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Résultats d’unicité de Cauchy instable dans des situations où la condition de pseudo-convexité dégénère. (Uniqueness results of the Cauchy problem nonstable in situations where the pseudo-convexity conditions degenerates). (French) Zbl 0619.35005
The article contains uniqueness and non-uniqueness results in the Cauchy problem for \(C^{\infty}\) second order operators of real principal type across the noncharacteristic oriented surface \(S=\{\phi (x)=0\}\). It is an attempt to prove that the local uniqueness property at \(x_ 0\) is equivalent to the following assertion: no bicharacteristic curve through \(x_ 0\) of the principal symbol of the operator is locally contained in the half-space determined by \(\phi\) (x)\(\geq 0\) (this equivalence is still a conjecture).
The article deals with bicharacteristic curves which meet S with a large order of contact since the result is known for small orders of contact (and due to L. Hörmander [”Linear partial differential operators”, (1963; Zbl 0108.093)] and to S. Alinhac [Ann. Math., II. Ser. 117, 77-108 (1983; Zbl 0516.35018)]). For a comparison with N. Lerner and L. Robbiano’s recent result in J. Anal. Math. 44, 32-66 (1985; Zbl 0574.35003), the assumptions are made here only at \(x_ 0\) or in a half neighbourhood of \(x_ 0\), and the conclusion is the actual uniqueness in the Cauchy problem instead of the weaker ”compact uniqueness”.
More recently, further results of this type have been obtained by the author for general principally normal operators.

MSC:
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
32T99 Pseudoconvex domains
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Full Text: Numdam
References:
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