Cauchy Problem for Differential Operators with Double Characteristics: Non-Effectively Hyperbolic Characteristics (1st ed. 2017) (Lecture Notes in Mathematics #2202)

By: Tatsuo Nishitani

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Synopsis: 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 −Pµj and Pµj , 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.