Asymptotic analysis for non-local problems in composites with different imperfect contact conditions

We consider a composite material made up of a hosting medium containing an $\eps$-periodic array of perfect thermal conductors. Comparing with the previous contributions in the literature, in the present paper, the inclusions are completely disconnected and form two families with dissimilar physical behaviour. More specifically, the imperfect contact between the hosting medium and the inclusions obeys two different laws, according to the two different types of inclusions. The contact conditions involve the small parameter $\eps$ and two positive constants $\contuno,\contdue$. We investigate the homogenization limit $\eps\to 0$ and the limits for $\contuno,\contdue$ going to $0$ or $+\infty$, taken in any order, with the aim to find out the cases in which the two limits commute.