Let $\Omega\subset\mathbb{R}^n$ be open and connected. Given any function $V\in L^2(\Omega,\mathbb{R}^n)=:H$, is there an $u\in W^{1,2}(\Omega)$ such that $\nabla u = V$, i.e. $\|\nabla u-V\|_H = 0$? Here $W^{1,2}(\Omega)$ denotes a Sobolev space and the coordinates of the gradient $\nabla u$ are the weak partial derivatives $\partial_i u$ for $1\le i \le n$.
I feel that these question might be quite difficult. I'd be glad for any hints or pointers to literature on this or anything that might help me answer this question.
Thank you!