Cauchy Problem for Differential Operators with Double Characteristics: Non-Effectively Hyperbolic Characteristics: 2202 - Brossura

Nishitani, Tatsuo

Brossura

ISBN 10: 3319676113 ISBN 13: 9783319676111

Casa editrice: Springer, 2017

Riassunto

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.

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Dalla quarta di copertina

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 eigenvalues. 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 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 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.

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2017
• Lingua: Inglese
• ISBN 10: 3319676113
• ISBN 13: 9783319676111
• Rilegatura: Copertina flessibile
• Numero edizione: 1
• Numero di pagine: 224
• Contatto del produttore: non disponibile
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