Representation of Global Viscosity Solutions for Tonelli Hamiltonians

TL;DR

This work advances the non-autonomous weak-KAM framework for Tonelli Hamiltonians by analyzing the non-wandering set $\Omega(\mathcal{T})$ of the Lax–Oleinik operator $\mathcal{T}$ and proving that $\Omega(\mathcal{T})$ coincides with the set of global viscosity solutions, with $\mathcal{T}$ acting as an isometry on this set. It introduces a generalized Peierls barrier $\underline{k}$ and a generalized static class structure $\underline{\mathbb{M}}$ to obtain a canonical representation formula $u(x) = \inf_{y\in \underline{\mathbb{M}}} \{u(y) + \underline{k}(y,x)\}$ for $u \in \Omega(\mathcal{T})$, and applies this to recover Fathi’s autonomous convergence results and to characterize $n$-periodic viscosity solutions via $h^{n\infty}$. The results connect the long-time dynamics of NW solutions to the Lagrangian flow on the Mather set, yielding periodicity and recurrence implications when the Mather set has periodic or recurrent structure. Together, these results provide robust representation and structural insights into global viscosity solutions in non-autonomous Tonelli Hamiltonian systems, with potential implications for SEO-relevant terms like weak-KAM, barrier methods, and action-minimizing dynamics.

Abstract

We consider the Lax-Oleinik operator $\mathcal{T}$ associated with the non-stationary Hamilton-Jacobi equation $\partial_tu + H(t,x,\partial_xu) = α_0$ for a Tonelli Hamiltonian $H$ and its \Mane critical value $α_0$. It is known from the work of A. Fathi and J.N. Mather \cite{MR1792479} that the convergence of this semigroup fails in the non-autonomous framework. In this context, we study the action of $\mathcal{T}$ on its non-wandering set $Ω(\mathcal{T})$. First, we show that $\mathcal{T}$ acts as an isometry on this set, and then we characterize $Ω(\mathcal{T})$ as the set of global viscosity solutions of the Hamilton-Jacobi equation, i.e. solutions that are defined for all real times. Next, we introduce a generalized Peierls barrier $\underline{k}$ and a set of generalized static classes $\underline{\mathbb{M}}$ within the Mather set. Using these, we represent elements $u$ of $Ω(\mathcal{T})$ as \begin{equation*} u(x) = \inf_{y \in \underline{\mathbb{M}}} \{ u(y) + \underline{k}(y,x) \} \end{equation*} We apply this representation formula to prove Fathi's convergence theorem for autonomous systems and provide a representation formula for $n$-periodic viscosity solutions. Additionally, we establish that the dynamics of non-wandering viscosity solutions are governed by the Lagrangian flow on the Mather set. Specifically, we show that if the Mather set consists solely of $N$-periodic orbits for some integer $N$, then all non-wandering viscosity solutions are $N$-periodic. Furthermore, we show that if the restriction of the Lagrangian flow to the Mather set is uniformly recurrent for a time sequence $p_n$, then all non-wandering viscosity solutions are uniformly recurrent for the same time sequence $p_n$.

Theorems & Definitions (131)

• Proposition 1.1
• Theorem 1.2
• Proposition 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Corollary 1.8
• Corollary 1.9
• Definition 2.1