Cauchy Problem in the Gevrey Classes
In Chap. 6 we showed that there exists a second order differential operator of spectral type 2 on Σ with bicharacteristics tangent to the double characteristic manifold for which the Cauchy problem is ill-posed in the Gevrey class of order s for any s > 5 even though the Levi condition is satisfied. The best we can expect is the well-posedness in the Gevrey class of order 5 under the Levi condition. This is indeed the case. We prove that for general second order differential operator of spectral type 2 on Σ which may have tangent bicharacteristics, the Cauchy problem is well-posed in the Gevrey class of order 5 under the Levi condition.
- 34.L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Grundlehren Math. Wiss., vol. 274 (Springer, Berlin, 1985)Google Scholar