Homogenization of Hamilton-Jacobi equations on networks

TL;DR

This paper establishes a homogenization result for time-dependent Hamilton-Jacobi equations on periodic networks, deriving a limiting equation in Euclidean space that reflects the network's topological complexity.

Contribution

It extends homogenization theory to networks and graphs, generalizes Mather's asymptotic analysis, and proves well-posedness of problems on non-compact networks.

Findings

Derived a limiting Hamilton-Jacobi equation in Euclidean space

Extended Mather's asymptotic behavior results to network settings

Proved well-posedness of homogenized problems on non-compact networks

Abstract

We prove a homogenization result for a family of time-dependent Hamilton-Jacobi equations, rescaled by a parameter $\varepsilon$ tending to zero, posed on a periodic network, with a suitable notion of periodicity that will be defined. As $\varepsilon$ becomes infinitesimal, we derive a limiting Hamilton-Jacobi equation in a Euclidean space, whose dimension is determined by the topological complexity of the network and is independent of the ambient space in which the network is embedded. Among the key contributions of our analysis, we extend to the setting of networks and graphs Mather's result on the asymptotic behavior of the average minimal action functional, as time tends to infinity. Additionally, we establish the well-posedness of the approximating problems, representing a nontrivial generalization of existing results for finite networks to a non-compact setting.