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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. BourgeatA. 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. BourgeatA. 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

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