Cauchy Problem for Differential Operators with Double Characteristics Non-Effectively Hyperbolic Characteristics /

• Κύριος συγγραφέας: Nishitani, Tatsuo (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Cham : Springer International Publishing : Imprint: Springer, 2017.
• Σειρά: Lecture Notes in Mathematics, 2202
• Θέματα: Mathematics.; Differential equations.; Partial differential equations.; Partial Differential Equations.; Ordinary Differential Equations.

 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. 

 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.