Homogenization and Phase Separation with Fixed Wells -- The Supercritical Case

A variational model for the interaction between homogenization and phase separation is considered in the regime where the former happens at a finer scale than the latter. The first order $\mathrm{\Gamma }-$limit is proven to exhibit a separation of scales which has been previously conjectured in a paper by Bradies and Zeppieri and in the PhD thesis of Hagerty.