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On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems
1. | Dip. di Matematica Pura e Applicata, Univ. dell'Aquila, loc. Monteluco di Roio, 67040 l'Aquila, Italy |
2. | Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova |
References:
[1] |
O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control, SIAM J. Control Optim., 40 (2001), 1159-1188.
doi: 10.1137/S0363012900366741. |
[2] |
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. |
[3] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equation, Mem. Amer. Math. Soc., 204 (2010). |
[4] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenizations for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387.
doi: 10.1016/j.jde.2007.05.027. |
[5] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbation and homogenization of optimal control problems, in "Geometric Control and Nonsmooth Analysis" (F. Ancona, A. Bressan, P. Cannarsa, F. Clarke, P.R. Wolenski; Eds.), World Scientific, Singapore, 2008, 1-27. |
[6] |
M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. |
[7] |
G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN Math. Model. Numer. Anal., 36 (2002), 33-54.
doi: 10.1051/m2an:2002002. |
[8] |
A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,'' Clarendon Press, Oxford, 1998. |
[9] |
A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'' North-Holland, Amsterdam, 1978. |
[10] |
L. A. Caffarelli, P. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), 319-361.
doi: 10.1002/cpa.20069. |
[11] |
F. Camilli and C. Marchi, Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs, Nonlinearity, 22 (2009), 1481-1498.
doi: 10.1088/0951-7715/22/6/011. |
[12] |
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. |
[13] |
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. |
[14] |
L. Evans, The perturbed test function method for viscosity solutions of nonlinear P.D.E., Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. |
[15] |
L. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. |
[16] |
W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' Springer-Verlag, Berlin, 1993. |
[17] |
W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-players, zero-sum stochastic differential games, Indiana Univ. Math. J., 38 (1989), 293-314.
doi: 10.1512/iumj.1989.38.38015. |
[18] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd edition, Springer, Berlin, 1983. |
[19] |
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,'' Springer, Berlin, 1994. |
[20] |
N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probab. Theory Related Fields, 117 (2000), 1-16.
doi: 10.1007/s004400050264. |
[21] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,'' Academic Press, New York, 1968. |
[22] |
P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, 1986. |
[23] |
P.L. Lions and P. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations, 30 (2005), 335-375.
doi: 10.1081/PDE-200050077. |
[24] |
P. L. Lions and P. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincarè Anal. Non Linèaire, 22 (2005), 667-677.
doi: 10.1016/j.anihpc.2004.10.009. |
[25] |
C. Marchi, Rate of convergence for multiscale homogenization of Hamilton-Jacobi equations, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 519-539.
doi: 10.1017/S0308210507000704. |
[26] |
M. V. Safonov, Classical solution of nonlinear elliptic equations of second-order, Math. USSR-Izv., 33 (1989), 597-612. (Engl. transl. of Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988)). |
show all references
References:
[1] |
O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control, SIAM J. Control Optim., 40 (2001), 1159-1188.
doi: 10.1137/S0363012900366741. |
[2] |
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. |
[3] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equation, Mem. Amer. Math. Soc., 204 (2010). |
[4] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenizations for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387.
doi: 10.1016/j.jde.2007.05.027. |
[5] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbation and homogenization of optimal control problems, in "Geometric Control and Nonsmooth Analysis" (F. Ancona, A. Bressan, P. Cannarsa, F. Clarke, P.R. Wolenski; Eds.), World Scientific, Singapore, 2008, 1-27. |
[6] |
M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. |
[7] |
G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN Math. Model. Numer. Anal., 36 (2002), 33-54.
doi: 10.1051/m2an:2002002. |
[8] |
A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,'' Clarendon Press, Oxford, 1998. |
[9] |
A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'' North-Holland, Amsterdam, 1978. |
[10] |
L. A. Caffarelli, P. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), 319-361.
doi: 10.1002/cpa.20069. |
[11] |
F. Camilli and C. Marchi, Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs, Nonlinearity, 22 (2009), 1481-1498.
doi: 10.1088/0951-7715/22/6/011. |
[12] |
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. |
[13] |
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. |
[14] |
L. Evans, The perturbed test function method for viscosity solutions of nonlinear P.D.E., Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. |
[15] |
L. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. |
[16] |
W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' Springer-Verlag, Berlin, 1993. |
[17] |
W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-players, zero-sum stochastic differential games, Indiana Univ. Math. J., 38 (1989), 293-314.
doi: 10.1512/iumj.1989.38.38015. |
[18] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd edition, Springer, Berlin, 1983. |
[19] |
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,'' Springer, Berlin, 1994. |
[20] |
N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probab. Theory Related Fields, 117 (2000), 1-16.
doi: 10.1007/s004400050264. |
[21] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,'' Academic Press, New York, 1968. |
[22] |
P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, 1986. |
[23] |
P.L. Lions and P. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations, 30 (2005), 335-375.
doi: 10.1081/PDE-200050077. |
[24] |
P. L. Lions and P. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincarè Anal. Non Linèaire, 22 (2005), 667-677.
doi: 10.1016/j.anihpc.2004.10.009. |
[25] |
C. Marchi, Rate of convergence for multiscale homogenization of Hamilton-Jacobi equations, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 519-539.
doi: 10.1017/S0308210507000704. |
[26] |
M. V. Safonov, Classical solution of nonlinear elliptic equations of second-order, Math. USSR-Izv., 33 (1989), 597-612. (Engl. transl. of Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988)). |
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