On reconstruction in the inverse conductivity problem with one measurement

Abstract

We consider an inverse problem for electrically conductive material occupying a domain $\Omega$ in ${\mathbb{R}}^2$. Let $\gamma$ be the conductivity of $\Omega$, and $D$ a subdomain of $\Omega$. We assume that $\gamma$ is a positive constant $k$ on $D$, $k\ne 1$ and is $1$ on $\Omega \setminus D$; both $D$ and $k$ are unknown. The problem is to find a reconstruction formula of $D$ from the Cauchy data on $\partial \Omega$ of a non-constant solution $u$ of the equation $\nabla \cdot \gamma \nabla u=0$ in $\Omega$. We prove that if $D$ is known to be a convex polygon such that $\text{diam}D<\text{dist}(D,\partial \Omega )$, there are two formulae for calculating the support function of $D$ from the Cauchy data.

arXiv comment10 pages

Cited by in corpus (17)

• Monotonicity and enclosure methods for the p-Laplace equation
• Radiating and non-radiating sources in elasticity
• Enclosure method for the p-Laplace equation
• An inverse source problem for the heat equation and the enclosure method
• Monotonicity Principle in Tomography of Nonlinear Conducting Materials
• Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval
• The Monotonicity Principle for Magnetic Induction Tomography
• Extracting discontinuity using the probe and enclosure methods
• Inverse obstacle scattering problems with a single incident wave and the logarithmic differential of the indicator function in the Enclosure Method
• On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operator
• The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Data
• Extraction formulae for an inverse boundary value problem for the equation $\nabla \cdot (\sigma -i\omega \epsilon )\nabla u=0$
• Uniqueness for time-dependent inverse problems with single dynamical data
• A note on the enclosure method for an inverse obstacle scattering problem with a single point source
• The $p_0$-Laplace "Signature" for Quasilinear Inverse Problems
• The inverse obstacle problem for nonlinear inclusions
• Inverse time-harmonic electromagnetic scattering from coated polyhedral scatterers with a single far-field pattern