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