Homogenization of energies defined on pairs set-function

In the present work, we deal with the problem of the asymptotic behavior of a sequence of non-homogeneous energies depending on a pair set-function, with E regular open set and the bulk and surface energy densities and both 1-periodic in the first variable; this leads, in the Gamma-limit, to a problem of homogenization. We prove a Gamma-convergence result for the sequence of functionals, showing that there is no interaction between the homogenized bulk and surface energy density; that is, even though the effect of the bulk and surface energies are at the same energy scale, oscillations in the bulk term can be neglected close to the boundary of E and to the jump set S(u), where surface oscillations are dominant.