    September  2015, 14(5): 1841-1863. doi: 10.3934/cpaa.2015.14.1841

## Approximation schemes for non-linear second order equations on the Heisenberg group

 1 Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo., Padre Contreras 1300, Parque Gral. San Martin. M5502JMA Mendoza, Argentina

Received  September 2014 Revised  March 2015 Published  June 2015

In this work, we propose and analyse approximation schemes for fully non-linear second order partial differential equations defined on the Heisenberg group. We prove that a consistent, stable and monotone scheme converges to a viscosity solution of a second order PDE on the Heisenberg group provided that comparison principles exists for the limiting equation. We also provide examples where this technique is applied.
Citation: Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
##### References:
  Y. Achdou and I. Capuzzo-Dolcetta, Approximation of solutions of Hamilton-Jacobi equations on the Heisengerb group,, \emph{ESAIM: Mathematical Modelling and Numerical Analysis}, 42 (2008), 565. doi: 10.1051/m2an:2008017.  Google Scholar  Y. Achdou and N. Tchou, A finite difference scheme on a non commutative group,, \emph{Numer. Math.}, 89 (2001), 401. doi: 10.1007/PL00005472.  Google Scholar  G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully non-linear second order equations,, \emph{Asymptotic Analysis}, 4 (1991), 271. Google Scholar  T. Bieske, On $\infty$-harmonic functions on the Heisenberg group,, \emph{Comm. in PDE}, 27 (2002), 727. doi: 10.1081/PDE-120002872.  Google Scholar  M. Crandall, Viscosity Solutions: A Primer,, lecture notes in Mathematics 1660, (1660). doi: 10.1007/BFb0094294.  Google Scholar  M. Crandall, H. Ishii and P-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. of Amer. Soc.}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar  M. Crandall and P-L. Lions, Two approximations of solutions of Hamilton equations,, \emph{Math. Comp.}, 43 (1984), 1. doi: 10.2307/2007396.  Google Scholar  F. Ferrari, Q. Liu and J. Manfredi, On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group,, \emph{Communications in Contemporary Mathematics}, 16 (2014). doi: 10.1142/S0219199713500272.  Google Scholar  F. Ferrari, Q. Liu and J. Manfredi, On the characterization of $p$-Harmonic functions on the Heisenberg group by mean value properties,, \emph{Discrete and Continuous Dynamical Systems}, 34 (2014), 2779. doi: 10.3934/dcds.2014.34.2779.  Google Scholar  Y. Giga, Surface Evolution Equations: A Level Set Method,, Monographs in Mathematics 99, (2006). Google Scholar  H. Ishii and P-L. Lions, Viscosity solutions of fully non-linear second order elliptic partial differential equations,, \emph{Journal of Differential Equations}, 83 (1990), 26. doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar  D. Jerison, The Poincaré inequalities for vector fields satisfying Hormander's condition,, \emph{J. Duke Math.}, 53 (1986), 503. doi: 10.1215/S0012-7094-86-05329-9.  Google Scholar  J. 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, (2003). Google Scholar  S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, \emph{J. Comput. Phys.}, 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar  M. Rudd, Statistical exponential formulas for homogeneous diffusions,, preprint, (). doi: 10.3934/cpaa.2015.14.269.  Google Scholar  J. Sethian, Level Set Methods and Fast Marching Methods,, 2$^{nd}$ edition, (1999). Google Scholar  R. Vargas, Matrix Iterative Analysis,, Springer-Verlag, (2000). doi: 10.1007/978-3-642-05156-2.  Google Scholar

show all references

##### References:
  Y. Achdou and I. Capuzzo-Dolcetta, Approximation of solutions of Hamilton-Jacobi equations on the Heisengerb group,, \emph{ESAIM: Mathematical Modelling and Numerical Analysis}, 42 (2008), 565. doi: 10.1051/m2an:2008017.  Google Scholar  Y. Achdou and N. Tchou, A finite difference scheme on a non commutative group,, \emph{Numer. Math.}, 89 (2001), 401. doi: 10.1007/PL00005472.  Google Scholar  G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully non-linear second order equations,, \emph{Asymptotic Analysis}, 4 (1991), 271. Google Scholar  T. Bieske, On $\infty$-harmonic functions on the Heisenberg group,, \emph{Comm. in PDE}, 27 (2002), 727. doi: 10.1081/PDE-120002872.  Google Scholar  M. Crandall, Viscosity Solutions: A Primer,, lecture notes in Mathematics 1660, (1660). doi: 10.1007/BFb0094294.  Google Scholar  M. Crandall, H. Ishii and P-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. of Amer. Soc.}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar  M. Crandall and P-L. Lions, Two approximations of solutions of Hamilton equations,, \emph{Math. Comp.}, 43 (1984), 1. doi: 10.2307/2007396.  Google Scholar  F. Ferrari, Q. Liu and J. Manfredi, On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group,, \emph{Communications in Contemporary Mathematics}, 16 (2014). doi: 10.1142/S0219199713500272.  Google Scholar  F. Ferrari, Q. Liu and J. Manfredi, On the characterization of $p$-Harmonic functions on the Heisenberg group by mean value properties,, \emph{Discrete and Continuous Dynamical Systems}, 34 (2014), 2779. doi: 10.3934/dcds.2014.34.2779.  Google Scholar  Y. Giga, Surface Evolution Equations: A Level Set Method,, Monographs in Mathematics 99, (2006). Google Scholar  H. Ishii and P-L. Lions, Viscosity solutions of fully non-linear second order elliptic partial differential equations,, \emph{Journal of Differential Equations}, 83 (1990), 26. doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar  D. Jerison, The Poincaré inequalities for vector fields satisfying Hormander's condition,, \emph{J. Duke Math.}, 53 (1986), 503. doi: 10.1215/S0012-7094-86-05329-9.  Google Scholar  J. 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, (2003). Google Scholar  S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, \emph{J. Comput. Phys.}, 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar  M. Rudd, Statistical exponential formulas for homogeneous diffusions,, preprint, (). doi: 10.3934/cpaa.2015.14.269.  Google Scholar  J. Sethian, Level Set Methods and Fast Marching Methods,, 2$^{nd}$ edition, (1999). Google Scholar  R. Vargas, Matrix Iterative Analysis,, Springer-Verlag, (2000). doi: 10.1007/978-3-642-05156-2.  Google Scholar
  Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053  L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107.  Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775  Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461  Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209  Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905  Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495  Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619  Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113  Luis F. López, Yannick Sire. Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2639-2656. doi: 10.3934/dcds.2014.34.2639  Patrizia Pucci. Critical Schrödinger-Hardy systems in the Heisenberg group. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 375-400. doi: 10.3934/dcdss.2019025  Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779  M. DeDeo, M. Martínez, A. Medrano, M. Minei, H. Stark, A. Terras. Spectra of Heisenberg graphs over finite rings. Conference Publications, 2003, 2003 (Special) : 213-222. doi: 10.3934/proc.2003.2003.213  Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286  Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155  Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139  Guozhen Lu and Juncheng Wei. On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups. Electronic Research Announcements, 1997, 3: 83-89.  Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977  Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637  John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7.

2018 Impact Factor: 0.925