Conformal mapping and an inverse impedance boundary value problem

Abstract

Akduman, Haddar and Kress [1, 12, 15] proposed a conformal mapping approach for solving inverse boundary value problems for the two-dimensional Laplace equation in a doubly connected domain D with interior boundary curve Γ0 and exterior boundary curve Γ1. The inverse problem consists of reconstructing the interior boundary Γ0 from the Cauchy data on Γ1 of a harmonic function satisfying a homogeneous Dirichlet or Neumann boundary condition on the unknown interior boundary curve Γ0. The reconstruction method consists of two parts: In the first step, by successive approximations a nonlocal and nonlinear ordinary differential equation is solved to determine the boundary values of a holomorphic function Ψ on the outer boundary circle C1 of an annulus B. Then in the second step via regularizing a Laurent expansion in the sense of Tikhonov an ill-posed Cauchy problem is solved to determine Ψ in the annulus and the unknown Γ0 as the image Ψ(C0) of the interior boundary circle C0 of B. The present paper extends this approach to the case of a homogeneous impedance boundary condition. The analysis and the numerical implementation of the method differ from the limiting cases of the Dirichlet and Neumann conditions since the impedance problem in the annulus B that is associated with the impedance problem in the original domain D depends on the conformal map Ψ.