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The Journal of Geometric Analysis

, Volume 15, Issue 1, pp 83–121 | Cite as

Balls have the worst best Sobolev inequalities

  • Francesco MaggiEmail author
  • Cédric Villani
Article

Abstract

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝn. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].

Math Subject Classifications

26D15 46E35 

Key Words and Phrases

Isoperimetric inequality Sobolev inequality mass transportation trace 

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Copyright information

© Mathematica Josephina, Inc. 2005

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität Duisburg-EssenDuisburgGermany
  2. 2.UMPALyon Cedex 07France

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