Parametrices, singularities, and high frequency asymptotics in the theory of sound waves

  • Hans-Dieter Alber
Part of the International Centre for Mechanical Sciences book series (CISM, volume 277)


Let Ω ≤ R3 be an unbounded domain with bounded complement and boundary ∂Ω ∈ C. Let with C1 ≤ a(x), b(x) ≤ C2 for x ∈ Ω and for suitable constants C1,C2 > 0. In the next section I want to study the singularities of solutions v(x,t) of the following Dirichlet and Neumann problems
where f, v0,v1 are given functions, and where ∂n denotes the derivative of v with respect to the normal n at the boundary ∂Ω × R+. To do this, I need results from the solution theory, and in particular, from the regularity theory of these problems. These results are well known and can be proved, for example, by semi-group theory. Therefore, in this introductory section I only give some definitions and state the results needed later on.


Wave Equation Neumann Problem Wave Operator Infinite Order Riemann Function 
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  1. 1.
    Alber, H.D., Justification of geometrical optics for non-convex obstacles, J. Math. Anal. Appt. 80, 372, 1981.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alber, H.D., Zur Hochfrequenzasymptotik der Lösungen der Schwingungs-gleichung - Verhalten auf Tangentialstrahlen, Habilitationsschrift Universität Bonn, 1982.Google Scholar
  3. 3.
    Chazarin, J., Construction de la paramétrix du probléme mixte hyperbolique pour l’ equation des ordes, C.R. Acad. Sci. Paris 276, 1213, 1973.Google Scholar
  4. 4.
    Duistermaat, J.J., Hörmander, L., Fourier integral operators II, Acta Math. 128, 183, 1972.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Hörmander, L., Fourier integral operators, Acta Math. 127, 79, 1971.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Majda, A., High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering, Comm. Pure Appl. Math. 29, 261, 1976.ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Melrose, R., Sjöstrand, J., Singularities of boundary value problems I, Comm. Pure Appl. Math. 31, 593, 1978.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Morawetz, C.S., Ralston, J.V., Strauss, W.A., Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math. 30, 447, 1977.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Taylor, M.E., Grazing rays and reflection of singularities of solutions to wave equations, Comm. Pure Appl. Math. 29, 1, 1976.ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1983

Authors and Affiliations

  • Hans-Dieter Alber
    • 1
  1. 1.Department of MathematicsUniversity of BonnBonn 1West-Germany

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