Homogenization of the Maxwell equations. Case I: Linear theory. (Q1771816)

The paper deals with the Maxwell system of equations with spatially periodic coefficients in the linear constitutive relations. The homogenization problem is studied, i.e. a sequence of problems with coefficients with diminishing period is studied. The corresponding solutions converge to solution of the so-called homogenized problem with constant coefficient. The problem was studied in 1978 by the method of asymptotic analysis in [\textit{A. Bensoussan, J. L. Lions} and \textit{G. Papanicolaou}, Asymptotic analysis for periodic structure, North Holland, Amsterdam (1978; Zbl 0404.35001)]. In the paper the homogenization result is proved by a new two-scale convergence method, see [\textit{G. Allaire}, SIAM J. Math. Anal. 23, 1482--1518 (1992; Zbl 0770.35005)] which simplifies the proofs and yields simultaneously both formulae for the homogenized coefficients and proof of the convergence. Further corrector results are proved. Part II, cf. ibid. 47, No. 3, 255--283 (2002).