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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully non-linear second order equations, Asymptotic Analysis, 4 (1991), 271-283.  Google Scholar

[4]

T. Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. in PDE, 27 (2002), 727-761. doi: 10.1081/PDE-120002872.  Google Scholar

[5]

M. Crandall, Viscosity Solutions: A Primer, lecture notes in Mathematics 1660, Springer-Verlag, 1997. doi: 10.1007/BFb0094294.  Google Scholar

[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.  Google Scholar

[7]

M. Crandall and P-L. Lions, Two approximations of solutions of Hamilton equations, Math. Comp., 43 (1984), 1-19. doi: 10.2307/2007396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

Y. Giga, Surface Evolution Equations: A Level Set Method, Monographs in Mathematics 99, Birkhauser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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/. Google Scholar

[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.  Google Scholar

[15]

M. Rudd, Statistical exponential formulas for homogeneous diffusions,, preprint, ().  doi: 10.3934/cpaa.2015.14.269.  Google Scholar

[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.  Google Scholar

[17]

R. Vargas, Matrix Iterative Analysis, Springer-Verlag, 2000. doi: 10.1007/978-3-642-05156-2.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully non-linear second order equations, Asymptotic Analysis, 4 (1991), 271-283.  Google Scholar

[4]

T. Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. in PDE, 27 (2002), 727-761. doi: 10.1081/PDE-120002872.  Google Scholar

[5]

M. Crandall, Viscosity Solutions: A Primer, lecture notes in Mathematics 1660, Springer-Verlag, 1997. doi: 10.1007/BFb0094294.  Google Scholar

[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.  Google Scholar

[7]

M. Crandall and P-L. Lions, Two approximations of solutions of Hamilton equations, Math. Comp., 43 (1984), 1-19. doi: 10.2307/2007396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

Y. Giga, Surface Evolution Equations: A Level Set Method, Monographs in Mathematics 99, Birkhauser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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/. Google Scholar

[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.  Google Scholar

[15]

M. Rudd, Statistical exponential formulas for homogeneous diffusions,, preprint, ().  doi: 10.3934/cpaa.2015.14.269.  Google Scholar

[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.  Google Scholar

[17]

R. Vargas, Matrix Iterative Analysis, Springer-Verlag, 2000. doi: 10.1007/978-3-642-05156-2.  Google Scholar

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