Let $\Omega$ be a bounded $C^1$ domain satisfying the exterior sphere condition at every boundary point and $f$ be a bounded continuous function in $\Omega$. Suppose $u \in C^2(\Omega) \cap C^1(\overline{\Omega})$ solves the equation
$$ \Delta u = f \text{ in } \Omega \\ u = 0 \text{ on } \partial \Omega $$ Then there is a constant $C = C(n, \Omega)$ such that there holds $$\sup_{\partial \Omega}|\frac{\partial u}{\partial\nu}| \leq C \sup_{\Omega} |f|$$
I am stuck in this problem, I believe the solution involves the Maximum principle but I don't see how to use it and I haven't been able to advance much. Any hint is appreciated.
