# On regularity of the logarithmic forward map of electrical impedance tomography

## TL;DR

This paper investigates the mathematical properties of the logarithmic forward map in electrical impedance tomography, establishing its differentiability and boundedness characteristics, which are crucial for understanding inverse problems in this field.

## Contribution

It proves the continuous Fréchet differentiability of the logarithmic forward map and its inverse, providing new insights into the stability and structure of the inverse problem in EIT.

## Key findings

- The logarithmic forward map is continuously Fréchet differentiable.
- The difference between logs of two Neumann-to-Dirichlet maps is bounded.
- Results hold for both Neumann-to-Dirichlet and Dirichlet-to-Neumann maps.

## Abstract

This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of the Neumann-to-Dirichlet map is continuously Fr\'echet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fr\'echet derivative defines a bounded linear operator on the space of square integrable functions living on the domain boundary, although the logarithm of the Neumann-to-Dirichlet map itself is unbounded in that topology. In particular, it follows from the fundamental theorem of calculus that the difference between the logarithms of any two Neumann-to-Dirichlet maps is always bounded on the space of square integrable functions. All aforementioned results also hold if the Neumann-to-Dirichlet boundary map is replaced by its inverse, i.e. the Dirichlet-to-Neumann map.