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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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), 597-638. doi: 10.1002/cpa.20274.  Google Scholar

[8]

L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Rational Mech. Anal., 180 (2010), 301-360.  Google Scholar

[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.  Google Scholar

[10]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Financial Mathematics Series, Chapman & Hall/CRC, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.  Google Scholar

[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.  Google Scholar

[17]

C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176. doi: 10.4171/IFB/207.  Google Scholar

[18]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications, preprint, 2009, arXiv:0907.5524v1. 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), 137-165. doi: 10.1007/s00030-005-0031-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," vol. 69 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass., 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. Spencer, Balancing games, J. Combinatorial Theory Ser. B, 23 (1977), 68-74. doi: 10.1016/0095-8956(77)90057-0.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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), 597-638. doi: 10.1002/cpa.20274.  Google Scholar

[8]

L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Rational Mech. Anal., 180 (2010), 301-360.  Google Scholar

[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.  Google Scholar

[10]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Financial Mathematics Series, Chapman & Hall/CRC, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.  Google Scholar

[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.  Google Scholar

[17]

C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176. doi: 10.4171/IFB/207.  Google Scholar

[18]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications, preprint, 2009, arXiv:0907.5524v1. 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), 137-165. doi: 10.1007/s00030-005-0031-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," vol. 69 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass., 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. Spencer, Balancing games, J. Combinatorial Theory Ser. B, 23 (1977), 68-74. doi: 10.1016/0095-8956(77)90057-0.  Google Scholar

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