# Density of space in a Sobolev space

An exercise from Gilbarg-Trudinger Elliptic Partial Differential Equations states the following :

"Using Lemma 9.12, show that for a $C^{1,1}$ domain $\Omega$ the subspace $$\{u \in C^2{(\bar{\Omega})} | u = 0 \ \text{on} \ \partial \Omega\}$$ is dense in $W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$ for $1<p<\infty$."

I don't see how the lemma quoted helps in the proof. Here is the lemma

Lemma 9.12 Let $u \in W^{1,1}_0(\Omega^{+}), f\in L^p(\Omega^{+}), 1<p<\infty$ satisfy $\Delta u= f$ weakly in $\Omega^{+}$ with $u=0$ near $(\partial \Omega)^{+}$. Then $u \in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega^{+})$ and $$||D^2u||_{p;\Omega^{+}} \leq C||f||_{p;\Omega^{+}}.$$

Here $\Omega^{+}$ means $\{x \in \partial \Omega | x_n >0\}$.

Could you provide some help please? Thank you.

-

-