First eigenvalue and maximum principle for fully nonlinear singular operators

Advances in Differential Equations

Volumn 11, Issue 1, 2006, Pages 91-119

  Birindelli, Isabeau ^a   Demengel, Françoise ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34247141781     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (23)

1. L. Ambrosio and N. Dancer, Calculus of variations and partial differential equations. Topics on geometrical evolution problems and degree theory, Papers from the Summer School held in Pisa, September 1996. Edited by G. Buttazzo, A. Marino, and M.K.V. Murthy. Springer-Verlag, Berlin, 2000.
2. A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, (French) [Simplicity and isolation of the first eigenvalue of the p-Laplacian with weight] C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728.
3. H. Berestycki, L. Nirenberg, and S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
4. I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Annales de la faculté des sciences de Toulouse, Vol. XIII, n'2 (2004), 261-287.
5. I. Birindelli, F. Demengel, and J. Wigniolle, Strict maximum principle, to appear in the Proceedings ofWorkshop on Second Order Subelliptic Equations and Applications Cortona, (2003).
6. J. Busca, M.J. Esteban, and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operator, to appear in Annales de l'Institut H. Poincaré, Analyse nonlin éaire.
7. L. Caffarelli and X. Cabré, "Fully Nonlinear Equations," Colloquium Publications 43, American Mathematical Society, Providence, RI,1995.
8. Y.G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
9. M.G. Crandall, H. Ishii, and P.L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.
10. A. Cutrì and F. Leoni, On the Liouville property for fully-nonlinear equations, Annales de l'Institut H. Poincaré, Analyse non-linéaire, (2000), 219-245.
11. H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.
12. R. L. Foote, Regularity of the distance function, Proc. Amer. Math. Soc., 92 (1984), 153-155.
13. D. Gilbarg and N.S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
14. H. Ishii, "Viscosity solutions of non-linear partial differential equations," Sugaku Expositions, 9 (1996).
15. H. Ishii and P.L. Lions, Viscosity solutions of fully-nonlinear second order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.
16. P. Juutinen, P. Lindqvist, and J. Manfredi, On the equivalence o f viscosity solutions and weak solutions for a quasi linear equation, SIAM J. Math. Anal., 33 (2001), 699-717.
17. P. Juutinen and J.J. Manfredi, Viscosity solutions of the p-Laplace equation, Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), 273-284, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, (2000).
18. k hypersurfaces, J. Differential Equations, 40 (1981), 116-120.
19. P. Lindqvist, On a nonlinear eigenvalue problem, Fall School in Analysis (Jyväskylä, 1994), 33-54, Report, 68, Univ. Jyväskylä, Jyväskylä, 1995.
20. P.L. Lions, Bifurcation and optimal stochastic control Nonlinear Ana. T.M.A., 7 (1983).
21. M.H. Protter, H.F. Weinberger, "Maximum principles in differential equations," Prentice-Hall, Inc., Englewood Cliffs, N.J. 1967.
22. A. Quaas, Existence of positive solutions to a "semilinear" equation involving the Pucci's operators in a convex domain, submitted.
23. J. L. Vazquez A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.