On Duality Principles and Concerned Convex Dual Formulations Applied to Models in Superconductivity and Phase Transition

This article develops duality principles applicable to some originally non-convex primal variational formulations. More specifically, in a first step, we develop applications to a full complex Ginzburg-Landau system in superconductivity, including a magnetic field and respective magnetic potential. The results are obtained through basic tools of functional analysis, calculus of variations, duality and optimization theory in infinite dimensional spaces. It is worth emphasizing we have obtained convex dual variational formulations which may be applied to a large class of similar models in the calculus of variations. In the subsequent sections we also present a procedure for improving the convexity conditions of an originally non-convex primal formulation which is also applied to a Ginzburg-Landau type equation. Finally, in the last sections, we develop duality principles and related numerical examples for models in phase transition.