Let $\Omega$ be an open bounded set of $R^n$, and let $\omega$ be an open subset of $\Omega$ s.t $\overline{\omega} \subset \Omega.$
For $f\in H_0^1(\omega)$, it is known that the extension of $f$ to $\Omega$ by $0$ is an element of $H_0^1(\Omega).$
I wonder if the result remains true when we replace $H_0^1$ with $H_0^1\cap H^2$.
No, it will not be in $H^2(\Omega)$ in general.
A one-dimensional example: let $\omega=(-1,1)$, $\Omega=(-2,2)$, and $f(x) = x^2-1$. Then $f\in H_0^1(\omega)\cap H^2(\omega)$, but its zero extension is not in $H^2(\Omega)$ (for one thing, $H^2\subset C^1$ in this case).
A similar example works in higher dimensions: take two concentric balls, with $f(x)=|x|^2-1 $ on the smaller one.