An example in the vanishing discount problem for monotone systems of Hamilton-Jacobi equations

TL;DR

This paper presents a counterexample demonstrating that the convergence of solutions as the discount factor vanishes does not always occur in nonlinear monotone systems of Hamilton-Jacobi equations with convex Hamiltonians, extending previous scalar results.

Contribution

It provides the first known example of a monotone Hamilton-Jacobi system with convex Hamiltonians where full solution convergence fails as the discount parameter approaches zero.

Findings

Counterexample for monotone systems with convex Hamiltonians

Full convergence of solutions does not always occur

Extends scalar non-convergence results to systems

Abstract

In recent years, there have been many contributions to the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto [Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample. J. Math. Pures Appl. (9) 128 (2019), 330-338] has shown an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give an example of the nonlinear monotone system of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the whole family convergence of the solutions does not hold.