TY  - BOOK
AU  - Nishitani,Tatsuo
ED  - SpringerLink (Online service)
TI  - Cauchy Problem for Differential Operators with Double Characteristics: Non-Effectively Hyperbolic Characteristics 
T2  - Lecture Notes in Mathematics,
SN  - 9783319676128
AV  - QA370-380 
U1  - 515.353 23
PY  - 2017///
CY  - Cham
PB  - Springer International Publishing, Imprint: Springer
KW  - Partial differential equations
KW  - Differential equations
KW  - Partial Differential Equations
KW  - Ordinary Differential Equations
N1  - 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
N2  - 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
UR  - https://doi.org/10.1007/978-3-319-67612-8
ER  -