# Schauder Estimates for a Class of Potential Mean Field Games of Controls

## TL;DR

This paper proves the existence of classical solutions for a class of potential mean field games of controls, where agents' costs depend on collective controls and congestion, using a potential formulation and Leray-Schauder theorem.

## Contribution

It introduces a new existence proof for potential mean field games of controls with congestion effects, employing a potential formulation and a priori bounds.

## Key findings

- Existence of classical solutions established.
- A potential formulation aids in deriving a priori bounds.
- Application of Leray-Schauder theorem confirms solutions.

## Abstract

An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray-Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.