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

TL;DR

This paper investigates electromagnetic wave diffraction by a half-plane with a Lipschitz inclusion, establishing solvability and solutions in generalized Sobolev spaces using potential operators and integral equations.

Contribution

It introduces a novel approach to boundary value problems with Lipschitz inclusions, proving solvability and representing solutions via potential operators.

Findings

Proved solvability of Dirichlet and Neumann problems.

Derived solutions using potential type operators.

Reduced boundary problems to second-kind integral equations.

Abstract

This paper considers to the problems of diffraction of electromagnetic waves on a half-plane, which has a finite inclusion in the form of a Lipschitz curve. The diffraction problem formulated as boundary value problem for Helmholtz equations and boundary conditions Dirichlet or Neumann on the boundary, as well as the radiation conditions at infinity. We carry out research on these problems in generalized Sobolev spaces. We use the operators of potential type, that by their properties are analogs of the classical potentials of single and double layers. We proved the solvability of the boundary value problems of Dirichlet and Neumann. We have obtained solutions of boundary value problems in the form of operators of potential type. Boundary problems are reduced to integral equations of the second kind.