The Scattering Problem Of Mixed Transmission Obstacles

In this paper, we mainly study the scattering problem of electromagnetic waves by two disjoint perfectly conducting infinite cylinders in R2.LetD1 and D2denote the cross section with smooth bounded boundary such that R2\(D1 U D2) is connected. D1is transmission and D2 is not transmission,In the end. The scattering problem can be formulated as a mixed boundary value by Helmholtz equation in R2, Given ,find functions u ∈ H1(R2\(D1 ∪ D2)) and v ∈ H1(D1) such that and the scattered field u satisfy the well-known Sommerfeld Radiation condition:uniformly in x =x/|x| with r=|x|. For the above problem, we mainly research the existence and uniqueness of the solution to the scattering problem. Also,we research the inverse scattering problem. The uniqueness of the solution to the scattering problem can be directly obtained by Rellich lemma. We obtain the existence of the solution to the problem by boundary integral equation methods. That is, we establish a boundary integral system by using the Green’s representation formula and potential theory. Then by the Fredholm theory, we obtain the existence and uniqueness of the solution to the boundary integral system, and then the existence of the solution to the original direct scattering problem are obtained. For the inverse scattering problem, we can use the liner sampling method to solve it.