# American Institute of Mathematical Sciences

• Previous Article
Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network
• NHM Home
• This Issue
• Next Article
Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours
March  2018, 13(1): 27-45. doi: 10.3934/nhm.2018002

## Stochastic homogenization of maximal monotone relations and applications

 1 Dipartimento di Scienze Matematiche "G.L. Lagrange", Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129 Torino, Italy 2 Dipartimento di Matematica e Fisica "N. Tartaglia", Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy 3 Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Via Ferrata 5, I-27100 Pavia, Italy

Received  March 2017 Revised  September 2017 Published  March 2018

We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity.

Citation: Luca Lussardi, Stefano Marini, Marco Veneroni. Stochastic homogenization of maximal monotone relations and applications. Networks & Heterogeneous Media, 2018, 13 (1) : 27-45. doi: 10.3934/nhm.2018002
##### References:
 [1] N. W. Ashcroft and N. D. Mermin, Solide State Physics, Holt, Rinehart and Winston, Philadelphia, PA, 1976. Google Scholar [2] A. Bourgeat, A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51.   Google Scholar [3] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North Holland, 1973.  Google Scholar [4] P. G. Ciarlet, Mathematical Elasticity. Vol. Ⅰ, In Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar [5] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.  Google Scholar [6] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 386 (1986), 28-42.   Google Scholar [7] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.  Google Scholar [8] S. Fitzpatrick, Representing monotone operators by convex functions, in Workshop/Miniconference on Functional Analysis and Optimization, vol. 20 (eds. Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University), Canberra, (1988), 59–65.  Google Scholar [9] M. Heida and S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, preprint, arXiv: 1701.03505. Google Scholar [10] V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, 1994. Google Scholar [11] S. M. Kozlov, The averaging of random operators, Math. Sb., 109 (1979), 188-202.   Google Scholar [12] L. Landau and E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960.  Google Scholar [13] K. Messaoudi and G. Michaille, Stochastic homogenization of nonconvex integral functionals. Duality in the convex case, Sém. Anal. Convexe, 21 (1991), Exp. No. 14, 32 pp.  Google Scholar [14] K. Messaoudi and G. Michaille, Stochastic homogenization of nonconvex integral functionals, RAIRO Modél. Math. Anal. Numér., 28 (1994), 329-356.  doi: 10.1051/m2an/1994280303291.  Google Scholar [15] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 489-507.   Google Scholar [16] A. Pankov, Strong $G$ -convergence of nonlinear elliptic operators and homogenization, Constantin Carathéodory: An International Tribute: (In 2 Volumes) (eds. World Scientific), Ⅰ/Ⅱ (1991), 1075-1099.   Google Scholar [17] A. Pankov, G-convergence and Homogenization of Nonlinear Partial Differential Operators, Kluwer Academic Publisher, Dordrecht, 1997.  Google Scholar [18] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Random fields, vol. Ⅰ and Ⅱ, Colloq. Math. Soc. János Bolyai, North Holland, Amsterdam., 27 (1981), 835-873.   Google Scholar [19] F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann., 97 (1927), 737-755.  doi: 10.1007/BF01447892.  Google Scholar [20] M. Sango and J. L. Woukeng, Stochastic two-scale convergence of an integral functional, Asymptotic Anal., 73 (2011), 97-123.   Google Scholar [21] B. Schweizer, Averaging of flows with capillary hysteresis in stochastic porous media, European J. Appl. Math., 18 (2007), 389-415.  doi: 10.1017/S0956792507007000.  Google Scholar [22] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, volume 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997.  Google Scholar [23] L. Tartar, Cours Peccot au College de France, Partially written by F. Murat in Séminaire d'Analyse Fonctionelle et Numérique de l'Université d'Alger, unpublished, 1977. Google Scholar [24] M. Veneroni, Stochastic homogenization of subdifferential inclusions via scale integration, Intl. J. of Struct. Changes in Solids, 3 (2011), 83-98.   Google Scholar [25] A. Visintin, Scale-integration and scale-disintegration in nonlinear homogenization, Calc. Var. Partial Differential Equations, 36 (2009), 565-590.  doi: 10.1007/s00526-009-0245-2.  Google Scholar [26] A. Visintin, Scale-transformations and homogenization of maximal monotone relations with applications, Asymptotic Anal., 82 (2013), 233-270.   Google Scholar [27] A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations., 47 (2013), 273-317.  doi: 10.1007/s00526-012-0519-y.  Google Scholar

