STOCHASTIC HOMOGENIZATION: CONVEXITY AND NONCONVEXITY

摘要

In the determination of effective properties of heterogeneous media with random distribution of microinhomogeneities we distinguish, grosso modo, two approaches (similarly to the deterministic case). The first approach is typical for physical and engineering papers where one usually does not introduce a small parameter ε > 0 characterizing mircoinhomogeneities and the notion of effective properties is rather intuitive. The second approach is mathematically rigorous and effective properties, for instance elastic moduli, are derived by performing the limit passage ε → 0 (in an appropriate sense). Just this case will be considered in the present paper. More precisely, we will consider application of G, H - and Γ - convergence to stochastic homogenization problems. We will also present the stochastic two - and multi-scale convergence in the mean as well as comments on application of stochastic partial differential equations.