A duality theory for non-convex problems in the Calculus of Variations

TL;DR

This paper introduces a novel duality framework for non-convex variational problems with mixed boundary conditions, establishing no duality gap and providing optimality conditions, with applications to free boundary problems.

Contribution

It develops a new duality theory for non-convex variational problems that can be reformulated as linear programming problems, enabling better analysis and numerical methods.

Findings

No duality gap in the proposed framework

Necessary and sufficient optimality conditions derived

Application demonstrated on a free boundary problem

Abstract

We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no duality gap. Further, we provide necessary and sufficient optimality conditions, and we show that our duality principle can be reformulated as a min-max result which is quite useful for numerical implementations. As an example, we illustrate the application of our method to a celebrated free boundary problem. The results were announced in \cite{BoFr}.