Existence and uniqueness in anisotropic conductivity reconstruction with Faraday's law

Yong-Jung Kim; Min-Gi Lee

doi:10.3934/ipi.2022051

Article Contents

2023, Volume 17, Issue 2: 441-462. Doi: 10.3934/ipi.2022051

This issue Previous Article Source and metric estimation in the eikonal equation using optimization on a manifold Next Article Scattering in a partially open waveguide: The inverse problem

Existence and uniqueness in anisotropic conductivity reconstruction with Faraday's law

Yong-Jung Kim^1, and
Min-Gi Lee^2, ,

1.
Department of Mathematical Sciences, KAIST, Republic of Korea
2.
Department of Mathematics, Kyungpook National University, Republic of Korea

^*Corresponding author: Min-Gi Lee
^*Corresponding author: Min-Gi Lee

Received: April 2021

Revised: September 2022

Early access: October 2022

Published: April 2023

This research was supported by Kyungpook National University Research Fund, 2018.

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Abstract

We show that three sets of internal current densities are the right amount of data that give the existence and the uniqueness at the same time in reconstructing an anisotropic conductivity in two space dimensions. The curl free equation of Faraday's law is taken instead of the usual divergence free equation of the electrical impedance tomography. Boundary conditions related to given current densities are introduced which complete a well determined problem for conductivity reconstruction together with Faraday's law.

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Mathematics Subject Classification: Primary: 35R30, 94A08; Secondary: 35L04.

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Figure 1. Configuration of $ \Gamma^-_j $ $ j = 1,2 $ on $ \partial U $

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Figure 2. Causal structure on $ T_z $ is illustrated. The closure of the open positive sector is the nonnegative non-timelike sector $ K_z $

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Figure 3. An example of a monotone and non-timelike curve that consists of portions of characteristic curves $ C^+_1(z_1) $, $ C^-_2(z_2) $, $ C^+_1(z_3) $, $ C^-_2(z_4) $. Red arrows are parallel to $ {{\bf N}}_1 $ and blue arrows are parallel to $ {{\bf N}}_2 $ at respective points

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Figure 4. Configurations for the Goursat problem and the Triangle problem. The red curves are in $ 1 $-characteristic family and the blue curves are in $ 2 $-characteristic family

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