$\Gamma$-Convergence of the Ginzburg-Landau Functional with tangential boundary conditions

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.