Global Results for Eikonal Hamilton-Jacobi Equations on Networks

TL;DR

This paper investigates Eikonal Hamilton-Jacobi equations on networks, establishing the existence of a unique critical value for global solutions and linking the PDE to a discrete graph-based functional equation.

Contribution

It introduces a novel approach by associating the network with an abstract graph to analyze the equations and proves comparison principles and formulas in the supercritical case.

Findings

Existence of a unique critical value for global solutions.

Development of an Hopf-Lax type formula for solutions.

Establishment of comparison principles in the supercritical case.

Abstract

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.