关于声散射中的正则化全斗篷和部分斗篷
On regularized full- and partial-cloaks in acoustic scattering
摘要：
这项工作的目的是通过变换光学方法获得[28]中正则化全和部分斗篷的定性收敛结果的精确定量估计。设$Γ_0$是$\mathbb{R}^3$中的紧集，$Γ_δ$是$δ\in\mathbb{R}_+$的$Γ_0$的$δ$ -邻域. $Γ_δ$表示用于爆破构造的虚拟域。通过适当设计的有损层，它表明，如果生成集$Γ_0$是一个通用的曲线，那么一个将有一个近似的全斗篷$δ^2$内的完美的全斗篷;而如果$Γ_0$是一个开放的子集在一个平面上的封闭，那么一个将有一个近似的部分斗篷内$δ$，其完美的副本。估计得出的是独立的内容被掩盖，也就是说，隐身设备几乎能够掩盖任意内容。此外，作为一个重要的副产品，我们的论点允许放松[28]中对$Γ_0$的凸性要求，这对其中的Mosco收敛论点至关重要。
英文摘要：
The aim of this work is to derive sharp quantitative estimates of the qualitative convergence results developed in [28] for regularized full- and partial-cloaks via the transformation-optics approach. Let $Γ_0$ be a compact set in $\mathbb{R}^3$ and $Γ_δ$ be a $δ$-neighborhood of $Γ_0$ for $δ\in\mathbb{R}_+$. $Γ_δ$ represents the virtual domain used for the blow-up construction. By incorporating suitably designed lossy layers, it is shown that if the generating set $Γ_0$ is a generic curve, then one would have an approximate full-cloak within $δ^2$ to the perfect full-cloak; whereas if $Γ_0$ is the closure of an open subset on a flat surface, then one would have an approximate partial-cloak within $δ$ to its perfect counterpart. The estimates derived are independent of the contents being cloaked; that is, the cloaking devices are capable of nearly cloaking an arbitrary content. Furthermore, as a significant byproduct, our argument allows the relaxation of the convexity requirement on $Γ_0$ in [28], which is critical for the Mosco convergence argument therein.
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