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

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.

arXiv comment(2.3) on page 7, fixed the same as the final version: put $+$ in front of the first integral on line 8 down; replaced $u_1-u_2$ with $u_2-u_1$ in the first integral on line 10 down

References in corpus (1)

• On reconstruction in the inverse conductivity problem with one measurement

Cited by in corpus (1)

• Extracting discontinuity using the probe and enclosure methods