Condition on domain for existence of vector potential

Question

Let $\Omega \subset \mathbb R^3$ be open. I would like to have a reference for something like

Assume that $\Omega$ satisfies (*), then every solenoidal vector field possesses a vector potential, i.e., for every $F \in C^1(\Omega; \mathbb R^3)$ with $\operatorname{div}F = 0$, there exists $G \in C^2(\Omega; \mathbb R^3)$ such that $F = \operatorname{rot}G$.

Here, (*) should be as weak as possible.

For star-shaped domains, this is known as Poincaré's lemma. However, it should also hold for domains "without inner boundaries" (claimed here), but I am not able to find a reference for this stronger result.

Also clear: $\Omega$ being simply connected is not enough, see https://math.stackexchange.com/a/4272245.

Answer 1

I finally managed to find a nice reference specifically for the three-dimensional situation:

Cantarella, Jason; DeTurck, Dennis; Gluck, Herman (Mai 2002).
„Vector Calculus and the Topology of Domains in 3-Space“.
The American Mathematical Monthly 109.5, S. 409–442.
doi: 10.1080/00029890.2002.11919870.