Is is true that if $1 \leq p < \infty$, $k \geq 0$, and if $\Omega$ is an open subset of $\mathbf{R}^d$, then $C_c^\infty(\Omega)$ is dense in $W^{k,p}(\Omega)$? I seem to have a proof of this statement, but the care with which Evans handles the question in his textbook on Partial Differential Equations (assuming boundary conditions) makes me worry that I've made a naive error. I've included my proof below (here $\Subset$ means compactly contained, and for each function $u$ and $\varepsilon > 0$, $u^\varepsilon$ denotes the mollification of $u$ by a mollifier supported in a $\varepsilon$ ball):
Consider a family of open sets $\{ V_n \}$ such that $V_n \Subset \Omega$ for each $n$, and $U = \bigcup V_n$. Then we can consider a smooth partition of unity $\{ \xi_n \}$ subordinate to the cover $\{ V_n \}$. For each $u \in W^{k,p}(\Omega)$, we can write $u = \sum_n u \xi_n$. In particular, this means that for each $\varepsilon > 0$, there is $N$ such that $\| \sum_{n = N+1}^\infty u \xi_n \|_{W^{k,p}(\Omega)} \leq \varepsilon$. For each $n \in \{ 1, \dots, N \}$, we can find $\delta_n$ small enough that the $\delta_n$ thickening of $V_n$ is compactly contained in $\Omega$. If $\varepsilon_n$ is small enough, we find $(u \xi_n)^{\varepsilon_n}$ is supported on the $\delta_n$ thickening of $V_n$, and $\| (u \xi_n)^{\varepsilon_n} – u \xi_n \|_{W^{k,p}(V_n)} \leq \varepsilon / N$. But we then find
\begin{align*}
\| u – \sum_{n = 1}^N (u \xi_n)^{\varepsilon_n} \|_{W^{k,p}(\Omega)} \leq \varepsilon + \sum_{n = 1}^N \| u \xi_n – (u \xi_n)^{\varepsilon_n} \|_{W^{k,p}(\Omega)} \leq 2\varepsilon.
\end{align*}
Since $\sum_{n = 1}^N (u \xi_n)^{\varepsilon_n} \in C_c^\infty(\Omega)$, $C_c^\infty(\Omega)$ is dense in $W^{k,p}(\Omega)$.
Best Answer
PhoemueX already mentioned in the comments that this is not true. The problem is right at the beginning: You claim that the norm of $$ \sum_{n=N+1}^\infty u\xi_n=u-\sum_{n=1}^N u\xi_n $$ is small for $N$ sufficiently large without any argument. This is not true in general, and in fact would be all you'd have to prove.
The identity $u=\sum_n u\xi_n$ holds pointwise because there are only finitely many non-zero terms, but it does not necessarily hold in $W^{k,p}$.