The jacobian and the ginzburg-landau energy

Abstract

We study the Ginzburg-Landau functional Iε(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 uε satisfies sup Iε(uε) andlt; ∞, then their Jacobians Juε are precompact in the dual of Cc0,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 Iε (·), 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 Iε (uε) ≤ C then the Jacobians Juε are again precompact in (Cc0,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.