# Reference request: Density of testfunctions in sobolev space $W^{1}_{0}$

can someone help me out and name a good source where the following statement ist proven?

$$C^\infty_0(\Omega)$$ is dense in $$W^{1,p}_0(\Omega)$$ with $$\Omega \subset \mathbb{R}^n$$ being an open, bounded domain with continuous boundary $$\partial \Omega$$.

• Do you know Evans book on PDE? This proof is included in Chapter 4. – Joaquin San May 19 at 12:59
• Are you sure? In the chapter "4. other ways to represent solutions"? Without reading the whole chapter I am a bit skeptical since sobolev spaces are in only introduced in the following chapter 5... At least in my version of the book. – superdave99 May 19 at 13:04
• For me (and many others), this is the definition of $W_0^{1,p}(\Omega)$. Which definition do you use? – gerw May 19 at 14:40
• Ohh correct. Chapter 5 heh – Joaquin San May 20 at 14:09