Homogenization of energies defined on \(1\)-rectifiable currents

In this paper we study the homogenization of a class of energies concentrated on lines. In dimension \(2\) (i.e., in codimension \(1\)) the problem reduces to the homogenization of partition energies studied by \cite{AB}. There, the key tool is the representation of partitions in terms of \(BV\) functions with values in a discrete set. In our general case the key ingredient is the representation of closed loops with discrete multiplicity either as divergence-free matrix-valued measures supported on curves or with \(1\)-currents with multiplicity in a lattice. In the \(3\) dimensional case the main motivation for the analysis of this class of energies is the study of line defects in crystals, the so called dislocations.