Cauchy Problem for Differential Operators with Double Characteristics

Non-Effectively Hyperbolic Characteristics

  • Tatsuo Nishitani

Part of the Lecture Notes in Mathematics book series (LNM, volume 2202)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Tatsuo Nishitani
    Pages 1-23
  3. Tatsuo Nishitani
    Pages 25-42
  4. Tatsuo Nishitani
    Pages 43-70
  5. Tatsuo Nishitani
    Pages 71-93
  6. Tatsuo Nishitani
    Pages 95-127
  7. Tatsuo Nishitani
    Pages 129-147
  8. Tatsuo Nishitani
    Pages 149-179
  9. Tatsuo Nishitani
    Pages 181-201
  10. Back Matter
    Pages 203-213

About this book


Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for differential operators with non-effectively hyperbolic double characteristics. Previously scattered over numerous different publications, the results are presented from the viewpoint that the Hamilton map and the geometry of bicharacteristics completely characterizes the well/ill-posedness of the Cauchy problem.

A doubly characteristic point of a differential operator P of order m (i.e. one where Pm = dPm = 0) is effectively hyperbolic if the Hamilton map FPm has real non-zero eigenvalues. When the characteristics are at most double and every double characteristic is effectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms.

If there is a non-effectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between − Pµj and P µj , where iµj are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 × 4 Jordan block, the spectral structure of FPm is insufficient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.


Cauchy problem Well/ill-posedness Non-effectively hyperbolic IPH condition Microlocal energy estimates Tangent bicharacteristic Gevrey classes Transition of spectral type

Authors and affiliations

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

Bibliographic information

Industry Sectors
Finance, Business & Banking
Energy, Utilities & Environment