Infinitely differentiable function with given zero set and given one set

Consider $V\subseteq U\subseteq\mathbb{R}^n$, where $V$ and $U$ are both open sets and $\partial V\subset U$.

Is it possible to construct a $\mathcal C^\infty$ function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ such that the following conditions hold?

1. $g(x)=1$ for all $x\in V$
2. $g(x)\neq0$ for all $x\in U$
3. $g(x)=0$ for all $x\in \mathbb{R}^n\setminus U$

Note that this question is closely related to the one inInfinitely differentiable function with given zero set?. The difference is that $V=\emptyset$ in that question (i.e. we do not have condition 1 above).