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?



Possible way to define $H_0^1(\Omega)$ Sobolev spaces
The first two are equivalent, as the $H^1(\Omega)$ norm and $H^1(\mathbb{R}^d)$ norm coincide for $C^\infty\_c(\Omega)$ functions.


The third is in general different:


If you let $d = 1$ and $\Omega = \mathbb{R}\setminus \{0\}$,
you see that $H\_{\Omega}(\Omega) = H^1(\mathbb{R}) \supsetneq H^1\_0(\Omega)$. You can create similar examples in higher dimensions (by omitting a hyperplane instead of a point). On the other hand, if $\Omega$ is a Sobolev extension domain (see e.g. Leoni's *First Course in Sobolev Spaces*) then $u\in H^1\_0(\Omega) \iff$ extending $u$ by zero to the exterior gives an $H^1(\mathbb{R}^d)$ function. And in that case, the third is equivalent to the first two.


