Boundary-Value Problems for the Helmholtz Equation for a Half-Plane with a Lipschitz Inclusion

Abstract

I consider the problems of diffraction of electromagnetic waves on a half-plane, which has a finite inclusion in the form of a Lipschitz curve. Boundary value problems, modeling the process of wave diffraction, are constructed in the form of Helmholtz equations and boundary conditions on the boundary, formulated in terms of traces, as well as the radiation conditions at infinity. I carry out research on these problems in generalized Sobolev spaces. I proved the solvability of the boundary value problems of Dirichlet and Neumann. I have obtained solutions of boundary value problems in the form of functions that by their properties are analogs of the classical potentials of single and double layers. Boundary problems are reduced to integral equations of the second kind.