A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities.
type: Item
instance of: scholarly article
title: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities.
zbMATH Open document ID: 1048.26010
DOI: 10.1016/S0001-8708(03)00080-X
author: Bruno Nazaret
author: Cédric Villani
author: D. Cordero-Erausquin
published in: Advances in Mathematics
publication date: 2004-03-14T00:00:00Z
review text: The authors give a new proof of the sharp Sobolev embedding in \(R^n\), \[ \left( \int_{R^n} | f(x)| ^{p^*} \, dx \right)^{1/p^*} \leq S_n (p) \, \left( \int_{R^n} | \nabla f| ^p \, dx \right)^{1/p} \] where \[ 1<p<n, \quad 1- n/p = - n/p^*. \] The sharp constants \(S_n (p)\) and related extremal functions are calculated using new measure-theoretical arguments, called mass-transportation. Similar results are obtained for \(p=1\) (related to isoperimetrical problems). Euclidean norms are replaced by more general norms. Some results for related Gagliardo-Nirenberg inequalities are obtained.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 2055356
zbMATH Keywords: Sobolev inequalities
zbMATH Keywords: Gagliardo-Nirenberg inequalities
zbMATH Keywords: mass-transportation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2055356