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MiS Preprint

Balls Have the Worst Best Sobolev Inequalities

Francesco Maggi and Cédric Villani


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.

May 21, 2004
May 21, 2004
sobolev inequality, mass transportation, trace

Related publications

2005 Repository Open Access
Francesco Maggi and Cédric Villani

Balls have the worst best Sobolev inequalities

In: The journal of geometric analysis, 15 (2005) 1, pp. 83-121