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Does anyone know how the fundamental solution of the laplacian reads on a flat torus $T^2=R^2/Z^2$? I have found somewhere a formula involving the Fourier transform that looks like this

$-\Delta u=f$ for $f\in C^\infty(T^2)$, then there exists a solution $u\in C^\infty(T^2)$ iff $\int f =0$. Moreover \begin{equation} u(x)=\sum_{k\neq 0}e^{2\pi ik\cdot x}(4\pi|k|^2)^{-1}\hat{f}(k). \end{equation}

Is there any interpretation (an idea to obtain this from the usual fund sol on the plane?). Is it necessary to write this using Fourier transform, or there is an alternative description?

Duca_Conte
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1 Answers1

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The formula simply follows from Fourier inversion theorem. The zero-average restriction simply follows from the fact that the range of $\Delta$ of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ consists only of such functions.

As for the fundamental solution, the above observation makes it impossible to solve $\Delta \Psi = \delta$ on $\mathbb{T}^2$. A natural substitute is to consider the following compensated equation

$$ \Delta \Psi = \delta - 1 \tag{1}$$

where $1$ represents the constant function with the value $1$. Then this paper constructs such a solution using elliptic function. Specifically, if $\vartheta_1$ denotes the first Jacobi theta function, then with $z = x+iy$,

$$ \Psi(z) = - \frac{y^2}{8} + \frac{1}{2\pi} \log \left| \vartheta_1\left(\frac{\pi z}{2} \middle|i\right) \right|. \tag{2} $$

Perhaps a more transparent formula is

$$ \begin{aligned} \Psi(z) &= - \frac{|z|^2}{4} + \lim_{N\to\infty} \sum_{\substack{k \in \mathbb{Z}^2 \\ \|k\|\leq N}} \frac{1}{2\pi} \left( \log|z + k| - \log\max\{|k|,1\} \right) \\\ &= - \frac{|z|^2}{4} + \frac{1}{2\pi} \log |z| + \frac{1}{8\pi} \sum_{k \in \mathbb{Z}^2\setminus\{0\}} \log\left| 1 - \frac{z^4}{k^4} \right|. \end{aligned} \tag{3} $$

(In the last sum, $k$'s are regarded as elements of $\mathbb{C}$ so that division makes sense.) This formula shows how this $\Psi$ is cooked up; it is a periodic sum of translates of the fundamental solution of $\Delta$ on $\mathbb{R}^2$. Despite their looks, it turns out that both $\text{(2)}$ and $\text{(3)}$ are $\mathbb{Z}^2$-periodic.

Sangchul Lee
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