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I am learning Sobolev spaces. There seem to be a difference while approximating a function in $W^{k,P}(\Omega)$ by smooth function $C^\infty (\Omega)$ and $C^\infty ( \overline \Omega)$, where we use $\overline \Omega$ to refer to the closure of $\Omega$.

How does the closure make a difference , waiting for explanation.

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$C^{\infty}(\overline{\Omega})$ is the set of functions of $C^{\infty}(\Omega)$ such that their derivatives extend continuously to $\overline{\Omega}$. – Davide Giraudo May 28 '12 at 11:02
@DavideGiraudo , can you help me by explaining why should we deal approximation of $W_{loc}$ and global ? I didn't understand why would there be so much of difference ? – Theorem May 28 '12 at 11:17
For example, $1/x^{1/3}$ is in $L^2(0,1)$, but it is not even well defined in $[0,1]$. This condition excludes the function which have kind of behavior at the border. – guaraqe May 28 '12 at 11:37
@JuanSimões: What is the basic thing that we need to keep in mind when we go from $W_{loc}$ to the whole of $W$ ( but not upto the boundary ) ? – Theorem May 28 '12 at 12:11

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