A singularly perturbed non-ideal transmission problem and application to the effective conductivity of a periodic composite

We investigate 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. The diameter of each inclusion is assumed to be proportional to a positive real parameter \epsilon. Under suitable assumptions, we show that the effective conductivity can be continued real analytically in the parameter \epsilon around the degenerate value \epsilon=0, in correspondence of which the inclusions collapse to points.