Point Measurements for a Neumann-to-Dirichlet Map and the Calderón Problem in the Plane

Abstract

This work considers properties of the Neumann-to-Dirichlet map for the conductivity equation under the assumption that the conductivity is identically one close to the boundary of the examined smooth, bounded and simply connected domain. It is demonstrated that the so-called bisweep data, i.e., the (relative) potential differences between two boundary points when delta currents of opposite signs are applied at the very same points, uniquely determine the whole Neumann-to-Dirichlet map. In two dimensions, the bisweep data extend as a holomorphic function of two variables to some (interior) neighborhood of the product boundary. It follows that the whole Neumann-to-Dirichlet map is characterized by the derivatives of the bisweep data at an arbitrary point. On the diagonal of the product boundary, these derivatives can be given with the help of the derivatives of the (relative) boundary potentials at some fixed point caused by the distributional current densities supported at the same point, and thus such point measurements uniquely define the Neumann-to-Dirichlet map. This observation also leads to a new, truly local uniqueness result for the so-called Calderón inverse conductivity problem.2010 Mathematics Subject Classification. 35R30, 35Q60.

Recent progress in the Calderón problem with partial data

“…An early result for the linearized problem in an annular domain in R 2 with no measurements on the inner boundary is in [Hä98]. In the case when the conductivity is known near the boundary, the partial data problem can be reduced to the full data problem [Is88], [AU04], [Fa07], [Be09], [AK12], [HPS12]. Also, we remark that in the corresponding problem for the wave equation, it has been known for a long time (see [KKL01]) that measuring the Dirichlet and Neumann data of waves on an arbitrary open subset of the boundary is sufficient to determine the coefficients uniquely up to natural gauge transforms.…”

Point Electrode Problems in Piecewise Smooth Plane Domains

“…Theorem 2.1 allows one to extend the partial data result [21] for Calderón problem to piecewise smooth plane domains: This can be generalized as follows Theorem 2.3. Let Γ, Ξ be (possibly disjoint) countably infinite subsets of ∂D.…”

“…The following two lemmas state the relationship between the weak formulations presented above and a distributional Sobolev space formulation based on trace theorems, utilized in, e.g., [21]. Proof.…”

Construction of Indistinguishable Conductivity Perturbations for the Point Electrode Model in Electrical Impedance Tomography

“…Remark 4.1. In [17,31] it has been shown that M (σ ε ) = 0 if and only if σ ε = σ 0 (≡ 1) when the number of electrodes is countably infinite in two dimensions. Here, we have demonstrated that one can constructively find σ ε ≡ σ 0 such that M (σ ε ) = 0 in case the number of electrodes is finite.…”

“…All these articles assume the Cauchy data for (1) are known on all of ∂D, but the partial data problem of having access only to some subset(s) of ∂D has also been tackled by many mathematicians; see, e.g., [19,20] and the references therein. From our view point, the most important uniqueness results are presented in [17,31] where it is shown that any two-dimensional conductivity that equals a known constant in some interior neighborhood of ∂D is uniquely determined by the PEM measurements with countably infinite number of electrodes. This manuscript complements [17,31] by constructively showing that for an arbitrary, but fixed and finite, electrode configuration there exists a perturbation of the unit conductivity that is invisible for the EIT measurements in the framework of the PEM.…”