In Evans,



> 
> $\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there exist functions $u\_m \in C^{\infty}(\bar{U})$ such that
>  \begin{align\*}
> u\_m \rightarrow u \quad \textrm{ in } W^{k,p}(U)
> \end{align\*}
> 
> 
> 


$\textbf{Question}$ Although we change the boundary condition like
\begin{align\*}
\partial U=\bigcup\_{j=1}^n \Gamma\_j, \quad (\textrm{boundary is piecewise } C^{1})
\end{align\*}
where each $\Gamma\_j$ for $j=1, \cdots, n$ is a $C^1$, $\Gamma\_j$ and $\Gamma\_{j^{'}}$ do not intersect except at their endpoints if $j\neq j'$, then does the theorem still hold? 


Any help is appreciated!! 


I want to know references related that... 


Thank you!!



Extended Global approximation theorem
A standart assumption for the density of $C^{\infty}(\overline\Omega)$ in $W^{k,p}(\Omega)$ is that the domain $\Omega$ satisfies the segment condition, namely, for every point $x\in\partial\Omega$, there exists $c\_{x}>0$ and a nonzero vector $y\_{x}$ such that
$$\forall z\in B(x,c\_{x})\cap\overline\Omega,\quad\{z+ty\_{x},~0<t<1\}\subset\Omega.$$
In particular, $\Omega$ may contain corners and cusps. For a reference see Adams, Sobolev spaces, Condition 3.21 and Theorem 3.22, or Agmon, Lectures on Elliptic Boundary Value Problems, Section 2, Definition 2.1 and Theorem 2.1.


