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

di Tatsuo Nishitani (Autore)

Springer International Publishing AG, 2017

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.

• Autore: Tatsuo Nishitani
• Editore: Springer International Publishing AG
• Collana: Lecture Notes in Mathematics
• Anno: 2017
• Rilegatura: Paperback / softback
• Pagine: 213 p.
• Dimensioni: 235 x 155 mm
• EAN: 9783319676111

Testo in English