de Rham cohomology can be used to establish the existence of some PDEs. For example, for every smooth function $f:\mathbb R^3 \to \mathbb R,$ whether there exist a vector field $\mathbf X$ such that $\nabla \cdot \mathbf X = f$ depends on $H^3_{\text{dR}}(\mathbb R^3).$
However, we are often interested in problems with boundary data.
Can de Rham cohomology help with boundary data? To give a simple example, how to show that a solution to $$ \begin{cases} \nabla \cdot \mathbf X =f \text{ in } \Omega,\\ \mathbf X \cdot \mathbf n = 0 \text{ on }\partial \Omega. \end{cases} $$ Exists? (Of course this is closely related to the Newmann problem, but can de Rham Cohomology help?)
Also the uniqueness - in general, without the boundary condition, the solution is certainly not unique. We may write the general solution in the form $\mathbf X + \nabla \times \mathbf A$ for an arbitrary vector potential $\mathbf A,$ again when $H^2_{\text{dR}}(\mathbb R^3) = 0.$ But this does not take into account the boundary data. How can we include the boundary condition in de Rham cohomology?
Edit: I am thinking about something similar to turning the sobolev space (NOT cohomology) $H^1(U)$ into the sobolev space $H^1_0(U).$