Geodesic
Partial Differential Equations
Liouville-type theorems for certain degenerate and singular parabolic equations
[Théorèmes de type Liouville pour quelques équations paraboliques singulières dégénérées]

Présenté par : Pierre-Louis Lions

DiBenedetto, Emmanuele1 ; Gianazza, Ugo2 ; Vespri, Vincenzo3
1 Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
2 Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
3 Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
Comptes Rendus. Mathématique, Tome 348 (2010) no. 15-16, pp. 873-877.

Voir la notice de l'article provenant de la source Numdam

Relying on recent results on Harnack inequalities for equations of p-Laplacian type, we prove Liouville-type estimates for solutions to these equations, both in the degenerate (p>2), and in the singular (1<p<2) range.

En utilisant des résultats récents sur l'inégalité de Harnack pour les équations type p-laplacien, on établit des théorèmes de type Liouville pour les solutions de ces équations, dans le cas dégénéré p>2, ainsi bien que dans le cas singulier 1<p<2.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.06.019

DiBenedetto, Emmanuele 1 ; Gianazza, Ugo 2 ; Vespri, Vincenzo 3

1 Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
2 Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
3 Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy 
@article{CRMATH_2010__348_15-16_873_0,
     author = {DiBenedetto, Emmanuele and Gianazza, Ugo and Vespri, Vincenzo},
     title = {Liouville-type theorems for certain degenerate and singular parabolic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {873--877},
     publisher = {Elsevier},
     volume = {348},
     number = {15-16},
     year = {2010},
     doi = {10.1016/j.crma.2010.06.019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.06.019/}
}
TY  - JOUR
AU  - DiBenedetto, Emmanuele
AU  - Gianazza, Ugo
AU  - Vespri, Vincenzo
TI  - Liouville-type theorems for certain degenerate and singular parabolic equations
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 873
EP  - 877
VL  - 348
IS  - 15-16
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.06.019/
DO  - 10.1016/j.crma.2010.06.019
LA  - en
ID  - CRMATH_2010__348_15-16_873_0
ER  - 
%0 Journal Article
%A DiBenedetto, Emmanuele
%A Gianazza, Ugo
%A Vespri, Vincenzo
%T Liouville-type theorems for certain degenerate and singular parabolic equations
%J Comptes Rendus. Mathématique
%D 2010
%P 873-877
%V 348
%N 15-16
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.06.019/
%R 10.1016/j.crma.2010.06.019
%G en
%F CRMATH_2010__348_15-16_873_0
DiBenedetto, Emmanuele; Gianazza, Ugo; Vespri, Vincenzo. Liouville-type theorems for certain degenerate and singular parabolic equations. Comptes Rendus. Mathématique, Tome 348 (2010) no. 15-16, pp. 873-877. doi : 10.1016/j.crma.2010.06.019. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.06.019/

[1] DiBenedetto, E.; Gianazza, U.; Vespri, V. Harnack estimates for quasi-linear degenerate parabolic differential equation, Acta Math., Volume 200 (2008), pp. 181-209

[2] DiBenedetto, E.; Gianazza, U.; Vespri, V. Alternative forms of the Harnack inequality for non-negative solutions to certain degenerate and singular parabolic equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (9) Rend. Lincei Mat. Appl., Volume 20 (2009), pp. 369-377

[3] DiBenedetto, E.; Gianazza, U.; Vespri, V. Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume IX (2010), pp. 385-422

[4] Glagoleva, R.Ya. Liouville theorems for the solution of second order linear parabolic equations with discontinuous coefficients, Math. Zametki, Volume 5 (1969), pp. 599-606

[5] Glagoleva, R.Ya. Phragmén–Lindelöf type theorems and Liouville theorems for a linear parabolic equation, Math. Zametki, Volume 37 (1985), pp. 119-124

[6] Kogoj, A.E.; Lanconelli, E. Liouville theorems in halfspaces for parabolic hypoelliptic equations, Ric. Mat., Volume 55 (2006), pp. 267-282

[7] Moser, J. A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., Volume 17 (1964), pp. 101-134

Cité par Sources :