A note on two notions of unique continuation. (English) Zbl 0828.35035

Gossez, J.-P.; Loulit, A.

Bull. Soc. Math. Belg., Sér. B 45, No. 3, 257-268 (1993).

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