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The interior gradient estimate states that

If $u$ is harmonic in a unit ball $B_1$, then there holds $\mbox{sup}_{B_{1/2}}|Du|\leq c\mbox{sup}_{\partial B_1}|u|$. The standard proof follows the method due to Bernstein.


However, it seems that since $|Du(x_0)|\leq \frac{n}{R}\mbox{sup}_{\partial B_R}|u|$, we can use this inequality for all $x_0\in B_{1/2}$ and use maximum principle to get the desired result. I am wondering if it is right.

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Yes, considering the ball of radius $1/2$ centered at an arbitrary $x_0\in B_{1/2}$ is a typical approach here. It reduces the estimation to dealing with the center of a ball, which is simpler. For example, the proof in Interior gradient bound follows this.

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