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A non uniqueness result for operators of principal type. (English) Zbl 0851.35003
We present in this paper a detailed proof of a result announced in [the first author, Ann. Math., II. Ser. 117, 77-108 (1983; Zbl 0516.35018)], with only a sketch of the proof.
A typical example of our main theorem is the wave equation for a time-like surface; though Holmgren’s theorem applies to yield Cauchy uniqueness with respect to such a surface, our result shows that a zero order smooth perturbation may cause uniqueness to fail. In other words, there is no “stable” uniqueness. This theorem improves a previous result due to Hörmander, based on Cohen’s counterexample, and shows the optimality of the Hörmander-Lerner-Robbiano theorem in the weakly pseudoconvex case [see N. Lerner and L. Robbiano, J. Anal. Math. 44, 32-66 (1985; Zbl 0574.35003)].
The method of proof is based on geometrical optic techniques.

MSC:
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35G10 Initial value problems for linear higher-order PDEs
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References:
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