Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem

We consider the homogenization of a periodic interfacial energy, such as considered in recents papers by Caffarelli and De La Llave [14], or Dirr, Lucia and Novaga [16]. In particular, we include the case where an external forcing field (which is unbounded in the limit) is present, and suggest two different ways to take care of this additional perturbation. We provide a proof of a $\Gamma$-limit, however, we also observe that thanks to the coarea formula, in many cases such a result is already known in the framework of $BV$ homogenization. This leads to an interesting new construction for the plane-like minimizers in periodic media of Caffarelli and De La Llave, through a cell problem.