Tatsuo Nishitani

Cauchy Problem for Differential Operators with Double Characteristics (eBook, PDF)

Non-Effectively Hyperbolic Characteristics

• Format: PDF

Produktbeschreibung

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 insücient 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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Produktdetails

• Verlag: Springer International Publishing
• Seitenzahl: 213
• Erscheinungstermin: 24. November 2017
• Englisch
• ISBN-13: 9783319676128
• Artikelnr.: 53062007

Inhaltsangabe

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.

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