The potential method for the reactance wave diffraction problem in a scale of spaces

TL;DR: In this paper, a screen type boundary value prob- lem arising from the wave diraction problem with a reactance condition is considered, where both cases of a complex and a pure real wave number are ana- lyzed.

Abstract: This paper is concerned with a screen type boundary value prob- lem arising from the wave diraction problem with a reactance condition. We consider the problem in a weak formulation within Bessel potential spaces, and where both cases of a complex and a pure real wave number are ana- lyzed. Using the potential method, the boundary value problem is converted into a system of integral equations. The invertibility of the corresponding matrix pseudodierential operator is shown in appropriate function spaces which allows the conclusion about the existence and uniqueness of a weak solution to the original problem. Higher regularity properties of solutions are also proved to exist in some scale of Bessel potential spaces, upon the corresponding smoothness improvement of given data. In particular, the C fi -smoothness of solutions in a neighbourhood of the screen edge is estab- lished with arbitrary fi < 1 in the two-dimensional case and fi < 1=2 in the three-dimensional case.