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.


35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35G10 Initial value problems for linear higher-order PDEs
Full Text: DOI EuDML


[1] Alinhac, S.: Non unicité du problème de Cauchy. Annals of Math.117, 77–108 (1983) · Zbl 0516.35018 · doi:10.2307/2006972
[2] Alinhac, S., Baouendi, M.S.: Construction de solutions nulles et singulières pour des opérateurs de type principal. Seminaire Goulaouic-Schwartz, exposé no 22, Ecole Polytechnique, Paris 1979 · Zbl 0427.35082
[3] Bahouri, H.: Non unicité du problème de Cauchy pour des opérateurs à symbole principal réel. Comm. PDE8 (14), 1521–1547 (1983) · Zbl 0539.35002 · doi:10.1080/03605308308820314
[4] Cohen, P.: The non uniqueness of the Cauchy Problem. ONR Techn. Report 93. Stanford U. 1960 · Zbl 0151.38101
[5] Hörmander, L.: Non Uniqueness for the Cauchy Problem. Lect. Notes in Math.,459, 36–72, Springer Verlag (1975)
[6] Hörmander, L.: The Analysis of Linear Partial Differential Operators. Berlin: Springer Verlag 1983 · Zbl 0521.35002
[7] Hörmander, L.: Linear Partial Differential Operators. Berlin: Springer Verlag 1963 · Zbl 0108.09301
[8] Lerner, N., Robbiano, L.: Unicité de Cauchy pour des opérateurs de type principal. J. d’Analyse Math.44, 32–66 (1984) · Zbl 0574.35003 · doi:10.1007/BF02790189
[9] Métivier, G.: Counter examples to Holmgren’s uniqueness for analytic non linear Cauchy problems. Inv. Math.112, 217–222 (1993) · Zbl 0794.35026 · doi:10.1007/BF01232431
[10] Métivier, G.: Uniqueness and approximation of solutions of first order non linear equations. Invent. Math.82, 262–282 (1985)
[11] Pliš, A.: A smooth linear elliptic differential equation without any solution in a sphere. Comm. Pure Appl. Math.14, 599–617 (1961) · Zbl 0163.13103 · doi:10.1002/cpa.3160140331
[12] Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc.36, 63–89 (1934) · Zbl 0008.24902 · doi:10.1090/S0002-9947-1934-1501735-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.