Cauchy Problem for Differential Operators with Double Characteristics

Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for di¿erential operators with non-e¿ectively hyperbolic double characteristics. Previously scattered over numerous di¿erent 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 di¿erential operator P of order m (i.e. one where Pm = dPm = 0) is e¿ectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the characteristics are at most double and every double characteristic is e¿ectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms.
If there is a non-e¿ectively 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 insücient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.