June  2019, 14(2): 341-369. doi: 10.3934/nhm.2019014

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

1. 

Department of Mathematics, Faculty of Mathematical and Computer Sciences, University of Gezira, Wad Madani, P.O.Box 20, Sudan

2. 

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

* Corresponding author: Mogtaba Mohammed

Dedicated to the memory of Professor Salah-Eldin A. Mohammed (May 20, 1946 - Dec 21, 2016)

Received  June 2018 Revised  October 2018 Published  April 2019

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

Citation: Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
References:
[1]

A. Abdulle and M. J. Grote, Finite element heterogeneous multiscale method for the wave equation, SIAM, Multiscale Modeling and Simulation, 9 (2011), 766-792.  doi: 10.1137/100800488.  Google Scholar

[2]

A. AbdulleW. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer., 21 (2012), 1-87.  doi: 10.1017/S0962492912000025.  Google Scholar

[3]

A. Abdulle and G. A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys., 231 (2012), 2482-2497.  doi: 10.1016/j.jcp.2011.11.039.  Google Scholar

[4]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[5]

G. Allaire, Two-scale convergence: A new method in periodic homogenization. Nonlinear partial differential equations and their applicationsapplications, Collge de France Seminar, Vol. XII (Paris, 1991—1993), 1-14, Pitman Res. Notes Math. Ser., 302, Longman Sci. Tech., Harlow, 1994.  Google Scholar

[6]

N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials, Translated from the Russian by D. Lei(tes. Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2247-1.  Google Scholar

[7]

A. Bensoussan, Some existence results for stochastic partial differential equations., Stochastic Partial Differential Equations and Applications (Trento, 1990), 37—53, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992.  Google Scholar

[8]

A. Bensoussan, Homogenization of a class of stochastic partial differential equations. Composite Media and Homogenization Theory (Trieste, 1990), 47-65, Progr. Nonlinear Differential Equations Appl., 5, Birkhuser Boston, Boston, MA, 1991.  Google Scholar

[9]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Corrected reprint of the 1978 original., AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.  Google Scholar

[10]

H. BessaihY. Efendiev and F. Maris, Homogenization of Brinkman flows in heterogeneous dynamic media, Stoch. Partial Differ. Equ. Anal. Comput., 3 (2015), 479-505.  doi: 10.1007/s40072-015-0058-6.  Google Scholar

[11]

H. BessaihY. Efendiev and F. Maris, Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition, Netw. Heterog. Media, 10 (2015), 343-367.  doi: 10.3934/nhm.2015.10.343.  Google Scholar

[12]

P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

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A. BourgeatA. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51.   Google Scholar

[14]

A. Bourgeat and A. L. Piatnitski, Averaging of a singular random source term in a diffusion convection equation, SIAM J. Math. Anal., 42 (2010), 2626—2651. doi: 10.1137/080736077.  Google Scholar

[15]

S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM J. Math. Anal., 43 (2011), 2482-2518.  doi: 10.1137/100806710.  Google Scholar

[16]

S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.  Google Scholar

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D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999.  Google Scholar

[18]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J.Rei. Ang. Math. B., 368 (1986), 28-42.   Google Scholar

[19]

M. A. Diop and E. Pardoux, Averaging of a parabolic partial differential equation with random evolution. Seminar on Stochastic Analysis, Random Fields and Applications IV, Progr. Probab., 58, Birkh user, Basel, (2004), 111-128.  Google Scholar

[20]

Y. GorbF. Maris and B. Vernescu, Homogenization for rigid suspensions with random velocity-dependent interfacial forces, J. Math. Anal. Appl., 420 (2014), 632-668.  doi: 10.1016/j.jmaa.2014.05.015.  Google Scholar

[21]

N. Ichihara, Homogenization problem for stochastic partial differential equations of Zakai type, Stoch. Stoch. Rep., 76 (2004), 243-266.  doi: 10.1080/10451120410001714107.  Google Scholar

[22]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Translated from the Russian by G. A. Yosifian. Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[23]

