Lax-Oleinik formula for nonautonomous Hamilton-Jacobi equations on networks

TL;DR

This paper develops a Lax-Oleinik representation formula for nonautonomous Hamilton-Jacobi equations on complex networks, accommodating general geometries and ensuring well-posedness with flux limiters.

Contribution

It introduces a novel formula that accounts for network geometry, vertex constraints, and proves solution uniqueness without standard flux bounds.

Findings

The action functional admits Lipschitz continuous minimizers.

The representation formula provides unique solutions even with high flux limiters.

The approach handles networks with loops and countably many arcs.

Abstract

We provide a Lax-Oleinik-type representation formula for solutions to nonautonomous Hamilton-Jacobi equations posed on networks with a rather general geometry. The networks may possess countably many arcs and allow for the presence of loops. We consider Hamiltonians that are convex and superlinear in the momentum variable, and satisfy a Lipschitz-type condition in the time variable. The representation formula is constructed via an overall Lagrangian that accounts for both the arc-specific dynamics and vertex-based constraints, called flux limiters, which ensure the well-posedness of the problem. We prove that the corresponding action functional admits Lipschitz continuous minimizers without needing to rule out the Zeno phenomenon. Furthermore, we demonstrate that the formula yields the unique solution to the problem even when the flux limiters exceed standard upper bounds.