Exact Bounds on the Effective Behavior of a Conducting Discrete Polycrystal

In a recent paper by Braides and Francfort, the problem of the characterization of the overall properties of lattice energies describing networks with arbitrary mixtures of two types of linear conductors has been addressed in a two-dimensional setting. In this paper we investigate the connection between that discrete optimization process and the theory of bounds for mixtures of continuum energies, for which the choice of the relationships between the different phases of the mixture is unusual and leads to remarkably simple results in terms of G-closure.