Homogenization of energies defined on $1$-rectifiable currents

TL;DR

This paper investigates the homogenization of energies concentrated on lines, extending previous work on partition energies to higher dimensions using currents and measures, with applications to crystal dislocations.

Contribution

It introduces a novel approach to homogenize line-based energies in higher dimensions using currents and divergence-free measures, generalizing prior partition-based methods.

Findings

Established a representation of closed loops as divergence-free measures or currents.

Extended homogenization techniques to 3D line defect energies in crystals.

Connected geometric measure theory with materials science applications.

Abstract

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.