Uniqueness result of Cauchy type for operators of real principal type of order three. (Q1813472)

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 \(x_ 0\).