A Monotone--Operator Proof of Existence and Uniqueness for a Simple Stationary Mean Field Game

We study a stationary first--order mean field game on the $d$--dimensional torus. The system couples a Hamilton--Jacobi equation for the value function with a transport equation for the density of players. Our goal is to give a detailed and friendly exposition of the monotone--operator argument that yields existence and uniqueness of solutions. We first present a general framework in a Hilbert space and prove existence of a strong solution by adding a simple coercive regularisation and applying Minty's method. Then we specialise to the explicit Hamiltonian $$H(p,m)=|p|^2-m,$$ check all assumptions, and show how the abstract theorem gives existence and uniqueness for this concrete mean field game. The exposition is written in a slow and elementary way so that a motivated undergraduate can follow each step.