Exercise 10, § 3

Let $n\in \mathbf{N}$ and let U be an open subset of $\mathbf{R}^n$. Let $k\in \mathbf{N}$ and $p\in [1,+\infty [$. Let $V\subset U$ be a relatively compact open subset.

Let $\varphi \in \mathscr{D}(\mathbf{R}^n)$ be a positive function with support contained in the unit ball of $\mathbf{R}^n$ and integral 1. For every $\varepsilon >0$ and $x\in \mathbf{R}^n$, put $\varphi_{\varepsilon}(x) =\varepsilon^{-n}\varphi (x/\varepsilon )$.

Let $f\in W^{k,p}(U)$.

a) The restriction $f_V$ of $f$ to V belongs to $W^{k,p}(V)$.

b) Let $\widetilde{f}$ be the function on $\mathbf{R}^n$ which coincides with $f$ on U and is zero outside U. When $\varepsilon \rightarrow 0$, the restriction of $\widetilde{f}*\varphi_{\varepsilon}$ to V belongs to $W^{k,p}(V)$ and converges to $f_V$ in $W^{k,p}(V)$.