Discounted Hamilton-Jacobi equations on networks and asymptotic analysis

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