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Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems


Authors: L. P. Castro and D. Kapanadze
Journal: Proc. Amer. Math. Soc. 136 (2008), 2113-2123
MSC (2000): Primary 35J25; Secondary 35J05, 45E10, 47A20, 47G30, 47H50
DOI: https://doi.org/10.1090/S0002-9939-08-09288-5
Published electronically: February 14, 2008
MathSciNet review: 2383517
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Abstract: Boundary value problems originated by the diffraction of an electromagnetic (or acoustic) wave by a rectangular wedge with faces of possible different kinds are analyzed in a Sobolev space framework. The boundary value problems satisfy the Helmholtz equation in the interior (Lipschitz) wedge domain, and are also subject to different combinations of boundary conditions on the faces of the wedge. Namely, the following types of boundary conditions will be under study: Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, Impedance-Dirichlet, and Impedance-Neumann. Potential theory (combined with an appropriate use of extension operators) leads to the reduction of the boundary value problems to integral equations of Fredholm type. Thus, the consideration of single and double layer potentials together with certain reflection operators originate pseudo-differential operators which allow the proof of existence and uniqueness results for the boundary value problems initially posed. Furthermore, explicit solutions are given for all the problems under consideration, and regularity results are obtained for these solutions.


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Additional Information

L. P. Castro
Affiliation: Research Unit “Mathematics and Applications”, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Email: castro@ua.pt

D. Kapanadze
Affiliation: Research Unit “Mathematics and Applications”, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Email: david.kapanadze@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-08-09288-5
Received by editor(s): March 22, 2007
Published electronically: February 14, 2008
Additional Notes: This work was supported in part by Unidade de Investigação Matemática e Aplicações of Universidade de Aveiro, and the Portuguese Science Foundation (FCT–Fundação para a Ciência e a Tecnologia) through grant number SFRH/BPD/20524/2004.
Communicated by: Hart F. Smith
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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