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I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. There it is shown that:

If we have $f \in L^p(\mathbb{R}^n)$, then $(\rho_n *f) \to f $ in $L^p(\mathbb{R}^n)$, for a sequence $(\rho_n)$ of mollifiers. (*)

I would like to know if this result still holds for an open set $\Omega \subset \mathbb{R}^n$ and also how to prove it, i.e:

$f \in L^p(\Omega)$, then $(\rho_n *f) \to f $ in $L^p(\Omega)$, for a sequence $(\rho_n)$ of mollifiers and any open set $\Omega \subset \mathbb{R}^n$.

Note: Brezis relies on the density of $C_c(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$ to show the result stated here for $\mathbb{R}^n$ (*), and then shows this result that, indeed, $C_c(\Omega)$ is dense in $L^p(\Omega)$, for any open set $\Omega \subset \mathbb{R}^n$.

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  • $\begingroup$ But on Brezis there is this proof, you do not understand? $\endgroup$ – user288972 Apr 16 '16 at 13:11
  • $\begingroup$ There is no proof in Brezis for a general open set $\Omega \subset \mathbb{R}^n$. It is only stated (and proven) for $\mathbb{R^n}$ $\endgroup$ – D1X Apr 16 '16 at 13:12
  • $\begingroup$ See well, Corollary 4.23. $\endgroup$ – user288972 Apr 16 '16 at 13:15
  • $\begingroup$ It is conceivable that you need to assume something about the boundary of $\Omega$. $\endgroup$ – DisintegratingByParts Apr 18 '16 at 0:48
  • $\begingroup$ @JohnMartin No, Corolary 4.23 shows that $C_c(\Omega)$ is dense in $L^p(\Omega)$, for any open set $\Omega \subset \mathbb{R}^n$. $\endgroup$ – D1X Apr 18 '16 at 10:04
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Strictly speaking, the result you ask about cannot be true because if $f\in L^p(\Omega)$ then $\rho*f(x)$ is undefined for $x$ near the boundary (because the definition involves $f(x-t)$ where $x-t\notin\Omega$).

Given $f\in L^p(\Omega)$ you can make sense of $\rho*f$ by regarding $f$ as defined on all of $\Bbb R^n$ and vanishing off $\Omega$. But then you're not really talking about $L^p(\Omega)$, and in any case $\rho*f$ is not supported in $\Omega$.

Of course there's no problem using convolutions to show that $C_c(\Omega)$ is dense in $L^p(\Omega)$. First choose $g\in L^p(\Omega)$ so that $||f-g||_p<\epsilon$ and such that $g$ vanishes outside some compact set $K\subset\Omega$; now if the support of $\rho$ is small enough you will have $\rho*g\in C_c(\Omega)$.

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