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

Abstract

We consider the inverse problem to recover a part Γc 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 ∂D\Γc when u satisfies a homogeneous impedance boundary condition on Γc. 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.