The Jacobian and the Ginzburg-Landau Energy

We study the Ginzburg-Landau functional I (u) := 1 ln(1= ) Z U 1 2 jruj2 + 1 4 2 (1 juj2)2 dx ; for u 2 H(U ; IR), where U is a bounded, open subset of IR. We show that if a sequence of functions u satis es sup I (u ) < 1, then their Jacobians Ju are precompact in the dual of C c for every 2 (0; 1]. Moreover, any limiting measure is a sum of point masses. We also characterize the -limit I( ) of the functionals I ( ), in terms of the function space B2V introduced by the authors in [16, 17]: we show that I(u) is nite if and only if u 2 B2V (U ;S), and for u 2 B2V (U ;S), 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 (u ) C then the Jacobians Ju are again precompact in C c for all 2 (0; 1], and moreover we show that any limiting measure must be integer multiplicity recti able. We also show that the total variation of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional. Running title: Ginzburg-Landau Functional