The Jacobian and the Ginzburg-Landau energy

We study the Ginzburg-Landau functional I_ε(u):= 1/ln(1/ε) ∫_U 1/2|∇u|² + 1/4∈² (1- |u|²)² dx, for u ∈ H¹ (U; ℝ²), where U is a bounded, open subset of R². We show that if a sequence of functions u^ε satisfies sup I_ε(u^ε) < ∞, then their Jacobians Ju^ε are precompact in the dual of C_c^0,α for every α ∈ (0, 1]. Moreover, any limiting measure is a sum of point masses. We also characterize the Γ-limit I(·) of the functionals I_ε (·), 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 ∈ B2V(U; S¹), and for u ∈ B2V(U; S¹), 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_ε (u^ε) ≤ C then the Jacobians Ju^ε are again precompact in (C_c^0,α)* for all α ∈ (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 Γ limit of the Ginzburg-Landau functional.