Homogenization of composite media with non-standard transmission conditions

In this paper we study the homogenization limits for the steady state of a diffusion problem in a composite medium made up by two different materials: a host material and the inclusion material which is disposed in a periodic array and has a typical length scale $\eps$. On the interface separating the two phases two different sets of non-standard transmission conditions are assigned (thus originating two different systems of partial differential equations). As $\eps$ tends to zero a hierarchy of homogenization problem related to such interface conditions is studied and the physical properties of the various limits are discussed.