Cauchy Problem for Differential Operators with Double Characteristics: Non-Effectively Hyperbolic Characteristics

Description

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 eigen values. 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 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 insufficient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role. Front Matter ....Pages i-viii Introduction (Tatsuo Nishitani)....Pages 1-23 Non-effectively Hyperbolic Characteristics (Tatsuo Nishitani)....Pages 25-42 Geometry of Bicharacteristics (Tatsuo Nishitani)....Pages 43-70 Microlocal Energy Estimates and Well-Posedness (Tatsuo Nishitani)....Pages 71-93 Cauchy Problem: No Tangent Bicharacteristics (Tatsuo Nishitani)....Pages 95-127 Tangent Bicharacteristics and Ill-Posedness (Tatsuo Nishitani)....Pages 129-147 Cauchy Problem in the Gevrey Classes (Tatsuo Nishitani)....Pages 149-179 Ill-Posed Cauchy Problem, Revisited (Tatsuo Nishitani)....Pages 181-201 Back Matter ....Pages 203-213