Download PDF - Hyperbolic Cauchy problem and Leray's residue formulaAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Hyperbolic Cauchy problem and Leray's residue formula

Authors 1

Affiliations

  1. Moscow Independent University, Bol'shoĭ Vlasievskiĭ pereulok 11, 121002, Moscow, Russia

Abstract

We give an algebraic description of (wave) fronts that appear in strictly hyperbolic Cauchy problems. A concrete form of a defining function of the wave front issued from the initial algebraic variety is obtained with the aid of Gauss-Manin systems satisfied by Leray's residues.

Keywords

ENG
Leray's residue formula, Gauss-Manin connexion, Bonn, hyperbolic Cauchy problem

Bibliography

  1. A. G. Aleksandrov and S. Tanabé, Gauss-Manin connexions, logarithmic forms and hypergeometric functions, in: Geometry from the Pacific Rim (Singapore, 1994), de Gruyter, 1997, 1-21.
  2. P. Appel et J. Kampé de Fériet, Fonctions hypergéometriques et hypersphériques, Gauthier-Villars, Paris, 1926.
  3. E. Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970), 103-161.
  4. L. Gårding, Sharp fronts of paired oscillatory integrals, Publ. RIMS Kyoto Univ. 12 suppl. (1977), 53-68.
  5. G.-M. Greuel, Der Gauß-Manin Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann. 214 (1975), 235-266.
  6. G.-M. Greuel und H. Hamm, Invarianten quasihomogener vollständiger Durchschnitten, Invent. Math. 49 (1978), 67-86.
  7. Y. Hamada, The singularities of the solutions of the Cauchy problem, Publ. RIMS Kyoto Univ. 5 (1969), 21-40.
  8. Y. Hamada, J. Leray et C. Wagschal, Systèmes d'équations aux dérivées partielles à caractéristiques multiples: problème de Cauchy ramifié, hyperbolicité partielle, J. Math. Pures Appl. 55 (1976), 297-352.
  9. L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I, Springer, 1984.
  10. E. Leichtnam, Le problème de Cauchy ramifié linéaire pour des données à singularités algébriques, Mem. Soc. Math. France 53 (1993).
  11. Kh. M. Malikov, Overdeterminacy of the differential systems for versal integrals of type A, D, E, Differentsial'nye Uravneniya 18 (1982), 1394-1400 (in Russian).
  12. V. P. Palamodov, Deformations of complex spaces, in: Several Complex Variables IV, Encyclopedia Math. Sci. 10, Springer, 1990, 105-194.
  13. I. G. Petrovskiĭ, On the diffusion of waves and the lacunas for hyperbolic equations, Mat. Sb. 17 (1945), 289-370.
  14. F. Pham, Introduction à l'étude topologique des singularités de Landau, Gauthier-Villars, 1967.
  15. S. Tanabé, Lagrangian variety and the condition for the presence of sharp front of the fundamental solution to Cauchy problem, Sci. Papers College Arts Sci. Univ. Tokyo, 42 (1992), 149-159.
  16. S. Tanabé, Transformée de Mellin des intégrales fibres de courbe espace associées aux singularités isolées d'intersection complète quasihomogènes, Compositio Math., to appear.
  17. S. Tanabé, Connexion de Gauss-Manin associée à la déformation verselle des singularités isolées d'hypersurface et son application au XVIe problème de Hilbert, preprint.
  18. S. Tanabé, On geometry of fronts in wave propagations, in: Geometry and Topology of Caustics-Caustics'98, Banach Center Publ. 50, Inst. Math., Polish Acad. Sci., 1999, 287-304.
  19. V. A. Vassiliev, Ramified Integrals, Singularities and Lacunas, Kluwer, Dordrecht, 1995.
  20. B. Ziemian, Leray residue formula and asymptotics of solutions to constant coefficient PDEs, Topol. Methods Nonlinear Anal. 3 (1994), 257-293.
Annales Polonici Mathematici
2000/74/1
Export to PBN, Opens in new tab
Pages:
275-290
Main language of publication
English
Received
1999-07-01
Accepted
2000-09-12
Published
2000
Exact and natural sciences