On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres
type: Item
instance of: scholarly article
title: On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres
zbMATH Open document ID: 1372.35291
author: Yannick Sire
author: Vincent Millot
published in: Archive for Rational Mechanics and Analysis
publication date: 2015-03-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1307.7015
review text: 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.
reviewed by: Baasansuren Jadamba
Mathematics Subject Classification ID: 35Q56
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 6409890
zbMATH Keywords: fractional Ginzburg-Landau equation
zbMATH Keywords: 1/2-harmonic maps
zbMATH Keywords: asymptotic analysis
zbMATH Keywords: square-root Laplacian
MaRDI profile type: Publication
OpenAlex ID: W3106452197
arXiv ID: 1307.7015
cites work: Q4312199
cites work: Phase transition with the line-tension effect
cites work: Q4954179
cites work: Q4538000
cites work: Harmonic mappings with partially free boundary
cites work: Some constructions for the fractional Laplacian on noncompact manifolds
cites work: Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling
cites work: On the singular set of stationary harmonic maps
cites work: Ginzburg-Landau vortices
cites work: Harmonic maps with defects
cites work: Degree theory of BMO. I: Compact manifolds without boundaries
cites work: Degree theory and BMO. II: Compact manifolds with boundaries. (Appendix with Petru Mironescu)
cites work: Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian
cites work: Sharp energy estimates for nonlinear fractional diffusion equations
cites work: Nonlinear equations for fractional Laplacians. I: Regularity, maximum principles, and Hamiltonian estimates
cites work: Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions
cites work: Layer solutions in a half‐space for boundary reactions
cites work: Nonlocal minimal surfaces
cites work: An Extension Problem Related to the Fractional Laplacian
cites work: Existence and partial regularity results for the heat flow for harmonic maps
cites work: Q3418329
cites work: Three-term commutator estimates and the regularity of \(^{1}/_{2}\)-harmonic maps into spheres
cites work: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps
cites work: Strongly Nonlocal Dislocation Dynamics in Crystals
cites work: Hitchhiker's guide to the fractional Sobolev spaces
cites work: Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting
cites work: A mixed boundary value problem for energy minimizing harmonic maps
cites work: Q3477695
cites work: An optimal estimate for the singular set of a harmonic map in the free boundary.
cites work: Partial regularity for stationary harmonic maps into spheres
cites work: Q3134551
cites work: Minimal surfaces and eigenvalue problems
cites work: Potential Space Estimates for Green Potentials in Convex Domains
cites work: A variational model for dislocations in the line tension limit
cites work: Elliptic partial differential equations of second order
cites work: \(\Gamma\) convergence of an energy functional related to the fractional Laplacian
cites work: Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one
cites work: Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties
cites work: Q3470840
cites work: Q3755914
cites work: Harmonic maps of manifolds with boundary
cites work: Partially constrained boundary conditions with energy minimizing mappings
cites work: Q3351776
cites work: Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory.
cites work: A nonlocal singular perturbation problem with periodic well potential
cites work: Boundary vortices in thin magnetic films
cites work: Gradient estimates and blow-up analysis for stationary harmonic maps
cites work: Harmonic and quasi-harmonic spheres
cites work: Harmonic and quasi-harmonic spheres. II.
cites work: Harmonic and quasi-harmonic spheres. III: Rectifiability of the parabolic defect measure and generalized varifold flows
cites work: Fractional generalization of the Ginzburg-Landau equation: an unconventional approach to critical phenomena in complex media
cites work: A variational problem with lack of compactness for \(H^{1/2}(S^{1};S^{1})\) maps of prescribed degree
cites work: The gradient theory of phase transitions and the minimal interface criterion
cites work: Intrinsic semiharmonic maps
cites work: Local and global minimizers for a variational energy involving a fractional norm

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cites work: Everywhere discontinuous harmonic maps into spheres

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cites work: Lines vortices in the U(1) - Higgs model

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cites work: Regularity and perturbation results for mixed second order elliptic problems

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cites work: \(\Gamma \)-convergence for nonlocal phase transitions

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cites work: Density estimates for a variational model driven by the Gagliardo norm

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cites work: Partial regularity for stationary harmonic maps at a free boundary

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cites work: A regularity theory for harmonic maps

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cites work: Weak and viscosity solutions of the fractional Laplace equation

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cites work: Q3336268

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cites work: Q4896042

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cites work: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result

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cites work: Problemi al contorno ellittici, con dati discontinui, dotati di soluzioni hölderiane

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cites work: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus

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cites work: Extension Problem and Harnack's Inequality for Some Fractional Operators

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cites work: On a free boundary problem for minimal surfaces

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cites work: Some applications of fractional equations.

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cites work: Weakly Differentiable Functions

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