-
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
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. |
[1] |
Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 |
[2] |
Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 |
[3] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
[4] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[5] |
Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 |
[6] |
A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121. |
[7] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[8] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[9] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[10] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[11] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[12] |
Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 |
[13] |
Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 |
[14] |
Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 |
[15] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
[16] |
Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 |
[17] |
M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 |
[18] |
Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017 |
[19] |
Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20. |
[20] |
Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]