The Master Equation in a Bounded Domain with Neumann Conditions

TL;DR

This paper investigates the well-posedness of the Master Equation in Mean Field Games within a bounded domain, focusing on Neumann boundary conditions and establishing regularity results crucial for solution existence.

Contribution

It introduces a solution framework for the Master Equation with Neumann boundary conditions and proves regularity results essential for well-posedness in bounded domains.

Findings

Established well-posedness of the Master Equation with Neumann boundary conditions.

Proved global regularity of the linearized system at the boundary.

Developed boundary condition analysis for parabolic equations in this context.

Abstract

In this article we study the well-posedness of the Master Equation of Mean Field Games in a framework of Neumann boundary condition. The definition of solution is closely related to the classical one of the Mean Field Games system, but the boundary condition here leads to two Neumann conditions in the Master Equation formulation, for both space and measure. The global regularity of the linearized system, which is crucial in order to prove the existence of solutions, is obtained with a deep study of the boundary conditions and the global regularity at the boundary of a suitable class of parabolic equations.