Beyond uniqueness: Relaxation calculus of junction conditions for coercive Hamilton-Jacobi equations
Nicolas Forcadel, Regis Monneau
TL;DR
The paper studies evolution Hamilton–Jacobi equations on junctions formed by $N$ half-lines with coercive, possibly nonconvex Hamiltonians and a general junction condition $F_0$. It introduces a relaxation operator $ rak R F_0$ to yield an effective junction condition, proving that weak $F_0$-solutions are equivalent to strong $ rak R F_0$-solutions and that this relaxation can be computed via three equivalent approaches: viscosity inequalities (via Godunov fluxes), Godunov semi-flux compositions, and Riemann-problem constructions. It also shows that different original junction conditions that share the same relaxed form lead to the same problem, providing a robust unifying framework beyond uniqueness. The work develops the relaxation operator through Godunov fluxes, analyzes its properties, and establishes a comparison principle in this relaxed setting. A third relaxation formula based on Riemann problems is presented and shown to coincide with the others, offering a versatile toolkit for effective boundary conditions on networks of Hamilton–Jacobi equations.
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 $\frak R F_0=\frak R F_1$.