Mathematical basis for a mixed inverse scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R2R2 as cross section. To this end, we solve a mixed scattering problem for the Helmholtz equation in R2R2 where the scattering object is a combination of a crack ΓΓ and a bounded obstacle DD, and we set suitable boundary conditions on ΓΓ and ∂D∂D (∂D∈C2∂D∈C2). The boundary integral method is employed to study the direct scattering problem. The mathematical basis is given to reconstruct the shape of the crack and the obstacle by using the linear sampling method. The numerical examples are given to show the viability of the method.