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
[Sur les valuers propres et le problème de Dirichlet pour des opérateurs complètement non-linéaires]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2005-08-31
Accepté le : 2005-11-07
Publié le : 2005-12-07

DOI : 10.1016/j.crma.2005.11.003

Affiliations des auteurs :

Alexander Quaas ¹ ; Boyan Sirakov ^{2, 3}

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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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     doi = {10.1016/j.crma.2005.11.003},
     language = {en},
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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/

Bibliographie
Cité par

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

[2] H. Berestycki; L. Nirenberg; S.R.S. Varadhan 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] J. Busca; M. Esteban; A. Quaas 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] L.A. Caffarelli; M.G. Crandall; M. Kocan; A. Świech On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., Volume 49 (1996), pp. 365-397

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

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

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

[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

Cité par Sources :

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

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