Given some bounded domain $\Omega\subset \mathbb{R}^n$ with sufficiently regular boundary (e.g. smooth boundary). Then I saw two slightly different definitions for the Dirichlet-Laplacian. Some books consider the Laplacian on the initial domain $\lbrace f\in C^{\infty}(\Omega) \vert f_{\vert\partial\Omega}=0 \rbrace$, i.e. smooth functions which vanish on the boundary. Then the Laplacian is positive and symmetric, therefore there exists a self-adjoint extension (Friedrichs extension). On the other hand, the domain is sometimes $C^{\infty}_0(\Omega)$, i.e. the set of smooth functions which have compact support in $\Omega$. The Laplacian is also symmetric and positive on the domain $C^{\infty}_0(\Omega)$ and therefore for the same reason has a self-adjoint extension. But both initial domains are different. Therefore I'm a little bit worried about the self-adjoint extensions I end up with. Are both self-adjoint extensions the same? Are they different? Is there a reason, why the people like to define it the one or the other way around?

Thanks a lot!