Dual variational formulations for a large class of non-convex models in the calculus of variations

TL;DR

This paper introduces dual variational formulations for a broad class of non-convex models in the calculus of variations, utilizing functional analysis and duality theory, with applications to superconductivity and numerical methods.

Contribution

It develops new dual convex variational formulations with extensive convexity regions around critical points for non-convex models in physics and engineering.

Findings

New dual formulations with large convexity regions around critical points

Application to Ginzburg-Landau system in superconductivity

Numerical results using generalized method of lines

Abstract

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. In the first part final sections, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section we present some numerical results concerning the generalized method of lines applied to a Ginzburg-Landau type equation.