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

Abstract: Let $$H\in C^{\infty(\mathbb} R^{n\times\mathbb} T<sup>n)$$ 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 $({A}<em>M,δ_M)$ satisfies $H¹⁽ A_M,δ_M)=0$, then $u</em>\infty$ 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 $\mathbb T<sup>n$ when $n\leq3$. 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.