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Fréchet differentiability of the far field operator for scattering from a crack

The inverse problem we consider is to determine the shape of a two-dimensional crack from a knowledge of the far field pattern for the scattering of incident time-harmonic acoustic or electromagnetic plane waves. In particular, we will investigate the Frechet differentiability of the far field operator with respect to the boundary as a prerequisite for the foundation of Newton type methods for the approximate solution of this nonlinear and improperly posed problem.