Energy integral of the solution of a non-ideal transmission problem in a singularly perturbed periodic domain

In this paper, the behavior of the energy integral of the solution of a non-ideal transmission problem is investigated. Such problem appears in the study of the effective thermal conductivity of a two-phase composite with thermal resistance at the interface. The composite is 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 $epsilon$. Under suitable assumptions, we show that the energy integral of the solution can be continued real analytically in the parameter $epsilon$ around the degenerate value $epsilon=0$, in correspondence of which the inclusions collapse to points.