Ambient smooth restrictions are dense on bounded C^k domains

Fix $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$. Since $\mathrm{\Omega }$ is bounded, choose a bounded open $V$ with $\bar{\mathrm{\Omega }}\subseteq V$, and let $E$ be the extension operator supplied by [F1] for this $V$; put $F:=Eu\in W^{k,p}({\mathbb{R}}^n;\mathbb{K})$. Then $F{\mathrm{\mid }}_{\mathrm{\Omega }}=u$ almost everywhere on $\mathrm{\Omega }$ and $\mathrm{supp}F$ is a compact subset of $V$.

Let $\eta$ be a bump as in [F2] with $r=1\mathrm{/}2$, $R=1$, so that $\eta =1$ on ${\bar{B}}_{1\mathrm{/}2}(0)$ and $\mathrm{supp}\eta \subseteq B_1(0)$; then $0<{\int }_{{\mathbb{R}}^n}\eta <\mathrm{\infty }$ and $\rho :=\eta \mathrm{/}\int \eta$ is nonnegative, of class $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$, of unit mass, with $\mathrm{supp}\rho \subseteq B_1(0)\subseteq {\bar{B}}_1(0)$. Put ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x\mathrm{/}\epsilon )$ for $\epsilon >0$, so ${\rho }_{\epsilon }$ is nonnegative with $\int {\rho }_{\epsilon }=1$ and $\mathrm{supp}{\rho }_{\epsilon }\subseteq {\bar{B}}_{\epsilon }(0)$.

For every multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, the class $D^{\alpha }F\in L^p({\mathbb{R}}^n;\mathbb{K})$ exists, and the commutation clause of [F3] applied with $\mathrm{\Omega }={\mathbb{R}}^n$ (so that ${\mathrm{\Omega }}_{\epsilon }={\mathbb{R}}^n$ for every $\epsilon$) gives $D^{\alpha }({\rho }_{\epsilon }\ast F)={\rho }_{\epsilon }\ast (D^{\alpha }F)$ as almost-everywhere classes on ${\mathbb{R}}^n$.

For each $\epsilon >0$ the convolution $v_{\epsilon }:={\rho }_{\epsilon }\ast F$ is of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^n$ by the smoothness clause of [F3], and $\mathrm{supp}{v}_{\epsilon }\subseteq \mathrm{supp}F+\mathrm{supp}{\rho }_{\epsilon }\subseteq \mathrm{supp}F+{\bar{B}}_{\epsilon }(0)$ is compact, first because the support of a convolution is contained in the sum of the supports and then because $\mathrm{supp}F$ is compact; hence $v_{\epsilon }\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{K})$ and its restriction is an admissible approximant.

For each $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, $\mathrm{\parallel }{\rho }_{\epsilon }\ast (D^{\alpha }F)-D^{\alpha }F{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\to 0$ as $\epsilon \to 0^{+}$: for $\mathbb{K}=\mathbb{R}$ this is the real convergence of [F5] applied to the $L^1$ approximate identity of [F4] and the class $D^{\alpha }F\in L^p$; for $\mathbb{K}=\mathbb{C}$ the complex interface of [F6] applies to $D^{\alpha }F$ directly, a real class being a complex class.

By the norm formula of [F8] and step 3.1, the smooth convolutions converge in the whole-space Sobolev norm: $\mathrm{\parallel }{\rho }_{\epsilon }\ast F-F{\mathrm{\parallel }}_{W^{k,p}({\mathbb{R}}^n)}^p={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le k}\mathrm{\parallel }{\rho }_{\epsilon }\ast (D^{\alpha }F)-D^{\alpha }F{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}^p\to 0$.

By the restriction contraction [F7] applied to $v_{\epsilon }-F\in W^{k,p}({\mathbb{R}}^n;\mathbb{K})$ and the identity $F{\mathrm{\mid }}_{\mathrm{\Omega }}=u$ of step 1.1, $\mathrm{\parallel }{v}_{\epsilon }{\mathrm{\mid }}_{\mathrm{\Omega }}-u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}\le \mathrm{\parallel }{v}_{\epsilon }-F{\mathrm{\parallel }}_{W^{k,p}({\mathbb{R}}^n)}$, which tends to zero by step 4.1; given $\delta >0$ choose $\epsilon >0$ with $\mathrm{\parallel }{v}_{\epsilon }{\mathrm{\mid }}_{\mathrm{\Omega }}-u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}<\delta$.

Therefore the restrictions of $C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{K})$ functions are dense in $W^{k,p}(\mathrm{\Omega };\mathbb{K})$ for $1\le p<\mathrm{\infty }$, while nothing is asserted at $p=\mathrm{\infty }$: the convergence inputs [F5] and [F6] are stated for finite exponents only, and step 3.1 fails for the essential-supremum norm.