Low energy configurations of topological singularities in two dimensions: A convergence analysis of dipoles

This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg-Landau energy. Denoting by the length scale parameter in such models, we focus on the |log | energy regime. It is well known that, for configurations whose energy is bounded by c|log |, the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying I|log | energy, plus a measure supported on small zero-average sets. Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses. Here, we perform a compactness and -convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale s, for 0 < s < 1), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical -convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order c|log | with c < I.