Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices

Leonid Berlyand, Petru Mironescu

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Let Ω ⊂ R2 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 Eλ 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 H1-convergence. We show that the attainability of the minimum of Eλ over J is determined by the value of cap (A)-the H1-capacity of the domain A. In contrast, it is known, that the existence of minimizers of Eλ 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 Eλ exist for each λ > 0 and they are vortexless when λ is large. Assuming that λ → ∞, we demonstrate that the minimizers of Eλ converge in H1 (A) to an S1-valued harmonic map which we explicitly identify. When cap (A) < π (A is supercritical), we prove that either (i) there is a critical value λ0 such that the global minimizers exist when λ < λ0 and they do not exist when λ > λ0, 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 languageEnglish (US)
Pages (from-to)76-99
Number of pages24
JournalJournal of Functional Analysis
Issue number1
StatePublished - Oct 1 2006

All Science Journal Classification (ASJC) codes

  • Analysis


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