Possible way to define $H_0^1(\Omega)$ Sobolev spaces

Question

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?

Answer 1

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.