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

Let $H\in C^{\mathrm{\infty }}({\mathbb{R}}^n\times {\mathbb{T}}^n)$ 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], $$I_k[w]={\int }_{{\mathbb{T}}^n}{e}^{kH(Dw,x)}dx,{\int }_{{\mathbb{T}}^n}wdx=0.$$ If $u_{\mathrm{\infty }}$ 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 $u_{\mathrm{\infty }}$ is a critical subsolution that is strict outside $A$ and $$A=\{x\in {\mathbb{T}}^n:D{u}_{\mathrm{\infty }}(x)\text{ exists and }H(D{u}_{\mathrm{\infty }}(x),x)=c\}=\{x\in {\mathbb{T}}^n:u_{\mathrm{\infty }}(x)=u_{-}(x)\},$$ where $A$ is the projected Aubry set and $u_{-}$ is the backward weak KAM solution associated with $u_{\mathrm{\infty }}$. 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.