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MiS Preprint
32/2004
Balls Have the Worst Best Sobolev Inequalities
Francesco Maggi and Cédric Villani
Abstract
Using transportation techniques in the spirit of Cordero-Erasquin, Nazaret and Villani "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 $\mathbb{R}^n$. We deduce a sharp variant of the trace Sobolev inequality due to Brèzis and Lieb ("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 Brèzis-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.
