Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Homogenization
  • Gamma convergence
  • integral currents
  • dislocations
  • BV

Mathematics Subject Classification

28C05, 74Q99, 32U40

References

  1. L. Ambrosio, A. Braides: Functionals defined on partitions in sets of finite perimeter. I: Integral representation and Γ-convergence, J. Math. Pures Appl. 69/3 (1990) 285–305.Google Scholar
  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.Google Scholar
  3. A. Braides: Γ-convergence for Beginners, Oxford University Press, Oxford (2002).https://doi.org/10.1093/acprof:oso/9780198507840.001.0001Google Scholar
  4. A. Braides, S. Conti, A. Garroni: Density of polyhedral partitions, Calc. Var. Partial Diff. Equations 56/2 (2017), art. no. 28.https://doi.org/10.1007/s00526-017-1108-xGoogle Scholar
  5. A. Braides, M. Maslennikov, L. Sigalotti: Homogenization by blow up, Applicable Analysis 87 (2008) 1341–1356.https://doi.org/10.1080/00036810802555458Google Scholar
  6. S. Conti, A. Garroni, A. Massaccesi: Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity, Calc. Var. Partial Diff. Equations 54/2 (2015) 1847–1874.https://doi.org/10.1007/s00526-015-0846-xGoogle Scholar
  7. S. Conti, A. Garroni, M. Ortiz: The line-tension approximation as the dilute limit of linear-elastic dislocations, Arch. Ration. Mech. Analysis 218/2 (2015) 699–755.https://doi.org/10.1007/s00205-015-0869-7Google Scholar
  8. H. Federer: Geometric Measure Theory, Springer, Berlin (1969).Google Scholar
  9. H. Federer, W. Fleming: Normal and integral currents, Annals of Math. 72 (1969) 458–520.https://doi.org/10.2307/1970227Google Scholar
  10. I. Fonseca, S. Müller: Quasiconvex integrands and lower semicontinuity in L1, SIAM J. Math. Analysis 23 (1992) 1081–1098.https://doi.org/10.1137/0523060Google Scholar
  11. L. Simon: Lectures on Geometric Measure Theory, Proceedings of the Center for Mathematical Analysis Volume 3, Australian National University, Sydney (1983).Google Scholar

Suggested citation

A. Garroni, P. Vermicelli. “Homogenization of Energies Defined on 1-Rectifiable Currents.” Journal of Convex Analysis 28 (2021), No. 2, 581–598. https://doi.org/10.68381/jca28033

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.