Let $1 \le p < \infty$, $U \subset \real^d$ be a bounded [[Open Set|open]] set such that $\partial U$ is [[Differentiable Boundary|of class]] $C^1$, and $W^{1, p}$ be the [[Sobolev Space|Sobolev space]]. If $f \in W^{1, p}$ with [[Sobolev Trace Theorem|trace]] zero, then $f \in W^{1, p}_0(U)$. *Proof*. For each $x \in \partial U$, let $(V_x, \varphi_x)$ be a [[Atlas|chart]] at $x$. Since $\partial U$ is [[Compactness|compact]], there exists $\seqf{x_j}$ such that $\seqf{V_{x_j}}$ covers $\partial U$. Let - $V_j = V_{x_j}$, $\varphi_j = \varphi_{x_j}$ for each $1 \le j \le n$. - $V_0 = U$. - $\bracs{\psi_j}_0^n$ be a smooth [[Partition of Unity|partition of unity]] subordinate to $\bracs{V_j}_0^n$. - $f_j = f \psi_j$ for each $0 \le j \le n$. - $\wh f_j = f_j \circ \varphi_j^{-1}$ for each $1 \le j \le n$. then each $f_j$ is compactly supported. Since composition with a diffeomorphism [[Sobolev Norm Over Charts|is a bounded linear map]], 1. $\wh f_j \in W^{1, p}(\wh V_j)$ for each $1 \le j \le n$. 2. There exists $C \ge 0$ such that $\norm{g \circ \varphi_j}_{W^{1, p}(V)} \le C \norm{g}_{W^{1, p}(\wh V_j)}$ for any $1 \le j \le n$ and $g \in W^{1, p}(\wh V_j)$. Let $\eps > 0$. By the [[Zero Trace Theorem on Half Plane]], for each $1 \le j \le n$, there exists $\wh g_j \in W^{1, p}(\wh V_j)$, compactly supported in the interior of $\wh V_j$ such that $\normn{\wh g_j - \wh f_j}_{W^{1, p}(\wh V_j)} < \eps/(Cn)$. Take $g_j = \wh g_j \circ \varphi_j$, then $ \norm{g_j - f_j}_{W^{1, p}(V_j)} \le C\normn{\wh g_j - \wh f_j}_{W^{1, p}(\wh V_j)} < \eps/n $ with each $g_j$ compactly supported in $U$. Now define $g_0 = f_0$ and $g = \sum_{j = 0}^n g_j$, then $ \norm{f - g}_{W^{1, p}(U)} \le \sum_{j = 0}^n\norm{f_j - g_j}_{W^{1, p}(V_j)} = \sum_{j = 1}^n\norm{f_j - g_j}_{W^{1, p}(V_j)} < \eps $ with $g$ compactly supported in $U$ as well. [[Approximation of Sobolev Space by Smooth Functions|Approximation]] of $g$ gives $h \in C_c^\infty(U)$ with $\norm{g - h}_{W^{1, p}(U)} < \eps$, which yields the desired approximation.