$$\Omega : \nabla\cdot \textbf{A} = 0$$ $$\partial\Omega : \textbf{A} = 0$$
The vector field $\mathbf{A}$ with known divergence and curl is unique if either normal or tangential boundary condition is assigned. If both boundary conditions are assigned, $\mathbf{A}$ seems overdetermined. I am thinking about can we determine $\mathbf{A}$ from only the info of $\nabla \cdot \mathbf{A}$ and both boundary conditions? (In this case, $\nabla \times \mathbf{A}$ is unknown and to be determined).