Weak solutions for first order mean field games with local coupling

TL;DR

This paper establishes the existence and uniqueness of weak solutions for first order mean field games with local coupling using variational methods, and connects these solutions to approximate Nash equilibria in large-player differential games.

Contribution

It introduces a variational approach to prove weak solution existence and uniqueness for first order MFGs with local coupling, and links solutions to $ ext{ε}$-Nash equilibria.

Findings

Weak solutions exist and are unique for the considered MFG systems.

The weak solution's first component satisfies a degenerate elliptic PDE in the viscosity sense.

The approach applies to smooth data, enabling analysis of large finite-player games.

Abstract

Existence and uniqueness of a weak solution for first order mean field game systems with local coupling are obtained by variational methods. This solution can be used to devise $\epsilon-$Nash equilibria for deterministic differential games with a finite (but large) number of players. For smooth data, the first component of the weak solution of the MFG system is proved to satisfy (in a viscosity sense) a time-space degenerate elliptic differential equation.