Nonlocal-interaction vortices

TL;DR

This paper studies the asymptotic behavior of nonlocal quadratic functionals approximating the Dirichlet integral, focusing on vortex energies and their convergence to integral currents, inspired by Ginzburg-Landau vortex analysis.

Contribution

It introduces a new convergence notion for nonlocal functionals on $S^1$-valued functions and demonstrates their convergence to vortex energies akin to Ginzburg-Landau models.

Findings

Establishes a notion of convergence to integral currents for nonlocal functionals.

Shows the scaled energies are equi-coercive.

Proves convergence to vortex energies similar to classical models.

Abstract

We consider sequences of quadratic non-local functionals, depending on a small parameter $\e$, that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis and Mironescu. Similarly to what is done for hard-core approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by $|\log\e|^{-1}$ and restrict them to $S^1$-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equi-coercive, and show the converge to a vortex energy, similarly to the limit behaviour of Ginzburg-Landau energies at the vortex scaling.