On the characterization of the Fréchet derivative with respect to a Lipschitz domain of the acoustic scattered field (Q1307267)

The characterization of the Fréchet derivative with respect to a domain of a given obstacle is presented. The results are formulated for acoustic scattering problems with a sound-soft obstacle, a sound-hard obstacle, or lossy boundary conditions. It is assumed that, the shape of the scatter is only Lipschitzian, and that the admissible perturbations are the elements of the space of functions with continuous derivatives in \(\mathbb{R}^3\). The proof of the theorem about the characterization of the derivative of the scattered field with respect to the domain as the solution of a particular direct acoustic problem is based on the validity of the chain rule in infinite-dimensional spaces and the classical trace theorems. The results of this paper can be used for reducing the complexity of the solution of inverse scattering problems by iterative methods.