Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$.

Is there any smooth function $f \in C^\infty(\overline{D})$ such that $f=\partial_n f=0$ on $b$ and $f=1$ on $\partial D \setminus b$?

If there is no direct answer, it would be more helpful to mention a reference where the existence of such a function is studied.

Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$.

Is there any smooth function $f \in C^\infty(\overline{D})$ such that $f=\partial_n f=0$ on $b$.

PS: It was further required the following condition $f=1$ on $\partial D \setminus b$, but it seems to be impossible in this case.

If there is no direct answer, it would be more helpful to mention a reference where the existence of such a function is studied.