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An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations

  • Fausto Ferrari (a1)


We prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of ℝn space. In particular, the result applies also to some operators in Carnot groups.



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1Bardi, M. and Mannucci, P.. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun. Pure Appl. Anal. 5 (2006), 709731.
2Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F.. Stratified Lie groups and potential theory for their Sub-Laplacians. Springer Monographs in Mathematics,vol. 26 (New York, NY: Springer-Verlag, 2007.
3Caffarelli, L. A. and Cabré, X.. Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications,vol. 43 (Providence, RI:AMS, 1995).
4Crandall, M.. Viscosity solutions: a primer. Viscosity solutions and applications (Montecatini Terme, 1995), 1–43, Lecture Notes in Math., 1660, Fond. CIME/CIME Found. Subser. (Berlin: Springer, 1997).
5Crandall, M. and Ishii, H.. The maximum principle for semicontinuous functions. Differ. Integral Equ. 3 (1990), 10011014.
6Crandall, M. G., Ishii, H. and Lions, P.-L.. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), 167.
7Ferrari, F.. Hölder regularity of viscosity solutions of some fully nonlinear equations in the Heisenberg group (2017) arXiv:1706.05705.
8Ferrari, F. and Vecchi, E.. Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group, preprint (2017).
9Hormander, L.. Hypoelliptic second order differential equations. Acta. Mathematica 119 (1968), 147171.
10Imbert, C. and Silvestre, L.. C 1, α regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233 (2013), 196206.
11Ishii, H.. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42 (1989), 1545.
12Ishii, H.. On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcialaj Ekvacioj 38 (1995), 101120.
13Ishii, H. and Lions, P.-L.. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83 (1990), 2678.
14Jensen, R.. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101 (1988), 127.
15Lions, P.-L. and Villani, C.. Régularité optimale de racines carrées. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 15371541.
16Liu, Q., Manfredi, J. J. and Zhou, X.. Lipschitz and convexity preserving for solutions of semilinear equations in the Heisenberg group. Calc. Var. Partial Differ. Equ. 55 (2016), 25, Art. 80.
17Luiro, H. and Parviainen, M.. Regularity for nonlinear stochastic games, preprint (2015) arXiv:1509.07263v2.
18Mannucci, P., Marchi, C. and Tchou, N.. Singular perturbations for a subelliptic operator, (2017) to appear in COCV.
19Mannucci, P., Marchi, C. and Tchou, N.. Asymptotic behaviour for operators of Grushin type: invariant measure and singular perturbations, preprit (2017) to appear in DCDS-S.
20Parmeggiani, A. and Xu, C.-J.. The Dirichlet problem for sub-elliptic second order equations. Ann. Mat. Pura Appl. 173 (1997), 233243.
21Sachs, G.. Regolarità delle soluzioni viscose di operatori uniformemente ellittici, Master thesis (tesi laurea Magistrale), Università di Bologna (2017).
22Stroock, D. W. and Varadhan, S. R. S.. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften,vol. 233 (Berlin-New York: Springer-Verlag, 1979), xii+338.
23Wang, L.. Hölder estimates for subelliptic operators. J. Funct. Anal. 199 (2003), 228242.


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