Calculus of Variations and Geometric Measure Theory

F. Maggi - C. Villani

Balls Have the Worst Best Sobolev Inequalities

created on 26 May 2004
modified by maggi on 19 Dec 2005


Published Paper

Inserted: 26 may 2004
Last Updated: 19 dec 2005

Journal: J. Geom. Anal.
Volume: 15
Number: 1
Pages: 83-121
Year: 2005

Preprint MPI-MIS 322004


Using transportation techniques in the spirit of Cordero-Erasquin, Nazaret and Villani {\it A Mass-Transportation Approach to Sharp Sobolev and Gagliardo-Nirenberg Inequalities} (to appear in Adv. Math.), we establish an optimal non-parametric trace Sobolev inequality, for arbitrary Lipschitz domains in $*R*^n$. We deduce a sharp variant of the trace Sobolev inequality due to Brézis and Lieb ({\it Sobolev Inequalities with a Remainder Term}, J. Funct. Anal. 62 (1985), 375-417), 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 B. and L. op. cit.. Many variants will be investigated in a forthcoming companion paper.

Keywords: Mass transportation, Sobolev inequality, Trace