Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T07:55:39.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Some interior and exterior boundary-value problems for the Helmholtz equation in a quadrant

Published online by Cambridge University Press:  14 November 2011

E. Meister ,
F. Penzel ,
F.-O. Speck  and
F. S. Teixeira
E. Meister
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany
F. Penzel
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany
F.-O. Speck
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, P-1096 Lisboa Codex, Portugal
F. S. Teixeira
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, P-1096 Lisboa Codex, Portugal
Get access Rights & Permissions [Opens in a new window]

Synopsis

The Dirichlet, Neumann and mixed boundary-value problems for the two-dimensional Helmholtz equation in the interior or exterior of a quadrant are considered in a Sobolev space setting. It is shown that the potential operators arising in the interior problems can be used to derive systems of boundary integral equations to the exterior problems, which can be solved explicitly.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 2 , 1993 , pp. 275 - 294
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Eskin, G. I.. Boundary Value Problems for Elliptic Pseudodifferential Equations (Providence R. I.: American Mathematical Society, 1981).Google Scholar
2Grisvard, P.. Elliptic Problems in Nonsmooth Domains (London: Pitman, 1985).Google Scholar
3Meister, E.. Integral Equations for the Fourier-Transformed Boundary Values for the Transmission Problems for Right-Angled Wedges and Octants. Math. Methods Appl. Sci. 8 (1986), 182–205.CrossRefGoogle Scholar
4Meister, E. and Penzel, R.. Einige Randwerttransmissionsprobleme der Beugungstheorie für Keile bei mehreren Medien. Kleinheubacher Berichte 30, FTZ-Darmstadt, SSN 0343–5729, 429–438.Google Scholar
5Meister, E. and Speck, F.-O.. A contribution to the quarter-plane problem in diffraction theory. J. Math. Anal. Appl. 130 (1988), 223–236.CrossRefGoogle Scholar
6Meister, E. and Speck, F.-O.. Modern Wiener–Hopf methods in diffraction theory. Proc. Conf. Dundee 1988. In Ordinary and Partial Differential Equations eds. Sleeman, B. and , Jarvis, 2 (1989), 130–171.Google Scholar
7Meister, E., Speck, F.-O. and Teixeira, F. S.. Wiener–Hopf–Hankel operators for some wedge diffraction problems with mixed boundary conditions. J. Integral Equations and Appl. 4 no. 2 (1992) (to appear).Google Scholar
8Meister, E., Penzel, F., Speck, F.-O. and Teixeira, F. S.. Two media scattering problems in a half-space. In Research Notes in Mathematics (London: Pitman, to appear).Google Scholar
9von Petersdorff, T.. Boundary integral equations for mixed Dirichlet, Neumann and Transmission conditions. Math. Methods Appl. Sci. 11 (1989), 185–213.Google Scholar
10Penzel, R.. Über eine Klasse von Integralgleichungsoperatoren im Fourier-Bildereich bei Mehrmedien-Rand-Transmissionsproblemen (Dipl. Thesis, Technische Hochschule Darmstadt, 1988).Google Scholar
11Schneider, R.. Reduction of order for Pseudodifferential Operators on Lipschitz Domains. Comm. Partial Differential Equations 16 (1991), 1263–1286.Google Scholar
12Speck, F.-O.. Mixed boundary value problems of the type of Sommerfeld's halfplane problem. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 261–277.Google Scholar
13Speck, F.-O.. Sommerfeld diffraction problems with first and second kind boundary conditions. SIAMJ. Math. Anal. 20 (1989), 396–407.Google Scholar
14Speck, F.-O. and Duduchava, R.. Blessel Potential Operators for the Quarter Plane. Appl. Anal. 45 (1992), 49–68.Google Scholar
15Teixeira, F. S.. Wiener-Hopf Operators in Sobolev Spaces and Applications to Diffraction Theory (Ph.D. Thesis, Instituto Superior Técnico, U.T.L., Lisbon, 1989 in Portuguese).Google Scholar
16Teixeira, F. S.. On a class of Hankel operators: Fredholm properties and invertibility. Integral Equations Operator Theory 12 (1989), 592–613.Google Scholar
17Teixeira, F. S.. Diffraction by a rectangular wedge: Wiener-Hopf-Hankel formulation. Integral Equations Operator Theory 14 (1991), 436–454.CrossRefGoogle Scholar