Calculus of Variations and Geometric Measure Theory

R. Alicandro - A. Braides - M. Solci - G. Stefani

Topological singularities arising from fractional-gradient energies

created by stefani on 20 Sep 2023
modified on 15 May 2024


Submitted Paper

Inserted: 20 sep 2023
Last Updated: 15 may 2024

Year: 2023

ArXiv: 2309.10112 PDF


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.

Keywords: Gamma-convergence, Fractional Gradient, Ginzburg-Landau Model, Riesz potential, topological singularity, vortex, flat convergence