I have a question about weak derivatives.

Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is the $\alpha$-th ($\alpha$ is a multi index) weak derivative of $u$ if the following equation holds:
\begin{align}
\int_{U}u D^{\alpha}\varphi=(-1)^{|\alpha|}\int_{U}v\varphi
\end{align}
for all $\varphi \in C_{0}^{\infty}(U)$ (the space of infinitely differentiable functions with compact support in $U$).

Let $\emptyset \neq U$ be a bounded open subset of $\mathbb{R}^{n}$ with $C^{1}$-boundary. I want to define weak derivative of $u \in L^{1}(\overline{U})$. How should I define derivatives? I think this notion enables us to define Sobolev spaces on $\overline{U}$.