Mathematische Zeitschrift

, Volume 237, Issue 4, pp 737–767

From best constants to critical functions

  • Emmanuel Hebey
  • Michel Vaugon
Original article

DOI: 10.1007/PL00004889

Cite this article as:
Hebey, E. & Vaugon, M. Math Z (2001) 237: 737. doi:10.1007/PL00004889

Abstract.

The study of sharp Sobolev inequalities starts with the notion of best constant and leads naturally to the question to know whether or not there exist extremal functions for these inequalities. We restrict ourselves in this paper to the \(H_1^2\)-Sobolev inequality. Then, we extend the notion of best constant to that of critical function, and, with the help of this notion, we answer the question to know whether or not there exist extremal functions for the sharp \(H_1^2\)-Sobolev inequality. Partial answers to the more general question to know whether or not an extremal function always comes with a critical function are also given.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Emmanuel Hebey
    • 1
  • Michel Vaugon
    • 2
  1. 1.Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France (e-mail: Emmanuel.Hebey@math.u-cergy.fr) FR
  2. 2.Université Pierre et Marie Curie, Département de Mathématiques, 4 place Jussieu, 75252 Paris cedex 05, France (e-mail: vaugon@math.jussieu.fr) FR

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