Laplace equation :mean value formula for gradient of weak solution

Question

Let $B(x_0,R)$ an open ball in $\mathbb{R}^n$ ($n \geq 2$). Let $u \in H^{1}(B(x_0,R))$ a weak solution of $\Delta u =0$ in $B(x_0,R)$. Let $u_{x_i}, i \in \{ 1,...,n\}$ a weak derivative of $u$. Is it true that

$$ u_{x_i}(x_0) = \frac{1}{|B(x_0,R)|} \int_{B(x_0,R)}u_{x_i} (y) dy ?$$

I am asking by curiosity. I did not find anything in this direction...

Thanks in advance

Answer 1

Differentiating the equation we find that $$\Delta u_{x_i}=\partial_{x_i}(\Delta u)=0,$$so the derivative is also harmonic, and thus the mean value property holds.