On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres
The authors of this work carried out 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. The convergence 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 the fractional Ginzburg-Landau equation. The authors introduced a square-root Laplacian/Dirichlet-to-Neumann operator and presented some key estimates related to harmonic extensions in the first section. The next section is dedicated to the qualitative analysis of the fractional Ginzburg-Landau equation for \(\varepsilon\) fixed. The authors presented interior regularity results for the equation in this section. In the section after that, they introduced the concept of 1/2-harmonic maps as well as boundary harmonic maps. The authors dedicate this part of the paper to the analysis of 1/2-harmonic maps. They gave the regularity theory and proved the interior regularity theorem presented earlier. A part of this section also contains explicit examples of 1/2-harmonic maps and remarks on their analogies with standard harmonic maps. The proof of the epsilon-regularity estimate for the system of Ginzburg-Landau boundary reactions and its asymptotic analysis as \(\varepsilon \to 0\) are given in the next two sections. The authors presented the asymptotic analysis of the fractional Ginzburg-Landau equation with and without exterior Dirichlet condition in the last section. The special case of energy minimizers under the exterior Dirichlet condition is treated in this part of the paper. Appendices provide the proofs of several statements from earlier sections as well as some elliptic estimates used in the proof of the epsilon-regularity result.
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