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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
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