Let $\Omega$ be an open set of the $\Bbb R$

-As usual denotes by $H_0^1(\Omega)$ the closure of $C_c^\infty(\Omega)$ in $H^1(\Omega)$

-Consider $H_*(\Omega)$ the closure of $C_c^\infty(\Omega)$ in $H^1(\Bbb R^d)$.

-Consider $H_{\Omega}(\Omega)=\{u\in H^1(\Bbb R^d) \text{ u= 0 a.e on } \Omega^c\}$.

Question does the above spaces coincide? If not when are they equal?

Let $\Omega$ be an open set of $\Bbb R^d$: consider the following function spaces

• $H_0^1(\Omega)$, i.e. the closure of $C_c^\infty(\Omega)$ in $H^1(\Omega)$
• $H_*(\Omega)$, i.e. the closure of $C_c^\infty(\Omega)$ in $H^1(\Bbb R^d)$.
• $H_{\Omega}(\Omega)=\{u\in H^1(\Bbb R^d):\ u= 0 \text{ a.e on } \Omega^c\}$.

Question: does the above spaces coincide? If not when are they equal?