Is $C_c(\mathbb{R}^2,\mathbb{R}^2)$ dense in the irrotational square integrable functions?

Question

Let $L_D(\mathbb{R}^n)^n$ be the set of square integrable functions which are the weak derivative of a locally square integrable function. That is $$L_D(\mathbb{R}^n)^n=\{Du\colon u\in H^1_{loc}(\mathbb{R}^n), Du\in L^2(\mathbb{R}^n)^n\}.$$ For $n>2$ it can be shown that $C_c(\mathbb{R}^n)^n\cap L_D(\mathbb{R}^n)^n$ is dense in $L_D(\mathbb{R}^n)^n$. Is this also true for $n=2$?

Answer 1

It is true even for n=1. Consider the Fourier transform of functions u which have $Du\in L_D^n$. For such functions we have $$\int |k|^2\hat u(k)^2\,dk<\infty.$$ The norm associated with this integral corresponds to the norm in $L_D^n$. In this norm, we can approximate $\hat u$ by a test function; first cut off near zero and infinity and then apply a mollifier. Transforming back, we find that gradients of functions in ${\cal S}$ are dense in $L_D^n$. It is easy to get from there to compact support.