For $v\in C^1(\Omega,\mathbb{R}^n)$ the following is well-known:
Let $\Omega \subset \mathbb{R}^n$ be simply connected. Then for every $v\in C^1(\Omega,\mathbb{R}^n)$ with $curl \;v = 0$ there exists a $u \in C^1(\Omega,\mathbb{R}$ with $v= \nabla u$. Here $curl \;v= \{\partial_j v^i-\partial_i v^j\}_{ij}$.
I have read, that this statement also holds for weakly differentiable functions. Precisely expressed: Let $\Omega \subset \mathbb{R}^n$ be simply connected. Then for every $v \in L²(\Omega, \mathbb{R}^n)$ with $curl\; v = 0$ there exists a $u \in H^1(\Omega,\mathbb{R})$ with $ grad\; u = v$. Here: $(curl\; v, \varphi)_{ij} = \int_\Omega v^i\partial_j\varphi-v^j\partial_i\varphi\text{ for } \varphi \in H^1_0$ .This means that curl v is a Matrix, whose elements belong to the space $H^{-1}(\Omega)$. $grad \;u= v $ means, that $ \int_\Omega u \partial_i \varphi = -\int_\Omega v^i \varphi\quad \forall \varphi \in C^\infty_c$ .
My question is: How can I prove the second statement. I have found a proof for the 3 dimensional case:http://perso.ensta-paristech.fr/~ciarlet/Resumes/07_CRASb.pdf. Look at Theorem 1 with m=0. Is it possible to generalize this proof to the n-dimensional case or does somebody know a good reference for this problem?