Extention of functions in Sobolev spaces

Question

Let $\omega$ a subset of a domain $\Omega\in R^n,$ and let $f\in H^2(\omega)\cap H_0^1(\omega)$.

It is known that a function $u\in H_0^1(\omega)$ admits an axtention $U\in H_0^1(\Omega)$.

Does exist any extention of $f$ to a function $F \in H^2(\Omega)\cap H_0^1(\Omega)$?.

Answer 1

Try to find a counter-example in $\Omega = (-1,1)$ and $\omega = (-1,0) \cup (0,1)$.