Limiting behavior of the ginzburg-landau functional

Abstract

We continue our study of the functional E-0(u) := integral(0)1/2\delu\(2) + 1/4epsilon(2) (1 - \u\(2))(2) dx, for u is an element of H-1(U;R-2), where U is a bounded, open subset of R-2. Compactness results for the scaled Jacobian of u(E) are proved under the assumption that E, (u(r)) is bounded uniformly by a function of epsilon. In addition, the Gamma limit of E(u(r))/(ln epsilon)(2) is shown to be E(upsilon) := 1/2parallel toupsilonparallel to(2)(2) + parallel todel x upsilonparallel to(.H), where upsilon is the limit of j(u(0))/\ln epsilon\, j(u(0)) = u(0) x Du(0), and parallel to(.)parallel to(.H) is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional F-0(u,A;h(ext)) := integral(upsilon)1/2 \del(A)u\(2) + 1/4epsilon(2) (1 - \u\(2))(2) + 1/2\del x A - h(ext)\ dx, with external magnetic field h(ext) approximate to H\ln epsilon\. The Gamma limit of F-0/(ln epsilon)(2) is calculated to be F(upsilon, a; H) := 1/2[parallel toupsilon - aparallel to(2)(2) +parallel todel x upsilonparallel to(.H) + parallel todel x a - Hparallel to(2)(2)], where upsilon is as before, and a is the limit of A(p)/\ln epsilon\. (C) Elsevier Science (USA).