Homogenization of two heat conductors with an interfacial contact resistance

In this paper we describe the asymptotic behavior of a problem depending on a small parameter ε > 0 and modelling the stationary heat diffusion in a two-component conductor. The flow of heat is proportional to the jump of the temperature field, due to a contact resistance on the interface. More precisely, we give an homogenization result for the stationary heat equation with oscillating coefficients in a domain Ω = Ωε1 ∪ Ωε2 of Rn, where Ωε1 is connected and Ωε2 is union of ε-periodic disconnected inclusions of size ε. These two sub-domains of Ω are separated by a contact surface Γε, on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a function of order ε^γ. We describe the limit problem for γ > −1. The two cases −1 < γ ≤ 1 (Theorem 2.1) and γ > 1 (Theorem 2.2) need to be treated separately, because of different a priori estimates.