Wave diffraction by a strip with first and second kind boundary conditions: the real wave number case

Abstract

We prove unique existence of solution for a class of plane wave diffraction problems by a strip with first and second kind boundary conditions. This is done in a Bessel potential framework, and for a real (non- complex) wave number. At the end, results about the regularity (and data dependence) of the solution are exhibited upon the initial setting and the boundary parameters. In view of existence and uniqueness of solutions, we provide in this paper a rigorous account of possible first and second kind boundary value problems in the interval/strip for the Helmholtz equation with a real wave number, and within Bessel potential spaces. We find out a subset of the initial general boundary parameters for which the system of boundary integral equations (originated by the class of problems in study) has exactly one solution in the appropriated Bessel potential spaces, and for all real (non-zero) wave numbers. Additionally, in the final part of the paper, an improvement of the smoothness space parameters is exhibited for which the existence and uniqueness of solution (and continuous dependence on the data) is still guaranteed. The class of problems considered in the present work includes the famous Rawlins problem for the strip. It should be mentioned that the Rawlins problem for the half-plane was considered in detail in (19), also in the real wave number case. In (19) the main strategy was to consider such a case as the limit of the complex wave number situation by using a sort of a limiting absorption principle together with some symmetry of the smoothness space parameters. This symmetry property was specially important in the K. Rottbrand process of obtaining a certain deformation of the Fourier integral paths in order to get Laplace transform representations of the generalized eigenfunctions of the problem. This detail lead to the exclusion of finite energy spaces in the final results of