DIFFRACTION BY A UNION OF STRIPS WITH IMPEDANCE CONDITIONS IN BESOV AND BESSEL POTENTIAL SPACES

Luis P Castro,David Kapanadze

doi:10.3846/1392-6292.2008.13.183-194

Abstract

We consider an impedance boundary‐value problem for the Helmholtz equation which models a wave diffraction problem with imperfect conductivity on a union of strips. Pseudo‐differential operators acting between Bessel potential spaces and Besov spaces are used to deal with this wave diffraction problem. In particular, these operators allow a reformulation of the problem into a system of integral equations. The main result presents impedance parameters which ensure the well-posedness of the problem in scales of Bessel potential spaces and Besov spaces.

Highlights

The diffraction of waves by regions in a plane is an important topic in scattering theory from both mathematical and engineering points of view, and has been subjected to numerous past investigations
We refer to the survey paper [6] and to the book [3] for a description on the background of these problems
From the mathematical point of view, different studies have been made about the type of the spaces which are more appropriate to deal with such kind of problems

Summary

Introduction

The diffraction of waves by regions in a plane is an important topic in scattering theory from both mathematical and engineering points of view, and has been subjected to numerous past investigations. A great part of the mathematical interest in this kind of problems is devoted to the question of finding out the largest set of possible spaces where the existence of a unique solution, and continuous dependence on the given data will be shown. Within this goal, it is relevant to mention that for the real (non-complex) wave number case some of the known techniques fail (e.g., certain integral representations constructed by the Wiener-Hopf method for the complex wave number case have no sense in the real wave number situation). We improve for instance the results of [2] where only the one strip geometry was considered and in Hilbert Bessel potential spaces

The Impedance Boundary Value Problem for the Union of Strips

Uniqueness of Solution

The Integral Form of the Solutions

Existence and Regularity of Solutions on Bessel Potential and Besov Spaces