Scattering Problems Of Transmissible Obstacle With A Mixed Crack

In this paper, we consider the scattering of a time harmonic electromagnetic plane wave penetrated through a perfectly conducting infinite cylinder with a pene-trable obstacle D(αD∈C2) in R2and a mixture of an open arc Г as a cross section. Г divided into two parts, Г1and F2. One of the two parts is coated on one side by a material with surface impedance λ.Different boundary conditions are given on Г and αD.Dirichlet-Impedance type boundary condition on Г1, Dirichle boundary value condition on Г2and transmission boundary condition on αD lead to the following problem:Given f1∈H1/2(αD),91∈H-1/2(αD),f2∈H1/2(Г2), g2∈H1/2(Г1),h∈H-1/2(Г1) such that The scattered field u is required to satisfy the well-known Sommerfeld Radiation condition uniformly in x=x/|x|A with r=|x|. We are dedicated to research whether there exists a unique solution for our problem. By the Rellich’s lemma, we can obtain the uniqueness. Our main contribution is to establish a boundary integral system ac-cording to Green’s representation formula and potential theory and then prove that the Fredholm theory can be applied to the integral operator.Then the solvability of original problem is obtained.