Submitted Paper

Inserted: 20 sep 2023
Last Updated: 20 sep 2023

Year: 2023

Abstract:

We consider suitably scaled $(s,2)$-Gagliardo seminorms on spaces of $S^1$-valued functions, and the equivalent $L^2$-norms of s-fractional gradients, and show that they Gamma-converge to vortex-type energies with respect to a suitable compact convergence. In order to obtain such a result, we first compare such functionals with Ginzburg-Landau energies depending on Riezs potentials, borrowing an argument from discrete-to-continuum techniques. This allows to interpret the parameter $s$ (or, better, $1-s$) as having the same role as the geometric parameter $\varepsilon$ in the Ginzburg-Landau theory. As a consequence the energies are coercive with respect to the flat-convergence of the Jacobians of the Riesz potentials, and we can give a lower bound by comparison. As for the upper bound, this is obtained in the spirit of a recent work by Solci, showing that indeed we have pointwise convergence for functions of the form $x/\vert x\vert$ close to singularities. Using the explicit form of the $(s,2)$-Gagliardo seminorms we show that we can neglect contributions close to singularities by the use of the Gagliardo-Nirenberg interpolation inequality, and use a direct computation far from the singularities. A variation of a classical density argument completes the proof.