On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres

TL;DR

This paper analyzes a fractional Ginzburg-Landau equation involving the square root Laplacian, showing solutions converge to 1/2-harmonic maps and establishing partial regularity results for these maps.

Contribution

It introduces a fractional Ginzburg-Landau model with the square root Laplacian and proves convergence to 1/2-harmonic maps, along with partial regularity results.

Findings

Solutions converge weakly to sphere-valued 1/2-harmonic maps

Convergence occurs away from a rectifiable singular set

Partial regularity results for stationary 1/2-harmonic maps

Abstract

This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. The proof relies on the representation of the square root Laplacian as a Dirichlet-to-Neumann operator in one more dimension, and on the analysis of a boundary version of the Ginzburg-Landau equation. Besides the analysis of the fractional Ginzburg-Landau equation, we also give a general partial regularity result for stationary 1/2-harmonic maps in arbitrary dimension.