Nishitani, Tatsuo. Cauchy Problem for Differential Operators with Double Characteristics : Non-Effectively Hyperbolic Characteristics / [electronic resource] : / by Tatsuo Nishitani.. — 1st ed. 2017.. — VIII, 213 p. 7 illus. : online resource. — (Lecture Notes in Mathematics,) 2202 0075-8434 ;. - Lecture Notes in Mathematics, 2202 .
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.
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Анотація: 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.