
Cauchy Problem for Differential Operators with Double Characteristics
Non-Effectively Hyperbolic Characteristics
Tatsuo Nishitani(Author)
Springer (Publisher)
Published on 26. November 2017
Book
Paperback/Softback
VIII, 213 pages
978-3-319-67611-1 (ISBN)
Description
Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for di?erential operators with non-e?ectively hyperbolic double characteristics. Previously scattered over numerous di?erent 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 di?erential operator P of order m (i.e. one where Pm = dPm = 0) is e?ectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the characteristics are at most double and every double characteristic is e?ectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms.
If there is a non-e?ectively 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 insu?cient 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 di?erential operator P of order m (i.e. one where Pm = dPm = 0) is e?ectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the characteristics are at most double and every double characteristic is e?ectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms.
If there is a non-e?ectively 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 insu?cient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.
More details
Series
Edition
1st ed. 2017
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Professional and scholarly
Illustrations
7 s/w Abbildungen
VIII, 213 p. 7 illus.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 13 mm
Weight
347 gr
ISBN-13
978-3-319-67611-1 (9783319676111)
DOI
10.1007/978-3-319-67612-8
Schweitzer Classification
Other editions
Additional editions

Tatsuo Nishitani
Cauchy Problem for Differential Operators with Double Characteristics
Non-Effectively Hyperbolic Characteristics
E-Book
11/2017
Springer
€53.49
Available for download
Content
1. Introduction.- 2 Non-effectively hyperbolic characteristics.- 3 Geometry of bicharacteristics.- 4 Microlocal energy estimates and well-posedness.- 5 Cauchy problem-no tangent bicharacteristics. - 6 Tangent bicharacteristics and ill-posedness.- 7 Cauchy problem in the Gevrey classes.- 8 Ill-posed Cauchy problem, revisited.- References.