On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators

Alexander Quaas; Boyan Sirakov

doi:10.1016/j.crma.2005.11.003

Partial Differential Equations

On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators

Presented by: Pierre-Louis Lions

Alexander Quaas ¹; Boyan Sirakov ^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

Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 115-118.

Abstract
Résumé

We study uniformly elliptic fully nonlinear equations of the type $F (D^{2} u, D u, u, x) = f (x)$ . 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 $F (D^{2} u, D u, u, x) = f (x)$ . 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: 2005-08-31
Accepted: 2005-11-07
Published online: 2005-12-07

DOI: 10.1016/j.crma.2005.11.003

Author's affiliations:

Alexander Quaas ¹; Boyan Sirakov ^{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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     title = {On the principal eigenvalues and the {Dirichlet} problem for fully nonlinear operators},
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     pages = {115--118},
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Alexander Quaas; Boyan Sirakov. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.003/

References
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[9] A. Quaas 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

Cited by Sources:

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

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