Representation formulas and large time behavior for solutions to some nonconvex Hamilton-Jacobi equations

TL;DR

This work studies nonconvex first-order Hamilton–Jacobi equations on the torus $\mathbb{T}^n$ by deriving a new representation formula for viscosity solutions via the nonlinear adjoint method and adjoint measures. It constructs Mather measures in the nonconvex setting through vanishing viscosity and large-time averages, and analyzes dissipative mechanisms that may arise along shocks. Under assumptions (A1)–(A2) with $\overline H(0)=0$, the authors prove large-time convergence of $u(\cdot,t)$ to a cell-problem solution and establish structural properties of $u$ and $u_t$ using adjoint formulations. By extending convex-Hamiltonian ideas to nonconvex cases, the paper links representation formulas, Mather measures, and long-time dynamics, and outlines several open questions and illustrative examples.

Abstract

We give a new representation formula for solutions to nonconvex first-order Hamilton--Jacobi equations in the periodic setting and present some applications. We then prove the large time behavior for solutions under some additional assumptions.

Figures (1)

• Figure 4.1: Graph of $H(p)=|p|^4-|p|^2$ in one dimension

Theorems & Definitions (57)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Remark 2
• Definition 1: Mather measures in the nonconvex setting
• Theorem 1.3
• Remark 3
• Theorem 1.4
• Remark 4
• proof : Proof of Theorem \ref{['thm:rep1']}