• ISBN13: 9783319676111
• Binding: Paperback
• Subject: Mathematics and Statistics
• Publisher: Springer Verlag
• Publisher Imprint: Springer
• Publication Date: January 1, 2017
• Pages: 213
• Original Price: EUR 49.99
• Language: English
• Edition: N/A
• Item Weight: 318 grams
• BISAC Subject(s): Differential Equations / General

From the Back Cover
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.