show all references

##### References:
 [1] N. W. Ashcroft and N. D. Mermin, Solide State Physics, Holt, Rinehart and Winston, Philadelphia, PA, 1976. Google Scholar [2] A. Bourgeat, A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51.   Google Scholar [3] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North Holland, 1973.  Google Scholar [4] P. G. Ciarlet, Mathematical Elasticity. Vol. Ⅰ, In Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar [5] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.  Google Scholar [6] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 386 (1986), 28-42.   Google Scholar [7] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.  Google Scholar [8] S. Fitzpatrick, Representing monotone operators by convex functions, in Workshop/Miniconference on Functional Analysis and Optimization, vol. 20 (eds. Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University), Canberra, (1988), 59–65.  Google Scholar [9] M. Heida and S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, preprint, arXiv: 1701.03505. Google Scholar [10] V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, 1994. Google Scholar [11] S. M. Kozlov, The averaging of random operators, Math. Sb., 109 (1979), 188-202.   Google Scholar [12] L. Landau and E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960.  Google Scholar [13] K. Messaoudi and G. Michaille, Stochastic homogenization of nonconvex integral functionals. Duality in the convex case, Sém. Anal. Convexe, 21 (1991), Exp. No. 14, 32 pp.  Google Scholar [14] K. Messaoudi and G. Michaille, Stochastic homogenization of nonconvex integral functionals, RAIRO Modél. Math. Anal. Numér., 28 (1994), 329-356.  doi: 10.1051/m2an/1994280303291.  Google Scholar [15] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 489-507.   Google Scholar [16] A. Pankov, Strong $G$ -convergence of nonlinear elliptic operators and homogenization, Constantin Carathéodory: An International Tribute: (In 2 Volumes) (eds. World Scientific), Ⅰ/Ⅱ (1991), 1075-1099.   Google Scholar [17] A. Pankov, G-convergence and Homogenization of Nonlinear Partial Differential Operators, Kluwer Academic Publisher, Dordrecht, 1997.  Google Scholar [18] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Random fields, vol. Ⅰ and Ⅱ, Colloq. Math. Soc. János Bolyai, North Holland, Amsterdam., 27 (1981), 835-873.   Google Scholar [19] F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann., 97 (1927), 737-755.  doi: 10.1007/BF01447892.  Google Scholar [20] M. Sango and J. L. Woukeng, Stochastic two-scale convergence of an integral functional, Asymptotic Anal., 73 (2011), 97-123.   Google Scholar [21] B. Schweizer, Averaging of flows with capillary hysteresis in stochastic porous media, European J. Appl. Math., 18 (2007), 389-415.  doi: 10.1017/S0956792507007000.  Google Scholar [22] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, volume 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997.  Google Scholar [23] L. Tartar, Cours Peccot au College de France, Partially written by F. Murat in Séminaire d'Analyse Fonctionelle et Numérique de l'Université d'Alger, unpublished, 1977. Google Scholar [24] M. Veneroni, Stochastic homogenization of subdifferential inclusions via scale integration, Intl. J. of Struct. Changes in Solids, 3 (2011), 83-98.   Google Scholar [25] A. Visintin, Scale-integration and scale-disintegration in nonlinear homogenization, Calc. Var. Partial Differential Equations, 36 (2009), 565-590.  doi: 10.1007/s00526-009-0245-2.  Google Scholar [26] A. Visintin, Scale-transformations and homogenization of maximal monotone relations with applications, Asymptotic Anal., 82 (2013), 233-270.   Google Scholar [27] A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations., 47 (2013), 273-317.  doi: 10.1007/s00526-012-0519-y.  Google Scholar
 [1] Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 [2] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385 [3] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [4] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383 [5] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [6] Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075 [7] Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269 [8] Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 [9] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168 [10] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [11] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [12] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [13] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [14] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [15] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [16] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [17] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [18] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [19] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [20] Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

2019 Impact Factor: 1.053