I am studying the Vlasov-Poisson system on a torus in three dimensions, $\mathbb{T}^3$. I consider a constant mass density $\rho = C$. What is a solution to the Poisson equation $\Delta U = 4\pi\rho $ for this mass density? Unfortunately, I am not used to working with a torus.
Is a solution just \begin{align} U = \frac{2\pi}{3}C \,x^2 \end{align} or is this just the case for $\mathbb{R^3}$? Please let me know if something about my question is unclear. Thank you!