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 intrinsic boundary. The salient point of our method is to associate to the network an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on the graph. Comparison principles and representation formulae are proven in the supercritical case as well.1 project :"Fenomeni asintotici e omogeneizzazione". A. Siconolfi acknowledges the Progetto Ateneo 2015-Rome La Sapienza University: "Asintotica e omogeneizzazione di dinamiche Hamiltoniane". A. Sorrentino acknowledges the PRIN-2012-74FYK7 grant: "Variational and perturbative aspects of nonlinear differential problems".
Preliminaries on Graph TheoryIn this section we recall some basic material on the theory of abstract graphs and on functions defined on them. For a more detailed presentation of these and other related topics, we refer the interested readers, for instance, to [30].2.1. Abstract graphs. A (abstract) graph X = (V, E) is an ordered pair of sets V and E, which are called, respectively, vertices and (directed) edges, plus two functions:with the latter assumed to be a fixed-point-free involution, namely satisfying e = e and e = e for any e ∈ E.We give the following geometric picture of the setting: o(e) is the origin (initial vertex) of e and e its reversed edge, namely the same edge but with the opposite orientation. Analogously we define t(e) = o(e) the terminal vertex of e. The following compatibility condition holds true t(e) = o(e) = o(e).We say that e links o(e) to t(e), observe that it might well happen that o(e) = t(e), and in this case e will be called a loop. An edge is also said to be incident on o(e) and t(e). Two vertices are called adjacent if there is an edge linking them or, in other terms, if there is an edge incident on both of them.We say that the graph is finite if the set E, and consequently V, has a finite number of elements. We denote by |V|, |E| the number of vertices and edges.