A variational model of interaction between continuum and discrete systems

We consider a variational model which describes a complex system composed, in its reference configuration, of a periodic distribution of ‘small’ interacting particles immersed in a continuous medium. We describe its macroscopic limit via Gamma-convergence, highlighting different regimes. In particular, we show how the interplay between the particles and the continuum leads, for a critical size of the particles, to a capacitary term. Eventually, we discuss how the presence of a continuum affects the properties of the ground states of the system of particles in terms of the validity or not of the so called ‘Cauchy-Born’ rule.