Characteristic Cauchy problems in the complex domain

  • Yasunori Okada
  • Hideshi Yamane


Gårding-Kotake-Leray showed that in a certain characteristic Cauchy problem
$$Pu = \upsilon \in o\;(the\;sheaf\;of\;holomorphic\;functions)$$
with zero Cauchy data on a hypersurface S, u can be ramified. Moreover, u is of the form
$$(*)\;\upsilon (x) = \sum\limits_{i = 0}^{q - 1} {\upsilon i(x){{[k(x)]}^{1/q}}} $$
where q is a positive integer ≥ 2 and u is ramified around K : k(x) = 0. Here K is tangent to S at characteristic points of S. Let us denote by \(N_{q,K}^m\) the class of functions which have the form (*) and whose first m traces on S vanish.


Differential Operator Characteristic Point Complex Domain Cauchy Data Essential Singularity 
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  1. [D]
    J. Dunau, Un Problème de Cauchy Caractéristique, J. Math. pures et appl. 69 (1990), 369–402.MathSciNetMATHGoogle Scholar
  2. [G-K-L]
    L. Gårding, T. Kotake and J. Leray, Problème de Cauchy, I bis et VI, Bull. Soc. Math, de France 92 (1964), 263–361.MATHGoogle Scholar
  3. [H]
    Y. Hamada, Les singularités des solutions du problème de Cauchy à données holomorphes, Recent developments in hyperbolic equations (Pisa, 1987) (L. Cattabriga et. al., eds.), Pitman Research Notes in Math. 183, Longman, 1988, pp. 82–95.Google Scholar
  4. [K-K-K]
    M. Kashiwara, T. Kawai and T. Kimura, Foundation of Algebraic Analysis, (in Japanese), Kinokuniya, 1980; English translation, Princeton Mathematical Series 37, Princeton Univ. Press, 1986.Google Scholar
  5. [N-S]
    G. Nakamura, T. Sasai, The singularities of the solutions of the Cauchy problem for second order equations in case the initial manifold includes characteristic points, Tôhoku Math. Journ. 28 (1976), 523–539.MathSciNetMATHCrossRefGoogle Scholar
  6. [O-Y]
    Y. Okada, H. Yamane, A characteristic Cauchy problem in the complex domain, J. Math, pures et appl, (to appear).Google Scholar

Copyright information

© Springer-Verlag Tokyo 1997

Authors and Affiliations

  • Yasunori Okada
    • 1
  • Hideshi Yamane
    • 2
  1. 1.Department of Mathematics and Informatics, Faculty of ScienceChiba UniversityYayoi-cho, Inage-ku, Chiba 263Japan
  2. 2.MathematicsChiba Institute of TechnologyShibazono, Narashino, Chiba 275Japan

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