Local regularity for mean-field games in the whole space

TL;DR

This paper establishes Sobolev regularity results for mean-field games in the entire space by combining integrability estimates and entropy dissipation techniques, overcoming challenges due to non-compactness.

Contribution

It introduces a novel approach using entropy dissipation and the non-linear adjoint method to obtain uniform Sobolev estimates for mean-field game solutions in unbounded domains.

Findings

Proves Sobolev regularity for solutions in the whole space

Develops entropy dissipation estimates for the adjoint variable

Provides uniform estimates in Sobolev spaces for Hamilton-Jacobi solutions

Abstract

In this paper, we investigate the Sobolev regularity for mean-field games in the whole space $\Rr^d$. This is achieved by combining integrability for the solutions of the Fokker-Planck equation with estimates for the Hamilton-Jacobi equation in Sobolev spaces. To avoid the mathematical challenges posed due to the lack of compactness, we prove an entropy dissipation estimate for the adjoint variable. This, together with the non-linear adjoint method, yields uniform estimates for solutions of the Hamilton-Jacobi equation in $W^{1,p}_{loc}(\Rr^d)$.