boundary and positive m-weighted Ricci curvatureor on compact manifolds with boundary under some condition on the weightedRicci curvature.

    Mathematics Subject Classification: 35P15, 53C20, 53C42.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math. , pages 177-206. Springer, Berlin, 1985. doi: 10.1007/BFb0075847.

    M. Batista, M. P. Cavalcante and J. Pyo, Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds, J. Math. Anal. Appl., 419 (2014), 617-626.

    doi: 10.1016/j.jmaa.2014.04.074.


    D. Chen, Q. M. Cheng, Q. Wang and C. Xia, On eigenvalues of a system of elliptic equations and of the biharmonic operator, J. Math. Anal. Appl., 387 (2012), 1146-1159.

    doi: 10.1016/j.jmaa.2011.10.020.


    X. Cheng, T. Mejia and D. Zhou, Eigenvalue estimate and compactness for closed f-minimal surfaces, Pacific J. Math., 271 (2014), 347-367.

    doi: 10.2140/pjm.2014.271.347.


    Q. M. Cheng and H. C. Yang, Universal inequalities for eigenvalues of a system of elliptic equations, Proc. Royal Soc. Edinburgh, 139A (2009), 273-285.

    doi: 10.1017/S0308210507000649.


    F. Du, C. Wu, G. Li and C. Xia, Universal inequalities for eigenvalues of a system of subelliptic equations on Heisenberg group, Kodai Math. J., 38 (2015), 437-450.

    doi: 10.2996/kmj/1436403899.


    F. Du, C. Wu, G. Li and C. Xia, Estimates for eigenvalues of the bi-drifting Laplacian operator, Z. Angew. Math. Phys., 66 (2015), 703-726.

    doi: 10.1007/s00033-014-0426-5.


    A. Futaki, H. Li and X. D. Li, On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons, Ann. Glob. Anal. Geom., 44 (2013), 105-114.

    doi: 10.1007/s10455-012-9358-5.


    Q. Huang and Q. H. Ruan, Applications of some elliptic equations in Riemannian manifolds, J. Math. Anal. Appl., 409 (2014), 189-196.

    doi: 10.1016/j.jmaa.2013.07.004.


    S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc., 318 (1990), 615-642.

    doi: 10.2307/2001323.


    H. Li and Y. Wei, f-minimal surface and manifold with positive m-Bakry-Émery Ricci curvature, J. Geom. Anal., 25 (2015), 421-435.

    doi: 10.1007/s12220-013-9434-5.


    H. A. Levine and M. H. Protter, Unretricted lower bounds for eigenvalues of elliptic equations and systems of equations with applications to problem in elasticity, Math. Methods Appl. Sci., 7 (1985), 210-222.

    doi: 10.1002/mma.1670070113.


    M. Levitin and L. Parnovski, Commutators, spectral trance identities, and universal estimates for eigenvalues, J. Funct. Anal., 192 (2002), 425-445.

    doi: 10.1006/jfan.2001.3913.


    L. Ma and S. H. Du, Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians, C. R. Math. Acad. Sci. Paris., 348 (2010), 1203-1206.

    doi: 10.1016/j.crma.2010.10.003.


    L. Ma and B. Y. Liu, Convex eigenfunction of a drifting Laplacian operator and the fundamental gap, Pacific J. Math., 240 (2009), 343-361.

    doi: 10.2140/pjm.2009.240.343.


    L. Ma and B. Y. Liu, Convexity of the first eigenfunction of the drifting Laplacian operator and its applications, New York J. Math., 14 (2008), 393-401.


    R. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., 26 (1977), 459-472.


    G. Wei and W. Wylie, Comparison geometry for the Bakry-Émery Ricci tensor, J. Diff. Geom., 83 (2009), 377-405.


    C. Xia and H. Xu, Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Ann. Glob. Anal. Geom., 45 (2014), 155-166.

    doi: 10.1007/s10455-013-9392-y.

  • 加载中

Article Metrics

HTML views(1227) PDF downloads(183) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint