Helmholtz decomposition

Question

I have a question about the Helmholtz decomposition as shown in Wikipedia: https://en.wikipedia.org/wiki/Helmholtz_decomposition . Wikipedia provides both a solution in free space, and a solution for bounded spaces. However, I think the solution for bounded spaces is incorrect. To see this, suppose we are interested in a vector field $F$ on $V$ with bounding surface $S$. Suppose in addition that $F\cdot \mathbf{n}=0$ and $F \times \mathbf{n}=0$ for all $x$ in $S$. In that case, the boundary terms in the first two equations on the Wikipedia site each equal zero, and we are left with the free space solution. I don't see how this can be correct, when the free-space solution vanishes at infinity, whereas the bounded problem should vanish at a bounded point. Shouldn't the Green Function you use in the decomposition match the space over which you integrate? Any help towards finding the correct answer would be highly appreciated.