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.
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.
Accepted:
Published online:
Adrien Blanchet 1; Matteo Bonforte 1; Jean Dolbeault 1; Gabriele Grillo 2; Juan-Luis Vázquez 3
@article{CRMATH_2007__344_7_431_0, 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}, volume = {344}, number = {7}, year = {2007}, doi = {10.1016/j.crma.2007.01.011}, language = {en}, }
TY - JOUR AU - Adrien Blanchet AU - Matteo Bonforte AU - Jean Dolbeault AU - Gabriele Grillo AU - Juan-Luis Vázquez TI - Hardy–Poincaré inequalities and applications to nonlinear diffusions JO - Comptes Rendus. Mathématique PY - 2007 SP - 431 EP - 436 VL - 344 IS - 7 PB - Elsevier DO - 10.1016/j.crma.2007.01.011 LA - en ID - CRMATH_2007__344_7_431_0 ER -
%0 Journal Article %A Adrien Blanchet %A Matteo Bonforte %A Jean Dolbeault %A Gabriele Grillo %A Juan-Luis Vázquez %T Hardy–Poincaré inequalities and applications to nonlinear diffusions %J Comptes Rendus. Mathématique %D 2007 %P 431-436 %V 344 %N 7 %I Elsevier %R 10.1016/j.crma.2007.01.011 %G en %F CRMATH_2007__344_7_431_0
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] Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., Volume 240 (2006), pp. 399-428
[3] Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity, Volume 15 (2002), pp. 565-580
[4] Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations, Volume 28 (2003), pp. 1023-1056
[5] 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] A review of Hardy inequalities, Rostock, 1998 (Oper. Theory Adv. Appl.), Volume vol. 110, Birkhäuser, Basel (1999), pp. 55-67
[8] Explicit constants for Rellich inequalities in , Math. Z., Volume 227 (1998), pp. 511-523
[9] Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), Volume 81 (2002), pp. 847-875
[10] Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology, Arch. Ration. Mech. Anal., Volume 175 (2005), pp. 301-342
[11] On Persson's theorem in local Dirichlet spaces, Z. Anal. Anwend., Volume 17 (1998), pp. 329-338
[12] 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)
[13] Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand., Volume 8 (1960), pp. 143-153
[14] Integral inequalities and theorems of Liouville type, J. Math. Anal. Appl., Volume 26 (1969), pp. 630-639
[15] Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Ser. Math. Appl., vol. 33, Oxford Univ. Press, 2006
Cited by Sources:
Comments - Policy