Scale-transformations and homogenization of maximal monotone relations with applications

In homogenization, two-scale models arise, e.g., by applying Nguetseng’s notion of two-scale convergence to nonlinear PDEs. A homogenized single-scale problem may then be derived via scale-transformations. A variational formulation due to Fitzpatrick is here used for the scale-integration of two-scale maximal monotone relations, and for the converse operation of scale-disintegration. These results are applied to the periodic homogenization of a quasilinear model of Ohmic electric conduction with Hall effect: E ∈ α( J ,x/ε) + h(x/ε) J × B(x/ε) + Ea(x/ε), ∇× E = g(x/ε), ∇ · J = 0 in Ω, with α(·,x/ε) maximal monotone, B, Ea,h, g prescribed fields. (This corresponds to a quasilinear second-order elliptic equation in curl form: ∇× β(∇× u,x/ε) = g(x/ε).) This result is also retrieved via De Giorgi’s Γ -convergence.