In the Wikipedia article about the Helmholtz-Decomposition it says in the section about Weak Formulation:
For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds:
$\mathbf{u}=\nabla\varphi+\mathbf{v}$ $\;\;\;$where $\;\;\;\;$φ ∈ $H^1(Ω)$, v ∈ $(H^1(Ω))^d$.
Does anybody know how to prove this statement or where I can find a proof?
Thanks