Homogenization of a parabolic model of ferromagnetism

This work deals with the homogenization of hysteresis-free processes in ferromagnetic composites. A degenerate, quasilinear, parabolic equation is derived by coupling the Maxwell–Ohm system without displacement current with a nonlinear constitutive law:∂B→∂t+curl{A(xε)⋅curlH→}=curlE→a,B→∈α→(H→,xε). Here A   is a periodic positive-definite matrix, α→(⋅,y) is maximal monotone and periodic in y  , E→a is an applied field, and ε>0ε>0. An associated initial- and boundary-value problem is represented by a minimization principle via an idea of Fitzpatrick. As ε→0ε→0 a two-scale problem is obtained via two-scale convergence, and an equivalent coarse-scale formulation is derived. This homogenization result is then retrieved via Γ  -convergence, and the continuity of the solution with respect to the operator α→ and the matrix A is also proved. This is then extended to some relaxation dynamics.