Obstacle Mean-Field Game Problem

TL;DR

This paper introduces a first-order mean-field game obstacle problem, analyzing existence, uniqueness, and properties of solutions under various nonlinearities, with a focus on non-differentiable obstacle operators.

Contribution

It develops a penalized approach to handle the non-differentiability of the obstacle operator and establishes existence and uniqueness of solutions for the mean-field game obstacle problem.

Findings

Unique solutions exist for the penalized problem.

Uniform bounds enable passing to the limit and characterizing solutions.

The approach applies to logarithmic and power-like nonlinearities.

Abstract

In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.