On the Frechet differentiability of boundary integral operators in the inverse elastic scattering problem

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Abstract
This paper is concerned with the study of the Frechet differentiability properties of the operator connecting the scattered field with scatterer`s surface in the framework of the inverse elastic scattering problem. We adopt the integral equation approach, which transfers the solution of the inverse problem to the solution of a boundary integral equation of the second kind. We study the behaviour of the appeared integral operators and prove that they constitute Frechet differentiable operators. As we show, this result leads to the conclusion that the scattered elastic field is Frechet differentiable with respect to the boundary of the scatterer. Finally we present a characterization of the Frechet derivative of the scattered field as the solution of a direct scattering elastic problem with suitable Dirichlet boundary conditions.