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Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks. (English) Zbl 1139.45003
Authors’ summary: For the problem to determine the shape of a perfectly conducting inclusion within a conducting homogeneous host medium from overdetermined Cauchy data on the accessible exterior boundary, that is, for an inverse Dirichlet boundary value problem, recently R. Kress and W. Rundell [Inverse Probl. 21, No. 4, 1207–1223 (2005; Zbl 1086.35139)] suggested a new inverse algorithm based on nonlinear integral equations arising from the reciprocity gap principle. The present paper extends this approach to the case of a perfectly insulating inclusion and the case of a perfectly conducting crack. The mathematical foundations of these extensions are provided and numerical examples illustrate the feasibility of the method.

45G10Nonsingular nonlinear integral equations
47G10Integral operators
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35R30Inverse problems for PDE