Exercise 12, § 3

a) For all integers $n\in \mathbf{N}$ and $k\in \mathbf{N}$, and for every real number $p\geqslant 1$, one has $W^{k,p}(\mathbf{R}^n) = W^{k,p}_0(\mathbf{R}^n)$.

b) Let $p\geqslant 1$ be a real number. Let $U\subset \mathbf{R}^2$ be the set of $(x, y)\in \mathbf{R}^2$ such that $0<|x|<1$ and $0< y <1$. The characteristic function $\varphi$ of $U\cap (\mathbf{R}^*_+\times \mathbf{R})$ belong to $W^{1,p}(U)$; if $\varepsilon >0$ is sufficiently small, there does not exist a function $f$ of class $C^1$ in a neighbourhood of $\overline{U}$ such that $\|f-\varphi \|_{1,p}< \varepsilon$.