Stochastic two-scale convergence of an integral functional

Abstract

In this paper we discuss the concept of stochastic two-scale convergence, which is appropriate to solve coupledperiodic and stochastic homogenization problems. This concept is a combination of both well-known two-scale convergence and stochastic two-scale convergence in the mean schemes, and is a generalization of the said previous methods. By way of illustration we apply it to solve a homogenization problem related to an integral functional with convex integrand. This problematic relies on the notion of dynamical system which is our basic tool.

Stochastic homogenization of the bending plate model

“…By using this mollifier one can show that there exists a countable dense subset of L 2 (Ω) and W 1,2 (Ω) (see [BMW94]). Following [SW11] we denote by • #,k,2 the seminorm on C ∞ (Ω) given by…”

Stochastic-periodic homogenization of Maxwell’s equations with linear and periodic conductivity

“…In [31,30] is used Tartar method ( [34]). In [37,38] Nguetseng two scale convergence is used ( [23]), and recently in [17], (Woukeng et al [32]) stochastic two sale convergence method is used for linear system. To the best of our knowledge, homogenization of the Maxwell's equations with non linear conductivity was first studied in [38].…”

Homogenization in algebras with mean value

“…One then uses its infinitesimal generators to build up a framework for solving random homogenization problems. We refer, e.g., to the works [14,26,60] (see also [43]) for an exposition of this idea.…”