Is this dense embedding true?

Question

Let $N\geq 1$ and $1\leq p<+\infty$. It is true that the space $\mathcal{C}^1_c(\mathbb{R}^N_{+})$ is dense in $W^{1, p}(\mathbb{R}^N_{+})$? And what about if $p = +\infty$?

Thank You in advance!

Answer 1

The case of $p=\infty$ is trivially untrue as the function $f=1$ is in $\mathrm{W}^{1,\infty}$ but can not be approximated in the norm by a function with compact support.

The other question is a bit more complicated as it depends on whether the functions in $\mathrm{C}^1_c(\mathbb{R}^N_+)$ have compact support in the open set $\mathbb{R}^N_+$ or in the closed $\overline{\mathbb{R}^N_+}$. In the first case every element of $\mathrm{C}^1_c(\mathbb{R}^N_+)$ has a zero trace on the boundary $\mathbb{R}^{N-1}$ but not every function in $\mathrm{W}^{1,p}$ does have a zero trace. So the statement is untrue. In the second case the statement holds true. For more information on this topic I recommend Sobolev Spaces from Adams and Fournier.