Topological singularities arising from fractional-gradient energies

Abstract

We prove that, on a planar regular domain, suitably scaled functionals of Ginzburg-Landau type, given by the sum of quadratic fractional Sobolev seminorms and a penalization term vanishing on the unitary sphere, $\Gamma$-converge to vortex-type energies with respect to the flat convergence of Jacobians. The compactness and the $\Gamma$-$\liminf$ follow by comparison with standard Ginzburg-Landau functionals depending on Riesz potentials. The $\Gamma$-$\limsup$, instead, is achieved via a direct argument by joining a finite number of vortex-like functions suitably truncated around the singularity.