Let $D = B_{2r}(0)\backslash \overline{B}_r(0)$. Assume $u$ is harmonic in $D$ with $u = 0$ on $\partial B_{2r}(0)$ and $u = 1$ on $\partial B_r(0)$.
Let's take a point $x\in \partial B_{2r}(0)$, then $\frac{x}{2}\in \partial B_r(0)$. In the 1D case, we can show that $u$ is linear and thus the midpoint of $\left[\frac{x}{2},x\right]$ satisfies $$ u\left(\frac{3x}{4}\right) = \frac{1}{2}. $$
In the general case, can we at least show that $$ u\left(\frac{3x}{4}\right) \geq C $$ for some $C$ independent on $r$?
In general, I want to replace the Laplacian with a uniform elliptic operator and show $$ \frac{\partial u}{\partial n}(x_0) \geq \frac{C}{r}$$ for every point $x_0\in \partial \Omega$. I can use Hopf Lemma to reduce the problem to bounding from below any point $u(x)\geq C$ where $\mathrm{dist}(x,\partial \Omega) = \mathrm{const}$ (this constant could be dependent to $r$).