American Institute of Mathematical Sciences

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May  2007, 1(2): 229-245. doi: 10.3934/ipi.2007.1.229

Integral equations for inverse problems in corrosion detection from partial Cauchy data

Fioralba Cakoni 1, and  Rainer Kress 2,

1. 

Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553, United States
2. 

Institute für Numerische und Angevandte Mathematik, Universität Göttingen, Germany

Received  August 2006 Revised  October 2006 Published  April 2007

We consider the inverse problem to recover a part $\Gamma_c$ of the boundary of a simply connected planar domain $D$ from a pair of Cauchy data of a harmonic function $u$ in $D$ on the remaining part $\partial D\setminus \Gamma_c$ when $u$ satisfies a homogeneous impedance boundary condition on $\Gamma_c$. Our approach extends a method that has been suggested by Kress and Rundell [17] for recovering the interior boundary curve of a doubly connected planar domain from a pair of Cauchy data on the exterior boundary curve and is based on a system of nonlinear integral equations. As a byproduct, these integral equations can also be used for the problem to extend incomplete Cauchy data and to solve the inverse problem to recover an impedance profile on a known boundary curve. We present the mathematical foundation of the method and illustrate its feasibility by numerical examples.
Keywords: Inverse boundary value problem, integral equations, partial boundary measurements, impedance boundary condition..
Mathematics Subject Classification: 65J20, 65J1.
Citation: Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229
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