Comptes Rendus
no. 7
Partial Differential Equations
Hardy–Poincaré inequalities and applications to nonlinear diffusions
[Inégalités de Hardy–Poincaré et applications]

Présenté par : Pierre-Louis Lions

Adrien Blanchet1 ; Matteo Bonforte1 ; Jean Dolbeault1 ; Gabriele Grillo2 ; Juan-Luis Vázquez3
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, Volume 344 (2007) no. 7, pp. 431-436.

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.

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Accepté le :
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DOI : 10.1016/j.crma.2007.01.011
Adrien Blanchet 1 ; Matteo Bonforte 1 ; Jean Dolbeault 1 ; Gabriele Grillo 2 ; Juan-Luis Vázquez 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 
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     author = {Adrien Blanchet and Matteo Bonforte and Jean Dolbeault and Gabriele Grillo and Juan-Luis V\'azquez},
     title = {Hardy{\textendash}Poincar\'e inequalities and applications to nonlinear diffusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {431--436},
     publisher = {Elsevier},
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     year = {2007},
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Adrien Blanchet; Matteo Bonforte; Jean Dolbeault; Gabriele Grillo; Juan-Luis Vázquez. Hardy–Poincaré inequalities and applications to nonlinear diffusions. Comptes Rendus. Mathématique, Volume 344 (2007) no. 7, pp. 431-436. doi : 10.1016/j.crma.2007.01.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.011/

[1] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vázquez, Asymptotics of the fast diffusion equation, in preparation

[2] M. Bonforte; J.L. Vázquez Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., Volume 240 (2006), pp. 399-428

[3] J.A. Carrillo; C. Lederman; P.A. Markowich; G. Toscani Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity, Volume 15 (2002), pp. 565-580

[4] J.A. Carrillo; J.L. Vázquez Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations, Volume 28 (2003), pp. 1023-1056

[5] F. Cipriani Sobolev–Orlicz imbeddings, weak compactness, and spectrum, J. Funct. Anal., Volume 177 (2000), pp. 89-106

[6] P. Daskalopoulos, N. Sesum, On the extinction profile of solutions to fast-diffusion, 2006

[7] E.B. Davies A review of Hardy inequalities, Rostock, 1998 (Oper. Theory Adv. Appl.), Volume vol. 110, Birkhäuser, Basel (1999), pp. 55-67

[8] E.B. Davies; A.M. Hinz Explicit constants for Rellich inequalities in Lp(Ω), Math. Z., Volume 227 (1998), pp. 511-523

[9] M. Del Pino; J. Dolbeault Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), Volume 81 (2002), pp. 847-875

[10] J. Denzler; R.J. McCann Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology, Arch. Ration. Mech. Anal., Volume 175 (2005), pp. 301-342

[11] G. Grillo On Persson's theorem in local Dirichlet spaces, Z. Anal. Anwend., Volume 17 (1998), pp. 329-338

[12] B. Muckenhoupt Hardy's inequality with weights, Studia Math., Volume 44 (1972), pp. 31-38 (Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I)

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