Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani , 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 , 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 . Many variants will be investigated in a companion article .
Math Subject Classifications
Key Words and Phrases
Isoperimetric inequality Sobolev inequality mass transportation trace
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