Function extension in a Sobolev space

Question

Let $\Omega$ be a domain of $R^n$ and let $H^2(\Omega)$ be the usual Sobolev space.

Let $\emptyset\ne \omega_1\subset\omega_2$ be open subsets of $\Omega$, and let $\theta \in H^2(\omega_1)$.

I am wondering about the existence of a function $\tilde{\theta} \in H^2(\Omega)$ such that :

1) $\tilde{\theta}=\theta$ on $\omega_1$,

2) $\tilde{\theta} $ is constant on $\Omega-\omega_2$.

Thanks!

Answer 1

The answer is yes, with the caveat indicated in the comment below. Consider an arbitrary extension $\hat{\theta}\in H^2(\Omega)$. Now choose a compactly supported smooth function $\eta$ such that

$${\rm supp}\;\eta\subset \Omega\setminus \omega_2,\;\;\eta\equiv 1 \;\;\mbox{on $\omega_1$}. $$

The function $\tilde{\theta}=\eta\cdot \hat{\theta}$ has the properties you asked for.