Uniqueness for the electrostatic inverse boundary value problem with piecewise constant anisotropic conductivities

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion $\Sigma$ of the boundary $\partial\Omega$. We prove that anisotropic conductivities that are \textit{a-priori} known to be piecewise constant matrices on a given partition of $\Omega$ with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map

Invisibility and Inverse Problems

This survey of recent developments in cloaking and transformation optics is an expanded version of the lecture by Gunther Uhlmann at the 2008 Annual Meeting of the American Mathematical Society.Comment: 68 pages, 12 figures. To appear in the Bulletin of the AM

Limiting Carleman weights and anisotropic inverse problems

In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic X-ray transform. Earlier results in dimension $n \geq 3$ were restricted to real-analytic metrics.Comment: 58 page

Full-wave invisibility of active devices at all frequencies

There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or "cloaking") from observation by electromagnetic (EM) waves. Here, we prove invisibility, with respect to solutions of the Helmholtz and Maxwell's equations, for several constructions of cloaking devices. Previous results have either been on the level of ray tracing [Le,PSS] or at zero frequency [GLU2,GLU3], but recent numerical [CPSSP] and experimental [SMJCPSS] work has provided evidence for invisibility at frequency $k\ne 0$. We give two basic constructions for cloaking a region $D$ contained in a domain $\Omega$ from measurements of Cauchy data of waves at \p \Omega; we pay particular attention to cloaking not just a passive object, but an active device within $D$, interpreted as a collection of sources and sinks or an internal current.Comment: Final revision; to appear in Commun. in Math. Physic