Cauchy Problem: No Tangent Bicharacteristics

  • Tatsuo Nishitani
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2202)

Abstract

The main purpose of this chapter is to prove two new results on C well-posedness mentioned in the end of Sect.  1.4. If there is no transition of spectral type and no tangent bicharacteristics then the Cauchy problem is C well-posed for P of order m under the strict IPH condition. If the positive trace is zero, the IPH condition is reduced to the Levi condition. In this case we prove that when p is of spectral type 2 on Σ and there is no tangent bicharacteristics, the Levi condition is necessary and sufficient in order that the Cauchy problem is C well-posed for P of order m. The same result holds for second order differential operators of spectral type 1 on Σ with 0 positive trace. To prove these assertions using the results in Chap.  2 we first derive microlocal energy estimates. Then making use of the idea developed in Chap.  4 we prove the C well-posedness of the Cauchy problem.

References

  1. 32.
    L. Hörmander, The Cauchy problem for differential equations with double characteristics. J. Anal. Math. 32, 118–196 (1977)MathSciNetCrossRefMATHGoogle Scholar
  2. 34.
    L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Grundlehren Math. Wiss., vol. 274 (Springer, Berlin, 1985)Google Scholar
  3. 40.
    V.Ja. Ivrii, The wellposed Cauchy problem for non-strictly hyperbolic operators, III. The energy integral. Trans. Moscow Math. Soc. (English transl.) 34, 149–168 (1978)Google Scholar
  4. 89.
    C. Parenti, A. Parmeggiani, On the Cauchy problem for hyperbolic operators with double characteristics. Commun. Partial Differ. Equ. 34, 837–888 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 90.
    C. Parenti, A. Parmeggiani, On the Cauchy problem for hyperbolic operators with double characteristics, in Evolution Equations of Hyperbolic and Schrödinger Type. Progress in Mathematics, vol. 301 (Birkäuser/Springer, Basel, 2012), pp. 247–266Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Tatsuo Nishitani
    • 1
  1. 1.Department of MathematicsOsaka UniversityToyonakaJapan

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