Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T01:20:03.317Z Has data issue: false hasContentIssue false
  • English
  • Français

On Sommerfeld radiation conditions for the diffraction problem with two unbounded media

Published online by Cambridge University Press:  14 November 2011

G. F. Roach  and
Bo Zhang
G. F. Roach
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow Gl 1XH, Scotland, U.K.
Bo Zhang
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow Gl 1XH, Scotland, U.K.
Get access Rights & Permissions [Opens in a new window]

Synopsis

A radiation condition is obtained, and is then used together with weighted Sobolev spaces and the limiting absorption method to establish the unique existence of solutions to the diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant, and the surface separating two media is unbounded.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 1-2 , 1992 , pp. 149 - 161
Copyright
Copyright © Royal Society of Edinburgh 1992

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

1Agmon, S. and Hormander, L.. Asymptotic properties of solutions of differential equations with simple characteristics. J. Analyse Math. 30 (1976), 1–37.CrossRefGoogle Scholar
2Beck, P.. Ein Eindeutigkeitssatz fur Strahlungslosungen eines Ubergangsproblems zur Schwingungsgleichung mit unbeschrankter Trennflache. Math. Methods Appl. Sci. 7 (1985), 290–308.CrossRefGoogle Scholar
3Constantin, P.. Scattering for Schrodinger operators in a class of domains with non-compact boundaries. J. Funct. Anal. 44 (1981), 87–119.CrossRefGoogle Scholar
4Eidus, D.. The principle of limiting amplitude. Russian Math. Surveys 24 (1969), 97–167.CrossRefGoogle Scholar
5Eidus, D.. The limiting absorption and amplitude principles for the diffraction problem with two unbounded media. Comm. Math. Phys. 107 (1986), 29–38.CrossRefGoogle Scholar
6Eidus, D. and Vinnik, A.. On radiation conditions for domains with infinite boundaries. Soviet Math. Dokl. 15 (1974), 12–15.Google Scholar
7Ikebe, T. and Saito, Y.. Limiting absorption method and absolute continuity for the Schrodinger operator. J. Math. Kyoto Univ. 7 (1972), 513–542.Google Scholar
8Jager, W.. Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Aubengebieten. Math. Z. 102 (1967), 62–68.CrossRefGoogle Scholar
9Kristensson, G., A uniqueness theorem for Helmholtz' equation: penetrable media with an infinite interface. SIAM J. Math. Anal. 11 (1980), 1104–1117.CrossRefGoogle Scholar
10Leis, R., Initial boundary value problems in mathematical physics (New York: John Wiley, 1986).CrossRefGoogle Scholar
11Minskii, V.. Sommerfeld radiation condition for second-order differential operator in a domain with infinite border. J. Differential Equations 48 (1983), 157–176.CrossRefGoogle Scholar
12Neittaanmaki, P. and Roach, G. F.. Weighted Sobolev spaces and exterior problems for the Helmholtz equation. Proc. Roy. Soc. London, Ser. A 410 (1987), 373–383.Google Scholar
13Odeh, F. M.. Uniqueness theorems for the Helmholtz equation in domains with infinite boundaries. J. Math. Mech. 12 (1963), 857–868.Google Scholar
14Roach, G. F. and A, Bo Zhang. transmission problem for the reduced wave equation in inhomogeneous media with an infinite interface. University of Strathclyde, Mathematics Department Research Report No. 10, 1991 (accepted for Proc. Roy. Soc. London, Ser. A).Google Scholar
15Saito, Y.. A remark on the limiting absorption principle for the reduced wave equation with two unbounded media. Pacific J. Math. 136 (1989), 183–208.CrossRefGoogle Scholar
16Vogelsang, V.. Das Ausstrahllengsproblem fur elliptische Differentialgleichungen in Gebieten mit unbeschrankten Rand. Math. Z. 144 (1975), 101–124.CrossRefGoogle Scholar
17Zhang, Bo. Uniqueness theorems for the reduced wave equation with two unbounded media. Chinese J. Enging. Math. (to appear).Google Scholar