Read more
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 insu cient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.
List 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.
Summary
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.
