Discrete-to-continuum limits of multibody systems with bulk and surface long-range interactions

We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via Gamma-convergence. We consider energy densities that may depend on interactions between all points of the lattice, and we give conditions that ensure compactness and integral representation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk and surface contributions decouple in the limit. The results are applied to long-range and multibody interactions in the setting of weak-membrane energies.