### Abstract

Let Ω ⊂ R^{2} be a simply connected domain, let ω be a simply connected subdomain of Ω, and set A = Ω {set minus} ω. Suppose that J is the class of complex-valued maps on the annular domain A with degree 1 both on ∂Ω and on ∂ω. We consider the variational problem for the Ginzburg-Landau energy E_{λ} among all maps in J. Because only the degree of the map is prescribed on the boundary, the set J is not necessarily closed under a weak H^{1}-convergence. We show that the attainability of the minimum of E_{λ} over J is determined by the value of cap (A)-the H^{1}-capacity of the domain A. In contrast, it is known, that the existence of minimizers of E_{λ} among the maps with a prescribed Dirichlet boundary data does not depend on this geometric characteristic. When cap (A) ≥ π (A is either subcritical or critical), we show that the global minimizers of E_{λ} exist for each λ > 0 and they are vortexless when λ is large. Assuming that λ → ∞, we demonstrate that the minimizers of E_{λ} converge in H^{1} (A) to an S^{1}-valued harmonic map which we explicitly identify. When cap (A) < π (A is supercritical), we prove that either (i) there is a critical value λ_{0} such that the global minimizers exist when λ < λ_{0} and they do not exist when λ > λ_{0}, or (ii) the global minimizers exist for each λ > 0. We conjecture that the second case never occurs. Further, for large λ, we establish that the minimizing sequences/minimizers in supercritical domains develop exactly two vortices-a vortex of degree 1 near ∂Ω and a vortex of degree -1 near ∂ω.

Original language | English (US) |
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Pages (from-to) | 76-99 |

Number of pages | 24 |

Journal | Journal of Functional Analysis |

Volume | 239 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2006 |

### All Science Journal Classification (ASJC) codes

- Analysis