TY - JOUR

T1 - Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices

AU - Berlyand, Leonid

AU - Mironescu, Petru

N1 - Funding Information:
The authors thank H. Brezis for valuable discussions. They are also grateful to D. Golovaty for careful reading of the manuscript and useful suggestions. The work of L.B. was supported by NSF grant DMS-0204637. The work of P.M. is part of the RTN Program “Fronts-Singularities”. This work was initiated while both authors were visiting Rutgers University; part of the work was done while L.B. was visiting Université Paris-Sud and P.M. was visiting Penn State University. They thank the Mathematics Departments at these universities for their hospitality.

PY - 2006/10/1

Y1 - 2006/10/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jfa.2006.03.006

DO - 10.1016/j.jfa.2006.03.006

M3 - Article

AN - SCOPUS:33745430293

VL - 239

SP - 76

EP - 99

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -