Abstract :
Let Ω ⊂ R2 be a simply connected domain, let ω be a simply connected subdomain of Ω, and set
A = Ω \ 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.
© 2006 Elsevier Inc. All rights reserved.
