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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=- i,j=1 N x j (a ij (x)u x i ) on a domain Ω N , N2. Let V(x) be a function on Ω satisfying a local growth or integrability conditions. The operator L+V is said to enjoy the strong unique continuation property if the only solution uH loc 1 (Ω) of

Lu+V(x)u=0inΩ(1)

which admits a zero of infinite order is u0. 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 EΩ of positive measure, then almost every point of E is a zero of infinite order for u. The case N3 is discussed. Comparisons between strong unique continuation property and unique continuation property are given.

Reviewer: J.Diblík (Brno)
MSC:
35J15Second order elliptic equations, general
35B60Continuation of solutions of PDE