Classical and weak solutions to local first-order mean field games through elliptic regularity

TL;DR

This paper investigates the regularity and existence of solutions for local first-order mean field games systems, employing elliptic PDE techniques and considering different regimes based on the cost function's bounds.

Contribution

It introduces a transformation leading to elliptic PDEs with oblique boundary conditions and establishes smoothness or weak solution existence depending on ellipticity degeneracy.

Findings

Solutions are smooth when the PDE is strictly elliptic.

Existence and uniqueness of weak solutions are proven in degenerate elliptic cases.

Weak solutions can be obtained as limits of classical solutions.

Abstract

We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in the density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P.J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems.