Repeated games for non-linear parabolic integro-differential equations and integral curvature flows

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October 2011, 29(4): 1517-1552. doi: 10.3934/dcds.2011.29.1517

Repeated games for non-linear parabolic integro-differential equations and integral curvature flows

Cyril Imbert ^1, and Sylvia Serfaty ^2,

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

Received January 2010 Revised September 2010 Published December 2010

Abstract
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The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations.
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.

Keywords: geometric flows., viscosity solutions, parabolic integro-differential equations, Repeated games, integral curvature flows.

Mathematics Subject Classification: 35C99, 53C44, 90D10, 49K2.

Citation: Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517

References:

[1]	N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data, Trans. Amer. Math. Soc., 361 (2009), 2527-2566. 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), 449-504. 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, Analyse Non Linéaire, 25 (2008), 567-585. 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), 439-469. 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), 271-283.
[6]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Non local minimal surfaces, 2009, arXiv:0905.1183.v1.
[7]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. 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), 301-360.
[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), 749-786.
[10]	R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Financial Mathematics Series, Chapman & Hall/CRC, 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), 1-67.
[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), 183-186.
[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), 1061-1104. 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), 773-797. 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), 635-681.
[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), 785-826. doi: 10.3934/dcds.2009.23.785.
[17]	C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176. doi: 10.4171/IFB/207.
[18]	C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications, preprint, 2009, arXiv:0907.5524v1.
[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), 137-165. 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), 344-407. 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), 1298-1350. doi: 10.1002/cpa.20336.
[22]	P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," vol. 69 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass., 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), 12-49. 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), 1057-1074.
[25]	H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions, in "Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, Minn., 1986)," vol. 10 of IMA Vol. Math. Appl., Springer, New York, (1988), 501-511.
[26]	P. E. Souganidis, Front propagation: Theory and applications, in "Viscosity Solutions and Applications (Montecatini Terme, 1995)," vol. 1660 of Lecture Notes in Math., Springer, Berlin, (1997), 186-242.
[27]	J. Spencer, Balancing games, J. Combinatorial Theory Ser. B, 23 (1977), 68-74. 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), 2527-2566. 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), 449-504. 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, Analyse Non Linéaire, 25 (2008), 567-585. 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), 439-469. 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), 271-283.
[6]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Non local minimal surfaces, 2009, arXiv:0905.1183.v1.
[7]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. 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), 301-360.
[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), 749-786.
[10]	R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Financial Mathematics Series, Chapman & Hall/CRC, 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), 1-67.
[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), 183-186.
[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), 1061-1104. 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), 773-797. 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), 635-681.
[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), 785-826. doi: 10.3934/dcds.2009.23.785.
[17]	C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176. doi: 10.4171/IFB/207.
[18]	C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications, preprint, 2009, arXiv:0907.5524v1.
[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), 137-165. 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), 344-407. 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), 1298-1350. doi: 10.1002/cpa.20336.
[22]	P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," vol. 69 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass., 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), 12-49. 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), 1057-1074.
[25]	H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions, in "Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, Minn., 1986)," vol. 10 of IMA Vol. Math. Appl., Springer, New York, (1988), 501-511.
[26]	P. E. Souganidis, Front propagation: Theory and applications, in "Viscosity Solutions and Applications (Montecatini Terme, 1995)," vol. 1660 of Lecture Notes in Math., Springer, Berlin, (1997), 186-242.
[27]	J. Spencer, Balancing games, J. Combinatorial Theory Ser. B, 23 (1977), 68-74. doi: 10.1016/0095-8956(77)90057-0.

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