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On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations
1. | Dipartimento di Matematica, Università di Padova, via Trieste, 63; I-35121 Padova |
2. | Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa |
References:
[1] |
Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Homogenization of Hamilton-Jacobi equations: numerical methods, Math. Models Methods Appl. Sci., 18 (2008), 1115-1143.
doi: 10.1142/S0218202508002978. |
[2] |
O. Alvarez, Homogenization of Hamilton-Jacobi equations in perforated sets, J. Differential Equations, 159 (1999), 543-577.
doi: 10.1006/jdeq.1999.3665. |
[3] |
O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61.
doi: 10.1007/s00205-003-0266-5. |
[4] |
O. Alvarez and M. Bardi, Ergodic problems in differential games, In "Advances in Dynamic Game Theory," volume 9 of Ann. Internat. Soc. Dynam. Games, pages 131-152. Birkhäuser Boston, Boston, MA, 2007.
doi: 10.1007/978-0-8176-4553-3_7. |
[5] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010).
doi: 10.1090/S0065-9266-09-00588-2. |
[6] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387.
doi: 10.1016/j.jde.2007.05.027. |
[7] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems, In "Geometric Control and Nonsmooth Analysis," volume 76 of Ser. Adv. Math. Appl. Sci., pages 1-27. World Sci. Publ., Hackensack, NJ, 2008. |
[8] |
M. Arisawa, P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217.
doi: 10.1080/03605309808821413. |
[9] |
M. Bardi, On differential games with long-time-average cost, In "Advances in Dynamic Games and Their Applications," volume 10 of Ann. Internat. Soc. Dynam. Games, pages 3-18. Birkhäuser Boston Inc., Boston, MA, 2009. |
[10] |
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," Birkhäuser Boston Inc., Boston, MA, 1997. |
[11] |
G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466.
doi: 10.1007/s00526-007-0097-6. |
[12] |
G. Barles and P. E. Souganidis, Some counterexamples on the asymptotic behavior of the solutions of Hamilton-Jacobi equations, C. R. Acad. Sci. Paris Sect. I Math., 330 (2000), 963-968. |
[13] |
K. Bhattacharya and B. Craciun, Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 773-805.
doi: 10.1017/S0308210500002675. |
[14] |
I. Birindelli and J. Wigniolle, Homogenization of Hamilton-Jacobi equations in the Heisenberg group, Commun. Pure Appl. Anal., 2 (2003), 461-479.
doi: 10.3934/cpaa.2003.2.461. |
[15] |
A. Braides and A. Defranceschi, Homogenization of multiple integrals, in volume 12 of "Oxford Lecture Series in Mathematics and its Applications," The Clarendon Press Oxford University Press, New York, 1998. |
[16] |
F. Camilli and A. Siconolfi, Effective Hamiltonian and homogenization of measurable eikonal equations, Arch. Ration. Mech. Anal., 183 (2007), 1-20.
doi: 10.1007/s00205-006-0001-0. |
[17] |
I. Capuzzo Dolcetta and H. Ishii, On the rate of convergence in homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J., 50 (2001), 1113-1129.
doi: 10.1512/iumj.2001.50.1933. |
[18] |
P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $R^2/Z^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856.
doi: 10.1016/j.anihpc.2009.11.015. |
[19] |
P. Cardaliaguet, P.-L. Lions and P. E. Souganidis, A discussion about the homogenization of moving interfaces, J. Math. Pures Appl., 91 (2009), 339-363.
doi: 10.1016/j.matpur.2009.01.014. |
[20] |
P. Cardaliaguet, J. Nolen and P. E. Souganidis, Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199 (2011), 527-561.
doi: 10.1007/s00205-010-0332-8. |
[21] |
A. Davini and A. Siconolfi, Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case, Math. Ann., 345 (2009), 749-782.
doi: 10.1007/s00208-009-0372-2. |
[22] |
W. E., A class of homogenization problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759.
doi: 10.1002/cpa.3160440702. |
[23] |
L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. |
[24] |
L. C. Evans, A survey of partial differential equations methods in weak KAM theory, Comm. Pure Appl. Math., 57 (2004), 445-480.
doi: 10.1002/cpa.20009. |
[25] |
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. |
[26] |
A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Lecture Notes, ().
