### Abstract

We study the minimizers of the GinzburgLandau free energy functional in the class (u, A) ∈ H^{1}(Ω; ) × H^{1}(Ω; ^{2}) with |u| = 1 on ∂Ω, where Ω is a bounded simply connected domain in ^{2}. We consider the connected components of this class defined by the prescribed topological degree d of u on the boundary ∂Ω. We show that for d ≠ 0 the minimizers exist if 0 < λ ≤ 1 and do not exist if λ > 1, where λ is the coupling constant ($\sqrt{\lambda/2}$ is the GinzburgLandau parameter). We also establish the asymptotic locations of vortices for λ → 1 - 0 (the critical value λ = 1 is known as the Bogomol'nyi integrable case).

Original language | English (US) |
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Pages (from-to) | 53-66 |

Number of pages | 14 |

Journal | Communications in Contemporary Mathematics |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2011 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Berlyand, L., Misiats, O., & Rybalko, V. (2011). Minimizers of the magnetic Ginzburg-Landau functional in simply connected domain with prescribed degree on the boundary.

*Communications in Contemporary Mathematics*,*13*(1), 53-66. https://doi.org/10.1142/S0219199711004130