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.

Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations

“…Since then there has been a huge activity on this problem and therefore an important subsequent literature. Let us state amongst others [6] for a discrete time version, [22] for a non compact version, [26] on networks, [4,28] for nonlinear discounting versions, [20,21] for second order PDE versions, [12] for weakly coupled systems, following ideas from [9] and then widely generalized in [18,19], finally let us mention [1] for a more geometric interpretation of the convergence result and references therein.…”