Ricerc@Sapienza

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.