  This article develops duality principles applicable to non-convex models in
the calculus of variations. The results here developed are applied to
Ginzburg-Landau type equations. For the first and second duality principles,
through an optimality criterion developed for the dual formulations, we
qualitatively classify the critical points of the primal and dual functionals
in question. We formally prove there is no duality gap between the primal and
dual formulations in a local extremal context.
  Finally, in the last sections, we present a global existence result, a
duality principle and respective optimality conditions for the complex
Ginzburg-Landau system in superconductivity in the presence of a magnetic field
and concerning magnetic potential.
