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I have seen the spaces $C_c^\infty(\Omega)$, $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$ a lot in theorems regarding PDE where $\Omega$ denotes some open subset of $\Bbb{R}^d$. There is no doubt about the definitions of $C_c^\infty(\Omega)$ and $C_c^\infty(\Bbb{R}^d)$. But I'm not very clear about the relationships among these three spaces.

Here are my questions:

  • What is the definition for $C_c^\infty(\overline{\Omega})$? If one says it consists of functions $f:\overline{\Omega}\to\Bbb{R}$ such that $f$ is smooth (infinitely differentiable) and with compact support, my question concerns the value of $f$ on $\partial\Omega\setminus\Omega$. (I would really appreciate if one could also come up with a reference for definiton of this space. )
  • Could one come up with an example illustrating the difference between $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Omega)$?
  • Are $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$ "essentially" the same? If so, how? (I've seen these two spaces in different books regarding a same theorem.)

[Added:] In the book Navier-Stokes Equations--- Theory and Numerical Analysis by Temam, the author defines (page 3)

$\mathcal{D}(\Omega)$ (or $\mathcal{D}(\overline{\Omega})$) be the space of $C^\infty$ functions with compact support contained in $\Omega$ (or $\overline{\Omega}$).

I have replaced the symbol $\mathcal{D}$ with $C_c^\infty$ here.

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    $\begingroup$ Can you please write where have you seen $C^{\infty}_c(\overline{\Omega})$ used? $\endgroup$ – levap Dec 5 '15 at 22:25
  • $\begingroup$ Are you sure about $C_c^\infty (\overline {\Omega}) $? Maybe you are confusing it with $C^\infty (\overline {\Omega}) $? $\endgroup$ – PhoemueX Dec 5 '15 at 22:30
  • $\begingroup$ @levap: I've edited the question. $\endgroup$ – Jack Dec 5 '15 at 22:44
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Since it seems that the author of the book doesn't explicitly define what it means for a function defined on a closed set to be differentiable, I can only guess that for him differentiable functions on a closed set are restrictions of differentiable functions defined on a larger, open set and so

$$ \mathcal{D}(\overline{\Omega}) = \{ f \colon \overline{\Omega} \rightarrow \mathbb{R} \, | \, \exists g \in C^{\infty}(\mathbb{R}^n) \text{ s.t } f = g|_{\overline{\Omega}}, \mathrm{supp}_{\overline{\Omega}} f \subset \subset \overline{\Omega} \}. $$

Using this definition, we have $\mathcal{D}(\Omega) \subset \mathcal{D}(\overline{\Omega}) \subset C^{\infty}(\Omega)$. To see the difference between $\mathcal{D}(\Omega)$ and $\mathcal{D}(\overline{\Omega})$, take $\Omega = (0,1)$ and $g \colon \mathbb{R} \rightarrow \mathbb{R}$ smooth with compact support such that $g(x) = 1$ for $x \in [0,1]$. Then $g \notin \mathcal{D}((0,1))$ but $g \in \mathcal{D}([0,1])$. More generally, one can characterize $\mathcal{D}(\overline{\Omega})$ as

$$ \mathcal{D}(\overline{\Omega}) = \{ f \colon \Omega \rightarrow \mathbb{R} \, | \, \exists g \in C^{\infty}(\mathbb{R}^n) \text{ s.t } f = g|_{\Omega}, \,\, \mathrm{supp}_{\mathbb{R}^n} g \subset \subset \mathbb{R}^n \} $$

so that functions in $\mathcal{D}(\overline{\Omega})$ are restrictions to $\Omega$ of functions in $\mathcal{D}(\mathbb{R}^n)$. I've found your notation $C^{\infty}_c(\overline{\Omega})$ used together with the latter definition in Elliptic Problems in Nonsmooth Domains (page 24) by Pierre Grisvard.

If $\Omega = \mathbb{R}^n$, then $\mathcal{D}(\Omega) = \mathcal{D}(\overline{\Omega})$ as the notation suggests.

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