Non-convex coercive Hamilton-Jacobi equations: Guerand's relaxation revisited

TL;DR

The paper studies evolution-type Hamilton-Jacobi equations posed in a domain with boundary data, focusing on coercive but non-convex Hamiltonians. It derives a new, general formula for the relaxation operator $\mathfrak{R}F_0$, proves its equality with Guerand's operator $\mathfrak{J}F_0$ in 1D, and extends the construction to multi-dimensional settings via tangential freezing. A key insight is the connection between boundary relaxation and Godunov flux from scalar conservation laws, yielding Neumann boundary relaxations tied to $G$ and Dirichlet relaxations that reduce to a boundary obstacle problem involving the lower envelope $H_-$. The work also develops a robust weak-to-strong correspondence for boundary conditions, proves existence and stability of weak solutions, and provides concrete formulations for Neumann and Dirichlet problems with implications for junctions and networks. Overall, the results unify relaxation concepts for non-convex HJ equations with classical conservation-law flux notions, offering precise, computable boundary conditions in complex geometries.$

Abstract

This work is concerned with Hamilton-Jacobi equations of evolution type posed in domains and supplemented with boundary conditions. Hamiltonians are coercive but are neither convex nor quasiconvex. We analyse boundary conditions when understood in the sense of viscosity solutions. This analysis is based on the study of boundary conditions of evolution type. More precisely, we give a new formula for the relaxed boundary conditions derived by J. Guerand (J. Differ. Equations, 2017). This new point of view unveils a connection between the relaxation operator and the classical Godunov flux from the theory of conservation laws. We apply our methods to two classical boundary value problems. It is shown that the relaxed Neumann boundary condition is expressed in terms of Godunov's flux while the relaxed Dirichlet boundary condition reduces to an obstacle problem at the boundary associated with the lower non-increasing envelope of the Hamiltonian.

Figures (3)

• Figure 1: Effects of $\underline R$, $\overline R$ and $\mathfrak R$ on $F_0$. The Hamiltonian $H$ is represented with a plain line, while a dashed line is used for the function $F_0$. The relaxation operators appear in red. We see that $\underline RF_0 \le F_0$ while $\overline RF_0 \ge F_0$. We can also observe that $\mathfrak R F_0 = \underline RF_0$ in $\{ F_0 \ge H \}$ and $\mathfrak R F_0 = \overline RF_0$ in $\{ F_0 \le H \}$.
• Figure 2: Characteristic points of $F_0$ (along $H$)
• Figure 3: Points $p^+$ and $p^-$ associated to $p$ (and $H$)

Theorems & Definitions (38)

• Example 1.2
• Remark 1.4
• Remark 1.8
• Remark 2.2
• proof
• proof
• proof : Proof of Lemma \ref{['lem:opti-loc']}
• proof
• Remark 2.10
• proof