We study the behavior of surface energies defined over couples (E,u) where E is a set and u is a density on the boundary of E. Such energies have been considered in the context of materials science for modelling surface diffusion in a way that takes into consideration explicitly the effect of the free atoms moving on the surface (adatoms) in a regime where the elastic energy is negligible. We discuss regular critical points, existence and uniqueness of minimizers and we characterize the relaxation of the energy functional in a suitable topology. Finally, we will present approximations with phase field and discrete energy.