I need to prove the estimate

$|\partial_{x_i}u(x_0)| \leq \frac{N}{R}u(x_0)$

where $u \in C(B(x_0,R))$ and $u$ is nonnegative and harmonic in $B(x_0,R)$, and $i=1, \ldots,N$

I found this proof (see answer), but it's the case where $u$ is continuous up to the boundary $\partial B$ of the ball of radius $R$. This hypothesis is crucial when I apply divergence theorem.

My attempt

Since $u$ is harmonic in the ball, then also $\partial x_i u(x)$ is harmonic for every $i$, as showed in the linked post.

I know that I have to consider smaller balls, so I choose $\varepsilon>0$ and set $r=(1-\varepsilon)R$. Of course, $u$ is continuous (and also harmonic) also at the boundary of the ball of radius $r$.

So I write \begin{align} \frac{\partial u}{\partial x_i}(x_0) = \frac{1}{|(B(x_0,r))|}\int_{B(x_0,r)}\frac{\partial u}{\partial x_i}(x)\, dx = \frac{1}{|(B(x_0,r))|}\int_{\partial B(x_0,r)}u(x)v_i\, dS(x) \end{align}

where $v_i= \frac{x-x_{0i}}{r}$ is the i-th component of the outer normal.

Then I bound the absolute value of the previous term and by using the fact that $u$ is nonnegative I get

\begin{align} |\frac{\partial u}{\partial x_i}(x_0) | \leq \frac{1}{|(B(x_0,r))|}\int_{\partial B(x_0,r)}u(x)\, dS(x)=\frac{N}{r}\frac{1}{|\partial B(x_0,r)|} \int_{\partial B(x_0,r)}u(x)\, dS(x)=\frac{N}{r} u(x_0) \end{align}

where the last equality comes from the MVP. So, up to now, I have

\begin{align} |\partial x_i u(x_0)| \leq \frac{N}{(1-\varepsilon)R} u(x_0) \end{align}

and by letting $\varepsilon \rightarrow 0^{+}$ I have the thesis

Could it be okay?


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