# $C_0^\infty(\overline \Omega)$ is dense in $H(\operatorname{div};\Omega)$

I've been looking for a while in different Functional Analysis books such as Luc Tartar, Jean Pierre Aubin and Brezis but couldn't find the proof of the density of $C_0^\infty(\overline \Omega)$ in $H(\operatorname{div};\Omega)$ when $\Omega$ is an open bounded subset of $\mathbb R^n$ with Lipschitz boundary $\Gamma:=\partial \Omega$. Can you recommend me a good source to check it?

• What precisely do you mean by $H($ div ; $\Omega)$? – Memeboy Inc. Aug 27 '18 at 15:48
• $H(\operatorname{div};\Omega) := \{ v \in [L^2(\Omega)]^n \mid \operatorname{div} v \in L^2(\Omega) \}$. – Gonzalo Benavides Aug 27 '18 at 15:50
• $n$ is the dimension of the space. I mean, $\Omega \subset \mathbb R^n$. – Gonzalo Benavides Aug 27 '18 at 15:50
• Are you sure you don't mean $C^{\infty}(\overline{\Omega})$? I don't think $C_{0}^{\infty}(\overline{\Omega})$ is dense in $H(div;\Omega)$. See here for why: math.stackexchange.com/questions/201586/… – Wraith1995 Aug 27 '18 at 18:02
• Absolutely sure. Notice that we are talking about $\overline \Omega$ and not about $\Omega$. – Gonzalo Benavides Aug 27 '18 at 18:12