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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?

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1 Answers1

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... does the theorem still hold?

Smooth boundary is not needed. The theorem is true provided that the domain does not lie on both sides of any part of its boundary, i.e., provided that $U$ satisfies the segment condition.

Reference: Adams book, p. 68.

Pedro
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