Cauchy problem for differential operators with double characteristics : non-effectively hyperbolic characteristics
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Bibliographic Information
Cauchy problem for differential operators with double characteristics : non-effectively hyperbolic characteristics
（Lecture notes in mathematics, 2202）
Springer, c2017
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
L/N||LNM||2202200037693681
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Note
Includes bibliographical references (p. 203-207) and index
Description and Table of Contents
Description
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 eigen values. 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 Pj and Pj , where ij are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 x 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.
Table of Contents
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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