Let $U\subset \mathbb{R}^n$ be a bounded open set. I know that, for example, $C^1(U)$ and $C_0^1(U)$ are dense in $W^{1,p}(U)$ and $W_0^{1,p}(U)$ respectively etc.
Is it true that $C^\infty(U)$ is always dense in $W^{k,p}(U)$ for all $k,p$? Is it true that $C_0^k(U)$ is dense in $W^{k,p}(U)$ for all $k,p$? How to remember them?
Supose $p\in [1,\infty)$. First note that $C^1(U)$ is not dense in $W^{1,p}(U)$, because there are functions in $C^1$ that aren't even integrable. But this is just a technical problem and can be easily solved by considering the space $C^1(U)\cap W^{1,p}(U)$.
The way I usually remember it is like this: Take the space $W^{k,p}(U)$. We have $k$ derivatives and the functions does not need to be zero on the boundary, hence, it is sufficient to consider the space $C^k(U)\cap W^{k,p}(U)$ because, in this space we have $k$ derivatives and the functions dont need to be zero on the boundary of $U$.
On the other hand, if you take the $W_0^{k,p}(U)$, then you have $k$ derivatives and your functions are zero on the boundary, so it is plausible to take an space like $C_0^k(U)$.
To finalize, note that $C^r\cap W^{k,p}$ is dense in $C^k\cap W^{k,p}$ (for $r\geq k$) with the norm of $W^{k,p}$ and $C_0^r$ is dense in $C_0^k$ (for $r\geq k$) in the norm of $W^{k,p}$.
