Crack impedance-Dirichlet boundary value problems of diffraction in a half-plane

We study two wave diffraction problems modeled by the Helmholtz equation in a half-plane with a crack characterized by Dirichlet and impedance boundary conditions. The existence and uniqueness of solutions is proved by an appropriate combination of general operator theory, Fredholm theory, potential theory and boundary integral equation methods. This combination of methods leads also to integral representations of solutions. Moreover, in Sobolev spaces, a range of smoothness parameters is obtained in which the solutions of the problems are valid