The vanishing discount problem for monotone systems of Hamilton–Jacobi equations: part 2—nonlinear coupling

Abstract

We study the vanishing discount problem for a nonlinear monotone system of Hamilton–Jacobi equations. This continues the first author’s investigation on the vanishing discount problem for a monotone system of Hamilton–Jacobi equations. As in part 1, we introduce by the convex duality Mather measures and their analogues for the system, which we call respectively Mather and Green–Poisson measures, and prove a convergence theorem for the vanishing discount problem. Moreover, we establish an existence result for the ergodic problem.