Smooth compact-support approximation in Euclidean Sobolev space

Statement

For an integer $k\ge0$ and $1\le p<\infty$ , the test functions $C_c^\infty(\mathbb R^n)$ are dense in $W^{k,p}(\mathbb R^n)$ : for every $u$ there are $u_j\in C_c^\infty$ with $\|u_j-u\|_{W^{k,p}}\to0$ .

Cut off, then smooth

Choose a smooth cutoff $\chi=1$ near zero and set $\chi_R(x)=\chi(x/R)$ . The Leibniz formula gives $\chi_Ru\to u$ in $W^{k,p}$ : the main terms converge by integrable tails, and terms with a derivative on $\chi_R$ carry a factor $R^{-1}$ or smaller. For fixed $R$ , mollification converges for every weak derivative through order $k$ and preserves compact support up to an $\varepsilon$ -neighborhood. A diagonal choice gives the sequence.

This assertion concerns the whole space. On a domain with boundary, test functions need not approximate arbitrary Sobolev boundary values. The restriction $p<\infty$ is also essential to this statement.

References

Hunter, Sobolev Spaces, §§3.5–3.7.