Topological singularities arising from fractional-gradient energies

TL;DR

This paper establishes the convergence of fractional-gradient Ginzburg-Landau functionals to vortex energies on planar domains, revealing topological singularities through a detailed Gamma-convergence analysis.

Contribution

It introduces a novel fractional-gradient energy framework and proves their Gamma-convergence to vortex-type energies, extending classical results to fractional Sobolev settings.

Findings

Gamma-convergence of fractional Ginzburg-Landau functionals to vortex energies

Compactness and lower bound established via comparison with Riesz potential functionals

Construction of recovery sequences using vortex-like functions around singularities

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.