Cauchy Problem for Differential Operators with Double Characteristics, 1st ed. 2017 Non-Effectively Hyperbolic Characteristics Lecture Notes in Mathematics Series, Vol. 2202
Langue : Anglais
Auteur : Nishitani Tatsuo
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.
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.
Features thorough discussions on well/ill-posedness of the Cauchy problem for di?erential operators with double characteristics of non-e?ectively hyperbolic type Takes a uni?ed approach combining geometrical and microlocal tools Adopts the viewpoint that the Hamilton map and the geometry of bicharacteristics characterizes the well/ill-posedness of the Cauchy problem
Date de parution : 11-2017
Ouvrage de 213 p.
15.5x23.5 cm
© 2024 LAVOISIER S.A.S.
