Comptes Rendus
Partial Differential Equations
Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations
Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1165-1168.

We prove the classical Alexandroff–Bakelman–Pucci estimate for fully nonlinear elliptic equations involving singular or degenerate operators having as models |p|αMλ,Λ±(X), where Mλ,Λ± are the Pucci extremal operators with parameters 0<λΛ and α>1.

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|αMλ,Λ±(X)Mλ,Λ± sont les opérateurs extremal de Pucci avec des paramètres 0<λΛ et α>1.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.09.009

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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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/

[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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