  A classical result in the study of Ginzburg-Landau equations is that, for
Dirichlet or Neumann boundary conditions, if a sequence of functions has energy
uniformly bounded on a logarithmic scale then we can find a subsequence whose
Jacobians are convergent in suitable dual spaces and whose renormalized energy
is at least the sum of absolute degrees of vortices. However, the corresponding
question for the case of tangential or normal boundary conditions has not been
considered. In addition, the question of convergence of up to the boundary is
not very well understood. Here, we consider these questions for a bounded,
connected, open set of $\mathbb{R}^{2}$ with $C^{2,1}$ boundary.