E. Y. Khruslov, Homogenized models of composite media, Composite Media and Homogenization Theory (Trieste, 1990), 159-182, Progr. Nonlinear Differential Equations Appl., 5, Birkhuser Boston, Boston, MA, 1991. doi: 10.1007/978-1-4684-6787-1_10.  Google Scholar

[24]

S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109(151) (1979), 188-202, 327.  Google Scholar

[25]

S. V. Lototsky, Small perturbation of stochastic parabolic equations: A power series analysis, J. Funct. Anal., 193 (2002), 94-115.  doi: 10.1006/jfan.2001.3923.  Google Scholar

[26]

D. LukkassenG. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86.   Google Scholar

[27]

M. Mohammed and M. Sango, Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method, Asymptotic Analysis, 91 (2015), 341-371.   Google Scholar

[28]

M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptotic Analysis, 97 (2016), 301-327.  doi: 10.3233/ASY-151355.  Google Scholar

[29]

M. Mohammed and M. Sango, A Tartar approach to periodic homogenization of linear hyperbolic stochastic partial differential equation, Int. J. Mod. Phys. B, 30 (2016), 1640020, 9 pp. doi: 10.1142/S0217979216400208.  Google Scholar

[30]

M. Mohammed, Homogenization of nonlinear hyperbolic stochastic equation via Tartar's method, J. Hyper. Differential Equations, 14 (2017), 323-340.  doi: 10.1142/S0219891617500096.  Google Scholar

[31]

F. Murat and L. Tartar, H-convergence in Topics in the mathematical Modelling of composite Materials. ed. A. Cherkaev and Kohn, Birkhauser. Boston, 31 (1997), 21-43.   Google Scholar

[32]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[33]

G. NguetsengM. Sango and J. L. Woukeng, Reiterated ergodic algebras and applications, Comm. Math. Phys., 300 (2010), 835-876.  doi: 10.1007/s00220-010-1127-3.  Google Scholar

[34]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in elasticity and Homogenization, Studies in Mathematics and its Applications, 26. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[35]

M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Mathematicae, 426 (2004), 1-63.  doi: 10.4064/dm426-0-1.  Google Scholar

[36]

A. Pankov, G-Convergence and Homogenization of Nonlinear Partial Differential Operators, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-94-015-8957-4.  Google Scholar

[37]

G. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. I, II (Esztergom, 1979), 835-873, Colloq. Math. Soc. Janos Bolyai, 27, North-Holland, Amsterdam-New York, 1981.  Google Scholar

[38]

E. Pardoux, Équations aux dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse, Université Paris XI, 1975. Google Scholar

[39]

B. L. Rozovskiĭ, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Translated from the Russian by A. Yarkho. Mathematics and its Applications (Soviet Series), 35. Kluwer Academic Publishers Group, Dordrecht, 1990. xviii+315 pp doi: 10.1007/978-94-011-3830-7.  Google Scholar

[40]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory (Lecture Notes in Physics), Springer, 1980.  Google Scholar

[41]

E. Sanchez-Palencia and A. Zaoui, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987. doi: 10.1007/3-540-17616-0.  Google Scholar

[42]

M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25 (2013), 931-965.  doi: 10.1515/form.2011.138.  Google Scholar

[43]

M. Sango, Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries, Commun. Math. Sci., 12 (2014), 345-382.  doi: 10.4310/CMS.2014.v12.n2.a7.  Google Scholar

[44]

M. Sango, Asymptotic behavior of a stochastic evolution problem in a varying domain, Stochastic Anal. Appl., 20 (2002), 1331-1358.  doi: 10.1081/SAP-120015835.  Google Scholar

[45]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[46]

I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Nauka, Moscow, 1990. English translation in: Translations of Mathematical Monographs, AMS, Providence, 1994.  Google Scholar

[47]

E. P. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Analysis. IOS Press, 20 (1999), 1-11.   Google Scholar

[48]

N. Svanstedt, Multiscale stochastic homogenization of monotone operators, Netw. Heterog. Media, 2 (2007), 181-192.  doi: 10.3934/nhm.2007.2.181.  Google Scholar

[49]

L. Tartar, Quelques remarques sur l'homogénésation, in Functional Analysis and Numerical Analysis, Proc. Japan-France Siminar 1976, ed. H. Fujitaa, Japanese Society for the Promotion of Science, (1977), 468-486. Google Scholar

