In Evans' book, the definition of the Sobolev space is
$W^{k,p}(U)$ consists of all locally summable functions $u : U \to \mathbb{R}$ such that for each multiindex $\vert \alpha\vert \le k$, $D^{\alpha}u$ exists in the weak sense and belongs to $L^{p}(U)$.
Here, we suppose that the domain $U$ is open regardless of its boundedness. Then, what about the Sobolev space on $n$-dimentional torus $\mathbb{T}^n$? The domain is $\mathbb{R}^n/ \mathbb{Z}^n \equiv [0,1]^n$, which is sort of a closed set.
Then, on the domain of a closet set, can we define the Sobolev space in the sense of Evans', in other words, to define a weak derivative, we need a test function, which is a compact supported smooth function and so on. Then, what is the compact supported function on a torus? Or, should I go for the definition with respect to Fourier Analysis I'd appreciate it if you'd give me any help with understanding this or any reference to this!