A uniqueness result for harmonic functions

Let $d\geq 2$, $D=\mathbb{R}^{d}\times (0,\infty )$, and suppose $u$ is harmonic in $D$ and $C^{2}$ on the closure of $D$. If the gradient of $u$vanishes continuously on a subset of $\partial D$ of positive $d$-dimensional Lebesgue measure and $u$ satisfies certain regularity conditions, then $u$ must be identically constant.