The p-Faber-Krahn Inequality Noted

  • Jie Xiao
Part of the International Mathematical Series book series (IMAT, volume 11)


When revisiting the Faber-Krahn inequality for the principal p- Laplacian eigenvalue of a bounded open set in R n with smooth boundary, we simply rename it as the p-Faber-Krahn inequality and interestingly find that this inequality may be improved but also characterized through Maz'ya's capacity method, the Euclidean volume, the Sobolev type inequality and Moser-Trudinger's inequality.


Sobolev Inequality Isoperimetric Inequality Euclidean Ball Laplacian Eigenvalue Nonnegative Ricci Curvature 
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  1. 1.
    Bhattacharia, T.: A proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian. Ann. Mat. Pura Appl. Ser. 4 177, 225–231 (1999)CrossRefGoogle Scholar
  2. 2.
    Carron, G.: In'egakut'es isop'erim'etriques de Faber-Krahn et cons'equences. Publications de l'Institute Fourier. 220 (1992)Google Scholar
  3. 3.
    Chavel, I.: Isoperimetric Inequalities. Cambridge Univ. Press (2001)MATHGoogle Scholar
  4. 4.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R. (ed.), Problems in Analysis, pp. 195–199. Princeton Univ. Press, Princeton, NJ (1970)Google Scholar
  5. 5.
    Colesanti, A., Cuoghi, P., Salani, P.: Brunn–Minkowski inequalities for two functionals involving the p-Laplace operator of the Laplacian. Appl. Anal. 85, 45–66 (2006)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Dacorogna, B.: Introduction to the Calculus of Variations. Imperical College Press (1992)MATHGoogle Scholar
  7. 7.
    Demengel, F.: Functions locally almost 1-harmonic. Appl. Anal. 83, 865–893 (2004)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    D'Onofrio, L., Iwaniec, T.: Notes on p-harmonic analysis. Contemp. Math. 370, 25–49 (2005)MathSciNetGoogle Scholar
  9. 9.
    Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press LLC (1992)MATHGoogle Scholar
  10. 10.
    Federer, H., Fleming, W. H.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Flucher, M.: Variational Problems with Concentration. Birkhäuser, Basel (1999)MATHGoogle Scholar
  12. 12.
    Fusco, N., Maggi, F., Pratelli, A.: A note on Cheeger sets. Proc. Am. Math. Soc. electron.: January 26, 1–6 (2009)Google Scholar
  13. 13.
    Fusco, N., Maggi, F., Pratelli, A.: Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) [To appear]Google Scholar
  14. 14.
    Grigor'yan, A.: Isoperimetric inequalities and capacities on Riemannian manifolds. In: The Maz'ya anniversary collection 1 (Rostock, 1998), pp. 139–153. Birkhäuser, Basel (1999)Google Scholar
  15. 15.
    Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Institute of Math. Sci. New York University. 5 (1999)Google Scholar
  16. 16.
    Hebey, E., Saintier, N.: Stability and perturbations of the domain for the first eigenvalue of the 1-Laplacian. Arch. Math. (Basel) (2007)Google Scholar
  17. 17.
    Juutinen, P., Lindqvist, P., Manfredi, J.: The ∞-eigenvalue problem. Arch. Ration. Mech. Anal. 148, 89–105 (1999)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kawohl, B., Fridman, V.: Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Commun. Math. Univ. Carol. 44, 659–667 (2003)MATHMathSciNetGoogle Scholar
  19. 19.
    Kawohl, B., Lindqvist, P.: Positive eigenfunctions for the p-Laplace operator revisited. Analysis (Munich) 26, 545–550 (2006)MATHMathSciNetGoogle Scholar
  20. 20.
    Lefton, L., Wei, D.: Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method. Numer. Funct. Anal. Optim. 18, 389–399 (1997)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Maz'ya, V.G.: Classes of domains and imbedding theorems for function spaces (Russian). Dokl. Akad. Nauk SSSR 3, 527–530 (1960); English transl.: Sov. Math. Dokl. 1, 882–885 (1961)Google Scholar
  22. 22.
    Maz'ya, V.: Sobolev Spaces. Springer, Berlin etc. (1985)Google Scholar
  23. 23.
    Maz'ya, V.: Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Contemp. Math. 338, 307–340 (2003)MathSciNetGoogle Scholar
  24. 24.
    Maz'ya, V.: Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev type imbeddings. J. Funct. Anal. 224, 408–430 (2005)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Maz'ya, V.: Integral and isocapacitary inequalities. arXiv:0809.2511v1 [math.FA] 15 Sep 2008Google Scholar
  26. 26.
    Sakaguchi, S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV 14 (1987), 403–421 (1988)MathSciNetGoogle Scholar
  27. 27.
    Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. London Math. Soc. LMS. 289, Cambridge Univ. Press, Cambridge (2002)Google Scholar
  28. 28.
    Xiao, J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007)MATHMathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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