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Repeated games for non-linear parabolic integro-differential equations and integral curvature flows
1. | CEREMADE, UMR CNRS 7534, université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16 |
2. | UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France |
  In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For parabolic integro-differential equations, players choose smooth functions on the whole space. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces.
References:
[1] |
N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data,, Trans. Amer. Math. Soc., 361 (2009), 2527.
doi: 10.1090/S0002-9947-08-04758-2. |
[2] |
O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution,, Arch. Ration. Mech. Anal., 181 (2006), 449.
doi: 10.1007/s00205-006-0418-5. |
[3] |
G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited,, Annales de l'Institut Henri Poincaré, 25 (2008), 567.
doi: 10.1016/j.anihpc.2007.02.007. |
[4] |
G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory,, SIAM J. Control Optim., 31 (1993), 439.
doi: 10.1137/0331021. |
[5] |
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Anal., 4 (1991), 271.
|
[6] |
L. Caffarelli, J.-M. Roquejoffre and O. Savin, Non local minimal surfaces,, 2009, (). Google Scholar |
[7] |
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.
doi: 10.1002/cpa.20274. |
[8] |
L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Rational Mech. Anal., 180 (2010), 301.
|
[9] |
Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, J. Differential Geom., 33 (1991), 749.
|
[10] |
R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Financial Mathematics Series, (2004).
|
[11] |
M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1.
|
[12] |
M. G. Crandall and P.-L. Lions, Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre,, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 183.
|
[13] |
F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics,, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061.
doi: 10.4171/JEMS/140. |
[14] |
L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations,, Indiana Univ. Math. J., 33 (1984), 773.
doi: 10.1512/iumj.1984.33.33040. |
[15] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, J. Differential Geom., 33 (1991), 635.
|
[16] |
N. Forcadel, C. Imbert, and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics,, Discrete Contin. Dyn. Syst., 23 (2009), 785.
doi: 10.3934/dcds.2009.23.785. |
[17] |
C. Imbert, Level set approach for fractional mean curvature flows,, Interfaces Free Bound., 11 (2009), 153.
doi: 10.4171/IFB/207. |
[18] |
C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications,, preprint, (2009). Google Scholar |
[19] |
E. R. Jakobsen and K. H. Karlsen, A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations,, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 137.
doi: 10.1007/s00030-005-0031-6. |
[20] |
R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344.
doi: 10.1002/cpa.20101. |
[21] |
R. V. Kohn and S. Serfaty, A deterministic-control-based approach to fully non-linear parabolic and elliptic equations,, Comm. Pure Appl. Math., 63 (2010), 1298.
doi: 10.1002/cpa.20336. |
[22] |
P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", vol. 69 of Research Notes in Mathematics, 69 (1982).
|
[23] |
S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12.
doi: 10.1016/0021-9991(88)90002-2. |
[24] |
A. Sayah, Équations de Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité,, Comm. Partial Differential Equations, 16 (1991), 1057.
|
[25] |
H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions,, in, 10 (1988), 501.
|
[26] |
P. E. Souganidis, Front propagation: Theory and applications,, in, 1660 (1997), 186.
|
[27] |
J. Spencer, Balancing games,, J. Combinatorial Theory Ser. B, 23 (1977), 68.
doi: 10.1016/0095-8956(77)90057-0. |
show all references
References:
[1] |
N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data,, Trans. Amer. Math. Soc., 361 (2009), 2527.
doi: 10.1090/S0002-9947-08-04758-2. |
[2] |
O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution,, Arch. Ration. Mech. Anal., 181 (2006), 449.
doi: 10.1007/s00205-006-0418-5. |
[3] |
G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited,, Annales de l'Institut Henri Poincaré, 25 (2008), 567.
doi: 10.1016/j.anihpc.2007.02.007. |
[4] |
G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory,, SIAM J. Control Optim., 31 (1993), 439.
doi: 10.1137/0331021. |
[5] |
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Anal., 4 (1991), 271.
|
[6] |
L. Caffarelli, J.-M. Roquejoffre and O. Savin, Non local minimal surfaces,, 2009, (). Google Scholar |
[7] |
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.
doi: 10.1002/cpa.20274. |
[8] |
L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Rational Mech. Anal., 180 (2010), 301.
|
[9] |
Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, J. Differential Geom., 33 (1991), 749.
|
[10] |
R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Financial Mathematics Series, (2004).
|
[11] |
M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1.
|
[12] |
M. G. Crandall and P.-L. Lions, Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre,, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 183.
|
[13] |
F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics,, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061.
doi: 10.4171/JEMS/140. |
[14] |
L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations,, Indiana Univ. Math. J., 33 (1984), 773.
doi: 10.1512/iumj.1984.33.33040. |
[15] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, J. Differential Geom., 33 (1991), 635.
|
[16] |
N. Forcadel, C. Imbert, and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics,, Discrete Contin. Dyn. Syst., 23 (2009), 785.
doi: 10.3934/dcds.2009.23.785. |
[17] |
C. Imbert, Level set approach for fractional mean curvature flows,, Interfaces Free Bound., 11 (2009), 153.
doi: 10.4171/IFB/207. |
[18] |
C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications,, preprint, (2009). Google Scholar |
[19] |
E. R. Jakobsen and K. H. Karlsen, A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations,, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 137.
doi: 10.1007/s00030-005-0031-6. |
[20] |
R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344.
doi: 10.1002/cpa.20101. |
[21] |
R. V. Kohn and S. Serfaty, A deterministic-control-based approach to fully non-linear parabolic and elliptic equations,, Comm. Pure Appl. Math., 63 (2010), 1298.
doi: 10.1002/cpa.20336. |
[22] |
P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", vol. 69 of Research Notes in Mathematics, 69 (1982).
|
[23] |
S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12.
doi: 10.1016/0021-9991(88)90002-2. |
[24] |
A. Sayah, Équations de Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité,, Comm. Partial Differential Equations, 16 (1991), 1057.
|
[25] |
H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions,, in, 10 (1988), 501.
|
[26] |
P. E. Souganidis, Front propagation: Theory and applications,, in, 1660 (1997), 186.
|
[27] |
J. Spencer, Balancing games,, J. Combinatorial Theory Ser. B, 23 (1977), 68.
doi: 10.1016/0095-8956(77)90057-0. |
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