Aubry Set of Eikonal Hamilton-Jacobi Equations on Networks

TL;DR

This paper generalizes the study of eikonal Hamilton-Jacobi equations on networks, introducing a broader framework that includes a new comparison principle and the concept of an Aubry set for solution representation.

Contribution

It extends previous work by relaxing restrictive conditions, allowing for a more general setting and establishing a new comparison principle for solutions.

Findings

Aubry set acts as a uniqueness set for solutions.

Established a new comparison principle between super and subsolutions.

Generalized the framework for Hamilton-Jacobi equations on networks.

Abstract

We extend the study of eikonal Hamilton-Jacobi equations posed on networks performed by Siconolfi and Sorrentino (Anal. PDE, 2018) to a more general setting. Their approach essentially exploits that such equations correspond to discrete problems on an abstract underlying graph. However, a specific condition they assume can be rather restricting in some settings, which motivates the generalization we propose. We still get an Aubry set, which plays the role of a uniqueness set for our problem and appears in the representation of solutions. Exploiting it we establish a new comparison principle between super and subsolutions to the equation.