Cauchy Problem for Differential Operators with Double Characteristics e-bog

Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for dii erential operators with non-ei ectively hyperbolic double characteristics. Previously scattered over numerous dii 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 dii erential operator P of order m (i.e. one where Pm = dPm = 0) is ei 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 ei ectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms.If there is a non-ei ectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between Puj and Puj , where iuj 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 insui cient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.