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Unique continuation of solutions of differential equations with weighted derivatives. (English. Russian original) Zbl 0971.35003

Sb. Math. 191, No. 3, 431-458 (2000); translation from Mat. Sb. 191, No. 3, 113-142 (2000).
The author studies the local uniqueness of the solution of the Cauchy problem for a differential operator \(P(x,D)= \sum_{\langle\rho, \alpha\rangle\leq m} a_\alpha(x) D^\alpha\), where \(\langle\rho, \alpha\rangle= \sum \rho_j\alpha_j\). Let \(p_k(x, \xi)= \sum_{\langle\rho, \alpha\rangle= k} a_\alpha(x) \xi^\alpha\), \(\mu= \min\rho_j\) and define \(H(x,\xi,h)= \sum^{\mu- 1}_{k=0} h^k \rho_{m-k}(x, \xi)\). Denote \(x'= (x_j|j\), \(\rho_j= \mu)\) and \(x''= (x_j|j, \rho_j> \mu)\). Let \(\phi(x)\) a real function satisfying \(p_m(x^0, \phi_{x'}(x^0), 0)\neq 0\) and assume that the polynomial \(p_m(x^0,\xi'+ z\phi_{x'}(x^0), \xi'')\) with respect to \(z\) has simple zeros for every \(\xi\neq 0\) and in a neighbourhood of \(x^0H(x,\xi+ h^{\rho=\mu}\phi_x(x), h)\) has the roots \(z_j(x,\xi,h)\) satisfying one of the following conditions: \(\text{Im }z_j(x, \xi,h)= 0\) or \(\text{Im }z_j(x,\xi,0)\neq 0\). Then it is proved that if \(u\in H^m\), \(Pu= 0\) in a neighbourhood of \(x^0\) and \(u= 0\) in \(\phi< 0\), then \(u= 0\) in a neighbourhood of \(x^0\).

MSC:

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