We study discounted Hamilton Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced in 11, namely we associate to the differential problem on the network, a discrete functional equation on an abstract underlying graph.
We study a one--parameter family of Eikonal Hamilton-Jacobi
equations on an embedded network, and prove that there exists a
unique critical value for which the corresponding equation admits
global solutions, in a suitable viscosity sense. Such a solution is
identified, via an Hopf--Lax type formula, once an admissible trace
is assigned on an {it intrinsic boundary}. The salient point of
our method is to associate to the network an {it abstract graph},
encoding all of the information on the complexity of the network,
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