Research On The Two Cracks Reconstruction With The Linear Sampling Method

In this paper, we consider the scattering of the time-harmonic acoustic and electromagnetic waves with two infinite cylinders (one cylinder includes the other one). we assume that the cylinders are perfect conductors, the outside cylinder is coated on one side by a material with the surface impedance λ, the horizontal sections are two open arcs Γ1and Γ2, finally it comes down to discuss the following problems:given f∈H1/2(Γ1), h∈H1/2(Γ2), j∈H-1/2{Γ2), find u∈Hloc1(R2\Γ1U Γ2)) satisfying the following condition:For the direct scattering problem, that is to give Γ1and Γ2, find the solution which satisfy the condition (*). we can arrive at existence and uniqueness of a solution by using the Green formula and boundary integral equation approach. For the inverse scattering problem, that is given far-field information, how to reconstruct the shape Γ1and Γ2. On the basis of the research results of the direct scattering problem, we give the mathematical basis which is developed by using the linear sampling method to reconstruct Γ1and Γ2, that is to study the related properties of the corresponding far field operator, to provide the theoretical support for the implementation of the linear sampling method.