# American Institute of Mathematical Sciences

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 and Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
##### References:
 [1] Y. Achdou and I. Capuzzo-Dolcetta, Approximation of solutions of Hamilton-Jacobi equations on the Heisengerb group, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), 565-591. doi: 10.1051/m2an:2008017. [2] Y. Achdou and N. Tchou, A finite difference scheme on a non commutative group, Numer. Math., 89 (2001), 401-424. doi: 10.1007/PL00005472. [3] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully non-linear second order equations, Asymptotic Analysis, 4 (1991), 271-283. [4] T. Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. in PDE, 27 (2002), 727-761. doi: 10.1081/PDE-120002872. [5] M. Crandall, Viscosity Solutions: A Primer, lecture notes in Mathematics 1660, Springer-Verlag, 1997. doi: 10.1007/BFb0094294. [6] M. Crandall, H. Ishii and P-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. of Amer. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [7] M. Crandall and P-L. Lions, Two approximations of solutions of Hamilton equations, Math. Comp., 43 (1984), 1-19. doi: 10.2307/2007396. [8] F. Ferrari, Q. Liu and J. Manfredi, On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group, Communications in Contemporary Mathematics, 16 (2014), 1350027 (41 pages). doi: 10.1142/S0219199713500272. [9] F. Ferrari, Q. Liu and J. Manfredi, On the characterization of $p$-Harmonic functions on the Heisenberg group by mean value properties, Discrete and Continuous Dynamical Systems, 34 (2014), 2779-2793. doi: 10.3934/dcds.2014.34.2779. [10] Y. Giga, Surface Evolution Equations: A Level Set Method, Monographs in Mathematics 99, Birkhauser Verlag, Basel, 2006. [11] H. Ishii and P-L. Lions, Viscosity solutions of fully non-linear second order elliptic partial differential equations, Journal of Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z. [12] D. Jerison, The Poincaré inequalities for vector fields satisfying Hormander's condition, J. Duke Math., 53 (1986), 503-523. doi: 10.1215/S0012-7094-86-05329-9. [13] 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, Trento, (2003), available at http://www.pitt.edu/ manfredi/. [14] S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 1 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [15] M. Rudd, Statistical exponential formulas for homogeneous diffusions,, preprint, ().  doi: 10.3934/cpaa.2015.14.269. [16] J. Sethian, Level Set Methods and Fast Marching Methods, 2nd edition, Volume 3 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1999. [17] R. Vargas, Matrix Iterative Analysis, Springer-Verlag, 2000. doi: 10.1007/978-3-642-05156-2.

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##### References:
 [1] Y. Achdou and I. Capuzzo-Dolcetta, Approximation of solutions of Hamilton-Jacobi equations on the Heisengerb group, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), 565-591. doi: 10.1051/m2an:2008017. [2] Y. Achdou and N. Tchou, A finite difference scheme on a non commutative group, Numer. Math., 89 (2001), 401-424. doi: 10.1007/PL00005472. [3] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully non-linear second order equations, Asymptotic Analysis, 4 (1991), 271-283. [4] T. Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. in PDE, 27 (2002), 727-761. doi: 10.1081/PDE-120002872. [5] M. Crandall, Viscosity Solutions: A Primer, lecture notes in Mathematics 1660, Springer-Verlag, 1997. doi: 10.1007/BFb0094294. [6] M. Crandall, H. Ishii and P-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. of Amer. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [7] M. Crandall and P-L. Lions, Two approximations of solutions of Hamilton equations, Math. Comp., 43 (1984), 1-19. doi: 10.2307/2007396. [8] F. Ferrari, Q. Liu and J. Manfredi, On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group, Communications in Contemporary Mathematics, 16 (2014), 1350027 (41 pages). doi: 10.1142/S0219199713500272. [9] F. Ferrari, Q. Liu and J. Manfredi, On the characterization of $p$-Harmonic functions on the Heisenberg group by mean value properties, Discrete and Continuous Dynamical Systems, 34 (2014), 2779-2793. doi: 10.3934/dcds.2014.34.2779. [10] Y. Giga, Surface Evolution Equations: A Level Set Method, Monographs in Mathematics 99, Birkhauser Verlag, Basel, 2006. [11] H. Ishii and P-L. Lions, Viscosity solutions of fully non-linear second order elliptic partial differential equations, Journal of Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z. [12] D. Jerison, The Poincaré inequalities for vector fields satisfying Hormander's condition, J. Duke Math., 53 (1986), 503-523. doi: 10.1215/S0012-7094-86-05329-9. [13] 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, Trento, (2003), available at http://www.pitt.edu/ manfredi/. [14] S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 1 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [15] M. Rudd, Statistical exponential formulas for homogeneous diffusions,, preprint, ().  doi: 10.3934/cpaa.2015.14.269. [16] J. Sethian, Level Set Methods and Fast Marching Methods, 2nd edition, Volume 3 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1999. [17] R. Vargas, Matrix Iterative Analysis, Springer-Verlag, 2000. doi: 10.1007/978-3-642-05156-2.
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