On the solvability of the scalar monochromatic wave diffraction problem on an inhomogeneous solid with specific transmission conditions

Abstract

Background. The purpose of this work is to study the 3-D scalar problem of scattering a plane wave from an inhomogeneous solid covered with an infinitely thin layer of graphene. Material and methods. The transmission problem for the Helmholtz equation with special boundary conditions is considered; this problem, which has a unique solution, is reduced to a weakly singular integral equation; the operator of the equation is investigated in a Sobolev space. Results. The diffraction problem is reduced to an integral equation; the equivalence of the integral equation and the transmission is proved; Fredholm property and continuous invertibility of the operator of the integral equation are proved. Conclusions. Important results on existence and uniqueness of the solution to the diffraction problem are obtained and can be used for validation of projection methods for numerical solving of the diffraction problem.

Keywords

• diffraction problem
• uniqueness of the solution of the transmission problem
• integral equation
• sobolev spaces
• fredholm invertible operator