Equality between functions when their Laplacians are equal

Question

If we know that for two functions $f(x)$ and $g(x)$ is true that \begin{equation} \Delta f=\Delta g \end{equation} what is the relation between $f(x)$ and $g(x)$ in general?

Answer 1

$\Delta f = \Delta g$ if and only if their difference $h=f-g$ is a harmonic function, i.e. $\Delta h = 0$: $$\Delta f = \Delta g$$ $$\iff \Delta f - \Delta g = 0$$ $$\iff \Delta (f-g) = \Delta h = 0$$ There are often nontrivial functions $h$ such that $\Delta h = 0$, so in general we can't (for instance) claim that $f$ and $g$ are off by a constant or linear factor.

Answer 2

$$ \Delta ( f - g) = 0 $$

so $f-g$ is harmonic. Depending on what space you are on and boundary conditions, tells you what are the possible harmonic functions. For example, maybe you have $f$ and $g$ are on a ball with boundary conditions $0$ at the boundary sphere. Then you have $f-g$ has vanishing boundary conditions and is harmonic. In this case you are getting $f=g$.