Discounted Hamilton-Jacobi Equations on Networks and Asymptotic Analysis

We study discounted Hamilton-Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians are continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced in [11]; specifically, we associate with the differential problem on the network a discrete functional equation on an abstract underlying graph. We perform some qualitative analysis and single out a distinguished subset of vertices, called a lambda-Aubry set, which shares some properties of the Aubry set for eikonal equations on compact manifolds. We finally study the asymptotic behavior of solutions and lambda-Aubry sets as the discount factor lambda becomes infinitesimal.