Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group
Keywords
Heisenberg group, viscosity solutions, Theorem on SumsAbstract
In this paper we prove the $C^{0,\alpha}$ regularity of bounded and uniformly continuous viscosity solutions of some degenerate fully nonlinear equations in the first Heisenberg group.References
F. Abedin, C.E. Gutiérrez and G. Tralli, Harnack’s inequality for a class of nondivergent equations in the Heisenberg group, Comm. Partial Differential Equations 42 (2017), no. 10, 1644–1658.
Z.M. Balogh, A. Calogero and A. Kristály, Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group, J. Funct. Anal. 269 (2015), 2669–2708.
M. Bardi and P. Mannucci, On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations, Commun. Pure Appl. Anal. 5 (2006), 709–731.
M. Bardi and P. Mannucci, Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge–Ampére type, Forum Math. 25 (2013), 1291–1330.
T. Bieske, On ∞-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), 727–761.
I. Birindelli and F. Demengel, C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var. 20 (2014), 1009–1024.
I. Birindelli, G. Galise and H. Ishii, A family of degenerate elliptic operators: maximum principle and its consequences, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, 417–441.
I. Birindelli and B. Stroffolini, Existence theorems for fully nonlinear equations in the Heisenberg group, Subelliptic PDE’s and Applications to Geometry and Finance, Semin. Interdiscip. Mat. (S.I.M.), vol. 6, Potenza, 2007, 49–55.
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie groups and Potential theory for their Sub-Laplacians, Springer Monographs in Mathematics, vol. 26, Springer–Verlag, New York, 2007.
L.A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, Amer. Math. Soc., Providence, RI, 1995.
L. Capogna, D. Danielli, S. Pauls and J.T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian Isoperimetric problem. Basel: Progress in Mathematics, (2007) Vol. 259, Birkhäuser-Verlag.
M.G. Crandall, Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 419–435.
M.G. Crandall, Viscosity solutions: a primer, Viscosity Solutions and Applications (Montecatini Terme, 1995), 1–43, Lecture Notes in Math., 1660, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1997.
M.G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Differential Integral Equations 3 (1990), 1001–1014.
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. (N.S.) 27 (1992), 1–67.
A. Cutrı̀ and N. Tchou, Fully nonlinear degenerate operators associated with the Heisenberg group: barrier functions and qualitative properties, C.R. Math. Acad. Sci. Paris 344 (2007), 559–563.
A. Cutrı̀ and N. Tchou, Barrier functions for Pucci–Heisenberg operators and applications, Int. J. Dyn. Syst. Differ. Equ. 1 (2007), 117–131.
D. Danielli, N. Garofalo and D.-M. Nhieu, On the best possible character of the LQ norm in some a priori estimates for non-divergence form equations in Carnot groups, Proc. Amer. Math. Soc. 131 (2003), 3487–3498.
N. Garofalo and F. Tournier, New properties of convex functions in the Heisenberg group, Trans. Amer. Math. Soc. 358 (2006), 2011–2055.
F. Ferrari, Some a priori estimates for a class of operators in the Heisenberg group, Ann. Mat. Pura Appl. 193 (2014), 1019–1040.
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.
C.E. Gutiérrez and E. Lanconelli, Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for X-elliptic operators, Comm. Partial Differ. Equ. 28 (2003), 1833–1862.
C.E Gutiérrez and A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differerential Equations 29 (2004), 1305–1334.
L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1968), 147–171.
C. Imbert and L. Silvestre, C 1,α regularity of solutions of some degenerate fully nonlinear elliptic equations, Adv. Math. 233 (2013), 196–206.
H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkc. Ekvac. 38 (1995), 101–120.
H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1990), 26–78.
E. Jakobsen and K.H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations, Electron. J. Differential Equations 2002, no. 39, 10 pp.
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), 1–27.
Q. Liu, J. Manfredi and X. Zhou, Lipschitz and convexity preserving for solutions of semilinear equations in the Heisenberg group, Calc. Var. Partial Differential Equations 55 (2016), Art. 80, 25 pp.
H. Luiro and M. Parviainen, Regularity for nonlinear stochastic games, preprint (2015) arXiv:1509.07263v2.
J. Manfredi, Non-linear subelliptic equations on Carnot groups: Analysis and geometry in metric spaces, Notes of a course given at the Third School on Analysis and Geometry in Metric Spaces, Trento (2003).
P. Mannucci, The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction, Commun. Pure Appl. Anal. 13 (2014), 119–133.
A. Montanari, C. Guidi and V. Martino, Nonsmooth viscosity solutions of elementary symmetric functions of the complex Hessian, J. Differential Equations 260 (2016), 2690–2703.
L. Wang, Hölder estimates for subelliptic operators, J. Funct. Anal. 199 (2003), 228–242.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0