i read in Brezis book that
1)the spaces $C_{0}^{\infty}(\omega)$ and $C^{\infty}(\omega)\cap H^{1}(\omega)$are dense in $H^{1}(\omega)$ so can we say that also $C_{0}^{\infty}(\overline{\omega})\cap H^{1}(\omega)$ is dense in $H^{1}(\omega)$? here$\omega$ is open,regular,bounded domain of $R^{n}$
2)iknow that is $\omega$ is lipshitz domain then $C_{0}^{\infty}(\overline{\omega})$ is dense in $H^{1}(\omega)$ but if $\omega$ is regular bounded open of $R^{n}$ we have thes ame result?
3)the space $C_{0}^{\infty}(\omega)$is dense in $L^{2}(\omega)$ so can i say that also $C^{\infty}(\overline{\omega})$ is also dense in $L^{2}(\omega)$? here $\omega$ is open,regular,bounded domain of $R^{n}$
- if $\Omega $ is lipshitz domain we have : $u\in C^\infty(\overline{\Omega}) \implies $ $u$ has continuous classical derivatives of orders up to $k$ $\implies $ $u$ has weak derivatives of orders up to $k$, which are in $L^2$ .Now if $\omega$ is bounded regular open domain of $R^{n}$ do we have the same result?
- if u is in $C_{0}^{\infty}(\omega)$ then $u \in C^{\infty}(\overline{\omega})$ ?i think it is true because $C_{0}^{\infty}(\omega)$ consistes of functions in$C_{0}^{\infty}(\omega)$ with a compact support included in $\omega$,where$\omega$ is open regular bounded domain of $R^{n}$ thanks