no. 11-12
Partial Differential Equations
Solvability of monotone systems of fully nonlinear elliptic PDE's
[Solvabilité de systèmes monotones d'EDP complètement non-linéaires]

Présenté par : Pierre-Louis Lions

Quaas, Alexander1 ; Sirakov, Boyan2, 3
1 Departamento de Matemática, Universidad Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile
2 UFR SEGMI, Université Paris 10, 92001 Nanterre cedex, France
3 CAMS, EHESS, 54 bd Raspail, 75270 Paris cedex 06, France
Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 641-644.

On étudie des systèmes quasi-monotones d'équations complètement non-linéaires, uniformément elliptiques, de type Isaac. On obtient des résultats d'existence de solutions du problème de Dirichlet et une condition nécessaire et suffisante pour qu'un tel système satisfasse le principe de comparaison.

We study quasimonotone weakly coupled systems of uniformly elliptic equations of Isaac type. We prove results on existence of viscosity solutions of such systems and give a necessary and sufficient condition for such a system to satisfy the comparison principle.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.04.008
Quaas, Alexander 1 ; Sirakov, Boyan 2, 3

1 Departamento de Matemática, Universidad Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile
2 UFR SEGMI, Université Paris 10, 92001 Nanterre cedex, France
3 CAMS, EHESS, 54 bd Raspail, 75270 Paris cedex 06, France 
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     title = {Solvability of monotone systems of fully nonlinear elliptic {PDE's}},
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Quaas, Alexander; Sirakov, Boyan. Solvability of monotone systems of fully nonlinear elliptic PDE's. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 641-644. doi : 10.1016/j.crma.2008.04.008. http://www.numdam.org/articles/10.1016/j.crma.2008.04.008/

[1] Busca, J.; Sirakov, B. Harnack type estimates for nonlinear elliptic systems and applications, Ann. Inst. H. Poincaré Anal. Nonlinéaire, Volume 21 (2004) no. 5, pp. 543-590

[2] Caffarelli, L.A.; Crandall, M.G.; Kocan, M.; Świech, A. On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., Volume 49 (1996), pp. 365-397

[3] Ishii, H.; Koike, S. Viscosity solutions for monotone systems of second-order elliptic PDE's, Comm. Partial Differential Equations, Volume 16 (1991) no. 6–7, pp. 1095-1128

[4] Quaas, A.; Sirakov, B. Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006) no. 2, pp. 115-118

[5] Quaas, A.; Sirakov, B. Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., Volume 218 (2008) no. 1, pp. 105-135

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