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𝒞1,β regularity for Dirichlet problems associated to fully nonlineardegenerate elliptic equations

Published online by Cambridge University Press:  13 August 2014

I. Birindelli  and
F. Demengel
I. Birindelli
Affiliation:
Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
F. Demengel
Affiliation:
Laboratoire AGM, Université de Cergy Pontoise, 2 rue Adolphe Chauvin, 95302 Cergy Pontoise, France. francoise.demengel@u-cergy.fr
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Abstract

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

Keywords

Regularityfully nonlinear equationssimplicity of the first nonlinear eigenvalue
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1009 - 1024
Copyright
© EDP Sciences, SMAI, 2014

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References

D. Araujo, G. Ricarte and E. Teixeira, Optimal gradient continuity for degenerate elliptic equations. Preprint arXiv:1206.4089.
Barles, G., Chasseigne, E. and Imbert, C., Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. 13 (2011) 1–26. Google Scholar
Birindelli, I. and Demengel, F., Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci Toulouse Math. 13 (2004) 261–287. Google Scholar
Birindelli, I. and Demengel, F., Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6 (2007) 335–366. Google Scholar
Birindelli, I. and Demengel, F., Regularity and uniqueness of the first eigenfunction for singular fully non linear operators. J. Differ. Eqs. 249 (2010) 1089–1110. Google Scholar
Birindelli, I. and Demengel, F., Regularity results for radial solutions of degenerate elliptic fully non linear equations. Nonlinear Anal. 75 (2012) 6237–6249. Google Scholar
Cabré, X. and Caffarelli, L., Regularity for viscosity solutions of fully nonlinear equations F(D 2u) = 0. Topological Meth. Nonlinear Anal. 6 (1995) 31–48. Google Scholar
Caffarelli, L., Interior a Priori Estimates for Solutions of Fully Nonlinear Equations. Ann. Math. Second Ser. 130 (1989) 189–213. Google Scholar
L. Caffarelli and X. Cabré, Fully-nonlinear equations Colloquium Publications. Amer. Math. Soc. Providence, RI 43 (1995).
Crandall, M.G., Ishii, H. and Lions, P.-L., Users guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. Google Scholar
Evans, L.C., Classical Solutions of Fully Nonlinear, Convex, Second-Order Elliptic Equations. Commun. Pure Appl. Math. 25 (1982) 333–363. Google Scholar
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).
Imbert, C., Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations. J. Differ. Eqs. 250 (2011) 1555–1574. Google Scholar
Imbert, C. and Silvestre, L., C 1,α regularity of solutions of degenerate fully non-linear elliptic equations. Adv. Math. 233 (2013) 196–206. Google Scholar
C. Imbert and L. Silvestre, Estimates on elliptic equations that hold only where the gradient is large. Preprint arxiv:1306.2429v2.
Ishii, H. and Lions, P.L., Viscosity solutions of Fully-Nonlinear Second Order Elliptic Partial Differential Equations. J. Differ. Eqs. 83 (1990) 26–78. Google Scholar
Patrizi, S., The Neumann problem for singular fully nonlinear operators. J. Math. Pures Appl. 90 (2008) 286–311. Google Scholar
L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations. Preprint arXiv:1306.6672v1.
Winter, N., W 2,p and W 1,p-Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations. J. Anal. Appl. 28 (2009) 129–164. Google Scholar
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.