Overview
- Features thorough discussions on well/ill-posedness of the Cauchy problem for di?erential operators with double characteristics of non-e?ectively hyperbolic type
- Takes a uni?ed 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
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2202)
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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.
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.
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Table of contents (8 chapters)
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Front Matter
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Back Matter
Authors and Affiliations
Bibliographic Information
Book Title: Cauchy Problem for Differential Operators with Double Characteristics
Book Subtitle: Non-Effectively Hyperbolic Characteristics
Authors: Tatsuo Nishitani
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-67612-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Softcover ISBN: 978-3-319-67611-1Published: 26 November 2017
eBook ISBN: 978-3-319-67612-8Published: 24 November 2017
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: VIII, 213
Number of Illustrations: 7 b/w illustrations
Topics: Partial Differential Equations, Ordinary Differential Equations