The Jacobian and the Ginzburg-Landau energy

We study the Ginzburg-Landau functional I-epsilon(u) := 1/ln(1/is an element of) integral(U) 1/2\delu\(2) + 1/4is an element of(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. We show that if a sequence of functions u(epsilon) satisfies sup 1(epsilon) (u(epsilon)) < infinity, then their Jacobians Ju(epsilon) are precompact in the dual of C-c(0,alpha) for every alpha is an element of [0, 1]. Moreover, any limiting measure is a sum of point masses. We also characterize the Gamma-limit I((.)) of the functionals I-epsilon ((.)), in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if u is an element of B2V(U; S-1), and for u is an element of B2V(U; S-1), I(u) is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if I-epsilon (u(epsilon)) less than or equal to C then the Jacobians Ju(epsilon) are again precompact in (C-c(0,alpha) for all alpha is an element of (0, 1), and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the Gamma limit of the Ginzburg-Landau functional.