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 Dtw − ∇ · ⃗z = g(x, t, x/ε) (0.1) w ∈ α(u, x/ε) (0.2) ⃗z ∈ ⃗γ (∇u, x/ε). (0.3) Here ε is a positive parameter; α and γ⃗ 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 → 0, a two-scale formulation is derived via Nguetseng’s notion of two-scale convergence, and a (single-scale) homogenized problem is then retrieved.