Aubry set of eikonal Hamilton–Jacobi equations on networks

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.