Lax--Oleinik formula on networks

TL;DR

This paper extends the Lax-Oleinik formula to Hamilton-Jacobi equations on complex networks with multiple vertices, providing a new representation formula that accounts for general geometries and flux conditions.

Contribution

It introduces a Lax-Oleinik-type representation for Hamilton-Jacobi equations on networks with general geometry, including multiple vertices and flux limiters, expanding beyond previous junction-only results.

Findings

Established a new action functional for network problems

Proved existence and Lipschitz continuity of minimizers

Extended the formula to complex network geometries

Abstract

We provide a Lax-Oleinik-type representation formula for solutions of time-dependent Hamilton-Jacobi equations, posed on a network with a rather general geometry, under standard assumptions on the Hamiltonians. It depends on a given initial datum at $t=0$ and a flux limiter at the vertices, which both have to be assigned in order the problem to be uniquely solved. Previous results in the same direction are solely in the frame of junction, namely network with a single vertex. An important step to get the result is to define a suitable action functional and prove existence as well as Lipschitz-continuity of minimizers between two fixed points of the network in a given time, despite the fact that the integrand lacks convexity at the vertices.