双曲Aubry集的消失折扣问题的Sharp收敛速率

中文摘要 AI 辅助

设$H\in C^2(T^*M)$为闭连通流形上的Tonelli哈密顿量，$u_\lambda$满足方程$\lambda u_\lambda+H(x,Du_\lambda)=c(H)$于$M$中。本文研究$u_\lambda$向选定临界解$u_0$的收敛速率，假设提升的Aubry集是有限并$\widetilde{A}=\Gamma_1\sqcup\cdots\sqcup\Gamma_N$，其中每个$\Gamma_i$要么是双曲平衡点，要么是临界能级中双曲的周期轨道。本文证明$-C\lambda\le u_\lambda-u_0\le C\lambda|\log\lambda|$；若对每个$i$都满足$\int_M u_0\\,\mathrm d\mu_i=0$（$\mu_i$是与$\Gamma_i$关联的投影Mather测度），则$\\|u_\lambda-u_0\\|_\infty\le C\lambda$，特别地，若提升的Aubry集由单个双曲平衡点或单个双曲周期轨道构成，收敛速率为$O(\lambda)$。本文给出的例子表明，$O(\lambda)$和$O(\lambda|\log\lambda|)$这两种收敛速率均为最优；若无双曲性，有限阶退化例子给出阶为$\lambda^{1/(2r-1)}$（$r\ge2$）的下界，本文还构造了收敛速率任意慢的无穷阶退化例子。综上，这些结果构成了（据作者所知）Tonelli框架下消失折扣问题的首个系统定量理论。

英文摘要

Let $H\in C^2(T^*M)$ be a Tonelli Hamiltonian on a closed connected manifold and let $u_λ$ solve \[ λu_λ+H(x,Du_λ)=c(H)\qquad\text{in }M. \] We study the convergence rate of $u_λ$ to the selected critical solution $u_0$. Assume that the lifted Aubry set is a finite union $\widetilde{A}=Γ_1\sqcup\cdots\sqcupΓ_N$, where each $Γ_i$ is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove \[ -Cλ\le u_λ-u_0\le Cλ|\logλ|. \] Let $μ_i$ be the projected Mather measure associated with $Γ_i$. If \[ \int_M u_0\,\mathrm dμ_i=0\qquad \text{for every }i, \] then \[ \|u_λ-u_0\|_\infty\le Cλ. \] In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is $O(λ)$. We give examples showing that both convergence rates $O(λ)$ and $O(λ|\logλ|)$ are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order $λ^{1/(2r-1)}$ with $r\ge2$. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.

发表机构

• Waseda University（早稻田大学）
• Fudan University（复旦大学）

机构由 AI 辅助整理，请以论文原文为准。

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39 pages, 1 figure