Let $\Omega \subseteq \mathbb R^n$ be a bounded region with a $C^1$-boundary. We presume that for each $x \in \partial \Omega$, there exists an open ball $B_r(a), r > 0$ around some $a \in \mathbb R^n$ so that $\overline{B_r(a)} \cap \overline{\Omega} = \{x\}$. Furthermore, let $u \in C^2(\Omega) \cap C^1(\overline \Omega)$ be a solution of the boundary problem
$$\begin{cases} \Delta u = f & \text{on } \Omega \\ u \mid_{\partial \Omega} = 0 & \end{cases}$$
for some bounded continuous function $f: \Omega \to \mathbb R$.
I now want to show that
$$\sup_{x \in \partial \Omega} |\partial_\nu u(x)| \leq C\|f\|_\infty$$
where $\nu$ is the (unit) outward normal vector at $x$, and where $C > 0$ is a constant that only depends on $\Omega$ (not on $u$ or $f$).
I'm given the hint that I might consider the Green function $G$ on $\Omega$, and use the fact that $|G(x, y)| < |N_x(y)|$ for all $x, y \in \overline \Omega, x \neq y$, where $N$ is the Newton potential.
Now my attempt was: let’s first take the $\partial_\nu u$-expression. Because of the definition of $u$, we have that
$$\partial_\nu u(\xi) = \lim_{h \to 0} \frac{u(\xi + h \nu_\xi) - \overbrace{u(\xi)}^{= 0}}{h} = \lim_{h \to 0} \frac{u(\xi + h \nu_\xi)}{h}$$
I also think I need the Green’s formula at some point which (should, if I’m not mistaken) give me that
$$u(x) = \int_{\partial \Omega} \Phi(\xi - x) \partial_\nu u(\xi) - u(\xi) \partial_\nu N(\xi - x) d S(\xi) - \int_\Omega N(y - x ) \Delta u(y) d y$$
And also, because of $\overline \Omega$ is compact, I know that there exists one radius $r$ for all points in the boundary of $\Omega$ so that $\overline{B_r(x_y)}$ intersects with $\overline \Omega$ only in $x$ (for some point $x_y$).
From there on, I’m a bit helpless. How could I continue? I know general things like the maximum principle for harmonic functions, the Poisson equation and so on, but I'm not sure if (and how) any of that could help me.