Homogenization of vector-valued partition problems and dislocation cell structures in the plane

We consider target-space homogenization for energies defined on partitions parametrized by a discrete lattice B ⊂ RN. For a smallσ > 0, the variable is a piecewise constant function taking values in σB, and the energy depends on the jumps and their orientation. In the limit as σ → 0 we obtain a homogenized functional defined on functions of bounded variation. This result is relevant in the study of dislocation structures in plastically deformed crystals. We review recent literature on the topic and propose our limiting effective energy as a continuum model for strain-gradient plasticity.