Let $\Omega\subset\mathbb{R}^n$ be open and let $f\in C^\infty(\Omega)$ be a smooth function. What examples can one come up with that distinguish the 3 criteria below?
1: f satisfies the Neumann boundary condition, i.e. is in the domain of the laplacian defined from its quadratic form on $H^1(\Omega)$.
2: If $x0\in\partial\Omega$ and $\nu_0$ is the normal at $x_0$, we have:
$\nabla f(x)\cdot\nu_0 \to 0$ for $x\to x_0$
3: f extends to a smooth function on $\overline{\Omega}$ and its gradient on the boundary is parallel with the boundary,