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Journal d’Analyse Mathématique

, Volume 32, Issue 1, pp 118–196 | Cite as

The Cauchy problem for differential equations with double characteristics

  • Lars Hörmander
Article

Keywords

Cauchy Problem Symplectic Form Pseudodifferential Operator Formal Power Series Principal Symbol 
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.
    J. J. Duistermaat and L. Hörmander,Fourier integral operators II, Acta Math.128 (1972), 183–269.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    L. Hörmander,Linear Partial Differential Operators, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963.MATHGoogle Scholar
  3. 3.
    L. Hörmander,Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math.83 (1966), 129–209.CrossRefMathSciNetGoogle Scholar
  4. 4.
    L. Hörmander,Fourier integral operators. I., Acta Math.127 (1971), 79–183.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    L. Hörmander,A class of hypoelliptic pseudodifferential operators with double characteristics, Math. Ann.217 (1975), 165–188.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    V. Ia. Ivrii,Sufficient conditions for regular and completely regular hyperbolicity, Trudy Moskov. Mat. Obšč.33 (1975), 1–65.MathSciNetGoogle Scholar
  7. 7.
    V. Ia. Ivrii,Energy integrals for non-strictly hyperbolic operators, Uspehi Mat. Nauk30 (6) (1975), 169–170.MathSciNetGoogle Scholar
  8. 8.
    V. Ia. Ivrii and V. M. Petkov,Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations, Uspehi Mat. Nauk29: 5 (1974), 3–70.MathSciNetGoogle Scholar
  9. 9.
    H. Kumano-go,Factorizations and fundamental solutions for differential operators of elliptic-hyperbolic type, Mimeographed manuscript.Google Scholar
  10. 10.
    A. Lax,On Cauchy's problem for partial differential equations with multiple characteristics, comm. Pure Appl. Math.9 (1956), 135–169.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. D. Lax,Asymptotic solutions of oscillatory initial value problems, Duke Math. J.24 (1957), 627–646.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    P. D. Lax and L. Nirenberg,On stability for difference schemes; a sharp form of Gårding's inequality, Comm. Pure Appl. Math.19 (1966), 473–492.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    E. E. Levi,Caratteristiche multiple e problema di Cauchy, Ann. di Matem. Ser. 3,16 (1909), 161–201.CrossRefGoogle Scholar
  14. 14.
    A. Melin,Lower bounds for pseudo-differential operators, Ark. Mat.9 (1971), 117–140.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    S. Mizohata,Some remarks on the Cauchy problem, J. Math. Kyoto Univ.1 (1961), 109–127.MATHMathSciNetGoogle Scholar
  16. 16.
    L. Nirenberg and F. Treves,On local solvability of linear partial differential equations. Part II, Sufficient conditions, Comm. Pure Appl. Math.23 (1970), 459–509;24 (1971), 279–288.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    O. A. Olejnik,On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math.23 (1970), 569–586.CrossRefMathSciNetGoogle Scholar
  18. 18.
    V. M. Petkov,The Cauchy problem for a class of non-strictly hyperbolic equations with double characteristics, Serdica, Bulg. Mat. Publ.1 (1975), 372–380.MATHMathSciNetGoogle Scholar
  19. 19.
    I. G. Petrowsky,Über das Cauchysche Problem für ein System linearer partieller Differentialgleichungen, Mat. Sb.2 (44) (1937), 815–868.MATHGoogle Scholar
  20. 20.
    N. Schwid,The asymptotic forms of the Hermite and Weber functions, Trans. Amer. Math. Soc.37 (1935), 339–362.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    I. Segal,Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand.13 (1963), 31–43.MATHMathSciNetGoogle Scholar
  22. 22.
    J. Sjöstrand,Parametrices for pseudodifferential operators with multiple characteristics Ark. Mat.12 (1974), 85–130.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    L. Svensson,Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat.8 (1969), 145–162.CrossRefMathSciNetGoogle Scholar
  24. 24.
    A. Weil,Sur certains groupes d'opérateurs unitaires, Acta Math.111 (1964), 143–211.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 1977

Authors and Affiliations

  • Lars Hörmander
    • 1
  1. 1.Institute of MathematicsUniversity of LundLundSweden

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