Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.


In this case, I am (hopefully) trying to find an explicit formula for the Green function for the Laplacian $-\Delta$.


That is, what would a function $G$ on $\mathbb{T}^3$ satisfying $-\Delta G(x)=\delta^3(x)$ look like explicitly?


I am aware that on whole $\mathbb{R}^3$, $G(x)= \frac{1}{4\pi \lvert x \rvert}$.


According to standard PDE references, I think I need to find the corrector function $\phi^y(x)$ defined by
\begin{equation}
-\Delta\_x \phi^y(x)=0 \text{ for } x \in (0,1)^3 \text{ and } \phi^y(x)=\frac{1}{4\pi \lvert x-y \rvert} \text{ for } x \in \partial \mathbb{T}^3
\end{equation}
together with $y \in \mathbb{T}^3$ and $y \neq x$.


However, I have great difficulty solving the above Poisson equation, and moreover, I am not sure if the resulting $\phi^y(x)$ will even be a periodic function.


Could anyone please help me?


Edit: I forgot the fact that the periodic boundary conditions allow zero modes and the Laplacian does not have an inverse under this condition. So, I am restricting the domain of the Laplacian to $L^2$ space over $\mathbb{T}^3$ with no zero modes. Then will it make a difference?



Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Well it depends on what you mean by "explicit". Let $(\varphi\_k)\_k \subset L^2(\mathbb{S}^1)$ be the eigenfunctions of the Laplacian on $\mathbb{S}^1$, these have an explicit form that comes by solving the relative ODE, and let $(\lambda\_k)\_k$ be the relative eigenvalues. Then $G\_{\mathbb{S}^1}(p,q)=\sum\_{k\ge 1} \frac{\varphi\_k(p)\varphi\_k(q)}{\lambda\_k} $ and I highly doubt this has a form more explicit that this, meaning some form that does not involve an infinite sum or other similar operations. Since $\mathbb{T^3}=\mathbb{S}^1\times \mathbb{S}^1\times \mathbb{S}^1 $ you can write the Green function of $\mathbb{T}^3$ as a (triple) sum involving only the eigenfunctions $(\varphi\_k)\_k$ and eigenvalues $(\lambda\_k)\_k$, I don't this there is a form more explicit that this sum.


