BRIEF COMMUNICATIONS On the Homogenization of the Periodic Maxwell System
Tatiana Aleksandrovna Suslina · 2004
We study the homogenization problem for the stationary periodic Maxwell system in R 3 in the small period limit. Each field is represented as a sum of two terms. For some terms, we obtain convenient approximations in the L2(R 3 )-norm. 1. We consider the homogenization problem for the stationary periodic Maxwell system in the small period limit. There is a vast literature on this problem. In particular, it was discussed in the books (1-3). However, known results provide only the weak convergence of solutions. In the present paper, the abstract approach developed in (4, 5) is applied. Note that the Maxwell operator can be included in the class of differential operators studied in (4, 5) only if one of the two periodic coefficients e and µ is constant. Here we study the general case, which requires a substantial modification of the technique. The comparison with the case µ = const is given in Sec. 5. We represent each of the fields as a sum of two terms. For some terms, we obtain uniform approximations in the L2(R 3 )-norm with an explicitly controlled remainder estimate. For other terms, we still obtain only the weak convergence to the corresponding fields in the homogenized medium. 2. Statement of the problem. Let Γ be a lattice in R 3 , and let Ω be the fundamental cell of Γ. Suppose that the permittivity e(x)and the permeability µ(x)are Γ-periodic measurable (3 × 3)-matrix-valued functions in R 3 with real entries such that c01 e(x) c11 ,c 01 µ(x) c11, x ∈ R 3 , 0 <c 0 c1 < ∞.