no. 4
𝒞 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1009-1024.

We prove Hölder regularity of the gradient, up to the boundary for solutions of some fully-nonlinear, degenerate elliptic equations, with degeneracy coming from the gradient.

DOI : 10.1051/cocv/2014005
Classification : 35J25, 35J60, 35P30
Mots clés : regularity, fully nonlinear equations, simplicity of the first nonlinear eigenvalue 
@article{COCV_2014__20_4_1009_0,
     author = {Birindelli, I. and Demengel, F.},
     title = {$\mathcal {C}^{1,\beta }$ regularity for {Dirichlet} problems associated to fully nonlinear degenerate elliptic equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1009--1024},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {4},
     year = {2014},
     doi = {10.1051/cocv/2014005},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2014005/}
}
TY  - JOUR
AU  - Birindelli, I.
AU  - Demengel, F.
TI  - $\mathcal {C}^{1,\beta }$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2014
SP  - 1009
EP  - 1024
VL  - 20
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2014005/
DO  - 10.1051/cocv/2014005
LA  - en
ID  - COCV_2014__20_4_1009_0
ER  - 
%0 Journal Article
%A Birindelli, I.
%A Demengel, F.
%T $\mathcal {C}^{1,\beta }$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2014
%P 1009-1024
%V 20
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2014005/
%R 10.1051/cocv/2014005
%G en
%F COCV_2014__20_4_1009_0
Birindelli, I.; Demengel, F. $\mathcal {C}^{1,\beta }$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1009-1024. doi : 10.1051/cocv/2014005. http://archive.numdam.org/articles/10.1051/cocv/2014005/

[1] D. Araujo, G. Ricarte and E. Teixeira, Optimal gradient continuity for degenerate elliptic equations. Preprint arXiv:1206.4089.

[2] G. Barles, E. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. 13 (2011) 1-26. | MR | Zbl

[3] I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci Toulouse Math. 13 (2004) 261-287. | Numdam | MR | Zbl

[4] I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6 (2007) 335-366. | MR | Zbl

[5] I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully non linear operators. J. Differ. Eqs. 249 (2010) 1089-1110. | MR | Zbl

[6] I. Birindelli and F. Demengel, Regularity results for radial solutions of degenerate elliptic fully non linear equations. Nonlinear Anal. 75 (2012) 6237-6249. | MR | Zbl

[7] X. Cabré and L. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations F(D2u) = 0. Topological Meth. Nonlinear Anal. 6 (1995) 31-48. | MR | Zbl

[8] L. Caffarelli, Interior a Priori Estimates for Solutions of Fully Nonlinear Equations. Ann. Math. Second Ser. 130 (1989) 189-213. | MR | Zbl

[9] L. Caffarelli and X. Cabré, Fully-nonlinear equations Colloquium Publications. Amer. Math. Soc. Providence, RI 43 (1995). | MR | Zbl

[10] M.G. Crandall, H. Ishii and P.-L. Lions, Users guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR | Zbl

[11] L.C. Evans, Classical Solutions of Fully Nonlinear, Convex, Second-Order Elliptic Equations. Commun. Pure Appl. Math. 25 (1982) 333-363. | MR | Zbl

[12] 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). | MR | Zbl

[13] C. Imbert, Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations. J. Differ. Eqs. 250 (2011) 1555-1574. | MR | Zbl

[14] C. Imbert and L. Silvestre, C1,α regularity of solutions of degenerate fully non-linear elliptic equations. Adv. Math. 233 (2013) 196-206. | MR | Zbl

[15] C. Imbert and L. Silvestre, Estimates on elliptic equations that hold only where the gradient is large. Preprint arxiv:1306.2429v2.

[16] H. Ishii and P.L. Lions, Viscosity solutions of Fully-Nonlinear Second Order Elliptic Partial Differential Equations. J. Differ. Eqs. 83 (1990) 26-78. | MR | Zbl

[17] S. Patrizi, The Neumann problem for singular fully nonlinear operators. J. Math. Pures Appl. 90 (2008) 286-311. | MR | Zbl

[18] L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations. Preprint arXiv:1306.6672v1. | MR

[19] N. Winter, W2,p and W1,p-Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations. J. Anal. Appl. 28 (2009) 129-164. | MR | Zbl

[20] N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations. In Partial differential equations and the calculus of variations. II, vol. 2 of Progr. Nonlinear Differ. Eqs. Appl. Birkhauser Boston, Boston, MA (1989) 939-957. | MR | Zbl

Cité par Sources :