Global Results for Eikonal Hamilton-Jacobi Equations on Networks

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, and to relate the differential equation to a {it discrete functional equation} on the graph. Comparison principles and representation formulae are proven in the supercritical case as well.