# Gradient estimate and divergence theorem in an open ball (PDE)

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?