# Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u=0$

## TL;DR

This paper develops extraction formulae for an inverse boundary value problem involving a complex conductivity equation, using novel methods based on exponentially growing solutions and Mittag-Leffler functions to identify discontinuities.

## Contribution

It introduces new formulas and methods for determining the location of discontinuities in conductivity and permittivity from boundary measurements, extending the enclosure method with Mittag-Leffler functions.

## Key findings

- Derived formulas for locating discontinuities in  and .
- Extended the enclosure method using Mittag-Leffler functions.
- Provided theoretical framework for inverse boundary problems.

## Abstract

We consider an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u=0$ in a given bounded domain $\Omega$ at a fixed $\omega>0$. $\sigma$ and $\epsilon$ denote the conductivity and permittivity of the material forming $\Omega$, respectively. We give some formulae for extracting information about the location of the discontinuity surface of $(\sigma,\epsilon)$ from the Dirichlet-to-Neumann map. In order to obtain results we make use of two methods. The first is the enclosure method which is based on a new role of the exponentially growing solutions of the equation for the background material. The second is a generalization of the enclosure method based on a new role of Mittag-Leffler's function.