# The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

## TL;DR

This paper proves a convergence theorem for the vanishing discount problem in weakly coupled Hamilton-Jacobi systems, introducing new measure concepts to handle linear coupling.

## Contribution

It introduces viscosity Mather and Green-Poisson measures for systems, advancing the analysis of the vanishing discount problem with linear coupling.

## Key findings

- Established convergence theorem for linear coupling systems
- Introduced viscosity Mather and Green-Poisson measures
- Utilized convex duality in the analysis

## Abstract

We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling.