Exercise 14, § 3

Let U be an open set of $\mathbf{R}^n$ and let $k\geqslant 1$ be an integer. Let $p\geqslant 1$ be a real number. Suppose that the complement $\mathbf{R}^n-$ U is not negligible for Lebesgue measure. Let $j$ denote the canonical inclusion of U into $\mathbf{R}^n$.

a) There exists a relatively compact open rectangle B in $\mathbf{R}^n$ such that $B\cap U$ and $B\cap (\mathbf{R}^n-$ U) are not negligible.

b) Let $\widetilde{f}\in \mathscr{D}(\mathbf{R}^n)$ be equal to 1 on $B\cap U$. Let $f$ be the restriction of $\widetilde{f}$ to U. Then $f$ does not belong to $W^{k,p}_0$ (U). (In the contrary case, the extension by zero $j_!f$ would belong to $W^{k,p}(\mathbf{R}^n)$; the restriction of $j_!f$ to B would be a distribution such that $\partial_i(j_!f) = 0$ for $1\leqslant i\leqslant n$, and would therefore be the distribution associated with a constant.)