Cauchy Problem for Differential Operators with Double Characteristics
Non-Effectively Hyperbolic Characteristics
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Beschreibung
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.
Features thorough discussions on well/ill-posedness of the Cauchy problem for differential operators with double characteristics of non-effectively hyperbolic type
Takes a unified 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
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
Autor*in
Tatsuo Nishitani
Themen in »Cauchy Problem for Differential Operators with Double Characteristics«
Cauchy problem Well/ill-posedness Non-effectively hyperbolic IPH condition Microlocal energy estimates Tangent bicharacteristic Gevrey classes Transition of spectral type partial differential equations ordinary differential equations