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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 |
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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