Partial Differential Equations
On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators
Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 115-118.

We study uniformly elliptic fully nonlinear equations of the type F(D2u,Du,u,x)=f(x). We

  • – show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects;
  • – obtain existence and uniqueness results for non-proper operators whose principal eigenvalues (in some cases, only one of them) are positive;
  • – obtain an existence result for non-proper Isaac's equations.

On étudie des équations complètement non-linéaires, uniformément elliptiques, du type F(D2u,Du,u,x)=f(x). On

  • – montre que les opérateurs convexes et positivement homogènes de degré 1 possèdent deux valeurs propres et deux fonctions propres principales. On étudie les propriétés de ces objets ;
  • – obtient des résultats d'existence et d'unicité pour des équations qui ne sont pas « propres », mais dont les valeurs propres (l'une ou les deux) sont positives ;
  • – obtient un résultat d'existence pour une équation de Isaac.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.11.003
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, boulevard Raspail, 75270 Paris cedex 06, France
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Quaas, Alexander; Sirakov, Boyan. On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 115-118. doi : 10.1016/j.crma.2005.11.003. http://www.numdam.org/articles/10.1016/j.crma.2005.11.003/

[1] Berestycki, H. On some nonlinear Sturm–Liouville problems, J. Differential Equations, Volume 26 (1977), pp. 375-390

[2] Berestycki, H.; Nirenberg, L.; Varadhan, S.R.S. The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., Volume 47 (1994) no. 1, pp. 47-92

[3] Busca, J.; Esteban, M.; Quaas, A. Nonlinear eigenvalues and bifurcation problems for Pucci's operator, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 22 (2005) no. 2, pp. 187-206

[4] 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

[5] Crandall, M.G.; Kocan, M.; Lions, P.L.; Świech, A. Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electronic J. Differential Equations, Volume 24 (1999), pp. 1-20

[6] Felmer, P.; Quaas, A. Positive solutions to ‘semilinear’ equation involving the Pucci's operator, J. Differential Equations, Volume 199 (2004) no. 2, pp. 376-393

[7] Jensen, R.; Swiech, A. Uniqueness and existence of maximal and minimal solutions of fully nonlinear PDE, Comm. Pure Appl. Anal., Volume 4 (2005) no. 1, pp. 187-195

[8] Lions, P.L. Bifurcation and optimal stochastic control, Nonlinear Anal., Volume 7 (1983) no. 2, pp. 177-207

[9] Quaas, A. Existence of positive solutions to a ‘semilinear’ equation involving the Pucci's operator in a convex domain, Differential Integral Equations, Volume 17 (2004), pp. 481-494

[10] A. Quaas, B. Sirakov, The principal eigenvalue and Dirichlet problem for fully nonlinear operators, in press

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Supported by FONDECYT, Grant No. 1040794, and ECOS grant C02E08.