On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres

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 limit ε → 0, 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 countably H 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.

Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

“…After our work was completed, we learned that the same result has also been obtained independently by V. Millot and Y. Sire [35].…”

Dynamics of the 3-D fractional complex Ginzburg–Landau equation

“…One encounters them in the theory of systems with chaotic dynamics [35,42]; pseudochaotic dynamics [44]; dynamics in a complex or porous medium [10,30,37]; random walks with a memory and flights [28,36,43]; obstacle problems [6,34] and many other situations. Recently, fractional partial differential equation versions of some of the classical equations of mathematical physics have been studied, including the fractional Schrödinger equation [9,12,13,16,[18][19][20]29], the fractional Landau-Lifshitz equation [17], the fractional Landau-Lifshitz-Maxwell equation [31] and the fractional GinzburgLandau equation [25,27,39]. Furthermore, many recent studies of fractional derivative problems arise from probabilistic or purely mathematical considerations (see [1][2][3]5,11], for instance).…”

Minimizing 1/2-harmonic maps into spheres

“…It has been achieved in [13,14], thus extending the famous regularity result of F. Hélein for harmonic maps from surfaces [26]. The notion of 1/2-harmonic maps has been extended in [30,33] to higher dimensions, and partial regularity for minimizing or stationary 1/2-harmonic maps established (again in analogy with minimizing/stationary harmonic maps [4,19,38]). Before going further, let us now describe in detail the mathematical framework.…”

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In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of [30,33] in the case where the target manifold is the (m − 1)-dimensional sphere. For m 3, we show that minimizing 1/2-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions.

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