Universita’ di Roma

In the theory of dislocations one is naturally led to consider energies of “line tension” type concentrated on lines. These lines may have a local vector-valued multiplicity, and the energy may depend on this multiplicity and on the orientation of the line. In the two-dimensional case this problem reduces to the classical problem of energies defined on partitions which arises in the sharp-interface models for phase transitions. 

I will introduce the main results concerning functionals in the calculus of variations that are defined on partitions. Such partitions are nicely characterized as level sets of function with bounded variations with a discrete set of values.  In this setting I will recall the characterization of the lower semicontinuity and the relaxation formula, which gives rise to the notion of BV-ellipticity. The case of dislocations in a three-dimensional crystal requires a formulation in the setting of 1-rectifiable currents with multiplicity in a lattice. In this context I will describe the main results and some examples of interest, in which relaxation is necessary and can be characterized.