Let $U$ be an open subset of $\mathbb{R}^{n}$. The following are two theorems taken from the chapter about Sobolev Spaces of the Evans' book.
Theorem 1 Assume $u\in W^{k,p}(U)$ for some $1\le p<\infty$ and set $u^{\varepsilon}=\eta_{\varepsilon}\ast u$ in $U_{\varepsilon}$, where $U_{\varepsilon}:= \{ x\in U: dist(x,\partial U)>\varepsilon\}$. Then
- $u^{\varepsilon}\in C^{\infty}(U_{\varepsilon})$ for each $\varepsilon >0$ and
- $u^{\varepsilon}\to u$ in $W_{loc}^{k,p}(U)$ as $\varepsilon\to 0. $
Theorem 2 Assume $U$ is bounded and suppose as well that $u\in W^{k,p}(U)$ for some $1\le p<\infty$. Then there exist functions $u_m\in C^{\infty}(U)\cap W^{k,p}(U)$ such that $$ u_m\to u\qquad\text{ in}\,\,W^{k,p}(U).$$
Our professor asked us to reflect about why Theorem 2 is more useful than Theorem 2. Theorem 1 states the conditions for a local approximation, while theorem 2 states the conditions for a global approximation.
I can't say anything else. Any ideas?