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Homogenization of energies defined on 1-rectifiable currents. (English) Zbl 1467.28003

Summary: 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, J. Math. Pures Appl. (9) 69, No. 3, 307–333 (1990; Zbl 0676.49029). 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.

MSC:

28A75 Length, area, volume, other geometric measure theory
49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
74Q99 Homogenization, determination of effective properties in solid mechanics

Citations:

Zbl 0676.49029
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References:

[1] L. Ambrosio, A. Braides:Functionals defined on partitions in sets of finite perimeter. I: Integral representation andΓ-convergence, J. Math. Pures Appl. 69/3 (1990) 285- 305. · Zbl 0676.49028
[2] L. Ambrosio, A. Braides:Functionals defined on partitions in sets of finite perimeter. II: Semicontinuity, relaxation and homogenization, J. Math. Pures Appl. 69/3 (1990) 307-333. · Zbl 0676.49029
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