Cauchy Problem for Differential Operators with Double Characteristics

Name: Cauchy Problem for Differential Operators with Double Characteristics
Brand: Springer Nature
SKU: 9783319676111
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Springer Nature

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9783319676111

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Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for dierential operators with non-eectively hyperbolic double characteristics. Previously scattered over numerous dierent 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 dierential operator P of order m (i.e. one where Pm = dPm = 0) is eectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the characteristics are at most double and every double characteristic is eectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms. If there is a non-eectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between Pj and Pj, where ij 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 insucient 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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