[Opérateurs completement non-linéaires associés au groupe de Heisenberg : fonctions barrière et propriétés qualitatives]
Nous déterminons des propriétés de continuité jusqu'au bord des solutions de viscosité pour le problème de Dirichlet pour des opérateurs completement non-linéaires associés au groupe de Heisenberg. Pour ces opérateurs nous provons aussi des propriétés de Hadamard, Liouville et Harnack. L'outil essentiel est la constructions de fonctions barrière pour les opérateurs de Pucci–Heisenberg.
The aim of this Note is to state some continuity property, up to the boundary, for viscosity solutions to fully nonlinear Dirichlet problems on the Heisenberg group and to obtain qualitative properties of the Hadamard, Liouville and Harnack type. For this purpose, a key ingredient is the construction of some barrier functions for the Pucci–Heisenberg operators.
Accepté le :
Publié le :
@article{CRMATH_2007__344_9_559_0,
author = {Cutr{\`\i}, Alessandra and Tchou, Nicoletta},
title = {Fully nonlinear degenerate operators associated with the {Heisenberg} group: barrier functions and qualitative properties},
journal = {Comptes Rendus. Math\'ematique},
pages = {559--563},
publisher = {Elsevier},
volume = {344},
number = {9},
year = {2007},
doi = {10.1016/j.crma.2007.03.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.03.003/}
}
TY - JOUR AU - Cutrì, Alessandra AU - Tchou, Nicoletta TI - Fully nonlinear degenerate operators associated with the Heisenberg group: barrier functions and qualitative properties JO - Comptes Rendus. Mathématique PY - 2007 SP - 559 EP - 563 VL - 344 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.03.003/ DO - 10.1016/j.crma.2007.03.003 LA - en ID - CRMATH_2007__344_9_559_0 ER -
%0 Journal Article %A Cutrì, Alessandra %A Tchou, Nicoletta %T Fully nonlinear degenerate operators associated with the Heisenberg group: barrier functions and qualitative properties %J Comptes Rendus. Mathématique %D 2007 %P 559-563 %V 344 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.03.003/ %R 10.1016/j.crma.2007.03.003 %G en %F CRMATH_2007__344_9_559_0
Cutrì, Alessandra; Tchou, Nicoletta. Fully nonlinear degenerate operators associated with the Heisenberg group: barrier functions and qualitative properties. Comptes Rendus. Mathématique, Tome 344 (2007) no. 9, pp. 559-563. doi : 10.1016/j.crma.2007.03.003. http://www.numdam.org/articles/10.1016/j.crma.2007.03.003/
[1] M. Bardi, F. Da Lio, Propagation of maxima and strong maximum principle for degenerate elliptic equations, in: Proc. of the Eighth Tokyo Conference on Nonlinear PDE, 1998
[2] On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations, Commun. Pure Appl. Anal., Volume 5 (2006) no. 4, pp. 709-731 35J60 (35J70)
[3] Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, Amer. Math. Soc., Providence, RI, 1995
[4] Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Comm. Contemp. Math., Volume 5 (2003) no. 3, pp. 435-448
[5] User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., Volume 27 (1992), pp. 1-67
[6] On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 17 (2000) no. 2, pp. 219-245
[7] A. Cutrì, N. Tchou, Barrier functions for Pucci–Heisenberg operators and applications, submitted for publication
[8] Fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., Volume 79 (1973), pp. 373-376
[9] Singular integrals on homogeneous spaces and some problems of classical analysis, Ann. Scuola Norm. Sup. Pisa (3), Volume 25 (1971) no. 4, pp. 575-648
[10] Operatori ellittici estremanti, Ann. Mat. Pura Appl., Volume 72 (1966), pp. 141-170
Cité par Sources :
