Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces

We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic N-function M, which may also depend on the spatial variable, i.e. the homogenization process will change the underlying function spaces and the nonlinear elliptic operator at each step. The problem of homogenization of nonlinear elliptic systems has been solved for the L-P-setting with restrictions either on constant exponent or variable exponent that is assumed to be additionally log-Holder continuous. These results correspond to a very particular case of N-functions satisfying both Delta(2) and del(2)-conditions. We show that for general M satisfying a condition of log-Holder type continuity, one can provide a rather general theory without any assumption on the validity of neither Delta(2) nor del(2)-conditions.