Beyond uniqueness: Relaxation calculus of junction conditions for coercive Hamilton-Jacobi equations

TL;DR

This paper develops a relaxation calculus for junction conditions in coercive Hamilton-Jacobi equations on networks, establishing an effective junction condition that encompasses weak solutions and unifies different formulations.

Contribution

It introduces a novel relaxation calculus for junction conditions, allowing computation of an effective condition via multiple methods and extending the uniqueness theory for Hamilton-Jacobi equations on networks.

Findings

Solutions satisfy a strongly relaxed junction condition $rak R F_0$

The relaxed condition can be computed via viscosity inequalities, Godunov fluxes, or Riemann problems

Solutions for different desired junction conditions coincide if their relaxed conditions match

Abstract

A junction is a particular network given by the collection of $N\ge 1$ half lines $[0,+\infty)$ glued together at the origin. On such a junction, we consider evolutive Hamilton-Jacobi equations with $N$ coercive Hamiltonians. Furthermore,we consider a general desired junction condition at the origin, given by some monotone function $F_0:\R^N\to \R$.There is existence and uniqueness of solutions which only satisfy weakly the junction condition (at the origin, they satisfy either the desired junction condition or the PDE).We show that those solutions satisfy strongly a relaxed junction condition $\frak R F_0$ (that we can recognize as an effective junction condition). It is remarkable that this relaxed condition can be computed in three different but equivalent ways: 1) using viscosity inequalities, 2) using Godunov fluxes, 3) using Riemann problems.Our result goes beyond uniqueness theory, in the following sense: solutions to two different desired junction conditions $F_0$ and $F_1$ do coincide if $\frak R F_0=\frak R F_1$.