Consider the following vector differential equation $$\vec\nabla\cdot(\vec\nabla \vec F(\vec x)) = \vec S(\vec x)\,.$$ We can write this equation as an operator equation $$D[\vec F] = \vec S\,,$$ where $$D = \vec\nabla\otimes\vec\nabla\,,$$ where $\otimes$ is the outer product. Since this equation is a linear differential equation, it seems natural to go to Fourier space, where it becomes $$\tilde D\tilde F = \tilde S\,,$$ and $$\tilde D =-\vec k\otimes\vec k\,.$$ However, $\tilde D$ appears to be singular... Any suggestions for how to `invert' this operator?
Edit: Perhaps there is a way to project out the singular subspace?