C1,α domains and unique continuation at the boundary

It is shown that the square of a nonconstant harmonic function u that either vanishes continuously on an open subset V contained in the boundary of a Dini domain or whose normal derivative vanishes on an open subset V in the boundary of a C^1,1 domain in ℝ^d satisfies the doubling property with respect to balls centered at points Q ∈ V. Under any of the above conditions, the module of the gradient of u is a B₂(dσ)-weight when restricted to V, and the Hausdorff dimension of the set of points {Q ∈ V : ∇u(Q) = 0} is less than or equal to d−2. These results are generalized to solutions to elliptic operators with Lipschitz second-order coefficients and bounded coefficients in the lower-order terms. © 1997 John Wiley & Sons, Inc.