重探 Neumann 筛问题
The Neumann sieve problem revisited
AI总结：
本文重审含周期窄通道的 Neumann 筛均质化问题，在更一般通道假设下给出强预解收敛的速率与谱距估计，并补齐二维情形的遗留开放问题。
摘要：
设 $Ω\subset\mathbb{R}^n$ 为一个区域，$Γ$ 是与其相交的超平面。令 $\varepsilon>0$，并令 $Ω_\varepsilon=Ω\setminus\overline{Σ_\varepsilon}$，其中 $Σ_\varepsilon$（“筛”）是 $Γ$ 的一个 $\varepsilon$-邻域，并被许多狭窄通道穿孔。当 $\varepsilon\to0$ 时，通道数量趋于无穷，而其横截面直径趋于零。对于相同、笔直、周期分布并经过适当缩放的通道情形，T. Del Vecchio（1987）证明了 $Ω_\varepsilon$ 上的 Neumann Laplace 算子在强预解意义下收敛于 $Ω\setminusΓ$ 上、在 $Γ$ 上满足所谓 $δ'$-条件的 Laplace 算子。我们将改进这一结果：以多种算子范数导出收敛速率估计，并给出谱之间距离的估计。我们对通道施加的假设相当一般。对于 $n=2$，T. Del Vecchio 的结果并不完整，某些情形仍是开放问题，而在本工作中我们将填补这些空白。
英文摘要：
Let $Ω\subset\mathbb{R}^n$ be a domain, $Γ$ be a hyperplane intersecting it. Let $\varepsilon>0$, and $Ω_\varepsilon=Ω\setminus\overline{Σ_\varepsilon}$, where $Σ_\varepsilon$ ("sieve") is an $\varepsilon$-neighbourhood of $Γ$ punctured by many narrow passages. When $\varepsilon\to0$, the number of passages tends to infinity, while the diameters of their cross-sections tend to zero. For the case of identical straight periodically distributed and appropriately scaled passages T. Del Vecchio (1987) proved that the Neumann Laplacian on $Ω_\varepsilon$ converges in a strong resolvent sense to the Laplacian on $Ω\setminusΓ$ subject to the so-called $δ'$-conditions on $Γ$. We will refine this result by deriving estimates on the rate of convergence in terms of various operator norms, and providing the estimate for the distance between the spectra. The assumptions we impose on the passages are rather general. For $n=2$ the results of T. Del Vecchio are not complete, some cases remain as open problems, and in this work we will fill these gaps.
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