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Journal of Convex Analysis 28 (2021), No. 2, 581--598
Copyright Heldermann Verlag 2021

Homogenization of Energies Defined on 1-Rectifiable Currents

Adriana Garroni
Dip. di Matematica "G. Castelnuovo", Sapienza UniversitÓ, Roma, Italy

Pietro Vermicelli
Dip. di Matematica "G. Castelnuovo", Sapienza UniversitÓ, Roma, Italy

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) 307--333.] 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.

Keywords: Homogenization, Gamma convergence, integral currents, dislocations, BV.

MSC: 28C05, 74Q99, 32U40.

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