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Unicité de Cauchy pour des opérateurs de type principal. (Cauchy uniqueness for operators of principal type). (French) Zbl 0574.35003
Let P be a second order differential operator, with smooth principal symbol p(x,$$\xi)$$ and bounded lower order term, in a neighborhood V of a point $$x_ 0\in {\mathbb{R}}^ n$$. Let S be a smooth non characteristic hypersurface through $$x_ 0:$$ $$S=\{x\in V:$$ $$\phi (x)=\phi (x_ 0)\}$$. We denote by $$H_ p$$ the Hamiltonian field of p.
The main result of this paper is the following remarkable extension of the well known Hörmander’s theorem:
Suppose P of principal type and that, for every $$(x,\xi)\in V\times {\mathbb{R}}^ n,\quad p(x,\xi)=0,\quad H_ p(\phi)(x,\xi)=0\Rightarrow H^ 2_ p(\phi)(x,\xi)\leq 0,$$ then every solution $$u\in C^{\infty}(V)$$ of $$Pu=0$$, whose support is contained in $$\{$$ $$x: \phi (x)\geq \phi (x_ 0)\}$$ and supp $$u\cap \{x:$$ $$\phi (x)=\phi (x_ 0)\}$$ is compact, must vanish near $$x_ 0.$$
Some corollary concerning the usual Cauchy problem is derived from this result.
Let us note that an improvement of the above theorem has been given by L. Hörmander [”The analysis of linear partial differential operators. IV: Fourier integral operators”, Chapter 28, Berlin etc.: Springer-Verlag (1985)].
Reviewer: C.Zuily

##### MSC:
 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35S10 Initial value problems for PDEs with pseudodifferential operators
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