[50]

L. Tartar, The General Theory of Homogenization, A personalized introduction. Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009. doi: 10.1007/978-3-642-05195-1.  Google Scholar

show all references

References:
[1]

A. Abdulle and M. J. Grote, Finite element heterogeneous multiscale method for the wave equation, SIAM, Multiscale Modeling and Simulation, 9 (2011), 766-792.  doi: 10.1137/100800488.  Google Scholar

[2]

A. AbdulleW. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer., 21 (2012), 1-87.  doi: 10.1017/S0962492912000025.  Google Scholar

[3]

A. Abdulle and G. A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys., 231 (2012), 2482-2497.  doi: 10.1016/j.jcp.2011.11.039.  Google Scholar

[4]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[5]

G. Allaire, Two-scale convergence: A new method in periodic homogenization. Nonlinear partial differential equations and their applicationsapplications, Collge de France Seminar, Vol. XII (Paris, 1991—1993), 1-14, Pitman Res. Notes Math. Ser., 302, Longman Sci. Tech., Harlow, 1994.  Google Scholar

[6]

N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials, Translated from the Russian by D. Lei(tes. Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2247-1.  Google Scholar

[7]

A. Bensoussan, Some existence results for stochastic partial differential equations., Stochastic Partial Differential Equations and Applications (Trento, 1990), 37—53, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992.  Google Scholar

[8]

A. Bensoussan, Homogenization of a class of stochastic partial differential equations. Composite Media and Homogenization Theory (Trieste, 1990), 47-65, Progr. Nonlinear Differential Equations Appl., 5, Birkhuser Boston, Boston, MA, 1991.  Google Scholar

[9]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Corrected reprint of the 1978 original., AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.  Google Scholar

[10]

H. BessaihY. Efendiev and F. Maris, Homogenization of Brinkman flows in heterogeneous dynamic media, Stoch. Partial Differ. Equ. Anal. Comput., 3 (2015), 479-505.  doi: 10.1007/s40072-015-0058-6.  Google Scholar

[11]

H. BessaihY. Efendiev and F. Maris, Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition, Netw. Heterog. Media, 10 (2015), 343-367.  doi: 10.3934/nhm.2015.10.343.  Google Scholar

[12]

P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[13]

A. BourgeatA. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51.   Google Scholar

[14]

A. Bourgeat and A. L. Piatnitski, Averaging of a singular random source term in a diffusion convection equation, SIAM J. Math. Anal., 42 (2010), 2626—2651. doi: 10.1137/080736077.  Google Scholar

[15]

S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM J. Math. Anal., 43 (2011), 2482-2518.  doi: 10.1137/100806710.  Google Scholar

[16]

S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.  Google Scholar

[17]

D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999.  Google Scholar

[18]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J.Rei. Ang. Math. B., 368 (1986), 28-42.   Google Scholar

[19]

M. A. Diop and E. Pardoux, Averaging of a parabolic partial differential equation with random evolution. Seminar on Stochastic Analysis, Random Fields and Applications IV, Progr. Probab., 58, Birkh user, Basel, (2004), 111-128.  Google Scholar

[20]

Y. GorbF. Maris and B. Vernescu, Homogenization for rigid suspensions with random velocity-dependent interfacial forces, J. Math. Anal. Appl., 420 (2014), 632-668.  doi: 10.1016/j.jmaa.2014.05.015.  Google Scholar

[21]

N. Ichihara, Homogenization problem for stochastic partial differential equations of Zakai type, Stoch. Stoch. Rep., 76 (2004), 243-266.  doi: 10.1080/10451120410001714107.  Google Scholar

[22]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Translated from the Russian by G. A. Yosifian. Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[23]

E. Y. Khruslov, Homogenized models of composite media, Composite Media and Homogenization Theory (Trieste, 1990), 159-182, Progr. Nonlinear Differential Equations Appl., 5, Birkhuser Boston, Boston, MA, 1991. doi: 10.1007/978-1-4684-6787-1_10.  Google Scholar

[24]

S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109(151) (1979), 188-202, 327.  Google Scholar

