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 a 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.
Keywords
Hamilton–Jacobi equation, embedded networks, graphs,
viscosity solutions, viscosity subsolutions, comparison
principle, discrete functional equation on graphs, Hopf–Lax
formula, discrete weak KAM theory