Frechet differentiability of a field operator for scattering from an open screen

The inverse problem considered is the determination of the shape of a two-dimensional open screen from knowledge of the field on a curve for electromagnetic plane wave scattering. We extend Kress´s approach (1995) to the inverse problem of determining the shape of a two-dimensional open scatterer from knowledge of the scattered field on a curve. In particular, we investigate the Frechet differentiability of a field operator for scattering from an open screen with a boundary, as a prerequisite for the theoretical foundation of gradient methods or Newton type methods for the approximate solution of this nonlinear, improperly posed problem. The aim of this paper is to provide a proof for Frechet differentiability with respect to the boundary of an operator, which maps the boundary of an open screen onto the scattered field and to obtain expression of the derivatives.