$L^{\infty }$ Variational Approximation of the Aubry Set

Abstract

Let $H\in C^{\infty }({\mathbb{R}}^n\times {\mathbb{T}}^n)$ be a periodic Tonelli Hamiltonian with critical value $c$. For each $k\in \mathbb{N}$, let $u_k$ be the normalized minimizer of the variational functional introduced by Evans[7], \[ I_k[w]=\int_{\mathbb T^n} e^{kH(Dw,x)}\,dx, \qquad \int_{\mathbb T^n}w\,dx=0. \] If $u_{\infty }$ is a uniform limit of a subsequence of $\{u_k\}$ and the Mather quotient $(A_M,{\delta }_M)$ satisfies $H^1(A_M,{\delta }_M)=0$, then $u_{\infty }$ is a critical subsolution that is strict outside $A$ and \[ {A} = \{x\in\mathbb T^n\,:\,Du_\infty(x)\ \text{exists and }H(Du_\infty(x),x)=c\}=\{x\in\mathbb T^n\,:\,u_\infty(x)=u_{-}(x)\}, \] where $A$ is the projected Aubry set and $u_{-}$ is the backward weak KAM solution associated with $u_{\infty }$. In particular, by the theorem of Fathi--Figalli--Rifford[10], this conclusion holds for all smooth Tonelli Hamiltonians on ${\mathbb{T}}^n$ when $n\le 3$. This characterization also suggests a natural numerical localization principle for approximating the entire Aubry set through near-contact sets between $u_k$ and its large-time backward Lax--Oleinik evolution.

The authors acknowledge the use of ChatGPT 5.6 Plus for suggesting some ideas and assisting with verifications. Some of these suggestions were developed together with our own ideas. All mathematical results, as well as the final writing and revision of the manuscript, are due to the authors