Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations

Gonzalo Dávila; Patricio Felmer; Alexander Quaas

doi:10.1016/j.crma.2009.09.009

Partial Differential Equations

Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations
[Inégalité d'Alexandroff–Bakelman–Pucci pour des équations elliptiques entièrement non linéaires singulières ou dégénérés]

Présenté par : Pierre-Louis Lions

Gonzalo Dávila ¹ ; Patricio Felmer ¹ ; Alexander Quaas ²

¹ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Santiago, Chile
² Departamento de Matemática, Universidad Técnica Federico Santa María, Av. Espana 1680, V-110 Valparaiso, Chile

Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1165-1168.

Résumé
Abstract

Nous prouvons l'inégalité classique d'Alexandroff–Bakelman–Pucci pour des équations elliptiques entièrement non linéaires avec des opérateurs singulières ou dégénérés ayant comme modèles ${| p |}^{α} M_{λ, Λ}^{\pm} (X)$ où $M_{λ, Λ}^{\pm}$ sont les opérateurs extremal de Pucci avec des paramètres $0 < λ ⩽ Λ$ et $α > - 1$ .

We prove the classical Alexandroff–Bakelman–Pucci estimate for fully nonlinear elliptic equations involving singular or degenerate operators having as models ${| p |}^{α} M_{λ, Λ}^{\pm} (X)$ , where $M_{λ, Λ}^{\pm}$ are the Pucci extremal operators with parameters $0 < λ ⩽ Λ$ and $α > - 1$ .

Reçu le : 2008-12-23
Accepté le : 2009-09-01
Publié le : 2009-09-24

DOI : 10.1016/j.crma.2009.09.009

Affiliations des auteurs :

Gonzalo Dávila ¹ ; Patricio Felmer ¹ ; Alexander Quaas ²

¹ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Santiago, Chile
² Departamento de Matemática, Universidad Técnica Federico Santa María, Av. Espana 1680, V-110 Valparaiso, Chile

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     author = {Gonzalo D\'avila and Patricio Felmer and Alexander Quaas},
     title = {Alexandroff{\textendash}Bakelman{\textendash}Pucci estimate for singular or degenerate fully nonlinear elliptic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1165--1168},
     publisher = {Elsevier},
     volume = {347},
     number = {19-20},
     year = {2009},
     doi = {10.1016/j.crma.2009.09.009},
     language = {en},
}

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Gonzalo Dávila; Patricio Felmer; Alexander Quaas. Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1165-1168. doi : 10.1016/j.crma.2009.09.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.009/

Bibliographie
Cité par

[1] 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

[2] H. Berestycki; L. Nirenberg On the method of moving planes and the sliding method, Boll. Soc. Brasil Mat. (N.S.), Volume 22 (1991), pp. 237-275

[3] I. Birindelli; F. Demengel Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci Toulouse Math. (6), Volume 13 (2004) no. 2, pp. 261-287

[4] I. Birindelli; F. Demengel Eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Partial Differ. Equ., Volume 11 (2006) no. 1, pp. 91-119

[5] I. Birindelli; F. Demengel Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators, Commun. Pure Appl. Anal., Volume 6 (2007), pp. 335-366

[6] I. Birindelli; F. Demengel The Dirichlet problem for singular fully nonlinear operators, Discrete Contin. Dyn. Syst. (special vol.) (2007), pp. 110-121

[7] I. Birindelli; F. Demengel Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains, J. Math. Anal. Appl., Volume 352 (2009) no. 2, pp. 822-835

[8] L. Caffarelli; X. Cabré Fully Nonlinear Elliptic Equations, Colloquium Publications, vol. 43, American Mathematical Society, 1995

[9] L. Caffarelli; M. Crandall; M. Kocan; A. Świech On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., Volume 49 (1996) no. 4, pp. 365-398

[10] Y.G. Chen; Y. Giga; S. Goto Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., Volume 33 (1991), pp. 749-786

[11] G. Dávila, P. Felmer, A. Quaas, Harnack inequality for singular fully nonlinear operators and some existence results, preprint

[12] C. Evans; J. Spruck Motion of level sets by mean curvature, J. Differential Geom., Volume 33 (1991), pp. 635-681

[13] D. Gilbarg; N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983

[14] C. Imbert, Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations, preprint

[15] T. Junges Miotto, The Aleksandrov–Bakelman–Pucci estimate for singular fully nonlinear operators, preprint

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

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