# Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Journal of the European Mathematical Society (1999)

- Volume: 001, Issue: 3, page 237-311
- ISSN: 1435-9855

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topLin, Fanghua, and Rivière, Tristan. "Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents." Journal of the European Mathematical Society 001.3 (1999): 237-311. <http://eudml.org/doc/277610>.

@article{Lin1999,

abstract = {There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^1$-valued function defined on the boundary of a bounded regular domain of $R^n$. When such extensions do not exist, we use the Ginzburg-Landau relaxation
procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology class induced
from the $S^1$-valued boundary data. The union of this harmonic map and the minimal
current is the natural generalization of the harmonic extension.},

author = {Lin, Fanghua, Rivière, Tristan},

journal = {Journal of the European Mathematical Society},

keywords = {Ginzburg-Landau relaxation; Ginzburg-Landau minimizers; unimodular harmonic map; Ginzburg-Landau minimizer; harmonic map; superconductivity},

language = {eng},

number = {3},

pages = {237-311},

publisher = {European Mathematical Society Publishing House},

title = {Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents},

url = {http://eudml.org/doc/277610},

volume = {001},

year = {1999},

}

TY - JOUR

AU - Lin, Fanghua

AU - Rivière, Tristan

TI - Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

JO - Journal of the European Mathematical Society

PY - 1999

PB - European Mathematical Society Publishing House

VL - 001

IS - 3

SP - 237

EP - 311

AB - There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^1$-valued function defined on the boundary of a bounded regular domain of $R^n$. When such extensions do not exist, we use the Ginzburg-Landau relaxation
procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology class induced
from the $S^1$-valued boundary data. The union of this harmonic map and the minimal
current is the natural generalization of the harmonic extension.

LA - eng

KW - Ginzburg-Landau relaxation; Ginzburg-Landau minimizers; unimodular harmonic map; Ginzburg-Landau minimizer; harmonic map; superconductivity

UR - http://eudml.org/doc/277610

ER -

## Citations in EuDML Documents

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- Didier Smets, Problèmes d’évolution liés à l’énergie de Ginzburg-Landau
- Jean Bourgain, Haim Brezis, Petru Mironescu, H1/2 maps with values into the circle : minimal connections, lifting, and the Ginzburg–Landau equation
- F. Bethuel, G. Orlandi, Uniform estimates for the parabolic Ginzburg–Landau equation
- Fabrice Bethuel, Giandomenico Orlandi, Didier Smets, Motion of concentration sets in Ginzburg-Landau equations
- F. Bethuel, G. Orlandi, Uniform estimates for the parabolic Ginzburg–Landau equation
- Fabrice Bethuel, Giandomenico Orlandi, Didier Smets, Improved estimates for the Ginzburg-Landau equation : the elliptic case

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