Homogenization of nonlinear elliptic systems in nonreflexive Musielak–Orlicz spaces
Miroslav Bulíček, Piotr Gwiazda, Martin Kalousek, Agnieszka Świerczewska-Gwiazda · Nonlinearity · 2019
Abstract 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 -function , 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 -setting with restrictions either on constant exponent or variable exponent that is assumed to be additionally log-Hölder continuous. These results correspond to a very particular case of -functions satisfying both and -conditions. We show that for general satisfying a condition of log-Hölder type continuity, one can provide a rather general theory without any assumption on the validity of neither nor -conditions.