Annali di Matematica Pura ed Applicata

, Volume 191, Issue 3, pp 539–550 | Cite as

Eigenvalue estimate for the weighted p-Laplacian

  • Lin Feng WangEmail author


Let M be an n-dimensional closed manifold with metric g, dμ = e h(x) dV(x) be the weighted measure and ∆ μ, p be the weighted p-Laplacian. In this article, we get the lower bound estimate of the first nonzero eigenvalue for the weighted p-Laplacian when the m-dimensional Bakry-Émery curvature has a positive lower bound.


Weighted p-Laplacian Bakry-Émery curvature Maximum principle Eigenvalue 

Mathematics Subject Classification (2000)



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  1. 1.
    Lichnerowicz A.: Géometrie des groupes de transformations. Dunod, Paris (1958)zbMATHGoogle Scholar
  2. 2.
    Obata M.: Certain conditions for a Riemannian manifold to be isometric to the sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Qian Z.M.: Estimates for weight volumes and applications. J. Math. Oxford Ser. 48, 235–242 (1987)CrossRefGoogle Scholar
  4. 4.
    Lott J.: Some geometric properties of the Bakry-Emery Ricci tensor. Comment. Math. Helv. 78, 865–883 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Serrin J.: Local behavior of solutions of quasi-linear equations. Acta. Math. 111, 247–302 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Tolksdorf P.: Regularity for a more general class of quasilinear elliptic equations. J. Diff. Equ. 51, 126–150 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Matei A.M.: First eigenvalue for the p-Laplace operator. Nonlinear Anal. Ser. A Theory Methods 39(8), 1051–1068 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Takeuchi H.: On the first eigenvalue of the p-Laplacian in a Riemannian manifold. Tokyo J. Math. 21, 135–140 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Moser R.: The inverse mean curvature flow and p-harmonic functions. J. Eur. Math. Soc 9, 77–83 (2007)zbMATHCrossRefGoogle Scholar
  10. 10.
    Huisken G., Ilmanen T.: The inverse mean curvature flow and Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kotschwar B., Ni L.: Local gradient estimates of p-harmonic functions, \({\frac{1}{H}}\)−flow, and an entropy formula. Ann. Sci. Ec. Norm. Sup. 42(1), 1–36 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Li X.D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. de Mathématiques Pures et Appliqués 84(10), 1295–1361 (2005)zbMATHCrossRefGoogle Scholar
  13. 13.
    Wang L.F.: The upper bound of the \({L_{\mu}^2}\) spectrum. Ann. Glob. Anal. Geom. 37(4), 393–402 (2010)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kinnunen J., Kuusi T.: Local behaviour of solutions to doubly nonlinear parabolic equation. Math. Ann. 337, 705–728 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hein B.: A homotopy approach to solving the inverse mean curvature flow. Calc. Var. 28, 249–273 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg, New York (1977)zbMATHGoogle Scholar
  17. 17.
    Evans L.C.: A new proof of local C 1,α regularity for solutions of certain degenerate elliptic P.D.E. J. Differ. Equ. 45, 356–373 (1982)zbMATHCrossRefGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of ScienceNantong UniversityNantongChina

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