Density of gradient of restricted-function in $L^p$.

Question

Let $A \subset Q$ be two regular bounded subset of $\mathbb{R}^3$ and denotes

$$V=\{\nabla (\varphi|_{A}) \ |\ \varphi \in C^\infty_0(Q) \}.$$

It is clear that $V \subset L^p(A)^3$ for any $p \in ]1,+\infty[$ and therefore $\bar{V} \subset L^p(A)^3$ with respect to the norm $||\cdot||_{L^p(A)^3}$.

Can we prove the opposite inclusion and that therefore $V$ is dense in $L^p (A)^3 $ ?

Answer 1

This is not true: We have $\nabla \times v=0$ for all $v\in V$. So these functions in $V$ cannot approximate non-curl-free functions.