We present two continuous models for the study of topological singularities in 2D: the core-radius approach and the Ginzburg-Landau theory.
It is well known that - at zero temperature and under suitable regimes - the energies associated to these models tend to concentrate, as the length scale parameter epsilon goes to zero, around a finite number of points, the so-called vortices.
We focus on low energy regimes that prevent the formation of vortices in the limit as epsilon tends to zero, but that are compatible (for positive epsilon) with configurations of short (in terms of epsilon) dipoles, and more in general with short clusters of vortices having zero average.
By using a Gamma-convergence approach, we provide a quantitative analysis of the energy induced by such configurations on a continuous range of length scales.
Joint work with M. Ponsiglione (Rome).