Heat conduction in composite media involving imperfect contact and perfectly conductive inclusions

We study the thermal properties of a composite material made up of a medium hosting an $\eps$-periodic array of perfect thermal conductors. The thermal potentials of the two phases are coupled across the interface through a non-standard imperfect contact transmission condition, involving the external thermal flux and a proportionality coefficient $\cont\eps^\alpha$, where $\alpha\in\R$ is a scaling parameter and $\cont>0$ accounts for the imperfect contact. We perform the homogenization for all the scalings $\alpha\in\R$ and we compare the resulting models with the perfect contact transmission case addressed in \cite{Amar:Andreucci:Gianni:Timofte:2019B,Amar:Andreucci:Gianni:Timofte:2019A}, letting $\cont\to 0$, where this is meaningful.