Point measurements for a Neumann-to-Dirichlet map and the Calder\'on problem in the plane

TL;DR

This paper demonstrates that point measurements of potential differences, called bisweep data, uniquely determine the Neumann-to-Dirichlet map and lead to a new local uniqueness result for the Calderón inverse conductivity problem in the plane.

Contribution

It introduces a novel approach using bisweep data and their derivatives to uniquely identify the Neumann-to-Dirichlet map and the conductivity in a local setting in two dimensions.

Findings

Bisweep data extend as a holomorphic function in 2D.

Derivatives of bisweep data at a point determine the Neumann-to-Dirichlet map.

Point measurements can uniquely identify the conductivity locally.

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 Calder\'on inverse conductivity problem.