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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 $$\Omega \subset \mathbb{R}^N$$, $$N \geq 2$$. 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 $$N \geq 3$$ is discussed. Comparisons between strong unique continuation property and unique continuation property are given.
Reviewer: J.Diblík (Brno)

##### MSC:
 35J15 Second-order elliptic equations 35B60 Continuation and prolongation of solutions to PDEs
##### Keywords:
strong unique continuation property