On General Duality Principles for Non-Convex Variational Optimization with Applications to the Ginzburg-Landau System in Superconductivity

Abstract

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.