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 by Siconolfi and Sorrentino (2018); specifically, we associate to 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 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.