Given a bounded area $\Omega \subset \mathbb{R}^n$ which is at least of class $C^3$. If now there is a function $u$ satisfying the PDE
\begin{align*} \Delta u &= f~~~\text{in }\Omega \\ u &= 0 ~~~\text{in } \partial\Omega, \end{align*} for some smooth function $f$. Is it then possible to show that $$\partial_\nu u \overset{?}{=} 0$$ on $\partial \Omega$?
It is not possible to show that result because it is in general not true. Think for instance in the PDE $$ \left\{ \begin{array}{rl} -\Delta u = 4,& \textrm{in } \Omega \\ u = 0,& \textrm{on } \partial \Omega \end{array}\right. $$
where $\Omega=\{(x,y)\in \mathbb{R}^2: x^2+y^2 < 1\}$.
The exact solution is $u=1-x^2-y^2$, which does not have zero normal derivative on $\partial \Omega$.