Exercise 11, § 3

Let $n\in \mathbf{N}$ and let U be an open subset of $\mathbf{R}^n$. Let $k\in \mathbf{N}$ and $p\in [1,+\infty [$. We denote by $H^{k,p}(U)$ the closure of $\mathscr{C}^{\infty}(U)$ in $W^{k,p}(U)$.

For $x\in U$, let $\delta (x)$ be the distance from $x$ to the boundary of U in $\mathbf{R}^n$; let $U_{-1}= U_0=\emptyset$ and

for every integer $k\geqslant 1$. Put $V_k= U_{k+1}\cap (\mathbf{R}^n-\overline{U}_{k-1})$ for $k\geqslant 1$. The open sets $V_k$ cover U; let $(\psi_k)_{k\geqslant 1}$ be a partition of unity subordinate to the covering $(V_k)_{k\geqslant 1}$ consisting of functions in $\mathscr{D}(U)$.

a) For $k\geqslant 1$, the function

Supp($\psi_j$)$\subset V_k$

belongs to $\mathscr{D}(U)$ and has support contained in $V_k$; one has

on U.

b) Let $f\in W^{k,p}(U)$ and $\varepsilon >0$. Let $\varphi$ and $\varphi_{\varepsilon}$ for $\varepsilon >0$ be functions as in exercise 10. For every $k\geqslant 1$, there exists $\varepsilon_k>0$ such that

c) The function

belongs to $\mathscr{C}^{\infty}(U)$ and $\|f-f_{\varepsilon}\|_{k,p}< \varepsilon$.

d) The space $H^{k,p}(U)$ is equal to $W^{k,p}(U)$.

e) The space $H^{k,p}(U)$ is identified with the completion of the space $\mathscr{C}^k(U)$ for the norm $f\mapsto \|f\|_{k,p}$.

f ) Let V be an open subset of $\mathbf{R}^n$ and let $\varphi : U\rightarrow V$ be a diffeomorphism of class $C^k$. Assume that the partial derivatives of order $\leqslant k$ of $\varphi$ (resp. of $\varphi^{-1}$) are bounded and uniformly continuous on U (resp. on V). The mapping $f\mapsto f\circ \varphi$ is an isomorphism of topological vector spaces of $W^{k,p}(V)$ onto $W^{k,p}(U)$. (Use the result of the preceding question.)