HOMOGENIZATION OF A STATIONARY PERIODIC MAXWELL SYSTEM IN A BOUNDED DOMAIN IN THE CASE OF CONSTANT MAGNETIC PERMEABILITY

In a bounded domain O subset of R-3 of class C-1,C-1, consider a stationary Maxwell system with the boundary conditions of perfect conductivity. It is assumed that the magnetic permeability is given by a constant positive (3 x 3)-matrix mu(0) and the dielectric permittivity is of the form eta(x/epsilon), where eta(x) is a (3 x 3)-matrix-valued function with real entries, periodic with respect to some lattice, bounded, and positive definite. Here epsilon > 0 is the small parameter. Suppose that the equation involving the curl of the magnetic field intensity is homogeneous, and the right-hand side r of the second equation is a divergence-free vector-valued function of class L-2. It is known that, as epsilon -> 0, the solutions of the Maxwell system, namely, the electric field intensity u(epsilon) the electric displacement vector w(epsilon), the magnetic field intensity v(epsilon) and the magnetic displacement vector z(epsilon) converge weakly in L-2 to the corresponding homogenized fields u(0), w(0), v(0), z(0) (the solutions of the homogenized Maxwell system with effective coefficients). Classical results are improved. It is shown that v(epsilon) and z(epsilon) converge to v(0) and z(0) respectively, in the L-2-norm, and the error terms do not exceed C epsilon parallel to r parallel to L-2. Approximations for v(epsilon) and z(epsilon) in the energy norm are also found with error C root epsilon parallel to r parallel to L-2. For u(epsilon) and w(epsilon) approximations in the L-2-norm with error C root epsilon parallel to r parallel to L-2 are also found.