
Journal of Convex Analysis 28 (2021), No. 2, 581598 Copyright Heldermann Verlag 2021 Homogenization of Energies Defined on 1Rectifiable Currents Adriana Garroni Dip. di Matematica "G. Castelnuovo", Sapienza Università, Roma, Italy garroni@mat.uniroma1.it Pietro Vermicelli Dip. di Matematica "G. Castelnuovo", Sapienza Università, Roma, Italy pietro@pietrovermicelli.com 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 L. Ambrosio and A. Braides [Functionals defined on partitions in sets of finite perimeter. II: Semicontinuity, relaxation and homogenization, J. Math. Pures Appl. 69 (1990) 307333.] 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 divergencefree matrixvalued measures supported on curves or with 1currents 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. Keywords: Homogenization, Gamma convergence, integral currents, dislocations, BV. MSC: 28C05, 74Q99, 32U40. [ Fulltextpdf (186 KB)] for subscribers only. 