Cauchy Problem for Differential Operators with Double Characteristics [electronic resource] : Non-Effectively Hyperbolic Characteristics / by Tatsuo Nishitani.
Material type: TextSeries: Lecture Notes in Mathematics ; 2202Publisher: Cham : Springer International Publishing : Imprint: Springer, 2017Edition: 1st ed. 2017Description: VIII, 213 p. 7 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783319676128Subject(s): Partial differential equations | Differential equations | Partial Differential Equations | Ordinary Differential EquationsAdditional physical formats: Printed edition:: No title; Printed edition:: No titleDDC classification: 515.353 LOC classification: QA370-380Online resources: Click here to access onlineCurrent library | Home library | Call number | Materials specified | URL | Status | Date due | Barcode |
---|---|---|---|---|---|---|---|
IMSc Library | IMSc Library | Link to resource | Available | EBK15747 |
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.
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.
There are no comments on this title.