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

TL;DR

This paper develops duality principles for non-convex variational problems and applies them to analyze the Ginzburg-Landau equations in superconductivity, establishing duality gaps, optimality conditions, and existence results.

Contribution

It introduces new duality principles for non-convex variational models and applies them to the Ginzburg-Landau system, including critical point classification and existence theorems.

Findings

No duality gap in local extremal cases

Classification of critical points for primal and dual functionals

Existence and optimality conditions for superconductivity models

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.