Hardy–Poincaré inequalities and applications to nonlinear diffusions

Blanchet, Adrien; Bonforte, Matteo; Dolbeault, Jean; Grillo, Gabriele; Vázquez, Juan-Luis

doi:10.1016/j.crma.2007.01.011

Partial Differential Equations

Hardy–Poincaré inequalities and applications to nonlinear diffusions
[Inégalités de Hardy–Poincaré et applications]

Présenté par : Pierre-Louis Lions

Blanchet, Adrien ¹ ; Bonforte, Matteo ¹ ; Dolbeault, Jean ¹ ; Grillo, Gabriele ² ; Vázquez, Juan-Luis ³

¹ CEREMADE, Université Paris Dauphine, place de Lattre de Tassigny, 75775 Paris cedex 16, France
² Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
³ Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Tome 344 (2007) no. 7, pp. 431-436.

Résumé
Abstract

Nous étudions des inégalités de Poincaré qui sont étroitement reliées à des inégalités de type Hardy et établissons la forme des constantes optimales dans certains cas. De telles inégalités sont ensuite utilisées pour relier l'entropie avec la production d'entropie et obtenir des résultats d'asymptotiques intermédiaires pour les équations à diffusion rapide.

We systematically study weighted Poincaré type inequalities which are closely connected with Hardy type inequalities and establish the form of the optimal constants in some cases. Such inequalities are then used to relate entropy with entropy production and get intermediate asymptotics results for fast diffusion equations.

Reçu le : 2006-10-20
Accepté le : 2007-01-16
Publié le : 2007-03-30

DOI : 10.1016/j.crma.2007.01.011

Affiliations des auteurs :

Blanchet, Adrien ¹ ; Bonforte, Matteo ¹ ; Dolbeault, Jean ¹ ; Grillo, Gabriele ² ; Vázquez, Juan-Luis ³

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     title = {Hardy{\textendash}Poincar\'e inequalities and applications to nonlinear diffusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {431--436},
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Blanchet, Adrien; Bonforte, Matteo; Dolbeault, Jean; Grillo, Gabriele; Vázquez, Juan-Luis. Hardy–Poincaré inequalities and applications to nonlinear diffusions. Comptes Rendus. Mathématique, Tome 344 (2007) no. 7, pp. 431-436. doi : 10.1016/j.crma.2007.01.011. http://www.numdam.org/articles/10.1016/j.crma.2007.01.011/

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