Recovering the boundary corrosion from electrical potential distribution using partial boundary data

Jijun Liu; Gen Nakamura

doi:10.3934/ipi.2017024

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2017, Volume 11, Issue 3: 521-538. Doi: 10.3934/ipi.2017024

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Recovering the boundary corrosion from electrical potential distribution using partial boundary data

Jijun Liu^1, , and
Gen Nakamura^2,

1.
School of Mathematics, Southeast University, Nanjing 210096, China
2.
Department of Mathematics, Inha University, Incheon 22212, Korea

* Corresponding author: Prof. Dr. Jijun Liu, email: jjliu@seu.edu.cn
* Corresponding author: Prof. Dr. Jijun Liu, email: jjliu@seu.edu.cn

Received: December 31, 2015

Revised: January 31, 2017

Published: August 2017

This work is supported by NSFC grant No.11531005, No.11421110002, and the Fundamental Research Funds for the Central Universities (3207017455).

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Abstract

We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specified on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Hölder type based on some a priori assumptions on the unknown impedance and the electrical current input specified in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data.

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Mathematics Subject Classification: 35J05, 35J25, 35R25, 35R30.

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Figure 1. Geometric configuration for our direct problem (left) and the extension for dealing with the corner point A (right).

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