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A note on two notions of unique continuation. (English) Zbl 0828.35035
Let $L$ be a second order elliptic operator of the form $Lu = - \sum^N_{i,j = 1} {\partial \over \partial x_j} (a_{ij} (x) {\partial u \over \partial x_i})$ on a domain $\Omega \subset \bbfR^N$, $N \ge 2$. Let $V(x)$ be a function on $\Omega$ satisfying a local growth or integrability conditions. The operator $L + V$ is said to enjoy the strong unique continuation property if the only solution $u \in H^1_{\text{loc}} (\Omega)$ of $$Lu + V(x)u = 0 \quad \text{in } \Omega \tag 1$$ which admits a zero of infinite order is $u \equiv 0$. Another notion of unique continuation has also been used in the study of several questions of existence. Under certain conditions it is proved (for $N = 2)$ that if there is a solution of (1) which vanishes on a set $E \subset \Omega$ of positive measure, then almost every point of $E$ is a zero of infinite order for $u$. The case $N \ge 3$ is discussed. Comparisons between strong unique continuation property and unique continuation property are given.
Reviewer: J.Diblík (Brno)

35J15Second order elliptic equations, general
35B60Continuation of solutions of PDE