奥布里集的$L^\u221e$变分近似
$L^\infty$ Variational Approximation of the Aubry Set
浏览论文内容
中文总结 AI 辅助
该研究针对周期托内利哈密顿量，通过Evans引入的变分泛函的极小值序列的极限，刻画了投影奥布里集的$L^\u221e$变分近似，为近似奥布里集提供了数值定位原理。
中文摘要 AI 辅助
设$H\in C^\infty(\mathbb R^n\times\mathbb T^n)$为具有临界值$c$的周期托内利哈密顿量。对每个$k\in\mathbb N$，令$u_k$为Evans[7]引入的变分泛函的归一化极小值，该泛函为$I_k[w]=\int_{\mathbb T^n} e^{kH(Dw,x)}\\,dx$，约束为$\int_{\mathbb T^n}w\\,dx=0$。若$u_\infty$是序列$\{u_k\}$的某个子序列的一致极限，且马瑟商$({A}_M,\delta_M)$满足$H^1(A_M,\delta_M)=0$，则$u_\infty$是严格位于${A}$之外的临界下解，且${A}=\{x\in\mathbb T^n\\,:\\,Du_\infty(x)\text{存在且}H(Du_\infty(x),x)=c\}=\{x\in\mathbb T^n\\,:\\,u_\infty(x)=u_{-}(x)\}$，其中${A}$为投影奥布里集，$u_{-}$为与$u_\infty$相关的后向弱KAM解。特别地，根据Fathi--Figalli--Rifford[10]定理，当$n\leq3$时，该结论对$\mathbb T^n$上的所有光滑托内利哈密顿量成立。该刻画还提出了一种自然的数值定位原理，可通过$u_k$与其大时间后向拉克斯-奥莱尼克演化的近接触集来近似整个奥布里集。
英文摘要
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 $({A}_M,δ_M)$ satisfies $H^1( A_M,δ_M)=0$, 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 $n\leq3$. 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.
发表机构
- University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
- University of California at Irvine（加州大学尔湾分校）
机构由 AI 辅助整理，请以论文原文为准。
