Limiting Behavior of the Ginzburg–Landau Functional

We continue our study of the functional Eε(u)≔∫U12∣∇u∣2+14ε2(1−∣u∣2)2dx, for u∈H1(U;R2), where U is a bounded, open subset of R2. Compactness results for the scaled Jacobian of uε are proved under the assumption that Eε(uε) is bounded uniformly by a function of ε. In addition, the Gamma limit of Eε(uε)/(ln ε)2 is shown to be E(v)≔12∥v∥22+∥∇×v∥M, where v is the limit of j(uε)/∣ln ε∣, j(uε)≔uε×Duε, and ∥·∥M is the total variation of a Radon measure. These results are applied to the Ginzburg–Landau functionalFε(u,A;hext)≔∫U12∣∇Au∣2+14ε2(1−∣u∣2)2+12∣∇×A−hext∣ dx, with external magnetic field hext≈H∣ln ε∣. The Gamma limit of Fε/(ln ε)2 is calculated to beF(v,a;H)≔12[∥v−a∥22+∥∇×v∥M+∥∇×a−H∥22], where v is as before, and a is the limit of Aε/∣ln ε∣.