Cell Averaging Two-Scale Convergence: Applications to Periodic Homogenization

Abstract

The aim of the paper is to introduce an alternative notion of two-scale convergence which gives a more natural modeling approach to the homogenization of partial differential equations with periodically oscillating coefficients: while removing the bother of the admissibility of test functions, it nevertheless simplifies the proof of all the standard compactness results which made classical two-scale convergence very worthy of interest. Bounded sequences in $L^2_{\sharp} [ {Y}, L^2 (\Omega) ]$ and $L^2_{\sharp} [ {Y}, H^1 (\Omega) ]$ are proven to be relatively compact with respect to this new type of convergence. The strengths of the notion are highlighted on the classical homogenization problem of linear second-order elliptic equations for which first-order boundary corrector-type results are also established. Eventually, possible weaknesses of the method are pointed out on a nonlinear problem: the weak two-scale compactness result for ${\bf S}^2$-valued stationary harmonic maps.