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
."
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
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.
, be a strong solution of $Lu = f$ in $\Omega$ with $u = 0$ on $T$, in the sense of $W^{1,p}$, and $a^{ij} \in C^0{(\Omega \cup T)}$. Then for any domain $\Omega' \subset \subset \Omega \cup T$, we have $||u||_{2,p,\Omega'} \leq C(||u||_{p;\Omega} + ||f||_{p;\Omega})$ (In particular, if $T = \partial U$, take $\Omega'=\Omega$ for a global $W^{2,p}$ estimate)
– 2012-11-05