# Mean Field Games with state constraints: from mild to pointwise solutions of the PDE system

## TL;DR

This paper develops a mathematical framework for analyzing Mean Field Games with state constraints by establishing a global semiconcavity property of the value function, enabling a better understanding of the associated PDE system.

## Contribution

It introduces a novel approach to define solutions for the PDE system of constrained Mean Field Games using semiconcavity properties.

## Key findings

- Proves global semiconcavity of the value function with state constraints
- Provides a new interpretation of the PDE system for constrained Mean Field Games
- Bridges the gap between control problems and PDE analysis in this context

## Abstract

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the value function associated with optimal control problems with state constraints.