Variational approach to homogenization of doubly-nonlinear flow in a periodic structure

This work deals with the homogenization of an initial- and boundary-value problem for the doubly-nonlinear system equation(0.1)Dtw−∇⋅z→=g(x,t,x/ε)equation(0.2)w∈α(u,x/ε)w∈α(u,x/ε)equation(0.3)z→∈γ→(∇u,x/ε). Here εε is a positive parameter; αα and γ→ are maximal monotone with respect to the first variable and periodic with respect to the second one. The inclusions (0.2) and (0.3) are here formulated as null-minimization principles,   via the theory of Fitzpatrick [MR 1009594]. As ε→0ε→0, a two-scale formulation is derived via Nguetseng’s notion of two-scale convergence, and a (single-scale) homogenized problem is then retrieved.