I'm trying to understand the set of functions $\mathbb R^3 \rightarrow \mathbb R$ and how the Laplacian interacts with them. Recall that the laplacian $\Delta: (\mathbb R^3 \rightarrow \mathbb R)\rightarrow (\mathbb R^3 \rightarrow \mathbb R)$ is defined as
$$ \Delta(f) \equiv \frac{\partial^2 f}{\partial x^2} + \frac{\partial ^2 f}{\partial y^2} + \frac{\partial ^2 f}{\partial z^2} $$
Now, is there some convenient characterization of the image of the Laplacian, the set:
$$ Im(\Delta) \equiv \{ \Delta(f) \mid f: \mathbb R^3 \rightarrow \mathbb R \} $$
Searching for "image of Laplacian" is not useful since it gives me results about image processing that I don't care about! Is there a nice characterization of the set $Im(\Delta)$?
Search for codomain instead. You may like Sec. 4 here.