On regularized full- and partial-cloaks in acoustic scattering

Abstract

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.