Strong Uniqueness for Second Order Differential Operators

AbstractWe prove a strong unique continuation result for differential inequalities of the form |P(x,D)u|⩽C1|x|−2|u|+C2|x|−1|∇u|, whereP(x,D)=∑nj,k=1ajk(x)DjDkis an elliptic second order differential operator with Lipschitz coefficients such thatajk(0) is real.C1andC2are positive constants such thatC2is sufficiently small. Our assumption on the constantC2is justified by counterexamples due to Alinhac and Baouendi [2] and Wolff [6] showing that the strong unique continuation fails ifC2is not small