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Inventiones mathematicae

, Volume 156, Issue 2, pp 235–299 | Cite as

Propagation of singularities for the wave equation on conic manifolds

  • Richard MelroseEmail author
  • Jared WunschEmail author
Article

Abstract

For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, non-direct, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave.

Keywords

Manifold Wave Equation Compact Manifold Regularity Assumption Incoming Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Brüning, J., Guillemin , V.W.: Fourier integral operators. Berlin, Heidelberg, New York, Tokyo: Springer 1994 Google Scholar
  2. 2.
    Cheeger, J., Taylor, M.E.: On the diffraction of waves by conical singularities. I. Commun. Pure Appl. Math. 35, 275–331 (1982) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cheeger, J., Taylor, M.E.: On the diffraction of waves by conical singularities. II. Commun. Pure Appl. Math. 35, 487–529 (1982) zbMATHGoogle Scholar
  4. 4.
    Duistermaat, J.J., Hörmander, L.: Fourier integral operators, II. Acta Math. 128, 183–269 (1972) zbMATHGoogle Scholar
  5. 5.
    Friedlander, F.G.: Sound pulses. New York: Cambridge University Press 1958 Google Scholar
  6. 6.
    Gérard, P., Lebeau, G.: Diffusion d’une onde par un coin. J. Am. Math. Soc. 6, 341–424 (1993) MathSciNetGoogle Scholar
  7. 7.
    Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Berlin: Springer 1977 Google Scholar
  8. 8.
    Hörmander, L.: Fourier integral operators, I. Acta Math. 127, 79–183 (1971), See also [1] Google Scholar
  9. 9.
    Hörmander, L.: On the existence and the regularity of solutions of linear pseudo-differential equations. Enseign. Math. II. Sér. 17, 99–163 (1971) Google Scholar
  10. 10.
    Iagolnitzer, D.: Appendix: Microlocal essential support of a distribution and decomposition theorems – an introduction. In: Hyperfunctions and theoretical physics (Rencontre, Nice, 1973; dédié à la mémoire de A. Martineau). Lecture Notes in Math., Vol. 449, pp. 121–132. Berlin: Springer 1975 Google Scholar
  11. 11.
    Kalka, M., Menikoff, A.: The wave equation on a cone. Commun. Partial Differ. Equations 7, 223–278 (1982) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lebeau, G.: Propagation des ondes dans les variétés à coins. Séminaire sur les Équations aux Dérivées Partielles, 1995–1996, École Polytech., Palaiseau, 1996, pp. Exp. No. XVI Google Scholar
  13. 13.
    Lebeau, G.: Propagation des ondes dans les variétés à coins. Ann. Sci. Éc. Norm. Supér., IV. Sér. 30, 429–497 (1997) Google Scholar
  14. 14.
    Mazzeo, R.: Elliptic theory of differential edge operators I. Commun. Partial Differ. Equations 16, 1615–1664 (1991) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Melrose, R.B.: Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. In: Spectral and scattering theory (Sanda, 1992), ed. by M. Ikawa, pp. 85–130. Marcel Dekker 1994 Google Scholar
  16. 16.
    Melrose, R.B., Wunsch, J.: Singularities and the wave equation on conic spaces. In: National Research Symposium on Geometric Analysis and Applications, ed. by A. Isaev et al., Vol. 39. Australian National University: Centre for Mathematics and its Applications 2001 Google Scholar
  17. 17.
    Melrose, R.B.: The Atiyah–Patodi–Singer index theorem. Wellesley, MA: A K Peters Ltd. 1993 Google Scholar
  18. 18.
    Müller, D., Seeger, A.: Inequalities for spherically symmetric solutions of the wave equation. Math. Z. 218, 417–426 (1995) MathSciNetGoogle Scholar
  19. 19.
    Rouleux, M.: Diffraction analytique sur une variété à singularité conique. Commun. Partial Differ. Equations 11, 947–988 (1986) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Schulze, B.-W.: Boundary value problems and edge pseudo-differential operators. Microlocal analysis and spectral theory (Lucca, 1996), pp. 165–226. Dordrecht: Kluwer Acad. Publ. 1997 Google Scholar
  21. 21.
    Sjöstrand, J.: Singularités analytiques microlocales. Astérisque 95, 1–166 (1982) Google Scholar
  22. 22.
    Sommerfeld, A.: Mathematische Theorie der Diffraktion. Math. Ann. 47, 317–374 (1896) zbMATHGoogle Scholar
  23. 23.
    Taylor, M.E.: Pseudodifferential operators. Princeton, N.J.: Princeton University Press 1981 Google Scholar
  24. 24.
    Wunsch, J., Zworski, M.: The FBI transform on compact \(\mathcal{C}^{\infty}\) manifolds. Trans. Am. Math. Soc. 353, 1151–1167 (2001)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonUSA

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