Advanced search
Publication Cover
Communications in Partial Differential Equations Volume 17, 1992 - Issue 9-10
Submit an article Journal homepage
33
Views
10
CrossRef citations to date
0
Altmetric
Original Articles

Complete Low Frequency Analysis for the Reduced Wave Equation with Variable Coefficients in Three Dimensions *

Norbert Weck Universität GHS Essen Universitatsstraße 2, Essen 1, D 4300, GermanyView further author information
&
Karl J. Witsch Universität GHS Essen Universitatsstraße 2, Essen 1, D 4300, Germany
Pages 1619-1663 | Published online: 08 May 2007

  • Agmon , S. 1975 . Spectral properties of Schrodinger operators and scattering theory . Ann. Scuola Norm. Sup. Pisa , 2 ( 2 ) : 251 – 318 .
  • Alber , H.D. 1979 . Estimates for the asymptotic behavior of solutions of the Helmholtz equation with an application to second order elliptic differential operators with variable coefficients . Math.Z. , 167 : 213 – 226 .
  • Dassios , G. 1982 . Low frequency scattering theory for a penetrable body with an impenetrable core . SIAM J. Appl. Math. , 42 : 272 – 280 .
  • Dassios , G. and Kleinman , R. E. 1989 . On Kelvin inversion and low frequency scattering . SIAM Rev. , 31 : 565 – 585 .
  • Gilbarg , D. and Trudinger , N. S. 1977 . Elliptic partial differential equations of second order , New York : Springer, Berlin, Heidelberg .
  • Hörmander , L. 1985 . The analysis of linear partial differential operators IV , New York : Springer, Berlin, Heidelberg .
  • Ikebe , T. and Saito , Y. 1972 . Limiting absorption method and absolute continuity for the Schrodinger operators . J. Math. Kyoto Univ. , 12 : 513 – 542 .
  • Jäger , W. 1967 . Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten . Math. Z. , 102 : 62 – 88 .
  • Leis , R. 1986 . Initial boundary value problems in mathematical physics , Stuttgart : Teubner Wiley .
  • Magnus , W. , Oberhettinger , F. and Soni , R. P. 1966 . Formulas and theorems for the special functions of mathematical physics , Berlin, New York : Springer Heidelberg .
  • McCamy , R. C. 1965 . Low frequency acoustic oscillations . Quart. Appl. Math. , 23 : 247 – 256 .
  • McOwen , R. 1979 . Behavior of the Laplacian in weighted Sobolev spaces . Comm. Pure Appl. Math. , 32 : 783 – 795 .
  • McOwen , R. 1981 . Boundary value problems for the Laplacian in an exterior domain . Comm. PDE , 6 : 783 – 798 .
  • Muravei , L. A. 1973 . Asymptotic behavior for large values of time of the solutions of the second and third exterior boundary value problems for the wave equation with two spatial variables . Proc. Steklou Inst. Math. , 126 : 77 – 158 .
  • Muravei , L. A. 1979 . On the asymptotic behavior, for large values of time, of solutions of exterior boundary value problems for the wave equation with two space variables . Math. USSR Sb. , 35 : 377 – 423 .
  • Schechter , M. 1971 . Principles of Functional Analysis , New York : Academic Press .
  • Saito , Y. 1971 . The principle of limiting absorption for second order differential equations with operator valued coefficients . Publ. Res. Inst. Math. Sci., Ser. A , 7 : 581 – 619 .
  • Shibata , Y. 1983 . On the global existence theorem of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain . Tsukuba J. Math. , 7 : 1 – 68 .
  • Stuwe , H.-Chr. and Werner , P. 1989 . Remarks on the low frequency asymptotics for the reduced wave equation and the Schrodinger equation in low-dimensional spaces . Asymptotic Analysis , 2 : 179 – 202 .
  • Tsutsumi , Y. 1984 . Local energy decay of solutions to the free Schrodinger equation in exterior domains . J. Fac. Sci. Univ. Tokyo, sect. IA, Math. , 31 : 97 – 108 .
  • Vainberg , B. R. 1968 . On the analytic properties of the resolvent for a certain class of operator pencils . Math. USSR Sb. , 6 : 241 – 272 .
  • Vainberg , B. R. 1973 . On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behavior for large time of solutions of nonstationary problems . Math. USSR Sb. , 21 : 221 – 239 .
  • Vogelsang , V. 1983 . Die Hellichsche Abschatzung fiir die Schwingungsgleichung mit oszillierendem Hauptteil . J. reine angew. Math. , 338 : 109 – 120 .
  • Weck , N. and Witsch , K. J. 1991 . Exterior Dirichlet problem for the reduced wave equation: asymptotic analysis of low frequencies . Comm. PDE , 16 : 173 – 195 .
  • Weck , N. and Witsch , K. J. 1992 . Exact low frequency analysis for a class of exterior boundary value problems for the reduced wave equation in higher dimensions . Asymptotic Analysis , 6
  • Weck , N. and Witsch , K. J. Exact low frequency analysis for a class of exterior boundary value problems for the reduced wave equation in two dimensions . J. DiJ. Equ., ,
  • Werner , P. 1985 . Zur Asymptotik der Wellengleichung und der Wärmeleitungsgleichung in zweidimensionalen Außenraumen . Math. Meth. in the Appl. Sci. , 7 : 170 – 201 .
  • Werner , P. 1986 . Low frequency asymptotics for the reduced wave equation in twodimensional exterior spaces . Math. Meth. in the Appl. Sci. , 8 : 134 – 156 .
  • Wilcox , C. H. 1975 . “ Scattering theory for the d'Alembert equation in exterior domains ” . In Lecture Notes in Mathematics 442 , Berlin, New York : Springer .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.