Record Nr.

UNISALENTO991003576719707536

Autore

Nishitani, Tatsuo

Titolo

Cauchy Problem for Differential Operators with Double Characteristics [e-book]: Non-Effectively Hyperbolic Characteristics / Tatsuo Nishitani

ISBN

9783319676128

3319676121

9783319676111

3319676113

Descrizione fisica

1 online resource (viii, 211 pages) : illustrations

Collana

Lecture Notes in Mathematics, 0075-8434 ; 2202

Classificazione

AMS 35-02

LC QA370-380

Disciplina

515.353

Soggetti

Differential equations, Partial

Ordinary Differential Equations

Lingua di pubblicazione

Inglese

Formato

Materiale a stampa

Livello bibliografico

Monografia

Nota di bibliografia

Includes bibliographical references and index

Nota di contenuto

1. Introduction ; 2 Non-effectively hyperbolic characteristics.- 3 Geometry of bicharacteristics.- 4 Microlocal energy estimates and well-posedness.- 5 Cauchy problemno tangent bicharacteristics. - 6 Tangent bicharacteristics and ill-posedness.- 7 Cauchy problem in the Gevrey classes.- 8 Ill-posed Cauchy problem, revisited ; References

Sommario/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 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 Pæj and P æj , where iæj are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 ₉ 4 Jordan block, the spectral structure of FPm is insucient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role