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Uniqueness result of Cauchy type for operators of real principal type of order three. (Unicité de Cauchy pour les opérateurs de type principal réel d’ordre trois.) (French) Zbl 0742.35008
Let $$x_ 0$$ be a point of $$\mathbb{R}^ n$$, $$S$$ a hypersurface in $$\mathbb{R}^ n$$ defined by the equation $$\varphi(x)=0$$, where $$\varphi$$ is a $$C^ \infty$$ real valued function and satisfies $$d\varphi(x_ 0)\neq 0$$. Let $$P(x,D)$$ be a linear differential operator of order 3 with its real principal part $$p(x,\xi)$$ satisfying $$p(x_ 0,\xi)=H_ p\varphi(x_ 0,\xi)=H^ 2_ p\varphi(x_ 0,\xi)=0$$, further it is assumed that $$\xi\neq 0$$ implies $$d_ \xi p(x_ 0,\xi)\neq 0$$ and $$H^ 2_ p\varphi(x_ 0,\xi)\neq 0$$. Assume that $$S$$ is non-characteristic at $$x_ 0$$ for $$p$$ and there is a neighborhood $$\Omega$$ of $$x_ 0$$ such that for any $$x\in\Omega$$ and $$\xi\in\mathbb{R}^ n$$, $$p(x,\xi)=H_ p\varphi(x,\xi)=0$$ implies $$H^ 2_ p\varphi(x,\xi)\neq 0$$. Then any local solution $$u\in H^ 2_{loc}$$ of $$Pu=0$$ defined in a neighborhood of $$x_ 0$$ satisfies: $$\hbox{supp} u\subset\{x\in\mathbb{R}^ n;\varphi(x)>0\}\cup\{x_ 0\}$$ implies $$u=0$$ near $$x_ 0$$.

##### MSC:
 35B60 Continuation and prolongation of solutions to PDEs 35G05 Linear higher-order PDEs 35S05 Pseudodifferential operators as generalizations of partial differential operators
##### Keywords:
principal type; unique continuation; non-characteristic
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