Effective conductivity of a periodic dilute composite with perfect contact and its series expansion

We study the asymptotic behavior of the effective thermal conductivity of a periodic two-phase dilute composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material, each of them of size proportional to a positive parameter ϵ . We assume a perfect thermal contact at constituent interfaces, i.e., a continuity of the normal component of the heat flux and of the temperature. For ϵ small, we prove that the effective conductivity can be represented as a convergent power series in ϵ and we determine the coefficients in terms of the solutions of explicit systems of integral equations.