Prove that $C^{\infty}_ {c}(\mathbb{R}^n)$ is dense in $W^{k,p}(U)$ for any open $U\subset \mathbb{R}^n$ with $\partial U\in C^1.$ In which $p\in [1,\infty)$
Note: In Lawrence Evans's PDE text, the case where $U$ is bounded was proved as a Theorem. It thus only remained to prove the case when $U$ is unbounded.
Given $f \in W^{k,p}(U).$ I am thinking about partition $U$ into a sequence of annulli. Let $A_n:= \{x\in \mathbb{R}^n:n-1<|x|<n\},\forall n\in \mathbb{N}$ and let $U_n:=A_n\cap U.$ Then each $U_n$ satisfies the hypothesis of the Theorem for being a bounded set, and thus on each $U_n$ there exists a sequence $(f^{(n)}_j:j\in \mathbb{N}) \subset C^{\infty}_ {c}(U_n)$ which converges to the restriction $f\chi_{U_n}.$
But, how does one guarentee that the "glued-fuction" $\sum_{n=1}^{\infty}f^{(n)}_j\chi_{U_n}$ for each fixed $j$ is in $C^{\infty}_ {c}(U)$?? Is my idea plausible? If so, how to make it rigorous?
Thanks for any feedback!
There are two ways we could fix this.
The fastest way is noticing that $U$ is an extension domain and we could extend $u$ to $\bar{u}\in W^{1,p}(R^N)$ such that $\bar{u}=u$ inside $U$.
Next, we could use $(v_n)\subset C_c^{\infty}(R^N)$ to approximate $\bar{u}$ in $W^{1,p}$ by the fact that $W_0^{1,p}(R^N)=W^{1,p}(R^N)$. Then $v_n$ restrict to $U$ will do the job for you.
Note that generally we require $U$ to be bounded to be an extension domain, like the one you read from Evans or H. Brezis, but we do can do extension on unbounded domain, with more complicated approach. You could find one from Leoni's book, Theorem 12.15.
Moreover, you could also follows the idea in the comment I wrote yesterday, which only require $\partial U$ to be continuous. You could also find details in Theorem 10.29 in Leoni's book.