Cauchy Problem: No Tangent Bicharacteristics
The main purpose of this chapter is to prove two new results on C ∞ well-posedness mentioned in the end of Sect. 1.4. If there is no transition of spectral type and no tangent bicharacteristics then the Cauchy problem is C ∞ well-posed for P of order m under the strict IPH condition. If the positive trace is zero, the IPH condition is reduced to the Levi condition. In this case we prove that when p is of spectral type 2 on Σ and there is no tangent bicharacteristics, the Levi condition is necessary and sufficient in order that the Cauchy problem is C ∞ well-posed for P of order m. The same result holds for second order differential operators of spectral type 1 on Σ with 0 positive trace. To prove these assertions using the results in Chap. 2 we first derive microlocal energy estimates. Then making use of the idea developed in Chap. 4 we prove the C ∞ well-posedness of the Cauchy problem.
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