Un-curl operator?
As anon put it in a comment:
Essentially this is a vector boundary problem in $\mathbf G = \nabla \times \mathbf F$ defined by $$\begin{align} \nabla \cdot \mathbf G &=0\\ \mathbf G \vert_{\partial \Omega}\cdot \mathbf n &= 1 \end{align}$$ The vector field $\mathbf F$ can be recovered from $\bf G$ via the Helmholtz theorem.