Stochastic homogenization of maximal monotone relations and applications

Luca Lussardi; Stefano Marini; Marco Veneroni; Luca Lussardi; Stefano Marini; Marco Veneroni

doi:10.3934/nhm.2018002

Networks and Heterogeneous Media

2018, Volume 13, Issue 1: 27-45. doi: 10.3934/nhm.2018002

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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: 01 March 2017 Revised: 01 September 2017
Primary: 35B27, 47H05, 49J40; Secondary: 74B20, 74Q15, 78M40

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.
- Stochastic homogenization,
- random media,
- Fitzpatrick representation,
- maximal monotonicity,
- Ohm-Hall conduction,
- nonlinear elasticity
Citation: Luca Lussardi, Stefano Marini, Marco Veneroni. 2018: Stochastic homogenization of maximal monotone relations and applications, Networks and Heterogeneous Media, 13(1): 27-45. doi: 10.3934/nhm.2018002

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Abstract

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.

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