AI 中文总结
该研究证明了无穷维重排不变凸哈密顿-雅可比方程周期均匀化的最优收敛率为$O(\varepsilon)$，推广了有限维结果，并通过新方法克服了无穷维空间的拓扑限制，且验证了该收敛率的尖锐性。
AI 中文摘要
我们证明了环面上无穷多不可区分粒子构成的系统所对应的凸哈密顿-雅可比方程的周期均匀化最优收敛率为$O(\varepsilon)$，该结论在初始数据仅依赖平均构型的假设下成立。这一结果推广了有限维情形的成果[1]，后者基于拉格朗日作用度量的长时间行为和曲线手术论证。由于极小曲线存在于无穷维希尔伯特空间中，而该空间缺乏局部紧性和有限维拓扑，因此无法直接应用上述工具。我们通过切割极小曲线的有限维平均时间投影，并利用周期性和重排不变性诱导的紧商在希尔伯特空间中粘合提升后的片段，克服了这一困难。最后，我们给出一个例子表明该收敛率是尖锐的。
英文摘要
We prove the optimal convergence rate $O(\varepsilon)$ for periodic homogenization of convex Hamilton-Jacobi equations arising from infinite systems of indistinguishable particles on the torus, under the assumption that the initial data depend only on the mean configuration. This extends the finite-dimensional result [1], which is based on the large-time behavior of the Lagrangian action metric and a curve-surgery argument. Here, these tools cannot be applied directly because minimizing curves live in an infinite-dimensional Hilbert space, where local compactness and finite-dimensional topology are unavailable. We overcome this difficulty by cutting the finite-dimensional mean-time projection of a minimizing curve and gluing the lifted pieces in the Hilbert space using the compact quotient induced by periodicity and rearrangement invariance. We conclude with an example showing that this rate is sharp.
Comments29 pages, 1 figure
