Part VI Example Problems

An object is cloaked if its presence cannot be detected by an observer using electromagnetic or other forms of imaging. In fact, the observer should not notice that cloaking is even occurring. Cloaking has a long history in science fiction, but recent developments have put the idea on a firmer mathematical and physical basis. Suppose that an observer seeks to image some bounded region Ω in space. This is to be accomplished by injecting energy into Ω from the outside, then observing the response: that is, the energy that comes out of Ω. An example of this process is radar: the injected energy consists of electromagnetic waves and the observed response consists of the waves reflected by objects in the region. Many other types of energy can be used to form images, for example, acoustic (sonar), mechanical, electrical, or thermal. In each case energy is injected into Ω, response data is collected, and from this information an image may be formed by solving an inverse problem. To cloak an object the relevant physical properties of Ω must be altered so that the energy “flows around” the object, as if the object were not there. The challenge is to do this in a way that is mathematically rigorous and physically implementable. One successful approach to cloaking is based on the idea of transformation optics, which we will now discuss in the context of impedance imaging. In impedance imaging, the bounded region Ω ⊂ Rn to be imaged consists of an electrically conductive medium, with n = 2 or n = 3. Let the vector x represent position in a Cartesian coordinate system, let u = u(x) represent the electrical potential inside Ω, and let J = J(x) represent the electric current flux in Ω. We assume a linear relation J = −σ∇u (a form of Ohm’s law), where for each x the quantity σ = σ(x) is a symmetric positive-definite n×n matrix. The matrix σ is called the conductivity of Ω and dictates how variations in the potential induce current flow. If σ = γI for some scalar function γ(x) > 0 (I is the n×n identity matrix), then the conductivity is said to be isotropic : there is no preferred direction for current flow. Otherwise, σ is said to be anisotropic. In impedance imaging the goal is to recover the internal conductivity of Ω using external measurements. Specifically, to image a region Ω, the observer applies an electric current flux density g with ∫ ∂Ω g ds = 0 to the boundary ∂Ω. If charge is conserved inside Ω, then ∇ · J = 0 in Ω and the potential u satisfies the boundary-value problem ∇ · σ∇u = 0 in Ω, (1) (σ∇u) ·n = g on ∂Ω. (2)