Georgian Mathematical Journal 13 (2006), No. 2, 251--260
Copyright Heldermann Verlag 2006
The Potential Method for the Reactance Wave Diffraction Problem in a Scale of Spaces
Luis P. Castro
Dept. of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
lcastro@mat.ua.pt
David Natroshvili
Dept. of Mathematics, Georgian Technical University, 77 M.Kostava Street, Tbilisi 0175, Georgia
natrosh@hotmail.com
This paper is concerned with a screen type boundary value problem arising from the wave diffraction 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 analyzed. Using the potential method, the boundary value problem is converted into a system of integral equations. The invertibility of the corresponding matrix pseudodifferential 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 Cα-smoothness of solutions in a neighbourhood of the screen edge is established with arbitrary α < 1 in the two-dimensional case and α < 1/2 in the three-dimensional case.
Keywords: Helmholtz equation, wave diffraction, reactance condition, integral equations, potential theory.
MSC: 35J05, 35J25, 35P25
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