[25]

S. V. Lototsky, Small perturbation of stochastic parabolic equations: A power series analysis, J. Funct. Anal., 193 (2002), 94-115.  doi: 10.1006/jfan.2001.3923.  Google Scholar

[26]

D. LukkassenG. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86.   Google Scholar

[27]

M. Mohammed and M. Sango, Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method, Asymptotic Analysis, 91 (2015), 341-371.   Google Scholar

[28]

M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptotic Analysis, 97 (2016), 301-327.  doi: 10.3233/ASY-151355.  Google Scholar

[29]

M. Mohammed and M. Sango, A Tartar approach to periodic homogenization of linear hyperbolic stochastic partial differential equation, Int. J. Mod. Phys. B, 30 (2016), 1640020, 9 pp. doi: 10.1142/S0217979216400208.  Google Scholar

[30]

M. Mohammed, Homogenization of nonlinear hyperbolic stochastic equation via Tartar's method, J. Hyper. Differential Equations, 14 (2017), 323-340.  doi: 10.1142/S0219891617500096.  Google Scholar

[31]

F. Murat and L. Tartar, H-convergence in Topics in the mathematical Modelling of composite Materials. ed. A. Cherkaev and Kohn, Birkhauser. Boston, 31 (1997), 21-43.   Google Scholar

[32]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[33]

G. NguetsengM. Sango and J. L. Woukeng, Reiterated ergodic algebras and applications, Comm. Math. Phys., 300 (2010), 835-876.  doi: 10.1007/s00220-010-1127-3.  Google Scholar

[34]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in elasticity and Homogenization, Studies in Mathematics and its Applications, 26. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[35]

M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Mathematicae, 426 (2004), 1-63.  doi: 10.4064/dm426-0-1.  Google Scholar

[36]

A. Pankov, G-Convergence and Homogenization of Nonlinear Partial Differential Operators, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-94-015-8957-4.  Google Scholar

[37]

G. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. I, II (Esztergom, 1979), 835-873, Colloq. Math. Soc. Janos Bolyai, 27, North-Holland, Amsterdam-New York, 1981.  Google Scholar

[38]

E. Pardoux, Équations aux dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse, Université Paris XI, 1975. Google Scholar

[39]

B. L. Rozovskiĭ, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Translated from the Russian by A. Yarkho. Mathematics and its Applications (Soviet Series), 35. Kluwer Academic Publishers Group, Dordrecht, 1990. xviii+315 pp doi: 10.1007/978-94-011-3830-7.  Google Scholar

[40]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory (Lecture Notes in Physics), Springer, 1980.  Google Scholar

[41]

E. Sanchez-Palencia and A. Zaoui, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987. doi: 10.1007/3-540-17616-0.  Google Scholar

[42]

M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25 (2013), 931-965.  doi: 10.1515/form.2011.138.  Google Scholar

[43]

M. Sango, Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries, Commun. Math. Sci., 12 (2014), 345-382.  doi: 10.4310/CMS.2014.v12.n2.a7.  Google Scholar

[44]

M. Sango, Asymptotic behavior of a stochastic evolution problem in a varying domain, Stochastic Anal. Appl., 20 (2002), 1331-1358.  doi: 10.1081/SAP-120015835.  Google Scholar

[45]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[46]

I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Nauka, Moscow, 1990. English translation in: Translations of Mathematical Monographs, AMS, Providence, 1994.  Google Scholar

[47]

E. P. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Analysis. IOS Press, 20 (1999), 1-11.   Google Scholar

[48]

N. Svanstedt, Multiscale stochastic homogenization of monotone operators, Netw. Heterog. Media, 2 (2007), 181-192.  doi: 10.3934/nhm.2007.2.181.  Google Scholar

[49]

L. Tartar, Quelques remarques sur l'homogénésation, in Functional Analysis and Numerical Analysis, Proc. Japan-France Siminar 1976, ed. H. Fujitaa, Japanese Society for the Promotion of Science, (1977), 468-486. Google Scholar

[50]

L. Tartar, The General Theory of Homogenization, A personalized introduction. Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009. doi: 10.1007/978-3-642-05195-1.  Google Scholar

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