Fundamental solution of the Laplacian on flat tori and boundary value problems for the planar Poisson equation in rectangles

Abstract

The fundamental solution of the Laplacian on flat tori is obtained using Eisenstein’s approach to elliptic functions via infinite series over lattices in the complex plane. Most boundary value problems stated for the planar Poisson equation in a rectangle for which series-only representations of solution were known, may thus be solved explicitly in closed-form using the method of images. Moreover, the fundamental solution of n-Laplacian on flat tori may also be simply derived by a convolution power.