Geodesic
Homogenization of Energies Defined on 1-Rectifiable Currents
Journal of convex analysis, Tome 28 (2021) no. 2, pp. 581-598
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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.
Classification : 28C05, 74Q99, 32U40
Mots-clés : Homogenization, Gamma convergence, integral currents, dislocations, BV 
@article{JCA_2021_28_2_JCA_2021_28_2_a15,
     author = {A. Garroni and P. Vermicelli},
     title = {Homogenization of {Energies} {Defined} on {1-Rectifiable} {Currents}},
     journal = {Journal of convex analysis},
     pages = {581--598},
     year = {2021},
     volume = {28},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2021_28_2_JCA_2021_28_2_a15/}
}
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A. Garroni; P. Vermicelli. Homogenization of Energies Defined on 1-Rectifiable Currents. Journal of convex analysis, Tome 28 (2021) no. 2, pp. 581-598. http://geodesic.mathdoc.fr/item/JCA_2021_28_2_JCA_2021_28_2_a15/