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.

  • $\begingroup$ This might be of more use than the aforementioned "Lemma 9.12" It is the next theorem: Let $\Omega$ be a domain in $\mathbb{R}^n$ with a $C^{1,1}$ boundary portion $T\subset \partial \Omega$. Let $U\in W^{2,p}(\Omega), 1<p<\infty$, 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) $\endgroup$ – Euler....IS_ALIVE Nov 5 '12 at 1:35

I'm not very good at working with these results rigorously, but I'll try:

I think the idea is to take a dense subspace of $L^p$ and transfer it to $W^{2,p}$ using the Lemma (or some similar regularity result).

Let $u \in W^{2,p}(\Omega) \cap W^{1,p}_0(\Omega)$. Then $f:=\Delta u \in L^p(\Omega)$, and there exist $f_n \in C^\infty(\Omega)$ with $f_n \to f$ in $L^p(\Omega)$. Let $u_n \in C^2_0(\bar{\Omega})$ solve $\Delta u_n = f_n$ (should be possible with a $C^{1,1}$ boundary?). Then, by linearity and Lemma 9.17 (or the Lemma given if the $\Omega^+$ stuff can be ignored):

$\Delta(u_n-u)=f_n-f \implies ||u_n-u||_{W^{2,p}(\Omega)} \leq C ||f_n-f||_{L^p(\Omega)}$,

so $u_n \to u$ in $W^{2,p}(\Omega)$.


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