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Lecture Notes in Mathematics

Cauchy Problem for Differential Operators with Double Characteristics

Non-Effectively Hyperbolic Characteristics

Authors: Nishitani, Tatsuo

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  • Features thorough discussions on well/ill-posedness of the Cauchy problem for differential operators with double characteristics of non-effectively hyperbolic type
  • Takes a unified approach combining geometrical and microlocal tools
  • Adopts the viewpoint that the Hamilton map and the geometry of bicharacteristics characterizes the well/ill-posedness of the Cauchy problem
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  • ISBN 978-3-319-67612-8
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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 eigen values. 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 Pµj , 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.

Table of contents (8 chapters)

Table of contents (8 chapters)

Buy this book

eBook $44.99
price for Mexico (gross)
  • ISBN 978-3-319-67612-8
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.99
price for Mexico
  • ISBN 978-3-319-67611-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Cauchy Problem for Differential Operators with Double Characteristics
Book Subtitle
Non-Effectively Hyperbolic Characteristics
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2202
Copyright
2017
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing AG
eBook ISBN
978-3-319-67612-8
DOI
10.1007/978-3-319-67612-8
Softcover ISBN
978-3-319-67611-1
Series ISSN
0075-8434
Edition Number
1
Number of Pages
VIII, 213
Number of Illustrations
7 b/w illustrations
Topics