Sharp counterexamples in unique continuation for second order elliptic equations

Abstract

We construct nontrivial solutions with compact support for the elliptic equation ∆u = V u with V ∈ Lp, p < n/2 or V ∈ L w for n ≥ 3 and with V ∈ L1 for n = 2. The same method also yields nontrivial solutions with compact support for the elliptic equation ∆u = W∇u with W ∈ Lq, p < n or W ∈ Lw for n ≥ 2. For the second order elliptic equations ∆u = V u in R (1) respectively ∆u = W∇u in R (2) we define the (weak) unique continuation property (UCP) , respectively the strong unique continuation property (SUCP) as follows: Let u be a solution to (1) which vanishes in an open set. Then u = 0. (UCP) Let u be a solution to (1) which vanishes of infinite order at some point x0 ∈ R. Then u = 0. (SUCP) ∗Research partially supported by the DFG and DAAD. †Research partially supported by the NSF grants DMS-9970297, INT-9815286.