Integral equations for shape and impedance reconstruction in corrosion detection

Abstract

In a simply connected planar domain D a pair of Cauchy data of a harmonic function u is given on an accessible part of the boundary curve, and on the non-accessible part u is supposed to satisfy a homogeneous impedance boundary condition. We consider the inverse problems to recover the non-accessible part of the boundary or the impedance function. Our approach extends the method proposed by Kress and Rundell (2005 Inverse Problems 21 1207–23) for the corresponding problem to recover the interior boundary curve of a doubly connected planar domain and can be considered complementary to the potential approach developed by Cakoni and Kress (2007 Inverse Problems Imaging 1 229–45). It is based on a system of nonlinear and ill-posed integral equations which is solved iteratively by linearization. We present the mathematical foundation of the method and, in particular, establish injectivity for the linearized system at the exact solution when the impedance function is known. Numerical reconstructions will show the feasibility of the method.