American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals
  • DCDS Home
  • This Issue
  • Next Article
    Subshifts of finite type which have completely positive entropy
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 - A, 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.  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.  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, 25 (2008), 567.  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.  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.   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.  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.   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.   Google Scholar

[10]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Financial Mathematics Series, (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.   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.   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.  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.  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.   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.  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.  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).   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.  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.  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.  doi: 10.1002/cpa.20336.  Google Scholar

[22]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", vol. 69 of Research Notes in Mathematics, 69 (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.  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.   Google Scholar

[25]

H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions,, in, 10 (1988), 501.   Google Scholar

[26]

P. E. Souganidis, Front propagation: Theory and applications,, in, 1660 (1997), 186.   Google Scholar

[27]

J. Spencer, Balancing games,, J. Combinatorial Theory Ser. B, 23 (1977), 68.  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.  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.  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, 25 (2008), 567.  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.  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.   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.  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.   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.   Google Scholar

[10]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Financial Mathematics Series, (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.   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.   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.  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.  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.   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.  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.  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).   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.  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.  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.  doi: 10.1002/cpa.20336.  Google Scholar

[22]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", vol. 69 of Research Notes in Mathematics, 69 (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.  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.   Google Scholar

[25]

H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions,, in, 10 (1988), 501.   Google Scholar

[26]

P. E. Souganidis, Front propagation: Theory and applications,, in, 1660 (1997), 186.   Google Scholar

[27]

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

[1]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
[2]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[3]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160
[4]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015
[5]

Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053
[6]

Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069
[7]

Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17
[8]

Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793
[9]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
[10]

Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277
[11]

Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042
[12]

Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417
[13]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
[14]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17
[15]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541
[16]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[17]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[18]

Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693
[19]

Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119
[20]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences