# Mean field games under invariance conditions for the state space

## TL;DR

This paper studies mean field game systems with invariance conditions on the state space, establishing existence, uniqueness, and regularity of solutions, including Lipschitz and semiconcave properties, without requiring smooth domains.

## Contribution

It provides a novel analysis of mean field games under invariance conditions, demonstrating existence, uniqueness, and regularity results for solutions in non-smooth domains.

## Key findings

- Existence and uniqueness of solutions in $L^
abla$ and $L^1$ spaces.
- Value function is globally Lipschitz and semiconcave.
- Distribution density is bounded under certain conditions.

## Abstract

We investigate mean field game systems under invariance conditions for the state space, otherwise called {\it viability conditions} for the controlled dynamics. First we analyze separately the Hamilton-Jacobi and the Fokker-Planck equations, showing how the invariance condition on the underlying dynamics yields the existence and uniqueness, respectively in $L^\infty$ and in $L^1$. Then we apply this analysis to mean field games. We investigate further the regularity of solutions proving, under some extra conditions, that the value function is (globally) Lipschitz and semiconcave. This latter regularity eventually leads the distribution density to be bounded, under suitable conditions. The results are not restricted to smooth domains.