Displacement convexity and minimal fronts at phase boundaries

TL;DR

This paper demonstrates that certain non-convex free energy functionals become displacement convex under a natural change of variables, leading to uniqueness of critical points and new examples of displacement convex functionals, applicable to multi-component systems.

Contribution

It introduces a novel approach using displacement convexity to analyze free energy functionals, including multicomponent systems, and provides new examples of such functionals.

Findings

Certain free energy functionals are displacement convex after variable change.

Displacement convexity ensures critical points are minimizers.

New examples of displacement convex functionals, including multi-component cases.

Abstract

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition, are strictly convex in the sense of displacement convexity under a natural change of variables. We use this to show that in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals, and, in the multi-component setting, jointly displacement convex functionals.