# Analysis of Coupled Scalar Systems by Displacement Convexity

## TL;DR

This paper explores the properties of potential functionals in coupled scalar systems, demonstrating that they are displacement convex under certain conditions, which ensures uniqueness of the system's minimizer.

## Contribution

It introduces the concept of displacement convexity for potential functionals in coupled scalar systems and establishes conditions for strict convexity and uniqueness.

## Key findings

- Potential functional is displacement convex under mild assumptions.
- Conditions for strict displacement convexity are provided.
- Uniqueness of the minimizer is guaranteed under certain conditions.

## Abstract

Potential functionals have been introduced recently as an important tool for the analysis of coupled scalar systems (e.g. density evolution equations). In this contribution, we investigate interesting properties of this potential. Using the tool of displacement convexity, we show that, under mild assumptions on the system, the potential functional is displacement convex. Furthermore, we give the conditions on the system such that the potential is strictly displacement convex, in which case the minimizer is unique.