Abstract
In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks though non-degenerate are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.
Highlights
Introduction and preliminariesThe topic of cloaking has long been fascinating and has recently attracted a lot of attention within the mathematical and broader scientific community
Before we provide the definition of cloaking, since the problem of cloaking is essentially that of non-uniqueness, we digress a bit and mention some previous work regarding uniqueness in the inverse problem for the equation considered in (2)
We propose a change of variable scheme, similar to the one in [24, 25] and show how one can, in principle, obtain perfect cloaking using singular change of variables in the context of the equation considered in (2) and approximate cloaking using a regular change of variables
Summary
Volume 12, No 2 (2018), 461–491 where ν is the outer unit normal to ∂Ω. It is shown in Appendix A that one can define the Dirichlet-to-Neumann (DN) map for the equation (2) in a weak sense as a mapping. The inverse problem is to recover the quasi-linear coefficient matrix A(x, t), called the conductivity, from the knowledge of ΛA. That is, A(x, t) = a(x, t)IN×N where IN×N denotes the N × N identity matrix and a is a positive C2,γ(Ω × R) function having a uniform positive lower bound on Ω × [−s, s] for each s > 0, the Dirichlet to Neumann map. Λa determines uniquely the scalar coefficient a(x, t) on Ω × R This uniqueness result was first proved for the linear case (i.e when a is a function of x alone) in the fundamental paper [47] for N ≥ 3 for, in [41] for N = 2; and in [44] for the quasi-linear case. Aij(x, u) ∂yk ∂xi ∂yl ∂xj det ∂y dy
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