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 $H\_0^1\cap H^2$.



Extension by zero in Sobolev spaces
It does not remain true.
If $\omega=B(0,1)$ and $\Omega=B(0,2)$ and $f(x)=1-|x|^2$, then $f\in H^1\_0(\omega)\cap H^2(\omega)$ but the extension by zero is not in $H^2(\Omega)$.


