(This question is motivated by an equation I arrived at in electromagnetism. I am aware that the nature of the question might be against some site policies but I am in dire need of an answer. I also have no knowledge of differential / integral equations, apart from the miniscule half baked theorems found in DJ Griffiths)
Suppose I have a differential equation, (or I suppose an integral equation), given by: $$\vec E(r) = \vec E_{0}(r)+ K \nabla\iiint E(r')\cdot \frac{(\vec r-\vec r')}{|\vec r-\vec r'|^2}d\tau'$$ where K is a constant And: $\nabla \times \vec E_{0}(r)= \nabla \times \vec E(r)=0 $
Where $E_{0}(r)$ is known in principle, everywhere in space.
I know that there exists two concentric, closed surfaces $S_{1}$ and $S_{2}$ in which this vector function $\vec E_{0}$ is perpendicular to these two surfaces.
I also know that $\vec E(r) \to 0$ as $ r \to \infty.$ The question is, can I solve for $ \vec E(r) $ uniquely with these conditions?