Sobolev regularity for first order Mean Field Games

TL;DR

This paper establishes Sobolev regularity results for solutions of first-order Mean Field Game systems, providing global estimates in both stationary and time-dependent cases, and addressing an open question from prior research.

Contribution

It introduces Sobolev estimates for weak solutions of first-order MFG systems with local coupling, extending regularity results to both stationary and time-dependent problems.

Findings

First order Sobolev estimates for density variables

Second order Sobolev estimates for value functions

Global in time estimates for time-dependent problems

Abstract

In this paper we obtain Sobolev estimates for weak solutions of first oder variational Mean Field Game systems with coupling terms that are local function of the density variable. Under some coercivity condition on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity condition on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in [PS17]. Our methods apply to a large class of Hamiltonians and coupling functions.