Scale-transformations and homogenization of maximal monotone relations with applications

Abstract

In homogenization, two-scale models arise, e.g., by applying Nguetseng's notion of two-scale convergence to non- linear 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 opera- tion 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/e) + h(x/e) � J × � B(x/e) + � Ea(x/e), ∇× � E = � g (x/e), ∇· � J = 0i n Ω, with � α (·, x/e) maximal monotone, � B, � Ea, h, � g prescribed fields. (This corresponds to a quasilinear second-order elliptic equa- tion in curl form: ∇× �( ∇× u, x/e) = � g (x/e).) This result is also retrieved via De Giorgi's Γ -convergence.