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

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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

Pablo Ochoa ^1,

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

Abstract
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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.

Keywords: Heisenberg group, viscosity solutions, finite difference methods, approximation schemes., Partial differential equations on the Heisenberg group.

Mathematics Subject Classification: Primary: 35R03; Secondary: 65N0.

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, arXiv:math/1403.1853. 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.

show all references

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, arXiv:math/1403.1853. 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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