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

Part of: Partial differential equations Close-to-elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  26 January 2019

Fausto Ferrari
Fausto Ferrari*
Affiliation:
Dipartimento di Matematica dell'Università di Bologna, Piazza di Porta S. Donato, 5, Bologna40126, Italy ([email protected])
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Abstract

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.

Keywords

Degenerate elliptic operatorsnonlinear elliptic operatorscarnot groupsviscosity solutionstheorem on sumsHölder regularity

MSC classification

Primary: 35D40: Viscosity solutions
Secondary: 35B65: Smoothness and regularity of solutions 35H20: Subelliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 975 - 992
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Bardi, M. and Mannucci, P.. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun. Pure Appl. Anal. 5 (2006), 709–731.CrossRefGoogle Scholar
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.Google Scholar
3Caffarelli, L. A. and Cabré, X.. Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications,vol. 43 (Providence, RI:AMS, 1995).Google Scholar
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).Google Scholar
5Crandall, M. and Ishii, H.. The maximum principle for semicontinuous functions. Differ. Integral Equ. 3 (1990), 1001–1014.Google Scholar
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), 1–67.CrossRefGoogle Scholar
7Ferrari, F.. Hölder regularity of viscosity solutions of some fully nonlinear equations in the Heisenberg group (2017) arXiv:1706.05705.Google Scholar
8Ferrari, F. and Vecchi, E.. Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group, preprint (2017).Google Scholar
9Hormander, L.. Hypoelliptic second order differential equations. Acta. Mathematica 119 (1968), 147–171.CrossRefGoogle Scholar
10Imbert, C. and Silvestre, L.. C 1, α regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233 (2013), 196–206.CrossRefGoogle Scholar
11Ishii, H.. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42 (1989), 15–45.CrossRefGoogle Scholar
12Ishii, H.. On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcialaj Ekvacioj 38 (1995), 101–120.Google Scholar
13Ishii, H. and Lions, P.-L.. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83 (1990), 26–78.CrossRefGoogle Scholar
14Jensen, R.. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101 (1988), 1–27.CrossRefGoogle Scholar
15Lions, P.-L. and Villani, C.. Régularité optimale de racines carrées. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1537–1541.Google Scholar
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.CrossRefGoogle Scholar
17Luiro, H. and Parviainen, M.. Regularity for nonlinear stochastic games, preprint (2015) arXiv:1509.07263v2.Google Scholar
18Mannucci, P., Marchi, C. and Tchou, N.. Singular perturbations for a subelliptic operator, (2017) to appear in COCV.CrossRefGoogle Scholar
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.Google Scholar
20Parmeggiani, A. and Xu, C.-J.. The Dirichlet problem for sub-elliptic second order equations. Ann. Mat. Pura Appl. 173 (1997), 233–243.CrossRefGoogle Scholar
21Sachs, G.. Regolarità delle soluzioni viscose di operatori uniformemente ellittici, Master thesis (tesi laurea Magistrale), Università di Bologna (2017).Google Scholar
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.Google Scholar
23Wang, L.. Hölder estimates for subelliptic operators. J. Funct. Anal. 199 (2003), 228–242.CrossRefGoogle Scholar