Here is a lemma from Gilbarg-Trudinger Elliptic Partial Differential Equations states the following :
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 $$||D^2u||_{p;\Omega^{+}} \leq C||f||_{p;\Omega^{+}}.$$
Where $\Omega^{+}$ means $\{x \in \partial \Omega | x_n >0\}$.
At the end of the proof of the lemma, the authors state that
$\cdots\cdots$. Since $u_h(x',0)=0$, we also obtain $u\in W^{1,p}_0(\Omega^+)$.
Here $u_h$ is the regularization of $u$.
We know $u=0$ near $(\partial \Omega)^{+}$, so does $u_h$. But $u_h$ only vanish on $x_n=0$, how to obtain $u\in W^{1,p}_0(\Omega^+)$ ?