# An inverse conductivity problem in multifrequency electric impedance tomography

## TL;DR

This paper investigates the inverse problem of identifying an inclusion's shape within a medium using multifrequency electrical impedance tomography, establishing stability and uniqueness results through boundary measurement analysis.

## Contribution

It introduces a logarithmic stability estimate for the inverse shape problem and reduces it to boundary determination from a single measurement, advancing theoretical understanding.

## Key findings

- Established a logarithmic stability estimate.
- Proved uniqueness of the inclusion shape.
- Provided an eigenfunction expansion of the solution.

## Abstract

We deal with the problem of determining the shape of an inclusion embedded in a homogenous background medium. The multifre-quency electrical impedance tomography is used to image the inclusion. For different frequencies, a current is injected at the boundary and the resulting potential is measured. It turns out that the potential solves an elliptic equation in divergence form with discontinuous leading coefficient. For this inverse problem we aim to establish a logarithmic type stability estimate. The key point in our analysis consists in reducing the original problem to that of determining an unknown part of the inner boundary from a single boundary measurment. The stability estimate is then used to prove uniqueness results. We also provide an expansion of the solution of the BVP under consideration in the eigenfunction basis of the Neumann-Poincar{\'e} operator associated to the Neumann-Green function.