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

摘要

Let Omega subset of R-2 be a simply connected domain, let omega be a simply connected subdomain of Omega, and set A = Omega omega. Suppose that J is the class of complex-valued maps on the annular domain A with degree 1 both on partial derivative Omega and on a partial derivative omega. We consider the variational problem for the Ginzburg-Landau energy E-lambda 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-lambda 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-lambda among the maps with a prescribed Dirichlet boundary data does not depend on this geometric characteristic. When cap(A) >= pi (A is either subcritical or critical), we show that the global minimizers of E-lambda exist for each lambda > 0 and they are vortexless when lambda is large. Assuming that lambda --> infinity, we demonstrate that the minimizers of E-lambda converge in H-1 (A) to an S-1-valued harmonic map which we explicitly identify. When cap(A) < pi (A is supercritical), we prove that either (i) there is a critical value lambda(o) such that the global minimizers exist when lambda lambda(o), or (ii) the global minimizers exist for each lambda > 0. We conjecture that the second case never occurs. Further, for large lambda, we establish that the minimizing sequences/minimizers in supercritical domains develop exactly two vortices-a vortex of degree 1 near a partial derivative Omega and a vortex of degree - 1 near a partial derivative omega. (C) 2006 Elsevier Inc. All rights reserved.