|
[27] |
D. A. Gomes, Hamilton-Jacobi methods for vakonomic mechanics, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 233-257.
doi: 10.1007/s00030-007-5012-5. |
[28] |
K. Horie and H. Ishii, Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure, Indiana Univ. Math. J., 47 (1998), 1011-1058.
doi: 10.1512/iumj.1998.47.1385. |
[29] |
C. Imbert and R. Monneau, Homogenization of first-order equations with $(u/\epsilon)$-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89.
doi: 10.1007/s00205-007-0074-4. |
[30] |
H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In "International Conference on Differential Equations," Vol. 1, 2 (Berlin, 1999), pages 600-605. World Sci. Publ., River Edge, NJ, 2000. |
[31] |
P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, 1986. |
[32] |
C. Marchi, On the convergence of singular perturbations of Hamilton-Jacobi equations, Commun. Pure Appl. Anal., 9 (2010), 1363-1377.
doi: 10.3934/cpaa.2010.9.1363. |
[33] |
F. Rezakhanlou and J. E. Tarver, Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151 (2000), 277-309.
doi: 10.1007/s002050050198. |
[34] |
P. Soravia, Pursuit-evasion problems and viscosity solutions of Isaacs equations, SIAM J. Control Optim., 31 (1993), 604-623.
doi: 10.1137/0331027. |
[35] |
P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11. |
[36] |
B. Stroffolini, Homogenization of Hamilton-Jacobi equations in Carnot groups, ESAIM Control Optim. Calc. Var., 13 (2007), 107-119 (electronic).
doi: 10.1051/cocv:2007005. |
[37] |
G. Terrone, "Singular Perturbation and Homogenization Problems in Control Theory, Differential Games and Fully Nonlinear Partial Differential Equations," PhD thesis, University of Padova, 2008. |
[38] |
C. Viterbo, Symplectic homogenization, Preprint, 2008, arXiv:0801.0206v1">arXiv:0801.0206v1" target="_blank">arXiv:0801.0206v1. |
[39] |
J. Xin, "An Introduction to Fronts in Random Media," Springer, New York, 2009.
doi: 10.1007/978-0-387-87683-2. |
show all references
References:
[1] |
Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Homogenization of Hamilton-Jacobi equations: numerical methods, Math. Models Methods Appl. Sci., 18 (2008), 1115-1143.
doi: 10.1142/S0218202508002978. |
[2] |
O. Alvarez, Homogenization of Hamilton-Jacobi equations in perforated sets, J. Differential Equations, 159 (1999), 543-577.
doi: 10.1006/jdeq.1999.3665. |
[3] |
O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61.
doi: 10.1007/s00205-003-0266-5. |
[4] |
O. Alvarez and M. Bardi, Ergodic problems in differential games, In "Advances in Dynamic Game Theory," volume 9 of Ann. Internat. Soc. Dynam. Games, pages 131-152. Birkhäuser Boston, Boston, MA, 2007.
doi: 10.1007/978-0-8176-4553-3_7. |
[5] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010).
doi: 10.1090/S0065-9266-09-00588-2. |
[6] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387.
doi: 10.1016/j.jde.2007.05.027. |
[7] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems, In "Geometric Control and Nonsmooth Analysis," volume 76 of Ser. Adv. Math. Appl. Sci., pages 1-27. World Sci. Publ., Hackensack, NJ, 2008. |
[8] |
M. Arisawa, P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217.
doi: 10.1080/03605309808821413. |
[9] |
M. Bardi, On differential games with long-time-average cost, In "Advances in Dynamic Games and Their Applications," volume 10 of Ann. Internat. Soc. Dynam. Games, pages 3-18. Birkhäuser Boston Inc., Boston, MA, 2009. |
[10] |
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," Birkhäuser Boston Inc., Boston, MA, 1997. |
[11] |
G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466.
doi: 10.1007/s00526-007-0097-6. |
[12] |
G. Barles and P. E. Souganidis, Some counterexamples on the asymptotic behavior of the solutions of Hamilton-Jacobi equations, C. R. Acad. Sci. Paris Sect. I Math., 330 (2000), 963-968. |
[13] |
K. Bhattacharya and B. Craciun, Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 773-805.
doi: 10.1017/S0308210500002675. |
[14] |
I. Birindelli and J. Wigniolle, Homogenization of Hamilton-Jacobi equations in the Heisenberg group, Commun. Pure Appl. Anal., 2 (2003), 461-479.
