Multiscale Young measures in homogenization of continuous stationary processes in compact spaces and applications

Abstract

We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f ○ T : R n × Q → Q with f ○ T ( x , ω ) = f ( T ( x ) ω ) where Q is a compact (Hausdorff topological) space, f ∈ C ( Q ) and T ( x ) : Q → Q , x ∈ R n , is an n-dimensional continuous dynamical system endowed with an invariant Radon probability measure μ. It can be easily shown that for almost all ω ∈ Q the realization f ( T ( x ) ω ) belongs to an algebra with mean value, that is, an algebra of functions in BUC ( R n ) containing all translates of its elements and such that each of its elements possesses a mean value. This notion was introduced by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators, Mat. Zametki 33 (1983) 571–582, English transl. in Math. Notes 33 (1983) 294–300]. We then establish the existence of multiscale Young measures in the setting of algebras with mean value, where the compactifications of R n provided by such algebras plays an important role. These parametrized measures are useful in connection with the existence of correctors in homogenization problems. We apply this framework to the homogenization of a porous medium type equation in R n with a stationary continuous process as a stiff oscillatory external source. This application seems to be new even in the classical context of periodic homogenization.