NONLOCAL-INTERACTION VORTICES

We consider sequences of quadratic nonlocal functionals, depending on a small pa- rameter \varepsilon , that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis, and Mironescu. Similarly to what is done for core-radius approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by | log\varepsilon | - 1 and restrict them to S1-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equicoercive, and show the convergence to a vortex energy, similarly to the limit behavior of Ginzburg-Landau energies at the vortex scaling.