The classical Poincaré estimate establishes closedness of the range of the gradient in unweighted $L^2(\Omega)$-spaces as long as $\Omega\subset\mathbb{R}^3$ is contained in a slab, that is, $\Omega$ is bounded in one direction. Here, as a main observation, we provide closed range results for the rot-operator, if (and only if) $\Omega$ is bounded in two directions. Along the way, we characterise closed range results for all the differential operators of the primal and dual de Rham complex in terms of directions of boundedness of the underlying domain.
Our results are based on the validity of Gaffney's (in)equality and the transition of the same to unbounded (simple) domains as well as on the stability of closed range results under bi-Lipschitz regular transformations. For the results concerning Gaffney's estimate, we shall provide accessible, simple proofs using mere standard results.
Moreover, we shall present non-trivial examples and a closed range result for rot with mixed boundary conditions on a set bounded in one direction only.