Let $Omega_1$, $Omega_2$ be bounded,convex, open domains with smooth boundary in $mathbb{R}^2$ and $overlineOmega_1subsetOmega_2$. Suppose we are given a $C^1$ function $f:overlineOmega_1cup(mathbb{R}^2setminusOmega_2)rightarrowmathbb{R}$ satisfies the following properties:

(1)$fequiv 1$ on $partialOmega_1$ and $fequiv 0$ on $partialOmega_2$.

(2)$nabla fcdot nu_k< 0$ on $partialOmega_k$, where $nu_k$ is the unit outward normal vector to $Omega_k$, ($k=1,2$).

Now the question is that can we extend $f$ to $Omega_2setminusoverlineOmega_1$ such that $0leq fleq 1$ and $|nabla f|neq 0$ in $Omega_2setminusoverlineOmega_1$?

I think the answer must be yes because I can imagine its figure as a frustum of a cone. I have read some references about the Whitney’s extension theorem but they do not match. I would be very appreciate if anyone can provide the proof or references.