Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions

TL;DR

This paper proves the existence of weak solutions for first-order stationary mean-field games with Dirichlet boundary conditions by employing regularization, fixed-point theorems, and monotonicity methods.

Contribution

It introduces a novel regularized approach and applies fixed-point and monotonicity techniques to establish existence of solutions satisfying Dirichlet conditions.

Findings

Existence of weak solutions for the MFGs with Dirichlet boundary conditions.

Use of Schaefer's fixed-point theorem to construct solutions.

Application of Minty's method to pass to the limit and prove existence.

Abstract

In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. While for Hamilton--Jacobi equations Dirichlet conditions may not be satisfied, here, we establish the existence of solutions of MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and using Minty's method, we show the existence of weak solutions to the original MFG.