About this book
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 eigenvalues. 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 P µj , 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.
Cauchy problem Well/ill-posedness Non-effectively hyperbolic IPH condition Microlocal energy estimates Tangent bicharacteristic Gevrey classes Transition of spectral type
- DOI https://doi.org/10.1007/978-3-319-67612-8
- Copyright Information Springer International Publishing AG 2017
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
- Print ISBN 978-3-319-67611-1
- Online ISBN 978-3-319-67612-8
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
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