Interior oscillation controls the harmonic gradient

Work under [F2] and put $v:=u-u(x)$, which is harmonic on $\mathrm{\Omega }$ with the same derivatives as $u$, in particular $Dv(x)=Du(x)$; moreover $\mathrm{\mid }v(y)\mathrm{\mid }=\mathrm{\mid }u(y)-u(x)\mathrm{\mid }\le {\mathrm{osc}}_{B_r(x)}u$ for every $y\in B_r(x)$, so ${\sup }_{B_r(x)}\mathrm{\mid }v\mathrm{\mid }\le {\mathrm{osc}}_{B_r(x)}u$.

Apply [F1] with $\alpha =e_i$ to the harmonic function $v$ on $B_r(x)$: $\mathrm{\mid }{\mathrm{\partial }}_{i}u(x)\mathrm{\mid }=\mathrm{\mid }{\mathrm{\partial }}_{i}v(x)\mathrm{\mid }\le C_{n,e_i}^{\mathrm{\prime }}{r}^{-1}{\sup }_{B_r(x)}\mathrm{\mid }v\mathrm{\mid }\le C_{n,e_i}^{\mathrm{\prime }}{r}^{-1}{\mathrm{osc}}_{B_r(x)}u$, with a constant depending only on $n$ and the coordinate; taking $C_n:={\max }_i{C}_{n,e_i}^{\mathrm{\prime }}$ gives $\mathrm{\mid }{\mathrm{\partial }}_{i}u(x)\mathrm{\mid }\le C_n{r}^{-1}{\mathrm{osc}}_{B_r(x)}u$ for every $i$.

Summing the coordinate bounds, $\mathrm{\mid }Du(x)\mathrm{\mid }=({\sum }_i\mathrm{\mid }{\mathrm{\partial }}_{i}u(x){\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\le \sqrt{n}{\max }_i\mathrm{\mid }{\mathrm{\partial }}_{i}u(x)\mathrm{\mid }\le \sqrt{n}{C}_n{r}^{-1}{\mathrm{osc}}_{B_r(x)}u$; absorbing $\sqrt{n}$ into the constant gives the assertion with a constant depending only on $n$.