doi: 10.3934/cpaa.2003.2.461. |
[15] |
A. Braides and A. Defranceschi, Homogenization of multiple integrals, in volume 12 of "Oxford Lecture Series in Mathematics and its Applications," The Clarendon Press Oxford University Press, New York, 1998. |
[16] |
F. Camilli and A. Siconolfi, Effective Hamiltonian and homogenization of measurable eikonal equations, Arch. Ration. Mech. Anal., 183 (2007), 1-20.
doi: 10.1007/s00205-006-0001-0. |
[17] |
I. Capuzzo Dolcetta and H. Ishii, On the rate of convergence in homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J., 50 (2001), 1113-1129.
doi: 10.1512/iumj.2001.50.1933. |
[18] |
P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $R^2/Z^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856.
doi: 10.1016/j.anihpc.2009.11.015. |
[19] |
P. Cardaliaguet, P.-L. Lions and P. E. Souganidis, A discussion about the homogenization of moving interfaces, J. Math. Pures Appl., 91 (2009), 339-363.
doi: 10.1016/j.matpur.2009.01.014. |
[20] |
P. Cardaliaguet, J. Nolen and P. E. Souganidis, Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199 (2011), 527-561.
doi: 10.1007/s00205-010-0332-8. |
[21] |
A. Davini and A. Siconolfi, Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case, Math. Ann., 345 (2009), 749-782.
doi: 10.1007/s00208-009-0372-2. |
[22] |
W. E., A class of homogenization problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759.
doi: 10.1002/cpa.3160440702. |
[23] |
L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. |
[24] |
L. C. Evans, A survey of partial differential equations methods in weak KAM theory, Comm. Pure Appl. Math., 57 (2004), 445-480.
doi: 10.1002/cpa.20009. |
[25] |
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. |
[26] |
A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Lecture Notes, ().
|
[27] |
D. A. Gomes, Hamilton-Jacobi methods for vakonomic mechanics, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 233-257.
doi: 10.1007/s00030-007-5012-5. |
[28] |
K. Horie and H. Ishii, Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure, Indiana Univ. Math. J., 47 (1998), 1011-1058.
doi: 10.1512/iumj.1998.47.1385. |
[29] |
C. Imbert and R. Monneau, Homogenization of first-order equations with $(u/\epsilon)$-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89.
doi: 10.1007/s00205-007-0074-4. |
[30] |
H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In "International Conference on Differential Equations," Vol. 1, 2 (Berlin, 1999), pages 600-605. World Sci. Publ., River Edge, NJ, 2000. |
[31] |
P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, 1986. |
[32] |
C. Marchi, On the convergence of singular perturbations of Hamilton-Jacobi equations, Commun. Pure Appl. Anal., 9 (2010), 1363-1377.
doi: 10.3934/cpaa.2010.9.1363. |
[33] |
F. Rezakhanlou and J. E. Tarver, Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151 (2000), 277-309.
doi: 10.1007/s002050050198. |
[34] |
P. Soravia, Pursuit-evasion problems and viscosity solutions of Isaacs equations, SIAM J. Control Optim., 31 (1993), 604-623.
doi: 10.1137/0331027. |
[35] |
P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11. |
[36] |
B. Stroffolini, Homogenization of Hamilton-Jacobi equations in Carnot groups, ESAIM Control Optim. Calc. Var., 13 (2007), 107-119 (electronic).
doi: 10.1051/cocv:2007005. |
[37] |
G. Terrone, "Singular Perturbation and Homogenization Problems in Control Theory, Differential Games and Fully Nonlinear Partial Differential Equations," PhD thesis, University of Padova, 2008. |
[38] |
C. Viterbo, Symplectic homogenization, Preprint, 2008, arXiv:0801.0206v1">arXiv:0801.0206v1" target="_blank">arXiv:0801.0206v1. |
[39] |
J. Xin, "An Introduction to Fronts in Random Media," Springer, New York, 2009.
doi: 10.1007/978-0-387-87683-2. |
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